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+.RP
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+.TL
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+
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+
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+
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+Top-down Non-Correcting Error Recovery
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+ in LLgen
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+.AU
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+Arthur van Deudekom
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+Peter Kooiman
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+.AI
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+Department of Mathematics and Computer Science
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+Vrije Universiteit
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+Amsterdam
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+
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+
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+
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+
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+
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+Supervised by
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+.AU
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+dr. D. Grune
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+.AI
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+Department of Mathematics and Computer Science
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+Vrije Universiteit
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+Amsterdam
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+
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+.AB
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+This paper describes the design and implementation of a parser
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+generator with non-correcting error recovery based on the extended LL(1)
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+parser generator LLgen. It describes a top-down algorithm for implementing
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+this error recovery technique that can handle left-recursive grammars.
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+The parser generator has been tested with several existing ACK-compilers,
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+among which C and Modula-2. Various optimizations have been tried and are
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+discussed in this paper.
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+.AE
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+.LP
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+.nr PS 12
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+.nr VS 14
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+
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+.NH
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+Introduction
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+.EQ
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+delim $$
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+.EN
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+
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+.nr PS 10
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+.nr VS 12
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+.RS
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+.LP
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+One of the trickier problems in constructing parser-generators is what
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+to do when the input to the generated parser is not well formed. Several
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+approaches are known, most of which are `correcting', meaning that they
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+modify the input to make it correct. However, in most cases there are
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+several possible corrections, and often the one chosen will turn out
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+to be the wrong one. As a result of such an incorrect choice, spurious error
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+messages can occur. Every programmer knows from experience how the omission
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+of a single `)' can on occasion lead to pages of error messages.
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+
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+.LP
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+A radically different approach is to just discard all the input up to
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+and including the offending token, and start with a clean slate at the
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+token following the offending one. [RICHTER] describes how
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+this idea can be used to construct a non-correcting error recovery system
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+that will never introduce spurious error messages. It is, however,
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+possible that errors are overlooked.
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+
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+.LP
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+In this paper we describe the incorporation of this non-correcting error
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+recovery into LLgen, an existing LL(1) parser generator.
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+In this introduction, we will describe in detail this non-correcting error
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+recovery technique, give an overview of LLgen and how it handles
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+errors, and finally describe how we have incorporated noncorrecting
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+error recovery in LLgen.
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+.RE
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+
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+.NH 2
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+Non-correcting syntax error recovery
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+
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+.LP
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+Richter describes how syntax error recovery can be done
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+without making any corrections to the input text. Richter gives three
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+reasons why recovery without correction is desirable:
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+
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+.IP 1
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+In most cases there are many possible corrections, the choice among which
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+will severely influence the further processing of the input. Thus, the
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+probability of selecting the right correction is not high.
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+
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+.IP 2
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+The harm done by selecting the wrong correction is often unlimited.
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+
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+.IP 3
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+The loss of information to the user of a non-correcting recovery technique
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+need not be grave.
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+
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+.LP
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+The non-correcting technique described by Richter can be summarized as
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+follows: When a syntax-error has occurred, the input up to and including the
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+erroneous symbol is discarded; the remainder of the
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+input is processed by a substring parser of the input
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+language, that is a parser that recognizes any substring of a string in the input
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+language. When the substring parser detects a syntax error, the offending
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+symbol is reported as another error, and the input up to and including the
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+erroneous symbol is discarded. The process is then repeated with the remaining input, possibly
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+finding other syntax errors, until all the input is scanned.
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+This process yields what Richter calls a
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+.I
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+suffix analysis
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+.R
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+of an input string. Formally, given an input string
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+.I x
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+, suffix analysis produces a set of strings $w sub k$ and a set of symbols
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+$ a sub k$ such that
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+.br
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+
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+.IP
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+$x~ =~ w sub 0 a sub 0 w sub 1 a sub 1~...w sub n-1 a sub n-1 w sub n$
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+.LP
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+and such that:
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+.br
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+.IP
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+ $w sub 0$ is the longest prefix of $x$ that is a prefix of
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+a string in the input language L, formally: there is a string $y$ such that
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+$w sub 0 y$ is in L, but there is no string $z$ such that $w sub 0 a sub 0 z$
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+is in L;
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+.IP
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+For $0 < k < n$, $w sub k$ is a longest substring of $x$ that is also a
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+substring of a string in L, formally there are strings $u$ and $v$ such that
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+$u w sub k v$ is in L, but there are no strings $y$ en $z$ such that
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+$y w sub k a sub k z$ is in L;
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+.IP
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+$w sub n$ is a substring of $x$
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+that is a substring of a string in L, formally:
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+there exist $u$ and $v$, such that $u w sub n v$ is in L. Note that
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+$w sub n$ need not be a suffix of a string in L, if $x$ represents incomplete
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+input $w sub n$ is not a suffix of a string in L.
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+
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+.LP
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+Now, the $a sub k$ indicate points at which an error is detected. The
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+"real" error need not be at $a sub k$, it can have occurred anywhere
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+within $w sub k a sub k$.
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+In his paper, Richter shows that, although this method may miss errors, it
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+will never introduce spurious errors.
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+
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+.LP
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+For implementing the technique, a parser that recognizes any
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+substring of the input language is needed. If we confine ourselves to
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+syntactical analysis, it is sufficient to construct a substring
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+recognizer. Richter himself does not give a practical construction, but
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+[CORMACK] describes how a LR substring parser can be constructed
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+that handles BC-LR(1,1) grammars. In this paper, we describe the construction
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+of a LL substring recognizer that can handle any grammar. Furthermore,
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+our recognizer is actually a suffix-recognizer, that is, a recognizer that
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+recognizes any suffix of a string in the input language. Our suffix recognizer has the
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+correct-prefix property,
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+meaning that it detects the first syntax error as early as possible
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+in a left-to-right scan of the input. Specifically, if the input language
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+is L and the invalid input is $x$ , it finds a string $w$ and an input symbol
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+$a$ such that $x = way$ , there is a string $z$ such that $wz$
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+is in L, and there is no string $z$ such that $waz$ is in L.
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+Because the suffix parser has this correct-prefix property, it can be
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+used as a substring parser, because it will detect the first input symbol that
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+is not part of a substring of the language. Because it is a suffix-recognizer,
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+it additionally will detect incomplete input, because in that case
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+at the end of the input the parser will not be in an accepting state.
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+
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+.NH 2
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+Overview of LLgen
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+
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+.LP
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+LLgen is an extended LL(1) parser generator. For a complete description,
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+see [GRUNE].
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+LLgen can actually handle grammars that are not LL(1), because it allows
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+the use of conflict-resolvers. In case of an LL(1) conflict, these resolvers
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+are used to statically or dynamically decide which rule to use. As we will see
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+later, this feature makes it necessary for the suffix-recognizer to
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+handle grammars that are not LL(1). Semantic actions can occur anywhere
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+in the grammar rules, and they are executed when their position is
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+reached during parsing. A typical LLgen rule looks like
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+.br
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+.IP
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+S: A {
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+.I action
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+} B
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+.LP
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+where the action is a piece of C-code, that will be executed
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+when the parser is using the rule for S and has recognized A.
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+
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+.LP
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+LLgen-generated parsers use correcting syntax error recovery, based on a
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+scheme designed by R\*:ohrich [ROEHRICH], inserting or deleting symbols at the point of error detection
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+until correct input results. This means that actions in the parser will
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+always be executed in an order that could also have resulted from
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+syntactically correct input, and most importantly, once a grammar-rule
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+is started it is guaranteed to be completed. This means that syntactic
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+errors can never result in inconsistencies for the actions. Actions
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+only have to deal with syntactically correct input. In a nutshell, the
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+error recovery in LLgen-parsers works as follows: Suppose the parser is
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+presented with correct input that breaks off before the end. The error
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+recovery mechanism now provides a continuation path, chosen in such a
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+way that all active rules are left as soon as possible. Effectively, the
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+continuation path is the `shortest way out'. The symbols on this path are
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+called `acceptable', and end-of-file is also `acceptable'. Furthermore, at
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+each point along this `shortest path' there can be other terminals that
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+would be correct; these are `acceptable' as well. Now, when an
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+error occurs, all symbols that are not acceptable are discarded, until
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+an acceptable symbol appears in the input. The tokens on the path up to
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+but not including the acceptable input symbol are inserted.
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+From then on, normal parsing resumes.
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+
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+.NH 2
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+Incorporation of non-correcting error recovery in LLgen
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+
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+.LP
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+An important consideration in incorporating the non-correcting recovery
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+in LLgen was that correct programs should suffer as little as possible
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+in what regards compilation speed. Furthermore, the existing error
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+recovery method has the highly desirable property that rules that are
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+started will be finished too, thus ensuring that errors in the
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+input text will not cause inconsistencies in the semantic actions. We have
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+implemented the non-correcting error recovery in such a way that this
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+property is preserved.
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+
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+.LP
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+The way we have achieved these goals is by actually including
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+the suffix recognizer as a `second recognizer' in the generated parser.
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+Correct programs are handled in the usual way by the parser, but if an error
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+occurs the following happens: instead of going to the standard error
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+recovery routine, the parser starts executing the non-correcting error
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+handler. This process continues, reporting all errors, until the
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+end of the input text is reached. Then, control is handed back to
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+the standard error recovery routine. This routine will now think
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+there is no more input, and thus start inserting tokens so as to construct
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+a `shortest way out'. This ensures that all rules that were started are
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+also finished, and no inconsistencies can occur in the semantic actions.
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+However, this method does require some modifications to the error reporting
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+routine. Normally, if the generated parser inserts a token, it reports
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+this to the user, but in this case this is undesirable. The insertions only
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+serve to maintain consistency in the semantic actions
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+and do not signify errors, so reporting of insertions should be suppressed.
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+.bp
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+.nr PS 12
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+.nr VS 14
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+.PS
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+boxwid = boxwid / 1.5
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+boxht = boxht / 1.5
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+arcrad = arcrad / 1.5
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+movewid = movewid / 1.5
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+moveht = moveht / 1.5
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+arrowwid = arrowwid / 1.5
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+arrowht = arrowht / 1.5
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+arrowhead = arrowhead / 1.5
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+linewid = linewid / 1.5
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+lineht = lineht / 1.5
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+.PE
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+.NH
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+The LL suffix parser
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+
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+.nr PS 10
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+.nr VS 12
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+.RS
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+.LP
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+In this chapter, we describe the construction of the LL suffix parser.
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+The described parser is not restricted to LL(1) grammars, because the
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+presence of conflict resolvers in LLgen allows for more general grammars,
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+that may even be left-recursive. We start this chapter with a discussion
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+of the implications of conflict resolvers, and continue with descriptions
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+of the parser algorithm, the used data-structures,
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+the handling of left- and right recursion, and some possible optimizations.
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+.RE
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+
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+.NH 2
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+LLgen conflict resolvers and their implications
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+
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+.LP
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+In grammars that are nearly but not completely LL(1), conflicts
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+will arise in the two places where parsing decisions are made: the choice
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+of which alternative to start (`alternation conflicts') and the decision
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+to stop or continue a repeated item (`repetition conflicts'). In order to
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+allow LLgen to handle this type of grammar, the user can
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+specify conflict resolvers in those places where conflicts arise.
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+These resolvers are Boolean expressions labeling an alternative,
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+and are evaluated when a conflict arises during parsing. If the
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+expression evaluates to `true' the labeled alternative will be taken.
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+The Boolean expressions are expressions in C, and can consult
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+any information available at the point they occur.
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+However, if a syntactic error has occurred in the input, and the non-correcting
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+error recovery starts, we can no longer rely on the conflict resolvers to
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+guide parsing decisions. The suffix recognizer is only concerned with
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+syntax, and will not execute any semantic actions. It recognizes suffices
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+of correct input, but does not know or care what prefix would make
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+the suffix a correct program; as a result, the information that conflict
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+resolvers could use is not available, because the semantic actions
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+that would build this information have not been executed.
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+Therefore, the information used by the conflict resolvers is no longer
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+reliable, and the suffix parser needs to be able to handle the underlying
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+grammar without their help. In particular, it has to be able to handle
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+left-recursive grammars.
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+
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+.NH 2
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+The suffix parser algorithm
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+
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+.LP
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+Our algorithm needs easy access to the grammar rules; in the description
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+we assume there is an efficient way to access the grammar rules. In
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+the next chapter we will describe the details of the actual implementation.
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+For the moment, we will only consider grammars that are not left- or
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+right-recursive. In the next section, we will discuss how the algorithm has to be adapted
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+to handle left- and right recursion.
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+
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+.LP
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+Suppose the grammar is G, and the input to the suffix recognizer is
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+$a sub 0 a sub 1 ... a sub n-1 a sub n$. Remember that parsing is
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+always started by the `normal' LLgen generated parser. It's only after
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+a syntactic error has occurred that the suffix recognizer will be started.
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+The input to the suffix recognizer thus is the `tail' of the input, starting
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+at the first symbol after the position where the first syntax error was
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+found.
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+
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+.LP
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+Now, in order to get parsing going again, the parser scans the grammar
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+for rules which contain symbol $a sub 0$ in the right hand side:
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+.br
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+
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+ A: $alpha ~ a sub 0 ~ beta$
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+.br
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+
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+.LP
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+where $alpha$ and $beta$ represent a string of terminals and non-terminals,
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+possible empty. Now, for each of these rules found, and for any string
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+$b sub 0 b sub 1$...$ b sub m$ that can be generated by $beta$ it holds that
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+$a sub 0 b sub o b sub 1$...$b sub m$ is a substring of some string in L.
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+This can be shown as follows, supposing that the start symbol is S and
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+S $-> sup * gamma$ A $delta$:
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+.br
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+
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+S $-> sup * gamma$ A $delta$ $-> sup * gamma ~ alpha ~ a sub 0 beta ~ delta
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+-> sup * gamma ~ alpha ~ a sub 0 b sub 0 b sub 1$...$b sub m delta$
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+
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+.br
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+Of course, there may very well be more than one such string
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+$b sub 1 b sub 2$..$b sub m$, and one of these strings can be empty as well, if
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+$beta$ can produce empty. Now, in what we will call the
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+.I
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+predicting phase
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+.R
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+the algorithm will
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+produce all possible symbols $b sub 0$. Then, in what we will call the
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+.I
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+accepting phase
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+.R
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+these symbols are matched against
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+the input, and those not matching are discarded. Then, entering the next
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+predicting phase, the algorithm will produce
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+all symbols $b sub 1$, and match them against the next input symbol in
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+the subsequent accepting phase,
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+etc. In case one of the strings $b sub 0$...$b sub m$ is empty, or
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+the end of one of the strings is reached, some way to continue is
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+needed; we will discuss this later. First let's see how the
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+algorithm produces the strings $b sub 0$...$b sub m$ .
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+
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+.LP
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+For each rule in the grammar of the form
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+.br
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+
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+ A: $alpha a sub 0 W sub 1 W sub 2$...$W sub p$
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+.br
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+
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+with each $W sub k$ a terminal or nonterminal, a
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+.I
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+prediction graph
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+.R
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+is created that looks like this:
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+
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+.PS
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+down; box "$a sub 0$"; arrow; box "$W sub 1$"; arrow
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+box "$W sub 2$"; arrow dashed; box "$W sub p$"
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+arrow; box "END" "$[A]$"
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+.PE
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+
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+.LP
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+The bottom element of these prediction graphs is an end-marker containing the
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+left-hand side of the rule used. All these graphs have $a sub 0$ on top, and
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+this $a sub 0$ is matched against the $a sub 0$ in the input in the
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+accepting phase that follows, removing the
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+$a sub 0$ from the graph. If the prediction graph is now empty, we have to find a way
|
|
|
+to continue; this case is treated later. First we will consider what to do if
|
|
|
+the prediction graph is not empty. There are two possibilities: either $W sub 1$ is a
|
|
|
+terminal, or it is a nonterminal. If it is a terminal, we are finished for
|
|
|
+the moment; if not, the algorithm scans for rules of the form
|
|
|
+.br
|
|
|
+
|
|
|
+ $W sub 1$: $U sub 1 U sub 2$...$U sub i$
|
|
|
+.br
|
|
|
+
|
|
|
+.LP
|
|
|
+with each $U sub k$ a terminal or nonterminal. Now, the algorithm substitutes
|
|
|
+the top of the prediction graph with the right-hand sides
|
|
|
+of all the rules found. Because there can be more than one rule, the
|
|
|
+prediction graph can now become a DAG (Directed Acyclic Graph).
|
|
|
+Supposing there are two rules with $W sub 1$ in the LHS:
|
|
|
+
|
|
|
+.br
|
|
|
+
|
|
|
+ $W sub 1$: $U sub 1 U sub 2$...$U sub i$
|
|
|
+.br
|
|
|
+ $W sub 1$: $V sub 1 V sub 2$...$V sub j$
|
|
|
+
|
|
|
+.LP
|
|
|
+the prediction graph will now look like this:
|
|
|
+
|
|
|
+.PS
|
|
|
+B1: box "$U sub 1$"
|
|
|
+move
|
|
|
+B2: box "$V sub 1$"
|
|
|
+arrow dashed down from bottom of B1
|
|
|
+B3: box "$U sub i$"
|
|
|
+arrow dashed down from bottom of B2
|
|
|
+B4:box "$V sub j$"
|
|
|
+move to 0.5 <B3.se, B4.sw>
|
|
|
+down;move
|
|
|
+B5:box "$[W sub 1 ]$"
|
|
|
+arrow dashed;
|
|
|
+box "$W sub p$"
|
|
|
+arrow;
|
|
|
+box "END" "$[A]$"
|
|
|
+arrow from B3.bottom to B5.top
|
|
|
+arrow from B4.bottom to B5.top
|
|
|
+.PE
|
|
|
+
|
|
|
+.LP
|
|
|
+The graph element representing $W sub 1$ is left in the stack, the
|
|
|
+notation $[W sub 1 ]$ indicates it has been substituted. These substituted
|
|
|
+element will from now on be ignored by the algorithm. The elements
|
|
|
+$U sub 1$ and $V sub 1$ are now `on top' of the prediction graph.
|
|
|
+
|
|
|
+.LP
|
|
|
+If $W sub 1$ can also produce empty, its successor in the prediction graph
|
|
|
+has to be processed
|
|
|
+as well; the algorithm walks down the graph to this successor, and
|
|
|
+there the process is repeated; if it is a terminal we are finished, else we
|
|
|
+substitute it with the right hand sides of its grammar rule.
|
|
|
+However, the element that we want to substitute now, say $W sub k$, cannot
|
|
|
+be marked `substituted' just like that, because it can be on another
|
|
|
+path, on which it cannot be substituted yet. Therefore, a copy of element
|
|
|
+$W sub k$ is made, it is marked $[W sub k ]$, and an edge is created
|
|
|
+from $[W sub k ]$ to the successor of $W sub k$. This produces graphs like
|
|
|
+this:
|
|
|
+.br
|
|
|
+.PS
|
|
|
+B1: box "$U sub 1$"
|
|
|
+move
|
|
|
+B2: box "$V sub 1$"
|
|
|
+move
|
|
|
+X1:box "$X sub 1$"
|
|
|
+arrow dashed down from bottom of B1
|
|
|
+B3: box "$U sub m$"
|
|
|
+arrow dashed down from bottom of B2
|
|
|
+B4:box "$V sub j$"
|
|
|
+arrow dashed down from bottom of X1
|
|
|
+Xj: box "$X sub j$"
|
|
|
+move to 0.5 <B3.se, B4.sw>
|
|
|
+down;move
|
|
|
+B5:box "$[W sub 1 ]$"
|
|
|
+arrow dashed;
|
|
|
+B6: box "$W sub k$"
|
|
|
+arrow
|
|
|
+Wk1:box "$W sub k+1$"
|
|
|
+arrow dashed
|
|
|
+box "$W sub n$"
|
|
|
+arrow;
|
|
|
+box "END" "$[A]$"
|
|
|
+arrow from B3.bottom to B5.top
|
|
|
+arrow from B4.bottom to B5.top
|
|
|
+move down from Xj.top;move;move;move
|
|
|
+Wk: box "$[W sub k ]$"
|
|
|
+arrow from Xj.bottom to Wk.top
|
|
|
+arrow from Wk.bottom to Wk1.top
|
|
|
+.PE
|
|
|
+
|
|
|
+.LP
|
|
|
+This process of substituting is repeated with all nonterminals that are
|
|
|
+now on top of the prediction graph, until there are only terminals on top of
|
|
|
+the graph.
|
|
|
+This completes the prediction phase of the algorithm, not taking into account
|
|
|
+what to do if an END marker appears on top of the graph.
|
|
|
+Now, the algorithm enters its accepting phase, in which
|
|
|
+the terminals on top are compared with the next symbol in the input.
|
|
|
+If a terminal in the graph matches the input, its element is deleted
|
|
|
+from the graph, and the substitution process will continue with its
|
|
|
+successors, in the next prediction phase.
|
|
|
+If a terminal on top of the graph does not
|
|
|
+match the input, the path it is on represents a `dead-end', which
|
|
|
+does not need to be processed any further. The terminal is no longer
|
|
|
+a `top', and the algorithm will not visit it again.
|
|
|
+
|
|
|
+.LP
|
|
|
+There is one tricky situation: consider again this graph:
|
|
|
+
|
|
|
+.PS
|
|
|
+B1: box "$U$"
|
|
|
+move
|
|
|
+B2: box "$a$"
|
|
|
+move to 0.5 <B1.se, B2.sw>
|
|
|
+down;move
|
|
|
+B5:box "$W sub 1 $"
|
|
|
+arrow dashed;
|
|
|
+box "$W sub n$"
|
|
|
+arrow;
|
|
|
+box "END" "$[A]$"
|
|
|
+arrow from B1.bottom to B5.top
|
|
|
+arrow from B2.bottom to B5.top
|
|
|
+.PE
|
|
|
+
|
|
|
+.LP
|
|
|
+Here, the algorithm is processing $W sub 1$ in the predicting phase, and
|
|
|
+using some rule it has produced $a$ on top; there is another rule with
|
|
|
+$W sub 1$ in its LHS which has produced nonterminal $U$ on top.
|
|
|
+Now, suppose $U$ is a nonterminal that can
|
|
|
+produce empty. Now, the algorithm starts substituting $U$, and walks
|
|
|
+down $W sub 1$. What we definitely do not want
|
|
|
+is the algorithm to start substituting $W sub 1$ again, because then we
|
|
|
+would loop forever. Therefore, if the algorithm starts processing
|
|
|
+element $W sub 1$ it should make it $[W sub 1 ]$ before it does
|
|
|
+anything else. On entering the element
|
|
|
+for the second time in the prediction phase , it sees that it is already substituted,
|
|
|
+so there is nothing to do.
|
|
|
+It then just walks to the successor of $W sub 1$ and
|
|
|
+starts substituting it. This is correct, since the fact that the algorithm
|
|
|
+enters an element for the second time in a prediction phase means that the element
|
|
|
+indirectly can produce the empty string, and thus its successor must
|
|
|
+be substituted as well in the prediction phase.
|
|
|
+
|
|
|
+.LP
|
|
|
+It is easy to see that the substitution process will stop: the algorithm can
|
|
|
+only loop if it starts processing an element for the second time in a
|
|
|
+prediction phase,
|
|
|
+or if the processing of an element eventually yields a graph with that
|
|
|
+same element on top.
|
|
|
+The first case cannot occur because the algorithm marks elements it is
|
|
|
+processing as `substituted' before it does anything else, meaning that those elements will not
|
|
|
+be processed again; the second case can only occur if the grammar is
|
|
|
+left-recursive, which we assumed it was not.
|
|
|
+
|
|
|
+.LP
|
|
|
+The algorithm simulates
|
|
|
+left-most derivations of strings $a sub 0 b sub 0 b sub 1$..$b sub n$
|
|
|
+starting from $a sub 0 W sub 1$..$W sub p$; as we showed before, if
|
|
|
+the algorithm recognizes a string $a sub 0 b sub 0$..$b sub n$ that
|
|
|
+string is a substring of some string in L. Conversely, because the
|
|
|
+algorithm start out by using all rules of the form
|
|
|
+A: $alpha a sub 0 beta$, and then proceeds to simulate all
|
|
|
+possible left-most derivations, it will recognize all input
|
|
|
+$a sub 0 b sub 0$... $b sub n$ that can be produced starting from
|
|
|
+$a sub 0 beta$.
|
|
|
+
|
|
|
+.LP
|
|
|
+Now we will discuss what has to be done if an END marker appears as
|
|
|
+top of the prediction graph.
|
|
|
+When this happens, it means that starting from some rule
|
|
|
+.br
|
|
|
+
|
|
|
+ A: $alpha a sub 0 beta$
|
|
|
+
|
|
|
+.br
|
|
|
+the algorithm has produced a leftmost-derivation of a string
|
|
|
+$a sub 0 b sub 1 .. b sub n$ starting from $a sub 0 beta$, or that $beta$ can produce
|
|
|
+empty and the string so far is just $a sub 0$. The next step is to assume
|
|
|
+that the have recognized A and that that some string produced by $alpha$
|
|
|
+is part of the prefix that makes the suffix we are recognizing a
|
|
|
+correct string in L. Remember that in the END marker we kept record of
|
|
|
+the LHS of the rule that has started the graph, and we will now use this
|
|
|
+LHS to continue recognizing. What the algorithm does is scan for all
|
|
|
+rules of the form:
|
|
|
+.br
|
|
|
+
|
|
|
+ B: $gamma$ A $delta$
|
|
|
+.br
|
|
|
+
|
|
|
+with $gamma$ and $delta$ possibly empty strings of terminals and nonterminals.
|
|
|
+The algorithm now starts a new component in the prediction graph, and if $delta$ is
|
|
|
+$W sub 1 W sub 2$...$W sub n$ it looks like this:
|
|
|
+
|
|
|
+.PS
|
|
|
+down;box "$W sub 1$"; arrow
|
|
|
+box "$W sub 2$"; line dashed; box "$W sub n$"
|
|
|
+arrow; box "END" "$[B]$"
|
|
|
+.PE
|
|
|
+
|
|
|
+.LP
|
|
|
+Note that the END marker now contains B, because we have started to match
|
|
|
+a rule for B. If the $delta$ in the rule for B was empty, this just produces
|
|
|
+and END marker with B in it; in this case, the process is just repeated
|
|
|
+with all rules of the form:
|
|
|
+.br
|
|
|
+
|
|
|
+ C: $zeta$ B $eta$
|
|
|
+.br
|
|
|
+
|
|
|
+.LP
|
|
|
+etc, until we have a prediction graph with a nonterminal or terminal on top.
|
|
|
+Now, the substitution algorithm is again applied over all nonterminals on
|
|
|
+top, until every top contains a terminal. It is possible that during
|
|
|
+substitution again an END marker will turn up; if this happens
|
|
|
+we again scan for rules to continue with etc.
|
|
|
+This `continuation algorithm' can only loop if, when
|
|
|
+trying to build a new prediction graph for matched symbol A, it produces an empty
|
|
|
+graph with again matched symbol A. If this happens, the grammar was
|
|
|
+(directly or indirectly) right-recursive, and we assumed that it was not.
|
|
|
+Therefore, the algorithm will terminate. The terminals on top of the
|
|
|
+new graph after applying this `continuation' algorithm are exactly those
|
|
|
+that could follow the string $A sub 0 b sub 0$..$b sub n$ in a substring
|
|
|
+of a string in L.
|
|
|
+To see this, suppose we have `recognized' the rule
|
|
|
+.br
|
|
|
+
|
|
|
+ A: $alpha a sub 0 beta$
|
|
|
+
|
|
|
+.br
|
|
|
+and $a sub 0 b sub 0 b sub 1$...$b sub n$ is the string produced from
|
|
|
+$a sub 0 beta$ by the algorithm. Now, using rule:
|
|
|
+.br
|
|
|
+
|
|
|
+ B: $gamma$ A $delta$
|
|
|
+
|
|
|
+.br
|
|
|
+and supposing that S $->$ $zeta$ B $eta$ we get
|
|
|
+.br
|
|
|
+
|
|
|
+ S $->$ $zeta$ B $eta$ $->$ $zeta gamma$ A $delta$ $eta$ $->$ $zeta gamma a sub 0 b sub 0 b sub 1$ ... $b sub n$ $delta$ $eta$
|
|
|
+
|
|
|
+.br
|
|
|
+.LP
|
|
|
+and thus any string produced by a derivation starting from
|
|
|
+$delta$ can come right after $a sub 0 b sub 0$...$b sub n$ in a substring
|
|
|
+of some string in L. The algorithm will proceed to generate all these
|
|
|
+strings starting from $delta$. If $delta$ produces empty, the above
|
|
|
+is just repeated. Because in the `continuation' part
|
|
|
+all possible rules are considered, the whole algorithm will recognize
|
|
|
+all substrings of any string in L. In order to determine if we
|
|
|
+have actually recognized a suffix of some string in L, we need to
|
|
|
+remember if within a predicting phase the `continuation' part of the algorithm has been run
|
|
|
+on an END marker containing the start-symbol S;
|
|
|
+if this is the case, then the input seen until now is a suffix of some string in L.
|
|
|
+Formally, it means that there is a derivation starting from start symbol
|
|
|
+$S$ such that if the
|
|
|
+input seen until now is $a sub 0 a sub 1$..$a sub n$, then:
|
|
|
+.br
|
|
|
+
|
|
|
+ S $-> sup * alpha beta$ $-> sup * alpha a sub 0 a sub 1$..$a sub n$
|
|
|
+.br
|
|
|
+
|
|
|
+.LP
|
|
|
+where $alpha$ can be empty, $beta$ is not empty.
|
|
|
+
|
|
|
+.NH 2
|
|
|
+The prediction graph data structure
|
|
|
+
|
|
|
+.LP
|
|
|
+The graphs that are produced by the suffix recognizer may grow extremely
|
|
|
+large; to facilitate an efficient
|
|
|
+implementation we have devised a way of keeping the size of the
|
|
|
+data structure under control, in a way that is very similar to
|
|
|
+the way described in [TOMITA].
|
|
|
+
|
|
|
+.LP
|
|
|
+The basic idea is, that in a prediction phase of the algorithm, it is not
|
|
|
+necessary to explicitly substitute each nonterminal every time it
|
|
|
+turns up as a `top'; it is sufficient to do it once, because the
|
|
|
+second substitution will produce exactly the same subgraph starting at
|
|
|
+the substituted nonterminal. Here is an example:
|
|
|
+
|
|
|
+.PS
|
|
|
+down;box "$a$";arrow;box "A";arrow dashed;box "[B]";arrow
|
|
|
+box "C";arrow dashed;box "END" "[X]"
|
|
|
+move right from last box.e;
|
|
|
+box "END" "[Y]";
|
|
|
+arrow <- dashed up from last box.top;
|
|
|
+box "D";arrow <- up from last box.top
|
|
|
+box "B"
|
|
|
+.PE
|
|
|
+
|
|
|
+.LP
|
|
|
+Here, in the left component of the graph, nonterminal B has been
|
|
|
+substituted. Now, in the same prediction phase, the algorithm again runs into
|
|
|
+B, now in the right component. There is no need to compute again
|
|
|
+what the substitution will produce, it is exactly the part on top
|
|
|
+of B in the left component. Therefore, all that is needed is:
|
|
|
+
|
|
|
+.PS
|
|
|
+down;box "$a$";arrow;box "A";arrow dashed;
|
|
|
+B1: box "[B]";arrow
|
|
|
+box "C";arrow dashed;box "END" "[X]"
|
|
|
+move right from last box.e;
|
|
|
+box "END" "[Y]";
|
|
|
+arrow <- dashed up from last box.top;
|
|
|
+box "D"
|
|
|
+arrow from B1.bottom to last box.top
|
|
|
+.PE
|
|
|
+
|
|
|
+So, when, in a prediction phase of the algorithm, a nonterminal is substituted,
|
|
|
+the nonterminal is placed on a list, together with a pointer to
|
|
|
+the substituted nonterminal. If in the same prediction phase a nonterminal that
|
|
|
+is on the list becomes a top, all we need to do is place an edge
|
|
|
+between the already substituted one and the successor of the top we are currently
|
|
|
+processing. When a prediction phase is finished, the list is cleared.
|
|
|
+There is one catch: if we consider again the last picture,
|
|
|
+note that if nonterminal B can (directly or indirectly) produce empty,
|
|
|
+it is also necessary to substitute D. However, it is not difficult to
|
|
|
+determine if a nonterminal can produce empty. LLgen already computes
|
|
|
+this information for each nonterminal.
|
|
|
+
|
|
|
+.LP
|
|
|
+Without this `joining together' of graph components, each
|
|
|
+element in the graph has exactly one successor, except the END marker,
|
|
|
+which has none.
|
|
|
+Now that components get joined as described, an element can have any
|
|
|
+number of successors. The recognizer algorithm now has to consider all
|
|
|
+successors of a graph element instead of one.
|
|
|
+
|
|
|
+.NH 2
|
|
|
+Handling right recursion
|
|
|
+
|
|
|
+.LP
|
|
|
+The only problem right-recursive grammars cause in the algorithm is in the
|
|
|
+`continuation' part; they can cause this part of the algorithm to loop
|
|
|
+forever. As an example, consider:
|
|
|
+.br
|
|
|
+
|
|
|
+ A: $alpha$ B
|
|
|
+.br
|
|
|
+ B: $beta$ C
|
|
|
+.br
|
|
|
+ C: $gamma$ A
|
|
|
+
|
|
|
+.LP
|
|
|
+Now suppose the `substitution' part of the algorithm has turned up
|
|
|
+an END marker with nonterminal A in it. The continuation algorithm will
|
|
|
+now produce:
|
|
|
+
|
|
|
+.PS
|
|
|
+box "END" "[A]";move;box "END" "[C]";move;box "END" "[B]";move
|
|
|
+box "END" "[A]";move;box "END" "[C]"
|
|
|
+.PE
|
|
|
+
|
|
|
+.LP
|
|
|
+etc. etc. However, a slight modification to the algorithm suffices
|
|
|
+to eliminate this problem; within each prediction phase of the algorithm, we
|
|
|
+simply maintain a list of nonterminals that have turned up in an
|
|
|
+END marker. As soon as an END marker turns up whose nonterminal is
|
|
|
+already in the list, we stop the `continuation' algorithm; the part
|
|
|
+of the graph that would be produced by it already has been generated
|
|
|
+by an earlier invocation of the algorithm in the same prediction phase.
|
|
|
+At the end
|
|
|
+of a prediction phase, when all heads are terminals, we clear the list.
|
|
|
+This way, no looping can occur; even if the right recursion is
|
|
|
+indirect, for instance if in the above example the rule for A had been
|
|
|
+.br
|
|
|
+
|
|
|
+ A: $alpha$ B $delta$
|
|
|
+.br
|
|
|
+.LP
|
|
|
+where $delta$ can produce empty, the algorithm still works; the substitution
|
|
|
+of $delta$ will yield an END marker on top, and when trying to find
|
|
|
+a continuation for LHS A the algorithm notices A is already on the list.
|
|
|
+
|
|
|
+
|
|
|
+.NH 2
|
|
|
+Handling left recursion
|
|
|
+
|
|
|
+.LP
|
|
|
+Left-recursion is, unfortunately, a much tougher problem than
|
|
|
+right-recursion. The result of left-recursive grammar rules is that
|
|
|
+the substitution algorithm never stops, because it can keep on building
|
|
|
+the graph with the same set of rules without ever turning up a terminal.
|
|
|
+One course of action would be to pre-process the grammar rules to
|
|
|
+eliminate left-recursion; there are algorithms that eliminate direct
|
|
|
+and indirect left-recursion. However, we have taken another course; by
|
|
|
+allowing the produced graphs to contain loops, we can handle left
|
|
|
+recursion without any modifications to the grammar. As soon as
|
|
|
+we come to the point that we want to substitute a nonterminal
|
|
|
+which was already substituted earlier on the same path and in
|
|
|
+the same prediction phase, we can
|
|
|
+make a link from the `older' nonterminal to the successor of
|
|
|
+the `new' nonterminal. In this way we have constructed a loop
|
|
|
+in the graph. As an example, suppose we have the following rules:
|
|
|
+.br
|
|
|
+
|
|
|
+D: A
|
|
|
+
|
|
|
+A: B a
|
|
|
+
|
|
|
+B: A | x
|
|
|
+
|
|
|
+.br
|
|
|
+Suppose also that we have nonterminal `D' on top of a stack. We
|
|
|
+now start substituting `D':
|
|
|
+
|
|
|
+.PS
|
|
|
+A: box "A"
|
|
|
+move
|
|
|
+X: box "x"
|
|
|
+move to 0.5 <A.se, X.sw>
|
|
|
+down
|
|
|
+move
|
|
|
+B: box "[B]"
|
|
|
+arrow
|
|
|
+box "a"
|
|
|
+arrow
|
|
|
+box "[A]"
|
|
|
+arrow
|
|
|
+box "[D]"
|
|
|
+arrow dashed
|
|
|
+box "END" "[S]"
|
|
|
+
|
|
|
+arrow from A.s to B.n
|
|
|
+arrow from X.s to B.n
|
|
|
+
|
|
|
+.PE
|
|
|
+
|
|
|
+.LP
|
|
|
+We now have an `A' on top of of the stack which was already
|
|
|
+substituted on the same path and also in the same prediction phase. To avoid
|
|
|
+never ending substitution we make a loop as follows:
|
|
|
+
|
|
|
+.PS
|
|
|
+A: box "A" dashed
|
|
|
+move
|
|
|
+X: box "x"
|
|
|
+move to 0.5 <A.se, X.sw>
|
|
|
+down
|
|
|
+move
|
|
|
+B: box "[B]"
|
|
|
+arrow
|
|
|
+box "a"
|
|
|
+arrow
|
|
|
+A2: box "[A]"
|
|
|
+arrow
|
|
|
+box "[D]"
|
|
|
+arrow dashed
|
|
|
+box "END" "[S]"
|
|
|
+
|
|
|
+arrow dashed from A.s to B.n
|
|
|
+arrow from X.s to B.n
|
|
|
+arc <- from B.w to A2.w
|
|
|
+.PE
|
|
|
+
|
|
|
+.LP
|
|
|
+The dashed box with `A' in it means that it can be deleted, because
|
|
|
+there is already an occurrence of it in the loop.
|
|
|
+
|
|
|
+.LP
|
|
|
+The most beautiful result of loops in graphs is
|
|
|
+that the original parsing algorithm needs only one minor change.
|
|
|
+When the algorithm visits an element which has more than one
|
|
|
+outgoing edge the algorithm starts tracking down both paths,
|
|
|
+just like before, only now there may be one or more backedges among
|
|
|
+these edges, but the algorithm needs not to be aware of this fact.
|
|
|
+The only difficulty with loops is that the algorithm might go into
|
|
|
+a loop; it continues searching for terminals but it might happen
|
|
|
+that there are no valid terminals in the loop. The solution to this
|
|
|
+problem is not very difficult; just set a flag at all elements we
|
|
|
+visit. When we reach an element which has this flag turned on, we
|
|
|
+don't have to search any further. At the end of the prediction phase, when we
|
|
|
+have found all possible new heads, all flags are cleared.
|
|
|
+Even if there are no loops in the
|
|
|
+prediction graph, setting flags may be used as an optimization:
|
|
|
+it is possible that two paths come together at one point. In that situation
|
|
|
+it is useless to scan for the second time the part of the graph which
|
|
|
+both paths have in common.
|
|
|
+
|
|
|
+.NH 2
|
|
|
+Some optimizations using reference counts
|
|
|
+
|
|
|
+.LP
|
|
|
+As explained in section 2.2, it is sometimes necessary to copy a
|
|
|
+prediction graph element before substituting it. In order to determine
|
|
|
+if a certain element has to be copied, it is convenient to maintain
|
|
|
+a reference count in each graph element. This reference count keeps
|
|
|
+track of the number of edges that enter an element. Now, when we want
|
|
|
+to substitute an element with reference count not 0, we need to
|
|
|
+copy it, because there is another path in the prediction graph that
|
|
|
+contains the element we want to substitute, and on this other path
|
|
|
+the element cannot be substituted yet.
|
|
|
+
|
|
|
+.LP
|
|
|
+Maintaining reference counts also enables us to perform another
|
|
|
+optimization: remember that if, in a prediction phase, a terminal
|
|
|
+is predicted that does not match the current inputsymbol, we from
|
|
|
+then on just ignore the path in the graph starting at the terminal.
|
|
|
+However, we can safely delete the terminal from the graph; furthermore,
|
|
|
+all its successors in the prediction graph that have reference count
|
|
|
+0 can be deleted as well, as can their successors with reference
|
|
|
+count 0, etc. This way, we delete from the prediction graph
|
|
|
+most elements that are no longer accessible, but not all of them; as will
|
|
|
+be explained in the next section, loops in the prediction graph
|
|
|
+can cause problems.
|
|
|
+
|
|
|
+.NH 2
|
|
|
+The algorithm to delete inaccessible loops
|
|
|
+
|
|
|
+.LP
|
|
|
+Deleting graph elements which are no longer reachable is not as easy
|
|
|
+as it looks when there are loops in the graph, introduced by
|
|
|
+the extension to the algorithm that handles left recursive grammars.
|
|
|
+Suppose for example that we have a very simple loop as in the left
|
|
|
+picture below:
|
|
|
+
|
|
|
+.PS
|
|
|
+down
|
|
|
+X: box "x" "(0)"
|
|
|
+arrow
|
|
|
+box "[B]" "(2)"
|
|
|
+arrow
|
|
|
+box "a" "(1)"
|
|
|
+arrow
|
|
|
+box "[A]" "(1)"
|
|
|
+arrow
|
|
|
+box "[D]" "(1)"
|
|
|
+arc <- from 2nd box.w to 2nd last box.w
|
|
|
+
|
|
|
+move right from X.ne
|
|
|
+move
|
|
|
+move
|
|
|
+move
|
|
|
+move
|
|
|
+move
|
|
|
+move
|
|
|
+down
|
|
|
+box "x" "(0)" dashed
|
|
|
+arrow dashed
|
|
|
+B: box "[B]" "(1)"
|
|
|
+arrow
|
|
|
+box "a" "(1)"
|
|
|
+arrow
|
|
|
+box "[A]" "(1)"
|
|
|
+arrow
|
|
|
+box "[D]" "(1)"
|
|
|
+arc <- from B.w to 2nd last box.w
|
|
|
+.PE
|
|
|
+
|
|
|
+.LP
|
|
|
+The number below each symbol indicates the reference count of that element.
|
|
|
+Suppose now that we delete `x', then we have the situation depicted in the
|
|
|
+picture on the right. The loop consisting of `[B]', `a' and `[A]' is now
|
|
|
+unreachable, so all these elements can be deallocated.
|
|
|
+The reference count of `[B]' is 1, so it will not be deleted. To be precise
|
|
|
+all elements in the loop have their reference counts on 1, and
|
|
|
+consequently none of these will be deleted. But we stated earlier
|
|
|
+that all elements of the loop cannot be reached anymore and that the
|
|
|
+loop had to be deleted! In this example the reference counts of the
|
|
|
+loop elements are all 1, but in more complex situations it is also
|
|
|
+possible that some of the elements have a reference count of more
|
|
|
+than 1.
|
|
|
+
|
|
|
+.LP
|
|
|
+To solve this problem we present an algorithm, devised by E. Wattel, that
|
|
|
+determines whether a loop can be deleted or not.
|
|
|
+The algorithm consists of two parts. The first part of the algorithm goes as
|
|
|
+follows: it presumes that all elements of the loop will indeed be
|
|
|
+deleted. Every time it deletes an element it decreases the reference
|
|
|
+count of all the successors of the element that are also member of the same
|
|
|
+loop. How the algorithm knows which elements belong to the loop and which
|
|
|
+do not will be explained later. The situation of the example above will now
|
|
|
+look like this:
|
|
|
+
|
|
|
+.PS
|
|
|
+down
|
|
|
+box "[B]" "(0)"
|
|
|
+arrow
|
|
|
+box "a" "(0)"
|
|
|
+arrow
|
|
|
+box "[A]" "(0)"
|
|
|
+arrow
|
|
|
+box "[D]" "(1)"
|
|
|
+arc <- from 1st box.w to 2nd last box.w
|
|
|
+.PE
|
|
|
+
|
|
|
+.LP
|
|
|
+The number below each symbol indicates again the reference count
|
|
|
+after we have applied the first part of the algorithm.
|
|
|
+
|
|
|
+.LP
|
|
|
+The second part of the algorithm checks and restores the
|
|
|
+reference counts of all members of the loop . When it finds
|
|
|
+out that one or more reference counts are not 0, it concludes
|
|
|
+that it is still possible to enter the loop in some way, and
|
|
|
+that it cannot be
|
|
|
+deleted yet. In the other case it reports that the loop can be
|
|
|
+deleted, which is also true in our example.
|
|
|
+
|
|
|
+.LP
|
|
|
+We will now formally describe the first part of the algorithm
|
|
|
+that finds all directed circuits from a given vertex, and determines if
|
|
|
+the vertices on those circuits can be deleted.
|
|
|
+The algorithm works on prediction-graphs in which every edge that
|
|
|
+is in a circuit is marked. Note that a marked edge may be in more than one circuit.
|
|
|
+We will call this mark `C'.
|
|
|
+The input to the algorithm is such a prediction graph, and a start vertex,
|
|
|
+say A. The first part of the algorithm is:
|
|
|
+
|
|
|
+.IP 1
|
|
|
+Put the start vertex A on a list L; mark all edges `unused'
|
|
|
+.IP 2
|
|
|
+If L is empty, stop
|
|
|
+.IP 3
|
|
|
+For each vertex in list L, check if there are edges marked both C' and
|
|
|
+`unused'. For each edge found, mark it `used', and traverse it to its
|
|
|
+other endpoint; put this endpoint on a new list M, initially empty
|
|
|
+.IP 4
|
|
|
+Decrease the reference count of all vertices on M by 1
|
|
|
+.IP 5
|
|
|
+L := M; go to 2
|
|
|
+
|
|
|
+.LP
|
|
|
+It is clear that the algorithm will terminate: each edge is only traversed once,
|
|
|
+and the number of edges is finite. We will now prove some properties of this
|
|
|
+part of the algorithm.
|
|
|
+
|
|
|
+.LP
|
|
|
+.I
|
|
|
+An edge is traversed by the algorithm if and only if it is on some
|
|
|
+directed circuit $A ->$...$->A$.
|
|
|
+.R
|
|
|
+.br
|
|
|
+
|
|
|
+The if-part is easy; if an edge $e$ connecting vertices $W$ and $V$ is on some directed circuit starting in
|
|
|
+$A$, then there is a path $A ->$...$-> W -> V$; let $A ->$...$-> W -> V$ be a path
|
|
|
+of minimum length from $A$ to $V$. If the length of the path from $A$ to
|
|
|
+$W$ is $k$, then after turn $k$ of the algorithm $W$ will be on list L. To see
|
|
|
+that this is the case, suppose that $W$ is not on list L after turn $k$;
|
|
|
+this means that the edge entering $W$ was already marked used in a
|
|
|
+previous turn, but then there would be a shorter path from $A$
|
|
|
+to $W$, contradicting the assumption that the path is of
|
|
|
+minimum length. The edge
|
|
|
+$e$ is marked `C', because it is in a circuit; it is marked `unused', for if
|
|
|
+it were marked used, there would be a shorter path from $A$ to $V$. So,
|
|
|
+in turn $k + 1$, the edge $e$ will be traversed.
|
|
|
+
|
|
|
+.LP
|
|
|
+On the other hand, suppose that an edge $e$ is traversed by the algorithm;
|
|
|
+we will show by induction on the number of turns the algorithm has made
|
|
|
+that $e$ is on a directed circuit $A->$..$->A$. In the first turn, all
|
|
|
+edges from $A$ that are marked `C' are traversed, and clearly, if an edge
|
|
|
+from $A$ is part of a circuit then that edge is part of a circuit from $A$ to $A$.
|
|
|
+Now suppose that in turn $n+1$ an edge $e$ connecting vertices $W$ and
|
|
|
+$V$ is traversed. This means the edge is
|
|
|
+marked `C', so it is part of some circuit. If there is a path from $V$ to $A$,
|
|
|
+we can simply trace a circuit
|
|
|
+$A->$...$-> W -> V -> $...$-> A$, and clearly $e$ is on a circuit from
|
|
|
+$A$ to $A$. Now, suppose there is no path from $V$ to
|
|
|
+$A$. We can always trace a circuit $W -> V ->$...$-> W$ because the
|
|
|
+edge from $W$ to $V$ is part of a circuit; and by the
|
|
|
+induction hypothesis there is a circuit $A ->$...$-> W ->$...$-> A$. We can
|
|
|
+now make a `detour' at $W$, yielding a circuit $A->$...$-> W -> V$...
|
|
|
+$-> W ->$...$-> A$. This case is shown in the picture below.
|
|
|
+So in either case $e$ is on a circuit from $A$ to $A$.
|
|
|
+
|
|
|
+.PS
|
|
|
+down;
|
|
|
+B1: box "A";
|
|
|
+arrow dashed;
|
|
|
+B3: box dashed;
|
|
|
+arrow dashed;
|
|
|
+B2: box "W";
|
|
|
+arrow dashed; box dashed;
|
|
|
+arc <- from B1.w to last box.w
|
|
|
+arrow right "$e$" "C" from B2.e
|
|
|
+box "V"; arrow dashed; box dashed;
|
|
|
+arrow dashed -> from last box.n to B3.e
|
|
|
+.PE
|
|
|
+
|
|
|
+.LP
|
|
|
+.I
|
|
|
+A vertex appears on list L if and only if it is on some directed
|
|
|
+circuit from $A$ to $A$.
|
|
|
+.R
|
|
|
+.br
|
|
|
+
|
|
|
+.LP
|
|
|
+If a vertex is in such a circuit, there is an edge that enters it, which
|
|
|
+is part of a circuit form $A$ to $A$; we already showed that this edge
|
|
|
+is traversed by the algorithm, and thus the vertex will appear on list
|
|
|
+L. Conversely, if a vertex appears on list L, then an edge entering
|
|
|
+that vertex has been traversed by the algorithm; we showed that this
|
|
|
+edge is part of a circuit from $A$ to $A$, and thus the vertex is
|
|
|
+part of a circuit from $A$ to $A$.
|
|
|
+
|
|
|
+.LP
|
|
|
+.I
|
|
|
+When the algorithm is finished, each vertex that is part of some
|
|
|
+directed circuit from $A$ to $A$ has its reference count decreased by exactly
|
|
|
+the number of edges entering it that are part of a directed circuit from $A$ to $A$.
|
|
|
+.R
|
|
|
+.br
|
|
|
+
|
|
|
+.LP
|
|
|
+Each edge that is part of some circuit from $A$ to $A$ is traversed
|
|
|
+exactly once; the reference count of the endpoint is decreased
|
|
|
+by one after an edge has been traversed. Thus, if a vertex is endpoint
|
|
|
+of $k$ such vertices, its reference count is decreased by $k$.
|
|
|
+
|
|
|
+.LP
|
|
|
+.I
|
|
|
+If the reference count of each of the vertices visited by the algorithm
|
|
|
+is 0 after the algorithm has finised, all these vertices can be deleted;
|
|
|
+if the reference count is not zero for one or more of the visited
|
|
|
+vertices, then none of them can be deleted.
|
|
|
+.R
|
|
|
+.br
|
|
|
+
|
|
|
+.LP
|
|
|
+Suppose all visited vertices have reference count 0; this means that
|
|
|
+each of the vertices is only entered by edges that are on a circuit
|
|
|
+from $A$ to $A$. Therefore, it holds that any path leading to any
|
|
|
+of the visited vertices has to start in one of the visited vertices; there
|
|
|
+is no path starting in an unvisited vertex to a visited one. Thus,
|
|
|
+all the visited vertices are unreachable.
|
|
|
+Conversely, if one of the visited vertices has reference count not zero,
|
|
|
+then there is a path from an unvisited vertex to this vertex. Because from
|
|
|
+the vertex with reference count non zero, we can get to $A$, and from $A$
|
|
|
+we can get to any of the other vertices, all visited vertices are
|
|
|
+reachable.
|
|
|
+
|
|
|
+.LP
|
|
|
+The second part of the algorithm now checks if all reference counts are
|
|
|
+zero, and if they are, it deletes all visited vertices.
|
|
|
+
|
|
|
+
|
|
|
+.NH 2
|
|
|
+Marking loop elements
|
|
|
+
|
|
|
+.LP
|
|
|
+One point we have omitted so far is how the edges in the prediction
|
|
|
+graph that are part of a loop get marked.
|
|
|
+Basically, a loop can be detected:
|
|
|
+
|
|
|
+ a. when it is made;
|
|
|
+.br
|
|
|
+ b. when we want to know about it.
|
|
|
+
|
|
|
+.LP
|
|
|
+The first approach checks if a loop is constructed
|
|
|
+as soon as we join two paths in the graph, and if so, marks all
|
|
|
+edges of the loop. The other approach does not do any checking when two
|
|
|
+paths are joined together; it starts looking for loops when we want
|
|
|
+to delete an element with reference count not 0, marking all edges
|
|
|
+belonging to the loops it discovers. In practice it turns out that
|
|
|
+we very often encounter elements that we would like to delete, but that have
|
|
|
+reference count not 0, whereas the joining of paths occurs relatively
|
|
|
+infrequently. We therefore have chosen to check if a loop is created
|
|
|
+when two paths in a prediction graph are joined.
|
|
|
+
|
|
|
+.LP
|
|
|
+Now the question arises how to find and mark all edges of
|
|
|
+the loop. For this problem we devised also an algorithm.
|
|
|
+Because we already know that there is an edge from the element on which
|
|
|
+the new path is connected to the successor of the joined element, the
|
|
|
+algorithm only has to find a path from this last element back to the first one.
|
|
|
+This can be done by a backtracking depth first search; to find a path from
|
|
|
+one element to another we have to find a possible empty path
|
|
|
+from one of the successors of the first element to the last element. As
|
|
|
+soon as we have found a path, we can mark all the edges on the path and also
|
|
|
+the backedge as loop edges. In case that there is more than one path
|
|
|
+back to the first element it is necessary that the algorithm continues
|
|
|
+searching after it has found one path.
|
|
|
+
|
|
|
+.LP
|
|
|
+To avoid looping of this algorithm we have to set a flag at the elements
|
|
|
+which are on the path already. When the algorithm is backtracking it can
|
|
|
+clear the flags at the elements it is leaving.
|
|
|
+
|
|
|
+.LP
|
|
|
+To speed up the searching process we can set flags at the edges we have already
|
|
|
+visited but did not lead back to the first element. When the algorithm
|
|
|
+encounters such an edge it already knows that this edge is not worth
|
|
|
+searching again and can be skipped. At the end of the algorithm these
|
|
|
+flags have to be cleared again.
|
|
|
+
|
|
|
+.LP
|
|
|
+One might propose another optimization: as soon as
|
|
|
+we reach an edge that is already marked as a loop edge, we
|
|
|
+can stop searching for other loop edges. There is, however,
|
|
|
+a case in which this can go wrong. Imagine the following situation:
|
|
|
+
|
|
|
+.PS
|
|
|
+down
|
|
|
+E: box "[E]"
|
|
|
+arrow " C" ljust
|
|
|
+D: box "[D]"
|
|
|
+arrow " C" ljust
|
|
|
+C: box "c"
|
|
|
+arrow " C" ljust
|
|
|
+box "b"
|
|
|
+arrow " C" ljust
|
|
|
+A: box "[A]"
|
|
|
+arrow
|
|
|
+box "a"
|
|
|
+
|
|
|
+move right from D
|
|
|
+move right
|
|
|
+J: box "[J]"
|
|
|
+down
|
|
|
+arrow from J.s " C" ljust
|
|
|
+I: box "i"
|
|
|
+arrow " C" ljust
|
|
|
+H: box "[H]"
|
|
|
+arrow from H.s to A.e
|
|
|
+
|
|
|
+arc <- from E.w to A.w
|
|
|
+move left from C
|
|
|
+move left
|
|
|
+"C"
|
|
|
+arc -> from H.e to J.e
|
|
|
+move right from I
|
|
|
+move right
|
|
|
+"C"
|
|
|
+
|
|
|
+arrow dashed from E.s to J.n
|
|
|
+
|
|
|
+
|
|
|
+.PE
|
|
|
+
|
|
|
+What we have here is a prediction graph with two loops; all edges that belong
|
|
|
+to a loop are again marked with an `C'. Note that the edge between `[H]'
|
|
|
+and `[A]' is not a loop edge. Suppose that `[J]' is not yet
|
|
|
+completely substituted, i.e. there is another production rule for
|
|
|
+J:
|
|
|
+.br
|
|
|
+
|
|
|
+J: E
|
|
|
+
|
|
|
+.br
|
|
|
+The `E' on top of the right path is now joined with the `[E]'
|
|
|
+on the left path, which is depicted by the dashed arrow
|
|
|
+between `[E]' and `[J]'. When we take a good look at the graph
|
|
|
+we see that the two loops are merged into one. But that is not
|
|
|
+the most important observation we have to make: not only the
|
|
|
+edge between `[E]' and `[J]' must be marked as a loop edge, but
|
|
|
+also the edge between `[H]' and `[A]'! So it is not possible
|
|
|
+to stop searching for loop edges as soon as we have found an
|
|
|
+edge which was already marked as a loop edge. We have to continue
|
|
|
+until we reach the element at which we started: `[E]'. So the
|
|
|
+optimization proposed above is incorrect.
|
|
|
+
|
|
|
+
|
|
|
+.NH 2
|
|
|
+Optimizations using FIRST and FOLLOW sets
|
|
|
+
|
|
|
+.LP
|
|
|
+In the algorithm as we have described it, every nonterminal on top of the graph
|
|
|
+is substituted until only terminals remain on top; these terminals are
|
|
|
+then matched against the current input symbol. However, by using
|
|
|
+FIRST sets, we can save considerably on the number of computations
|
|
|
+necessary. Suppose one of the top elements of the graph is nonterminal A,
|
|
|
+and the current inputsymbol is $a$. Then, it is of no use to substitute
|
|
|
+A if terminal $a$ is not in FIRST(A), because then substituting A will
|
|
|
+never produce $a$ on top of the graph. So, before substituting a
|
|
|
+nonterminal we check if the current inputsymbol is in its FIRST set; if
|
|
|
+it is not, we can declare the path the nonterminal is on a dead end, and
|
|
|
+delete it, without having to perform the actual substitution. Of course, if
|
|
|
+A can produce empty, we still have to consider its successor in the graph.
|
|
|
+
|
|
|
+.LP
|
|
|
+Similarly, when we have an END marker on top, with nonterminal B in
|
|
|
+it, and we consider using rule
|
|
|
+.br
|
|
|
+
|
|
|
+ D: $alpha$ B C $gamma$
|
|
|
+
|
|
|
+.br
|
|
|
+We first check if the current inputsymbol is in FIRST(C); if this is
|
|
|
+not the case, there is no need to start a graph component with this
|
|
|
+rule, because it will never produce the next inputsymbol on top.
|
|
|
+Again, if C produces empty, we still have to evaluate the part of the
|
|
|
+rule following C.
|
|
|
+
|
|
|
+.LP
|
|
|
+To circumvent the problems caused in the FIRST set optimization by
|
|
|
+nonterminal that produce empty, we can also make use of FOLLOW-sets.
|
|
|
+When substituting, if we encounter a nonterminal whose FIRST set does
|
|
|
+not contain the current inputsymbol but which can produce empty,
|
|
|
+we check if the current inputsymbol is in its FOLLOW set. If it is not,
|
|
|
+there is no need to process its successor. Similarly, in case we
|
|
|
+are processing an END marker as explained above, there is no need
|
|
|
+to process the part of the rule following C if FIRST(C) does not
|
|
|
+contain the input symbol, or C produces empty but the inputsymbol
|
|
|
+is not in FOLLOW(C).
|
|
|
+.bp
|
|
|
+.nr PS 12
|
|
|
+.nr VS 14
|
|
|
+
|
|
|
+.NH
|
|
|
+Test results
|
|
|
+
|
|
|
+.nr PS 10
|
|
|
+.nr VS 12
|
|
|
+.RS
|
|
|
+
|
|
|
+.LP
|
|
|
+In this chapter, we discuss some test results that were obtained
|
|
|
+by recompiling existing ACK compilers with the modified LLgen.
|
|
|
+We tried several combinations of possible optimizations, including
|
|
|
+`dumb' ones, like no optimization at all, not even deleting unreachable
|
|
|
+prediction graph elements.
|
|
|
+The incorporation of LLgen with non-correcting error recovery went
|
|
|
+smoothly; only minor modifications to the Make-files were necessary.
|
|
|
+Specifically, these modifications consisted of passing an extra
|
|
|
+flag to LLgen, and including the new generated C-file Lncor.c in
|
|
|
+the list of generated C-files. Also, the LLmessage error reporting
|
|
|
+routine had to be adapted. We successfully recompiled the C, Modula-2
|
|
|
+and Occam compilers; in the next sections, we discuss some test results
|
|
|
+that were obtained with the Modula-2 and C compilers.
|
|
|
+
|
|
|
+.RE
|
|
|
+.LP
|
|
|
+.NH 2
|
|
|
+Performance
|
|
|
+
|
|
|
+.LP
|
|
|
+We will now present and discuss, with the aid of some
|
|
|
+diagrams, time and space measurements on the non-correcting error
|
|
|
+recovery. We have measured the effect of various optimizations.
|
|
|
+These optimizations include the first-set optimization and the follow-set
|
|
|
+optimization. We also measured the effect of leaving out the loop-deletion
|
|
|
+algorithm, regarding both time and space. We performed out measurements using
|
|
|
+C- and Modula-2-programs of three different sizes; one of approximately
|
|
|
+750 tokens, one of appr. 5000 tokens and one of appr. 15000 tokens. We have
|
|
|
+chosen to represent the sizes of programs in the number of tokens instead of
|
|
|
+number of lines, because the number of tokens more realistically
|
|
|
+reflects the load the programs put on the error recovery mechanism. Also we give
|
|
|
+our time measurements in usertime instead of realtime, because realtime
|
|
|
+depends heavily on the load of the system, which usertime does not.
|
|
|
+Our space measurements are based on the size of the prediction graphs.
|
|
|
+Note that all files are entirely recognized by the non-correcting error
|
|
|
+recovery technique. We achieved this by putting a `1' at the beginning
|
|
|
+of each file; because then each file starts with a syntax error LLgen
|
|
|
+is forced to continue with the non-correcting error recovery.
|
|
|
+
|
|
|
+.NH 3
|
|
|
+Time and space measurements on the effect of the first-set optimization
|
|
|
+
|
|
|
+.LP
|
|
|
+In the diagram below we show our time measurements we got from recognizing
|
|
|
+the C-programs both with and without first-set optimization.
|
|
|
+
|
|
|
+.G1
|
|
|
+coord x 0, 17000 y 0, 65
|
|
|
+ticks bot out at 750, 5000, 15000
|
|
|
+label bot "Number of tokens"
|
|
|
+label left "User Time" "(sec)" left .3
|
|
|
+draw no_opt dashed
|
|
|
+draw first_opt dashed
|
|
|
+
|
|
|
+copy thru X
|
|
|
+ times size +2 at $1, $2
|
|
|
+ times size +2 at $1, $3
|
|
|
+ next no_opt at $1, $2
|
|
|
+ next first_opt at $1, $3
|
|
|
+X until "XXX"
|
|
|
+
|
|
|
+742 2.5 .9
|
|
|
+5010 16.3 5.8
|
|
|
+14308 54.2 16.8
|
|
|
+XXX
|
|
|
+
|
|
|
+copy thru X "$1 $2" size -2 at 11000, $3 X until "XXX"
|
|
|
+No optimization 55
|
|
|
+First-set optimization 20
|
|
|
+XXX
|
|
|
+.G2
|
|
|
+
|
|
|
+.I
|
|
|
+.ce
|
|
|
+Time measurements of three C-programs with and without first-set optimization
|
|
|
+.R
|
|
|
+
|
|
|
+.LP
|
|
|
+Notice the considerable time savings we
|
|
|
+get when the first-set optimization is turned on; a factor of slightly more than
|
|
|
+3. Obviously this is an extremely useful optimization. On the other hand
|
|
|
+we found there were no measurable time savings when using the follow-set
|
|
|
+optimization; for that reason we did not chart the result of this optimization.
|
|
|
+It seems that the time savings gained by the optimization are
|
|
|
+waisted again by the extra processing time needed. We conclude that
|
|
|
+this optimization is of little or no use when we want to save on time.
|
|
|
+
|
|
|
+.LP
|
|
|
+In the following picture the time measurements of three Modula-2 programs
|
|
|
+are given, again with and without first-set optimization.
|
|
|
+
|
|
|
+.G1
|
|
|
+coord x 0, 17000 y 0, 65
|
|
|
+ticks bot out at 750, 5000, 15000
|
|
|
+label bot "Number of tokens"
|
|
|
+label left "User Time" "(sec)" left .3
|
|
|
+draw no_opt dashed
|
|
|
+draw first_opt dashed
|
|
|
+copy thru X
|
|
|
+ times size +2 at $1, $2
|
|
|
+ times size +2 at $1, $3
|
|
|
+ next no_opt at $1, $2
|
|
|
+ next first_opt at $1, $3
|
|
|
+X until "XXX"
|
|
|
+
|
|
|
+823 1.3 .6
|
|
|
+4290 7.6 3.5
|
|
|
+16530 30.5 14.3
|
|
|
+XXX
|
|
|
+
|
|
|
+copy thru X "$1 $2" size -2 at 13000, $3 X until "XXX"
|
|
|
+No optimization 30
|
|
|
+First-set optimization 15
|
|
|
+XXX
|
|
|
+.G2
|
|
|
+
|
|
|
+.I
|
|
|
+.ce
|
|
|
+Time measurements of three Modula-2-programs with and without first-set optimization
|
|
|
+.R
|
|
|
+
|
|
|
+.LP
|
|
|
+From this picture we can conclude mainly the same as above; considerable
|
|
|
+time savings when we use the first-set optimization;
|
|
|
+the factor is somewhat less, but still more than 2. Again we have omitted
|
|
|
+the results of the follow-set optimization, for the same reason as before.
|
|
|
+
|
|
|
+.LP
|
|
|
+There is however one remarkable difference between the two languages: parsing
|
|
|
+C-programs needs almost twice the time as parsing programs of comparable
|
|
|
+sizes written in Modula-2. This can be explained by the fact that the
|
|
|
+C-grammar is far more complicated than that of Modula-2, and also the
|
|
|
+production rules are longer in C, so building, deleting and definitely
|
|
|
+traversing the graph will consume more time.
|
|
|
+
|
|
|
+.LP
|
|
|
+Now we come to the space measurements of both C- and Modula-2 programs.
|
|
|
+In the picture below we present the maximum sizes of the prediction graphs,
|
|
|
+during the recognition of the three C-programs.
|
|
|
+
|
|
|
+.G1
|
|
|
+coord x 0, 17000 y 0, 18000
|
|
|
+ticks bot out at 750, 5000, 15000
|
|
|
+label bot "Number of tokens"
|
|
|
+label left "Maximum size of" "the prediction graph" "(bytes)"left .3
|
|
|
+draw no_opt dashed
|
|
|
+draw first_opt dashed
|
|
|
+copy thru X
|
|
|
+ times size +2 at $1, $2
|
|
|
+ times size +2 at $1, $3
|
|
|
+ next no_opt at $1, $2
|
|
|
+ next first_opt at $1, $3
|
|
|
+X until "XXX"
|
|
|
+
|
|
|
+742 5568 10444
|
|
|
+5010 7668 12664
|
|
|
+14308 13636 17308
|
|
|
+XXX
|
|
|
+
|
|
|
+copy thru X "$1 $2" size -2 at 8000, $3 X until "XXX"
|
|
|
+No optimization 16000
|
|
|
+First-set optimization 7000
|
|
|
+XXX
|
|
|
+.G2
|
|
|
+
|
|
|
+.I
|
|
|
+.ce
|
|
|
+Maximum sizes of the prediction graphs when recognizing three C-programs
|
|
|
+.R
|
|
|
+
|
|
|
+.LP
|
|
|
+From this diagram we see that, although the prediction graphs
|
|
|
+are smaller when the first-set optimization is used, the space savings are
|
|
|
+not as spectacular as the time savings achieved by this optimization.
|
|
|
+
|
|
|
+.LP
|
|
|
+In Modula-2 the first-set optimization also causes a decrease in memory
|
|
|
+usage. The savings are less than in C, but still about 1.5 Kb. Again
|
|
|
+this can be explained by the fact that the rules of the Modula-2 grammar
|
|
|
+are shorter than that of C.
|
|
|
+
|
|
|
+.G1
|
|
|
+coord x 0, 17000 y 0, 12000
|
|
|
+ticks bot out at 750, 5000, 15000
|
|
|
+label bot "Number of tokens"
|
|
|
+label left "Maximum size of" "the prediction graph" "(bytes)" left .3
|
|
|
+draw no_opt dashed
|
|
|
+draw first_opt dashed
|
|
|
+copy thru X
|
|
|
+ times size +2 at $1, $2
|
|
|
+ times size +2 at $1, $3
|
|
|
+ next no_opt at $1, $2
|
|
|
+ next first_opt at $1, $3
|
|
|
+X until "XXX"
|
|
|
+
|
|
|
+823 5056 3292
|
|
|
+4290 6420 4664
|
|
|
+16530 11388 9632
|
|
|
+XXX
|
|
|
+
|
|
|
+copy thru X "$1 $2" size -2 at 8000, $3 X until "XXX"
|
|
|
+No optimization 10000
|
|
|
+First-set optimization 4000
|
|
|
+XXX
|
|
|
+.G2
|
|
|
+
|
|
|
+.I
|
|
|
+.ce
|
|
|
+Maximum sizes of the prediction graphs when recognizing three Modula-2-programs
|
|
|
+.R
|
|
|
+
|
|
|
+.NH 3
|
|
|
+Input that is recognized in quadratic time
|
|
|
+
|
|
|
+.LP
|
|
|
+The measurements presented may suggest that the time required to
|
|
|
+recognize input depends linearly on the length of the input; however,
|
|
|
+this is not always the case. When there are recursive rules in the
|
|
|
+grammar, the time needed to recognize input that is produced by this
|
|
|
+rules can become proportional to the square of the input length.
|
|
|
+Consider this set of grammar rules:
|
|
|
+.br
|
|
|
+.nf
|
|
|
+
|
|
|
+ S: '{' A '}'
|
|
|
+ A: 'a' A | $epsilon$
|
|
|
+
|
|
|
+.fi
|
|
|
+.LP
|
|
|
+When the input is `{aaa....', the algorithm will produce the following
|
|
|
+prediction graphs:
|
|
|
+
|
|
|
+.PS
|
|
|
+up; B1: box "END" "S"; arrow <- ;box "}";arrow <- ;box "A";arrow <- ;box "{";
|
|
|
+move right from B1.se; move
|
|
|
+up; B2: box "END" "S"; arrow <-; box "}"; arrow <-; box "[A]";
|
|
|
+arrow <-; box "A"; arrow <-; box "a";
|
|
|
+move right from B2.se; move
|
|
|
+up; B3: box "END" "S"; arrow <-; box "}"; arrow <-; box "[A]";
|
|
|
+arrow <-; box "[A]"; arrow <-; box "A"; arrow <-; box "a";
|
|
|
+move right from B3.se;move
|
|
|
+up; B4: box "END" "S"; arrow <-; box "}"; arrow <-; box "[A]";
|
|
|
+arrow <-; box "[A]"; arrow <-; box "[A]"; arrow <- ; box "A"; arrow <-;box "a";
|
|
|
+.PE
|
|
|
+
|
|
|
+.LP
|
|
|
+In each prediction phase, a new [A] appears on the prediction graph. However,
|
|
|
+since A also produces empty, the prediction algorithm has to traverse all the
|
|
|
+elements [A] until it finds the element `}'. In the first prediction phase,
|
|
|
+there is one element [A], in the second there are two, etc, so in all
|
|
|
+1 + 2 + 3 + ... + k = $k(k+1) over 2$ elements have to be traversed if
|
|
|
+there are k prediction phases, making this proportional to the square
|
|
|
+of the input length. We constructed a parser with this simple input grammar
|
|
|
+and measured the processing time the error recovery mechanism used.
|
|
|
+In the following diagram the dashed line shows the processing time needed;
|
|
|
+the dotted line is the curve $t = 13 n sup 2$. Clearly the processing time
|
|
|
+is proportional to the square of the number of tokens.
|
|
|
+
|
|
|
+.G1
|
|
|
+coord x 0, 2100 y 0, 60
|
|
|
+ticks bot out at 500, 1000, 1500, 2000
|
|
|
+label bot "Number of tokens"
|
|
|
+label left "User Time" "(sec)" left .3
|
|
|
+draw quad dashed
|
|
|
+
|
|
|
+copy thru X
|
|
|
+ times size +2 at $1, $2
|
|
|
+ next quad at $1, $2
|
|
|
+X until "XXX"
|
|
|
+
|
|
|
+500 3.0
|
|
|
+1000 12.4
|
|
|
+1500 28.6
|
|
|
+2000 51.4
|
|
|
+XXX
|
|
|
+
|
|
|
+draw dotted
|
|
|
+for i from 0 to 2100 by 25 do { next at i, 0.000013 * i * i }
|
|
|
+.G2
|
|
|
+
|
|
|
+.LP
|
|
|
+In the grammar used for the C compiler, array initializations are handled by a recursive
|
|
|
+rule, so we would expect that the error recovery mechanism needs quadratic
|
|
|
+processing time to recognize such an initialization; we made measurements on
|
|
|
+the processing time and indeed, the
|
|
|
+processing time needed grows proportionally to the square of the size of the input, as the
|
|
|
+next figure shows. Here, the processing times are about half of those in
|
|
|
+the previous example; this is so because the recursion appears after two
|
|
|
+tokens are recognized. Note that the algorithm only takes quadratic time
|
|
|
+when it is recognizing input that is generated by a recursive grammar rule.
|
|
|
+Other input is still recognized in linear time, regardless of the fact that
|
|
|
+there are recursive grammar rules.
|
|
|
+
|
|
|
+.G1
|
|
|
+coord x 0, 5000 y 0, 85
|
|
|
+ticks bot out at 1150, 2400, 3600, 4800
|
|
|
+label bot "Number of tokens"
|
|
|
+label left "User Time" "(sec)" left .3
|
|
|
+draw quad dashed
|
|
|
+
|
|
|
+copy thru X
|
|
|
+ times size +2 at $1, $2
|
|
|
+ next quad at $1, $2
|
|
|
+X until "XXX"
|
|
|
+
|
|
|
+1150 5.1
|
|
|
+2400 20.3
|
|
|
+3600 43.7
|
|
|
+4800 78.6
|
|
|
+XXX
|
|
|
+.G2
|
|
|
+
|
|
|
+.LP
|
|
|
+Unfortunately, there is no easy way to speed up the recognition of these
|
|
|
+recursively defined language elements; they are caused by the substituted
|
|
|
+tokens that are left in the prediction graph, and we cannot just delete those
|
|
|
+`dummies' from the graph during a prediction phase because the `join' part of the
|
|
|
+prediction algorithm depends on them. One could traverse the graph after
|
|
|
+a prediction phase to delete the dummies, but then the processing
|
|
|
+time needed to recognize non-recursively defined language elements would
|
|
|
+increase dramatically. However, we feel that in practice things
|
|
|
+like large array initializations will not occur in hand-made programs; when
|
|
|
+they occur, it is probably in computer-generated programs, which normally
|
|
|
+will be correct anyway, meaning that the error recovery never sees them.
|
|
|
+When testing such generated programs, one is likely
|
|
|
+to use small test-cases, which are handled well by the error recovery.
|
|
|
+
|
|
|
+.NH 3
|
|
|
+Time measurements on the effect of leaving out the loop-deletion algorithm
|
|
|
+
|
|
|
+.LP
|
|
|
+We now show what effect the loop-deletion algorithm has on processing time.
|
|
|
+To put it another way: how much time can be saved when we turn off the
|
|
|
+loop-deletion algorithm. In the diagram below we give the measurements of
|
|
|
+the three C-programs; note that we do use the first-set optimization.
|
|
|
+
|
|
|
+.G1
|
|
|
+coord x 0, 17000 y 0, 22
|
|
|
+ticks bot out at 750, 5000, 15000
|
|
|
+label bot "Number of tokens"
|
|
|
+label left "User Time" "(sec)" left .3
|
|
|
+draw no_loop dashed
|
|
|
+draw loop dashed
|
|
|
+copy thru X
|
|
|
+ times size +2 at $1, $2
|
|
|
+ times size +2 at $1, $3
|
|
|
+ next no_loop at $1, $2
|
|
|
+ next loop at $1, $3
|
|
|
+X until "XXX"
|
|
|
+
|
|
|
+742 .9 .4
|
|
|
+5010 5.8 6.8
|
|
|
+14308 16.8 20.5
|
|
|
+XXX
|
|
|
+
|
|
|
+copy thru X "$1 $2" size -2 at 11300, $3 X until "XXX"
|
|
|
+With loop-deletion 20
|
|
|
+Without loop-deletion 9
|
|
|
+XXX
|
|
|
+.G2
|
|
|
+
|
|
|
+.I
|
|
|
+.ce
|
|
|
+Time measurements on processing three C-programs with and without the loop-deletion algorithm
|
|
|
+.R
|
|
|
+
|
|
|
+The diagram shows that the loop-deletion algorithm
|
|
|
+does not dramatically slow down the recognizing process. There is, however,
|
|
|
+a measurable time loss of \(+-25%. As we will see later, the loop-deletion
|
|
|
+algorithm will turn out to be extremely useful in efficient use of memory
|
|
|
+when there are many loops in the graph.
|
|
|
+
|
|
|
+The effect of the loop-detecion algorithm on parsing Modula-2 programs
|
|
|
+is even less than with C-programs; in fact there is no measurable
|
|
|
+time loss:
|
|
|
+
|
|
|
+.G1
|
|
|
+coord x 0, 17000 y 0, 15
|
|
|
+ticks bot out at 750, 5000, 15000
|
|
|
+label bot "Number of tokens"
|
|
|
+label left "User Time" "(sec)" left .3
|
|
|
+draw no_loop dashed
|
|
|
+draw loop dashed
|
|
|
+copy thru X
|
|
|
+ times size +2 at $1, $2
|
|
|
+ times size +2 at $1, $3
|
|
|
+ next no_loop at $1, $2
|
|
|
+ next loop at $1, $3
|
|
|
+X until "XXX"
|
|
|
+
|
|
|
+823 .6 .6
|
|
|
+4290 3.5 3.8
|
|
|
+16530 14.3 14.3
|
|
|
+XXX
|
|
|
+
|
|
|
+copy thru X "$1 $2" size -2 at 11800, $3 X until "XXX"
|
|
|
+With loop-deletion 13
|
|
|
+Without loop-deletion 7
|
|
|
+XXX
|
|
|
+.G2
|
|
|
+
|
|
|
+.I
|
|
|
+.ce
|
|
|
+Time measurements on processing three Modula-2-programs with and without a loop-deletion algorithm
|
|
|
+.R
|
|
|
+
|
|
|
+There are at least two reasons for this; both result from the relative
|
|
|
+simplicity of the Modula-2 grammar. The distance from a head to an
|
|
|
+end of stack marker is shorter than in C, and secondly Modula-2
|
|
|
+causes fewer joins to occur than C, meaning that the loop marking algorithm
|
|
|
+is run less often and when it is run it has fewer paths to search.
|
|
|
+
|
|
|
+
|
|
|
+.NH 3
|
|
|
+Space measurements on the effect of leaving out the loop-deletion algorithm
|
|
|
+
|
|
|
+.LP
|
|
|
+Clearly, to make any measurements on the space-usage effects of leaving out
|
|
|
+the loop-deletion algorithm we need a program that causes the prediction
|
|
|
+graph to contain loops; however, we have not been able to devise a C
|
|
|
+or Modula-2 program that does this. In order to be able to make measurements,
|
|
|
+we added an extra alternative to a rule of the C compiler grammar, making
|
|
|
+it directly left-recursive. To make LLgen accept this new grammar, we
|
|
|
+put a `%if' directive in the rule.
|
|
|
+
|
|
|
+.LP
|
|
|
+We have input our standard C test program consisting of 800 tokens to
|
|
|
+the error recovery routine for this `doctored' C compiler,
|
|
|
+and compared the storage needed for the prediction graphs with the
|
|
|
+loop deletion algorithm enabled with the storage needed when the
|
|
|
+algorithm is disabled. With the loop-deletion algorithm enabled, the
|
|
|
+maximum size of the prediction graph was 5576 bytes. When the loop
|
|
|
+algorithm was disabled, the maximum size of the prediction graph
|
|
|
+grew to 12676 bytes; furthermore, 12676 bytes of heap were allocated
|
|
|
+for the prediction graph, but not deallocated again, because they were
|
|
|
+in use by graph elements that were in inaccessible loops. The user-time
|
|
|
+the program needed decreased only slightly, from 0.9 to 1.0 seconds. Given the
|
|
|
+relatively small input program, this data suggests that when loops
|
|
|
+are actually being made, the loop deletion algorithm is definitely
|
|
|
+worth the extra overhead it costs, considering the space
|
|
|
+that would otherwise be occupied by inaccessible loops. To verify this,
|
|
|
+we input the C program consisting of 15000 tokens to the compiler;
|
|
|
+execution time increased from 17.3 to 21.1 seconds after enabling
|
|
|
+the loop deletion algorithm, while the maximum size of the prediction graph
|
|
|
+shrunk from 328664 to 13664 bytes. With the loop-deletion algorithm
|
|
|
+disabled, 326720 bytes allocated for the graph were not deallocated again.
|
|
|
+Again, given the relatively small increase in execution time and the
|
|
|
+large reduction of memory usage, we feel that the loop-deletion
|
|
|
+algorithm is useful enough to justify the overhead it creates.
|
|
|
+
|
|
|
+.NH 2
|
|
|
+Problems encountered
|
|
|
+
|
|
|
+.LP
|
|
|
+In this section we describe some of the problems we encountered
|
|
|
+while testing the non-correcting error recovery.
|
|
|
+
|
|
|
+.NH 3
|
|
|
+The LLgen error reporting mechanism.
|
|
|
+
|
|
|
+.LP
|
|
|
+The parsers generated by LLgen call a user-supplied error reporting
|
|
|
+routine, usually called LLmessage. This routine is called with an
|
|
|
+integer parameter that is positive, zero or negative. When the parameter
|
|
|
+is positive the parser has just inserted a token, whose
|
|
|
+number is equal to the parameter; if it is zero, the parser
|
|
|
+has deleted a token whose number is in a global variable called LLsymb; if
|
|
|
+it is negative, it means that LLgen expected end-of-file, but did not
|
|
|
+find it. The routine LLmessage is supposed to print an error message,
|
|
|
+and when a token is inserted, it should set all necessary attributes.
|
|
|
+
|
|
|
+.LP
|
|
|
+However, when non-correcting error recovery is used, the situation becomes slightly
|
|
|
+different; when the parser inserts a token, it is only to keep the
|
|
|
+semantic actions consistent, and does no longer signify an error.
|
|
|
+However, the LLmessage routine still has to be called because the
|
|
|
+attributes of the inserted token need to be set. Therefore, when
|
|
|
+non-correcting error recovery is used, the LLmessage routine should not
|
|
|
+print an error message when the parameter is positive, or else it will
|
|
|
+print highly confusing error messages indeed. Furthermore, the
|
|
|
+LLmessage routine will usually print a message like `token ... deleted' when
|
|
|
+it is called with parameter equal to zero; however, when the non-correcting
|
|
|
+error recovery is used, it is more appropriate to report something
|
|
|
+like `token ... illegal', as the non-correcting error recovery does
|
|
|
+not delete tokens. Finally, when an unexpected end-of-file is encountered,
|
|
|
+LLgen normally just inserts the missing tokens and calls
|
|
|
+LLmessage with the parameter equal to the token number;
|
|
|
+when non-correcting error recovery is used we need a way to
|
|
|
+actually report we have encountered an unexpected end-of-file. The
|
|
|
+way we achieved this is by calling LLgen with parameter 0 and the
|
|
|
+global variable LLsymb set to EOFILE when this situation occurs; the
|
|
|
+routine LLmessage should print something like `unexpected end of file'
|
|
|
+when it is called with parameter 0 and LLsymb is EOFILE. To facilitate
|
|
|
+switching between correcting and non-correcting error recovery, the
|
|
|
+file Lpars.h contains a statement `#define LLNONCORR' if non-correcting
|
|
|
+error recovery is used.
|
|
|
+
|
|
|
+
|
|
|
+.NH 3
|
|
|
+Parsers being started in semantic actions
|
|
|
+
|
|
|
+.LP
|
|
|
+LLgen allows the programmer to define more than one nonterminal as the
|
|
|
+start symbol of the input grammar; it will generate a parsing routine
|
|
|
+for each of the start symbols. However, the error recovery code
|
|
|
+is generated only once; it is shared by all parsers.
|
|
|
+The programmer is free to call any
|
|
|
+of the generated parsers whenever he wants; for instance, in the C-compiler
|
|
|
+a separate parser for expressions in #if and #elsif statements is used. Whenever
|
|
|
+the lexical analyzer encounters such a statement, it calls the expression
|
|
|
+parser. It is also possible to call a parser in a semantic action of
|
|
|
+another parser; in the MODULA-2 compiler a separate parser for
|
|
|
+definition modules is used. When the main parser encounters a
|
|
|
+FROM defmod IMPORT statement a semantic
|
|
|
+actions opens the definition module defmod and starts the parser for
|
|
|
+definition modules.
|
|
|
+
|
|
|
+.LP
|
|
|
+The fact that subparsers can be started just about anywhere causes
|
|
|
+problems when non-correcting error recovery is used.
|
|
|
+Suppose a parser calls another parser in a semantic action
|
|
|
+to parse a separate input file. In the Modula-2 compiler, after
|
|
|
+seeing the FROM defmod IMPORT statement a semantic action opens
|
|
|
+defmod and parses it; now, if a syntax error occurred before the
|
|
|
+FROM IMPORT statement, the non-correcting error recovery will not
|
|
|
+execute the action that opens and parses the definition module, but
|
|
|
+it will not report an error either, because the statement
|
|
|
+FROM defmod IMPORT is part of the input language of the main parser.
|
|
|
+However, suppose that during the parsing of a definition module
|
|
|
+an error occurs; then, some semantic actions that would normally
|
|
|
+be executed during parsing of the definition module will not have
|
|
|
+taken place. When normal parsing is now resumed by the main parser,
|
|
|
+after the non-correcting error recovery has finished with the
|
|
|
+definition module, a lot of spurious semantic errors are likely to be
|
|
|
+reported, because the semantic actions that would normally have been
|
|
|
+executed during the definition module parsing have not been executed
|
|
|
+by the error recovery. Therefore, it is desirable that the main parser
|
|
|
+does not resume normal parsing, but instead continues with the non-correcting
|
|
|
+error recovery as well. Any syntactic errors in the main program will
|
|
|
+still be reported, but no spurious semantic errors will be reported
|
|
|
+that way.
|
|
|
+
|
|
|
+.LP
|
|
|
+When the lexical analyzer calls other parsers, as is the case in
|
|
|
+the ACK C compiler, recursive invocations of the non-correcting error
|
|
|
+recovery routine can occur. This will happen if a parser starts the
|
|
|
+error recovery, the error recovery calls the lexical analyzer, which
|
|
|
+starts another parser that finds a syntax error and calls the
|
|
|
+error recovery again. This is not really a problem, but is has
|
|
|
+consequences for the implementation of the error recovery routine.
|
|
|
+
|
|
|
+.LP
|
|
|
+The worst case
|
|
|
+occurs when two parsers are involved in parsing one input file, and
|
|
|
+the secondary parser (e.g. an inline assembly parser) is called in a semantic
|
|
|
+action of the main parser. Suppose now that the input text contains
|
|
|
+a syntax error; after detecting this error, the parser starts the
|
|
|
+non-correcting error recovery. This recovery does not execute any
|
|
|
+semantic actions; therefore it will not start the subparser at those points
|
|
|
+where the original LLgen generated parser would. As a result, parts
|
|
|
+of the program that would be accepted by the subparser will now probably
|
|
|
+be rejected as illegal, because the error recovery does not know it
|
|
|
+should use another grammar to check these parts. This is a serious
|
|
|
+problem, and we have devised and implemented two ways to solve it.
|
|
|
+
|
|
|
+.LP
|
|
|
+The first solution is based on the assumption that whenever a semantic
|
|
|
+action occurs in the grammar, another parser can be started at that
|
|
|
+point. Obviously, we have no way of knowing which semantic actions start
|
|
|
+a parser and which don't, so we assume the worst.
|
|
|
+Now, assume that in the grammar there are k symbols defined as
|
|
|
+start symbols, say $W sub 1 , W sub 2 , ..., W sub k$. Each of these symbols
|
|
|
+will cause LLgen to generate a parser that can be called in any
|
|
|
+of the semantic actions of the grammar. We now introduce a new
|
|
|
+symbol $X$, and a new grammar rule $X -> W sub 1 X | W sub 2 X | ... |
|
|
|
+W sub k X |
|
|
|
+epsilon$.
|
|
|
+In the grammar the error recovery algorithm uses, we insert this symbol
|
|
|
+X at all positions where there are semantic actions in the original grammar,
|
|
|
+so a rule $A -> alpha$ { action } $beta$ becomes $A -> alpha X beta$. As a
|
|
|
+result, at each position in a grammar rule where a semantic action
|
|
|
+occurs, we now accept any input that would be accepted by any of the
|
|
|
+parsers. Clearly, this solution is somewhat of a kludge, as it will
|
|
|
+accept a lot of input that is not accepted by the original parser.
|
|
|
+However, it is guaranteed to never give spurious error messages, because
|
|
|
+whenever a parser would be started by the original parser, there now
|
|
|
+is an $X$ in the grammar that produces all the strings that would be
|
|
|
+accepted by that parser. We have implemented this solution, and found
|
|
|
+it to be extremely slow, which of course was to be expected given the
|
|
|
+number of semantic actions in the average grammar. Furthermore,
|
|
|
+because each time a semantic action occurs in the grammar
|
|
|
+a string accepted by any of the generated parsers is accepted, including
|
|
|
+strings recognized by the currently running parser, error messages
|
|
|
+become hard to interpret. As an example, consider the following
|
|
|
+C program:
|
|
|
+.br
|
|
|
+.nf
|
|
|
+
|
|
|
+
|
|
|
+ main()
|
|
|
+ {
|
|
|
+ int i, j;
|
|
|
+
|
|
|
+ while (i < j
|
|
|
+ j++;
|
|
|
+
|
|
|
+ i = 1;
|
|
|
+ j = 2;
|
|
|
+
|
|
|
+ }
|
|
|
+
|
|
|
+
|
|
|
+.fi
|
|
|
+.LP
|
|
|
+Clearly, there is a `)' missing in the while-statement;
|
|
|
+however, if this program is input to the error recovery it will complain
|
|
|
+"} illegal", since after recognizing the
|
|
|
+expression controlling the while the original parser starts a
|
|
|
+semantic action, so the non-correcting recovery will accept a valid
|
|
|
+C program at that point; after recognizing the three statements
|
|
|
+following the while-statement as a separate program the
|
|
|
+recognizer expects the missing `)', but gets `}' instead.
|
|
|
+
|
|
|
+.LP
|
|
|
+Our second solution is based on the observation that if we knew
|
|
|
+which semantic actions can start other parsers, we would only
|
|
|
+have to introduce the new symbol $X$ at those places where parsers
|
|
|
+can get started. We have therefore extended LLgen with a new directive
|
|
|
+%substart, which is used to indicate to the parser generator that
|
|
|
+another parser may be started. The %substart is followed by the
|
|
|
+startsymbols that will produce the parsers that can be called,
|
|
|
+so %substart A, B, C; indicates that in the semantic action
|
|
|
+following the directive the parsers produced by startsymbols
|
|
|
+A, B, en C can be started. In the grammar used by the error
|
|
|
+recovery, a new symbol $X$ will be introduced at this point,
|
|
|
+along with a new rule $X -> AX | BX | CX | epsilon$. Of course, this
|
|
|
+solution can still accept input that would not have been accepted
|
|
|
+by original parser, for instance if a parser is started
|
|
|
+conditionally, based on other semantic information. However, it
|
|
|
+is a big improvement over the first solution, both in performance
|
|
|
+and the input it accepts.
|
|
|
+
|
|
|
+.NH 3
|
|
|
+Syntactic errors being handled in semantic actions
|
|
|
+
|
|
|
+.LP
|
|
|
+A programmer may decide to handle certain syntactic errors
|
|
|
+in semantic actions, for instance because he is not satisfied with
|
|
|
+the standard error recovery. However, since the non-correcting error
|
|
|
+recovery does not execute semantic actions, this may cause errors
|
|
|
+to remain undetected. We encountered the following example in the ACK
|
|
|
+Modula-2 compiler, in the grammar rule for assignment statement:
|
|
|
+.br
|
|
|
+.nf
|
|
|
+
|
|
|
+
|
|
|
+ Assignment_statement: lvalue
|
|
|
+ [
|
|
|
+ '='
|
|
|
+ {
|
|
|
+ error(":= expected");
|
|
|
+ }
|
|
|
+
|
|
|
+ |
|
|
|
+
|
|
|
+ ':='
|
|
|
+ ]
|
|
|
+ expression
|
|
|
+ ;
|
|
|
+
|
|
|
+.fi
|
|
|
+.LP
|
|
|
+This works well in the original LLgen; however, statements like
|
|
|
+`j=9' are not treated as syntactic, but as semantic errors.
|
|
|
+The original LLgen generated parser
|
|
|
+will print the (semantic) error message, but the non-correcting recovery
|
|
|
+will not execute the semantic action and therefore the erroneous
|
|
|
+input will be accepted.
|
|
|
+
|
|
|
+.LP
|
|
|
+To facilitate the incorporation of non-correcting error recovery in parsers
|
|
|
+that use this kind of `trick', we extended LLgen with the %erroneous
|
|
|
+directive. The directive indicates to the non-correcting recovery
|
|
|
+mechanism that the token following it is not really part of the grammar.
|
|
|
+When recognizing input, the error recovery will ignore tokens in the
|
|
|
+grammar that have %erroneous in front of them. If in the example above,
|
|
|
+the '=' is replaced with %erroneous '=', the non-correcting mechanism will
|
|
|
+report an error when it sees a statement like 'j = 9'. See appendix B
|
|
|
+for details about the implementation of the %erroneous directive.
|
|
|
+
|
|
|
+.LP
|
|
|
+Another example is in the ACK C compiler. For some reason, the
|
|
|
+grammar accepts function definitions without `()', so according
|
|
|
+to the syntax a function definition can look like:
|
|
|
+.br
|
|
|
+.nf
|
|
|
+
|
|
|
+ int func
|
|
|
+ {
|
|
|
+ ....
|
|
|
+ }
|
|
|
+.fi
|
|
|
+
|
|
|
+.LP
|
|
|
+The absence of the `()', however, causes `func' to be entered in the
|
|
|
+symbol table as non-function, and when the parser encounters the body
|
|
|
+a semantic action will complain with the error message "Making function body
|
|
|
+for non-function". This again will cause the non-correcting error
|
|
|
+recovery to miss errors. Consider this piece of code:
|
|
|
+.br
|
|
|
+.nf
|
|
|
+
|
|
|
+int i int j = 1;
|
|
|
+{}
|
|
|
+
|
|
|
+.fi
|
|
|
+
|
|
|
+.LP
|
|
|
+where apparently there's a `;' missing between the declarations
|
|
|
+of i and j. The original LLgen-generated parser only gives semantic errors:
|
|
|
+.br
|
|
|
+.nf
|
|
|
+"Making function body for non-function"
|
|
|
+"j is not in parameter list"
|
|
|
+"Illegal initialization of formal parameter, ignored"
|
|
|
+.fi
|
|
|
+.LP
|
|
|
+As a result, the non-correcting error recovery will not report
|
|
|
+any errors in this piece of code, because it does not execute the
|
|
|
+semantic actions that recognize and report the error. Unfortunately,
|
|
|
+due to the way the C-grammar is written, it is not possible to solve
|
|
|
+this problem using a %erroneous directive; the part of the grammar
|
|
|
+that deals with declaratons would have to be rewritten so as to
|
|
|
+syntactically reject functions without `()'.
|
|
|
+
|
|
|
+.NH 3
|
|
|
+Semantic actions that read input
|
|
|
+
|
|
|
+.LP
|
|
|
+There are no restrictions on what a semantic action can do;
|
|
|
+there is nothing to stop the programmer from writing a parser in such
|
|
|
+a way that some of the input to the parser is processed by semantic
|
|
|
+actions. Obviously, because the non-correcting error recovery does not
|
|
|
+execute semantic actions, this kind of parser will not work at all
|
|
|
+with the new error recovery. Ironically, LLgen itself is written in
|
|
|
+such a fashion; {}-enclosed C-code in its input is processed by
|
|
|
+a semantic action in the LLgen grammar. We feel that it is bad
|
|
|
+practice to write parsers this way; the `eating' of parts of
|
|
|
+the input should be done in the lexical analyzer, not in the parser.
|
|
|
+After all, in the case of LLgen, one can regard a semantic action
|
|
|
+in the input as one token, and thus it should be handled by
|
|
|
+the lexical analyzer as such.
|
|
|
+
|
|
|
+.NH 2
|
|
|
+Examples of error recovery
|
|
|
+
|
|
|
+.LP
|
|
|
+We will now give some examples that compare non-correcting error
|
|
|
+recovery with the correcting error recovery used by parsers generated
|
|
|
+by `standard' LLgen.
|
|
|
+
|
|
|
+Consider the next C program, where there is a `)' missing in the
|
|
|
+header of function `test'.
|
|
|
+.br
|
|
|
+.nf
|
|
|
+
|
|
|
+ 1 int test(a,b
|
|
|
+ 2
|
|
|
+ 3 int a,b;
|
|
|
+ 4
|
|
|
+ 5 {
|
|
|
+ 6 if (a < b)
|
|
|
+ 7 return(1);
|
|
|
+ 8 else
|
|
|
+ 9 return(0);
|
|
|
+ 10 }
|
|
|
+.fi
|
|
|
+
|
|
|
+.LP
|
|
|
+This small error derails the `standard' parser; it produces the
|
|
|
+following error messages, where we have left out 7 messages reporting
|
|
|
+semantic errors:
|
|
|
+.br
|
|
|
+.nf
|
|
|
+
|
|
|
+ line 3: , missing before type_identifier
|
|
|
+ line 3: , missing before identifier
|
|
|
+ line 3: ) missing before ;
|
|
|
+ line 5: { deleted
|
|
|
+ line 6: if deleted
|
|
|
+ line 6: < deleted
|
|
|
+ line 6: ) missing before identifier
|
|
|
+ line 6: ) deleted
|
|
|
+ line 7: identifier missing before return
|
|
|
+ line 7: ; missing before return
|
|
|
+ line 7: { missing before return
|
|
|
+ line 8: else deleted
|
|
|
+
|
|
|
+.fi
|
|
|
+.LP
|
|
|
+In contrast, the parser using non-correcting error recovery produces
|
|
|
+only one error message:
|
|
|
+.br
|
|
|
+
|
|
|
+ line 3: type_identifier illegal
|
|
|
+
|
|
|
+This error message correctly pin-points the error: there should
|
|
|
+have been a `)' at the position where type-identifier `int' is.
|
|
|
+
|
|
|
+.LP
|
|
|
+Now, an example with Modula-2; consider this program:
|
|
|
+.br
|
|
|
+.nf
|
|
|
+
|
|
|
+ 1 MODULE test;
|
|
|
+ 2
|
|
|
+ 3 TYPES
|
|
|
+ 4 ElementRecordType = RECORD
|
|
|
+ 5 Element: ElementType;
|
|
|
+ 6 Next,
|
|
|
+ 7 Prior: ElementPointerType;
|
|
|
+ 8 END;
|
|
|
+ 9
|
|
|
+ 10 VARS a,b,c: ElementRecordType;
|
|
|
+ 11
|
|
|
+ 12
|
|
|
+ 13 BEGIN
|
|
|
+ 14
|
|
|
+ 15 a := b;
|
|
|
+ 16
|
|
|
+ 17 END test.
|
|
|
+
|
|
|
+.fi
|
|
|
+.LP
|
|
|
+There are two syntactic errors in this program; on line 3, TYPES should be TYPE, and
|
|
|
+on line 10, VARS should be VAR. We have left out the type declarations of
|
|
|
+ElementType and ElementPointerType; clearly this will generate semantic
|
|
|
+errors, but we are only interested in syntactic errors anyway.
|
|
|
+The correcting error recovery parser
|
|
|
+again derails on this program; it produces the following syntactic error messages:
|
|
|
+.br
|
|
|
+.nf
|
|
|
+
|
|
|
+ line 3: CONST missing before identifier
|
|
|
+ line 4: '=' missing before identifier
|
|
|
+ line 4: RECORD deleted
|
|
|
+ line 5: ':' deleted
|
|
|
+ line 5: ';' missing before identifier
|
|
|
+ line 5: '=' missing before ';'
|
|
|
+ line 5: number missing before ';'
|
|
|
+ line 6: ',' deleted
|
|
|
+ line 7: '=' missing before identifier
|
|
|
+ line 7: ':' deleted
|
|
|
+ line 7: ';' missing before identifier
|
|
|
+ line 7: '=' missing before ';'
|
|
|
+ line 7: number missing before ';'
|
|
|
+ line 8: ';' deleted
|
|
|
+ line 10: identifier deleted
|
|
|
+ line 10: ',' deleted
|
|
|
+ line 10: identifier deleted
|
|
|
+ line 10: ',' deleted
|
|
|
+ line 10: identifier deleted
|
|
|
+ line 10: ':' deleted
|
|
|
+ line 10: identifier deleted
|
|
|
+ line 10: ';' deleted
|
|
|
+ line 13: BEGIN deleted
|
|
|
+ line 15: identifier deleted
|
|
|
+ line 15: := deleted
|
|
|
+ line 15: identifier deleted
|
|
|
+ line 15: ';' deleted
|
|
|
+ line 17: END deleted
|
|
|
+ line 17: identifier deleted
|
|
|
+
|
|
|
+.fi
|
|
|
+.LP
|
|
|
+The error correction mechanism clearly makes the wrong guess by inserting
|
|
|
+CONST on line 3; as a result, all that follows is rejected as incorrect.
|
|
|
+In contrast, the non-correcting error recovery mechanism only produces
|
|
|
+two error messages:
|
|
|
+.br
|
|
|
+.nf
|
|
|
+
|
|
|
+ line 3: identifier illegal
|
|
|
+ line 10: identifier illegal
|
|
|
+
|
|
|
+.fi
|
|
|
+.LP
|
|
|
+This again exactly pin-points the errors: the identifiers TYPES and
|
|
|
+VARS constitute the only errors in the program. Note that the
|
|
|
+presence of more than one error does not cause any problems to the
|
|
|
+non-correcting recovery mechanism.
|
|
|
+
|
|
|
+.bp
|
|
|
+.nr PS 12
|
|
|
+.nr VS 14
|
|
|
+
|
|
|
+.NH
|
|
|
+Conclusion
|
|
|
+
|
|
|
+.nr PS 10
|
|
|
+.nr VS 12
|
|
|
+
|
|
|
+.LP
|
|
|
+After implementing and testing a non-correcting error recovery mechanism
|
|
|
+we have come to the conclusion that it indeed is superior to correcting
|
|
|
+mechanisms in what regards the error messages it produces;
|
|
|
+the examples we have given clearly show this. However, there is a
|
|
|
+clear loss of performance when errors are present in a program,
|
|
|
+although we have found this performance
|
|
|
+degradation to be acceptable. We feel that the benefits of
|
|
|
+better error messages outweigh the loss of performance. In any case,
|
|
|
+correct programs do not suffer at all from the incorporation
|
|
|
+of a non-correcting recovery mechanism.
|
|
|
+The error recovery mechanism we implemented does not make
|
|
|
+unreasonable demands on resources; the size of the prediction
|
|
|
+graphs stays within reasonable limits.
|
|
|
+
|
|
|
+.LP
|
|
|
+The main problems we encountered had to do with recognizing
|
|
|
+`languages within languages', and semantic actions that did
|
|
|
+unreasonable things like eating input. The more `well-behaved' a
|
|
|
+parser is, the better the results the non-correcting error recovery
|
|
|
+mechanism gives. This is also true for the input grammars: with a
|
|
|
+language like Modula-2, whose syntax has been designed with parser
|
|
|
+generators in mind, the performance of the non-correcting mechanism
|
|
|
+is better than with C, whose syntax is extremely hard, if not
|
|
|
+impossible to describe with a LL(1) grammar.
|
|
|
+
|
|
|
+.bp
|
|
|
+.nr PS 12
|
|
|
+.nr VS 14
|
|
|
+
|
|
|
+.NH
|
|
|
+Bibliography
|
|
|
+
|
|
|
+.nr PS 10
|
|
|
+.nr VS 12
|
|
|
+
|
|
|
+.IP [CORMACK] 12
|
|
|
+Gordon V. Cormack, `An LR substring parser for noncorrecting syntax error
|
|
|
+recovery', ACM SIGPLAN Notices, vol. 24, no. 7, p. 161-169, July 1989
|
|
|
+
|
|
|
+.IP [GRUNE] 12
|
|
|
+Dick Grune, Ceriel J.H. Jacobs, `A programmer friendly LL(1) parser
|
|
|
+generator', Softw. Pract. Exper., vol. 18, no. 1, p. 29-38, Jan 1988
|
|
|
+
|
|
|
+.IP [RICHTER] 12
|
|
|
+Helmut Richter, `Noncorrecting syntax error recovery', ACM Trans. Prog. Lang.
|
|
|
+Sys., vol.7, no.3, p. 478-489, July 1985
|
|
|
+
|
|
|
+.IP [ROEHRICH] 12
|
|
|
+Johannes R\*:ohrich, `Methods for the automatic construction of error
|
|
|
+correcting parsers', Acta Inform., vol. 13, no. 2, p. 115-139, Feb 1980
|
|
|
+
|
|
|
+.IP [TOMITA] 12
|
|
|
+Masaru Tomita, Efficient parsing for natural language, Kluwer Academic
|
|
|
+Publishers, Boston, p.210, 1986
|
|
|
+.bp
|
|
|
+.SH
|
|
|
+Appendix A: Implementation Issues
|
|
|
+
|
|
|
+.nr PS 10
|
|
|
+.nr VS 12
|
|
|
+.RS
|
|
|
+.LP
|
|
|
+In this appendix we will describe some implementation issues;
|
|
|
+the data structure used to store the grammar during non-correcting
|
|
|
+error recovery, postponing deletions of graph elements until after
|
|
|
+the prediction phase, and the implementation of the %substart directive .
|
|
|
+.RE
|
|
|
+
|
|
|
+.SH
|
|
|
+A.1 The grammar data structure
|
|
|
+
|
|
|
+.LP
|
|
|
+The grammar data structure used by the non-correcting error recovery technique has
|
|
|
+to meet two conditions: easy access to a rule as a whole to make
|
|
|
+substituting nonterminals efficient and easy access to each symbol in the RHS
|
|
|
+of a rule to make starting error recovery and finding continuations
|
|
|
+efficient. To fulfill these conditions we decided to construct the
|
|
|
+storage of the grammar as follows.
|
|
|
+
|
|
|
+.LP
|
|
|
+A rule in the grammar is divided in two
|
|
|
+parts: a LHS and a RHS. The LHS is represented by a struct `lhs' and
|
|
|
+for each symbol in the RHS a struct 'symbol' is constructed.
|
|
|
+A struct `lhs' contains the number of the
|
|
|
+nonterminal forming the LHS of the rule, a pointer to the RHS, the
|
|
|
+first- and follow-sets of the nonterminal and a flag 'empty' which
|
|
|
+indicates whether the nonterminal produces empty or not. A struct
|
|
|
+`symbol' contains a field indicating the type of the symbol, i.e.
|
|
|
+a terminal or a nonterminal, the number of the symbol, a `link' pointer
|
|
|
+to a struct `symbol' that represents the same symbol, a `next' pointer
|
|
|
+to the rest of the RHS and a pointer back to the LHS.
|
|
|
+
|
|
|
+.LP
|
|
|
+A special struct `symbol' is added to the end of the RHS to indicate
|
|
|
+the end of a rule. The type of this struct is LLEORULE, the number
|
|
|
+is set to -1 and the pointers 'link' and `next' are nil.
|
|
|
+
|
|
|
+.LP
|
|
|
+In case that there is more than one RHS for a LHS, all the RHS's
|
|
|
+are put after each other and separated by another special struct
|
|
|
+`symbol'. The type of this struct is LLALT, the number is set to
|
|
|
+-1 and the 'link' pointer is nil. After the last RHS a `LLEORULE'-struct
|
|
|
+marker is added.
|
|
|
+
|
|
|
+.LP
|
|
|
+Finally, to make searching efficient there are two arrays: `terminals'
|
|
|
+and `nonterminals'. `terminals' is indexed by the number of a terminal
|
|
|
+and contains for each terminal a struct containing a 'link' pointer
|
|
|
+to a symbol, representing this terminal, in the RHS of a rule. Because
|
|
|
+this symbol has again a 'link' pointer to another symbol representing
|
|
|
+the terminal, it is possible by following this chain of pointers
|
|
|
+to find all rules containing such a terminal. In a similar way `nonterminals'
|
|
|
+is indexed by the number of a nonterminal and contains for each
|
|
|
+nonterminal a struct. This struct not only contains a 'link' pointer
|
|
|
+linking all rules with this nonterminal, but also contains a 'rule'
|
|
|
+pointer. This pointer points to the RHS or RHS's of the rules of which
|
|
|
+the nonterminal forms the LHS.
|
|
|
+
|
|
|
+.LP
|
|
|
+As an example, consider the following grammar:
|
|
|
+
|
|
|
+.br
|
|
|
+A: a B
|
|
|
+.br
|
|
|
+B: a | $epsilon$
|
|
|
+.br
|
|
|
+
|
|
|
+This will result in the picture below. Note that `pointer' fields
|
|
|
+without an arrow indicate nil pointers.
|
|
|
+
|
|
|
+.PS
|
|
|
+dx = 0.05
|
|
|
+
|
|
|
+down
|
|
|
+A_a: box ht boxht/2 "link"
|
|
|
+box invis "a" ljust with .e at A_a.w
|
|
|
+
|
|
|
+move to A_a.s
|
|
|
+move
|
|
|
+move
|
|
|
+
|
|
|
+A: box "link" "rule"
|
|
|
+B: box "link" "rule"
|
|
|
+line dashed from A.w to A.e
|
|
|
+line dashed from B.w to B.e
|
|
|
+box invis "A" ljust with .e at A.w
|
|
|
+box invis "B" ljust with .e at B.w
|
|
|
+
|
|
|
+move to A.ne
|
|
|
+right
|
|
|
+move
|
|
|
+move
|
|
|
+down
|
|
|
+
|
|
|
+LHS_A: box wid 1.2 * boxwid ht 2.5 * boxht "`A'" "rhs" "first" "follow" "empty 0"
|
|
|
+line dashed from 0.2 <LHS_A.nw, LHS_A.sw> to 0.2 <LHS_A.ne, LHS_A.se>
|
|
|
+line dashed from 0.4 <LHS_A.nw, LHS_A.sw> to 0.4 <LHS_A.ne, LHS_A.se>
|
|
|
+line dashed from 0.6 <LHS_A.nw, LHS_A.sw> to 0.6 <LHS_A.ne, LHS_A.se>
|
|
|
+line dashed from 0.8 <LHS_A.nw, LHS_A.sw> to 0.8 <LHS_A.ne, LHS_A.se>
|
|
|
+
|
|
|
+move to LHS_A.ne + (1,0)
|
|
|
+
|
|
|
+RHS_a1: box wid 2.0 * boxwid ht 2.5 * boxht "LLTERM" "`a'" "link" "next" "lhs"
|
|
|
+line dashed from 0.2 <RHS_a1.nw, RHS_a1.sw> to 0.2 <RHS_a1.ne, RHS_a1.se>
|
|
|
+line dashed from 0.4 <RHS_a1.nw, RHS_a1.sw> to 0.4 <RHS_a1.ne, RHS_a1.se>
|
|
|
+line dashed from 0.6 <RHS_a1.nw, RHS_a1.sw> to 0.6 <RHS_a1.ne, RHS_a1.se>
|
|
|
+line dashed from 0.8 <RHS_a1.nw, RHS_a1.sw> to 0.8 <RHS_a1.ne, RHS_a1.se>
|
|
|
+
|
|
|
+move to RHS_a1.ne + (1,0)
|
|
|
+
|
|
|
+RHS_B: box wid 2.0 * boxwid ht 2.5 * boxht "LLNONTERM" "`B'" "link" "next" "lhs"
|
|
|
+line dashed from 0.2 <RHS_B.nw, RHS_B.sw> to 0.2 <RHS_B.ne, RHS_B.se>
|
|
|
+line dashed from 0.4 <RHS_B.nw, RHS_B.sw> to 0.4 <RHS_B.ne, RHS_B.se>
|
|
|
+line dashed from 0.6 <RHS_B.nw, RHS_B.sw> to 0.6 <RHS_B.ne, RHS_B.se>
|
|
|
+line dashed from 0.8 <RHS_B.nw, RHS_B.sw> to 0.8 <RHS_B.ne, RHS_B.se>
|
|
|
+
|
|
|
+move to RHS_B.ne + (1,0)
|
|
|
+
|
|
|
+RHS_END1: box wid 2.0 * boxwid ht 2.5 *boxht "LLEORULE" "-1" "link" "next" "lhs"
|
|
|
+line dashed from 0.2 <RHS_END1.nw, RHS_END1.sw> to 0.2 <RHS_END1.ne,RHS_END1.se>
|
|
|
+line dashed from 0.4 <RHS_END1.nw, RHS_END1.sw> to 0.4 <RHS_END1.ne,RHS_END1.se>
|
|
|
+line dashed from 0.6 <RHS_END1.nw, RHS_END1.sw> to 0.6 <RHS_END1.ne,RHS_END1.se>
|
|
|
+line dashed from 0.8 <RHS_END1.nw, RHS_END1.sw> to 0.8 <RHS_END1.ne,RHS_END1.se>
|
|
|
+
|
|
|
+
|
|
|
+move to LHS_A.s - (0,1)
|
|
|
+
|
|
|
+LHS_B: box wid 1.2 * boxwid ht 2.5 * boxht "`B'" "rhs" "first" "follow" "empty 1"
|
|
|
+line dashed from 0.2 <LHS_B.nw, LHS_B.sw> to 0.2 <LHS_B.ne, LHS_B.se>
|
|
|
+line dashed from 0.4 <LHS_B.nw, LHS_B.sw> to 0.4 <LHS_B.ne, LHS_B.se>
|
|
|
+line dashed from 0.6 <LHS_B.nw, LHS_B.sw> to 0.6 <LHS_B.ne, LHS_B.se>
|
|
|
+line dashed from 0.8 <LHS_B.nw, LHS_B.sw> to 0.8 <LHS_B.ne, LHS_B.se>
|
|
|
+
|
|
|
+move to LHS_B.ne + (1,0)
|
|
|
+
|
|
|
+RHS_a2: box wid 2.0 * boxwid ht 2.5 * boxht "LLTERM" "`a'" "link" "next" "lhs"
|
|
|
+line dashed from 0.2 <RHS_a2.nw, RHS_a2.sw> to 0.2 <RHS_a2.ne, RHS_a2.se>
|
|
|
+line dashed from 0.4 <RHS_a2.nw, RHS_a2.sw> to 0.4 <RHS_a2.ne, RHS_a2.se>
|
|
|
+line dashed from 0.6 <RHS_a2.nw, RHS_a2.sw> to 0.6 <RHS_a2.ne, RHS_a2.se>
|
|
|
+line dashed from 0.8 <RHS_a2.nw, RHS_a2.sw> to 0.8 <RHS_a2.ne, RHS_a2.se>
|
|
|
+
|
|
|
+move to RHS_a2.ne + (1,0)
|
|
|
+
|
|
|
+RHS_ALT: box wid 2.0 * boxwid ht 2.5 * boxht "LLALT" "-1" "link" "next" "lhs"
|
|
|
+line dashed from 0.2 <RHS_ALT.nw, RHS_ALT.sw> to 0.2 <RHS_ALT.ne, RHS_ALT.se>
|
|
|
+line dashed from 0.4 <RHS_ALT.nw, RHS_ALT.sw> to 0.4 <RHS_ALT.ne, RHS_ALT.se>
|
|
|
+line dashed from 0.6 <RHS_ALT.nw, RHS_ALT.sw> to 0.6 <RHS_ALT.ne, RHS_ALT.se>
|
|
|
+line dashed from 0.8 <RHS_ALT.nw, RHS_ALT.sw> to 0.8 <RHS_ALT.ne, RHS_ALT.se>
|
|
|
+
|
|
|
+move to RHS_ALT.ne + (1,0)
|
|
|
+
|
|
|
+RHS_END2: box wid 2.0 * boxwid ht 2.5 *boxht "LLEORULE" "-1" "link" "next" "lhs"
|
|
|
+line dashed from 0.2 <RHS_END2.nw, RHS_END2.sw> to 0.2 <RHS_END2.ne,RHS_END2.se>
|
|
|
+line dashed from 0.4 <RHS_END2.nw, RHS_END2.sw> to 0.4 <RHS_END2.ne,RHS_END2.se>
|
|
|
+line dashed from 0.6 <RHS_END2.nw, RHS_END2.sw> to 0.6 <RHS_END2.ne,RHS_END2.se>
|
|
|
+line dashed from 0.8 <RHS_END2.nw, RHS_END2.sw> to 0.8 <RHS_END2.ne,RHS_END2.se>
|
|
|
+
|
|
|
+# Next pointers upper row
|
|
|
+.ps 30
|
|
|
+circle radius .01 at 0.75 <A.ne, A.se> - (dx, 0)
|
|
|
+circle radius .01 at 0.3 <LHS_A.ne, LHS_A.se> - (dx, 0)
|
|
|
+circle radius .01 at 0.7 <RHS_a1.ne, RHS_a1.se> - (dx, 0)
|
|
|
+circle radius .01 at 0.7 <RHS_B.ne, RHS_B.se> - (dx, 0)
|
|
|
+.ps 10
|
|
|
+
|
|
|
+arrow from 0.75 <A.ne, A.se> - (dx, 0) to 0.3 <LHS_A.nw, LHS_A.sw>
|
|
|
+arrow from 0.3 <LHS_A.ne, LHS_A.se> - (dx, 0) to 0.3 <RHS_a1.nw,RHS_a1.sw>
|
|
|
+arrow from 0.7 <RHS_a1.ne, RHS_a1.se> - (dx, 0) to 0.7 <RHS_B.nw,RHS_B.sw>
|
|
|
+arrow from 0.7 <RHS_B.ne, RHS_B.se> - (dx, 0) to 0.7 <RHS_END1.nw, RHS_END1.sw>
|
|
|
+
|
|
|
+
|
|
|
+# Next pointers lower row
|
|
|
+.ps 30
|
|
|
+circle radius .01 at 0.75 <B.ne, B.se> - (dx, 0)
|
|
|
+circle radius .01 at 0.3 <LHS_B.ne, LHS_B.se> - (dx, 0)
|
|
|
+circle radius .01 at 0.7 <RHS_a2.ne, RHS_a2.se> - (dx, 0)
|
|
|
+circle radius .01 at 0.7 <RHS_ALT.ne, RHS_ALT.se> - (dx, 0)
|
|
|
+.ps 10
|
|
|
+
|
|
|
+arrow from 0.75 <B.ne, B.se> - (dx, 0) to 0.3 <LHS_B.nw, LHS_B.sw>
|
|
|
+arrow from 0.3 <LHS_B.ne, LHS_B.se> - (dx, 0) to 0.3 <RHS_a2.nw,RHS_a2.sw>
|
|
|
+arrow from 0.7 <RHS_a2.ne, RHS_a2.se> - (dx, 0) to 0.7 <RHS_ALT.nw,RHS_ALT.sw>
|
|
|
+arrow from 0.7 <RHS_ALT.ne, RHS_ALT.se> - (dx, 0) to 0.7 <RHS_END2.nw, RHS_END2.sw>
|
|
|
+
|
|
|
+
|
|
|
+# Link pointers
|
|
|
+.ps 30
|
|
|
+circle radius .01 at 0.5 <RHS_a1.ne, RHS_a1.se> - (2*dx, 0)
|
|
|
+circle radius .01 at 0.5 <A_a.ne, A_a.se> - (dx, 0)
|
|
|
+circle radius .01 at 0.25 <B.ne, B.se> - (dx, 0)
|
|
|
+.ps 10
|
|
|
+
|
|
|
+arrow dashed from 0.5 <RHS_a1.ne, RHS_a1.se> - (2*dx, 0) to RHS_a2.ne - (2*dx,0)
|
|
|
+line dashed from 0.5 <A_a.ne, A_a.se> - (dx, 0) right 4.0 * boxwid then to RHS_a1.ne - (2*dx, 0) ->
|
|
|
+line dashed from 0.25 <B.ne, B.se> - (dx, 0) right then up .75 then right 7.0 * boxwid then to RHS_B.ne - (2*dx, 0) ->
|
|
|
+
|
|
|
+
|
|
|
+# LHS pointers upper row
|
|
|
+.ps 30
|
|
|
+circle radius .01 at 0.9 <RHS_a1.ne, RHS_a1.se> - (3*dx, 0)
|
|
|
+circle radius .01 at 0.9 <RHS_B.ne, RHS_B.se> - (3*dx, 0)
|
|
|
+circle radius .01 at 0.9 <RHS_END1.ne, RHS_END1.se> - (3*dx, 0)
|
|
|
+.ps 10
|
|
|
+
|
|
|
+line from 0.9 <RHS_a1.ne, RHS_a1.se> - (3*dx, 0) down ->
|
|
|
+line from 0.9 <RHS_B.ne, RHS_B.se> - (3*dx, 0) down ->
|
|
|
+line from 0.9 <RHS_END1.ne, RHS_END1.se> - (3*dx, 0) down then left 8.0 * boxwid then to LHS_A.se ->
|
|
|
+
|
|
|
+
|
|
|
+# LHS pointers lower row
|
|
|
+.ps 30
|
|
|
+circle radius .01 at 0.9 <RHS_a2.ne, RHS_a2.se> - (3*dx, 0)
|
|
|
+circle radius .01 at 0.9 <RHS_ALT.ne, RHS_ALT.se> - (3*dx, 0)
|
|
|
+circle radius .01 at 0.9 <RHS_END2.ne, RHS_END2.se> - (3*dx, 0)
|
|
|
+.ps 10
|
|
|
+
|
|
|
+line from 0.9 <RHS_a2.ne, RHS_a2.se> - (3*dx, 0) down ->
|
|
|
+line from 0.9 <RHS_ALT.ne, RHS_ALT.se> - (3*dx, 0) down ->
|
|
|
+line from 0.9 <RHS_END2.ne, RHS_END2.se> - (3*dx, 0) down then left 8.0 * boxwid then to LHS_B.se ->
|
|
|
+
|
|
|
+
|
|
|
+# Text above structs
|
|
|
+box invis ht boxht/2 "terminals" with .s at A_a.n
|
|
|
+box invis ht boxht/2 "nonterminals" with .s at A.n
|
|
|
+box invis ht boxht/2 "lhs" with .s at LHS_A.n
|
|
|
+box invis ht boxht/2 "lhs" with .s at LHS_B.n
|
|
|
+box invis ht boxht/2 "symbol" with .s at RHS_a1.n
|
|
|
+box invis ht boxht/2 "symbol" with .s at RHS_B.n
|
|
|
+box invis ht boxht/2 "symbol" with .s at RHS_END1.n
|
|
|
+box invis ht boxht/2 "symbol" with .s at RHS_a2.n
|
|
|
+box invis ht boxht/2 "symbol" with .s at RHS_ALT.n
|
|
|
+box invis ht boxht/2 "symbol" with .s at RHS_END2.n
|
|
|
+.PE
|
|
|
+
|
|
|
+.LP
|
|
|
+Note that the empty alternative for `B' is represented in the
|
|
|
+data structure by the `LLEORULE-struct' immediately following
|
|
|
+the `LLALT'-struct. When there are still other alternatives
|
|
|
+the `LLEORULE'-struct is replaced by a `LLALT'-struct followed
|
|
|
+by the other alternatives and a `LLEORULE'-struct.
|
|
|
+Finally, when the empty rule is the only rule for a
|
|
|
+nonterminal the RHS will consist only of a `LLEORULE'-struct.
|
|
|
+
|
|
|
+.SH
|
|
|
+A.2 Delayed deletes
|
|
|
+
|
|
|
+.LP
|
|
|
+We encountered a problem with deleting elements during the
|
|
|
+prediction phase. Imagine that we have a nonterminal `B' on top of
|
|
|
+the graph, and `B' has two alternatives. Now suppose that we
|
|
|
+apply the first alternative and we find out that this alternative leads
|
|
|
+to a `dead end', i.e. a head that does not match the input symbol, so we want
|
|
|
+to get rid of it. When we delete it immediately the deletion algorithm
|
|
|
+will also deallocate `[B]' and possibly some elements below `[B]'.
|
|
|
+However, there was another alternative for `[B]' which was not yet
|
|
|
+developed and maybe this alternative leads to a head which is legal.
|
|
|
+But `[B]' has already been deleted and thus cannot be used anymore. A similar
|
|
|
+situation can occur when we want to delete a joined element;
|
|
|
+the substitution of a nonterminal
|
|
|
+that only produces empty and thus has no element above it in the graph
|
|
|
+can also lead to such a situation. We therefore decided to put `dead ends'
|
|
|
+on a list, `cleanup_arr[]', and after the prediction phase has
|
|
|
+finished we delete all elements on this list, and all their descendants
|
|
|
+that become unreachable of course.
|
|
|
+
|
|
|
+.SH
|
|
|
+A.3 Clearing flags
|
|
|
+
|
|
|
+.LP
|
|
|
+We implemented two different ways to clear the flags set by the prediction
|
|
|
+phase of the algorithm; the first recursively tracks down the whole graph
|
|
|
+following the flags, the second puts all elements visited by
|
|
|
+the prediction phase
|
|
|
+on a list; after the prediction phase has finished the algorithm walks
|
|
|
+through this list clearing the flags of all elements on it. We took measurements
|
|
|
+on both algorithms and found out that with small programs the times
|
|
|
+did not differ much but large programs were processed faster by the
|
|
|
+second algorithm. Therefore we decided to use the second algorithm.
|
|
|
+
|
|
|
+.LP
|
|
|
+To speed up the algorithm even more, we do not deallocate the list
|
|
|
+after a prediction phase has finished. We just set the number of
|
|
|
+elements on the list to 0. This saves considerably on the number
|
|
|
+of `Malloc'-calls.
|
|
|
+
|
|
|
+.SH
|
|
|
+A.4 Implementation of %erroneous directive
|
|
|
+
|
|
|
+.LP
|
|
|
+As explained in chapter 3, the user can put a %erroneous directive
|
|
|
+in front of a terminal, making the non-correcting error recovery
|
|
|
+mechanism ignore that terminal. However, implementing this directive
|
|
|
+was not entirely straightforward; consider, for example, the rule
|
|
|
+.br
|
|
|
+.nf
|
|
|
+
|
|
|
+ A: 'a' | %erroneous 'b' | 'c';
|
|
|
+
|
|
|
+.fi
|
|
|
+.LP
|
|
|
+Just leaving out terminal 'b' will not do, because then nonterminal
|
|
|
+A produces empty all of a sudden, which it did not before.
|
|
|
+The rule should become
|
|
|
+.br
|
|
|
+.nf
|
|
|
+
|
|
|
+ A: 'a' | 'c';
|
|
|
+
|
|
|
+.fi
|
|
|
+but this is hard to implement in LLgen. We took a different approach:
|
|
|
+we introduce a new terminal 'ERRONEOUS', and substitute it for all
|
|
|
+terminals with an %erroneous directive in front of them. Thus, the
|
|
|
+example rule becomes
|
|
|
+.br
|
|
|
+.nf
|
|
|
+
|
|
|
+ A: 'a' | ERRONEOUS | 'c';
|
|
|
+
|
|
|
+.fi
|
|
|
+.LP
|
|
|
+Since the terminal ERRONEOUS will never be in the input to the parser,
|
|
|
+this has exactly the desired effect; when a predicting phase produces
|
|
|
+ERRONEOUS as head of a prediction graph this head will never match the
|
|
|
+input. In particular, it will not match the terminal that was
|
|
|
+originally there (in this case 'b') so that terminal is no longer
|
|
|
+regarded as part of the input language at that point.
|
|
|
+.bp
|
|
|
+.SH
|
|
|
+Appendix B: Using the non-correcting error recovery
|
|
|
+
|
|
|
+.LP
|
|
|
+To use the new non-correcting error recovery mechanism, LLgen has to
|
|
|
+be called with the new flag -n. LLgen will then create an extra file
|
|
|
+called `Lncor.c' which contains the code for the non-correcting recovery
|
|
|
+mechanism. This file has to be compiled and linked with the rest
|
|
|
+of the program, just like the file `Lpars.c'.
|
|
|
+
|
|
|
+.LP
|
|
|
+The user-supplied error reporting routine `LLmessage' will have to be
|
|
|
+modified slightly; when it is called with a positive parameter, it
|
|
|
+should only set the attributes of the inserted token, but not report an
|
|
|
+error. Note that the lexical analyzer still must return the same token
|
|
|
+as it did the last time it was called. When LLmessage is called with
|
|
|
+parameter 0, it should report that the token in global variable LLsymb
|
|
|
+is illegal; if the value of LLsymb is `EOFILE', the routine should
|
|
|
+report an unexpected End-of-file. When LLmessage is called with parameter
|
|
|
+-1, it should report that end-of-file was expected. To facilitate
|
|
|
+switching between correcting and non-correcting error recovery,
|
|
|
+the file Lpars.h contains a statement `#define LLNONCORR'
|
|
|
+which indicates that the non-correcting
|
|
|
+mechanism is enabled.
|
|
|
+Here is a
|
|
|
+skeleton for the modified LLmessage routine:
|
|
|
+.nr PS 8
|
|
|
+.nr VS 10
|
|
|
+.LP
|
|
|
+.br
|
|
|
+.nf
|
|
|
+
|
|
|
+ #include "Lpars.h"
|
|
|
+ extern int LLsymb;
|
|
|
+
|
|
|
+ LLmessage(flag)
|
|
|
+ int flag;
|
|
|
+ {
|
|
|
+ if (flag < 0)
|
|
|
+ {
|
|
|
+ /* Error message "end-of-file expected" */;
|
|
|
+ }
|
|
|
+ else if (flag)
|
|
|
+ {
|
|
|
+ /* flag equals the number of the inserted token */
|
|
|
+#ifndef LLNONCORR
|
|
|
+
|
|
|
+ /* Error message "token inserted" */;
|
|
|
+#endif
|
|
|
+
|
|
|
+ /* Code to set attributes for inserted token */
|
|
|
+ /* Code to make lexical analyzer return same token as before */
|
|
|
+
|
|
|
+ else
|
|
|
+ {
|
|
|
+ /* The number of the illegal or deleted token is in LLsymb */
|
|
|
+#ifndef LLNONCORR
|
|
|
+
|
|
|
+ /* Error message "token deleted" */;
|
|
|
+#else
|
|
|
+
|
|
|
+ if (LLsymb == EOFILE)
|
|
|
+ {
|
|
|
+ /* Error message "unexpected end of file" */
|
|
|
+ }
|
|
|
+ else
|
|
|
+ {
|
|
|
+ /* Error message "token illegal" */;
|
|
|
+ }
|
|
|
+#endif
|
|
|
+
|
|
|
+ }
|
|
|
+
|
|
|
+ }
|
|
|
+
|
|
|
+.fi
|
|
|
+.nr PS 10
|
|
|
+.nr VS 12
|
|
|
+
|
|
|
+.LP
|
|
|
+For best results, one should check if the parser calls other parsers
|
|
|
+in semantic actions; if this is the case, and the called parser
|
|
|
+processes the same input file as the calling parser, then a %substart
|
|
|
+should be put in front of the semantic action that starts a parser.
|
|
|
+If a semantic action calls parsers defined by startsymbols say
|
|
|
+A and B, then `%substart A, B;' should be put in front of the action.
|
|
|
+As an alternative, one can use the -s flag of LLgen; this has the
|
|
|
+same effect as putting `%substart X, Y, ....;' in front of all
|
|
|
+semantic actions, where X, Y, .... are the startsymbols of the grammar.
|
|
|
+Clearly, it is preferable to analyze the grammar and put %substart
|
|
|
+directives only where appropriate.
|
|
|
+
|
|
|
+Finally, beware of syntactic errors being handled in semantic
|
|
|
+actions; eg, one could have a rule like
|
|
|
+.nr PS 8
|
|
|
+.nr VS 10
|
|
|
+.LP
|
|
|
+.br
|
|
|
+.nf
|
|
|
+
|
|
|
+ Assignment_statement: lvalue
|
|
|
+ [
|
|
|
+ '='
|
|
|
+ {
|
|
|
+ error(":= expected");
|
|
|
+ }
|
|
|
+
|
|
|
+ |
|
|
|
+
|
|
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+ ':='
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+ ]
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+ expression
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+ ;
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+.fi
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+
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+.nr PS 10
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+.nr VS 12
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+.LP
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+To ensure that the non-correcting mechanism will recognize the
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+`=' as a syntactic error, a `%erroneous' directive should be
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+put in front of it.
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