Ambiguity in Graphs and Expressions

Book, Ronald V.; Even, Shimon; Greibach, Sheila A.; Ott, Gene

doi:10.1109/t-c.1971.223204

Cited by 70 publications

(66 citation statements)

References 9 publications

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“…Otherwise, the language is called 1-ambiguous. 1-unambiguity is different from, though related with, unambiguity, as used to classify grammars in language theory, and studied for regular expressions by Book et al [1]. From [1]: "A regular expression is called unambiguous if every tape in the event can be generated from the expression in one way only".…”

Section: Definition 8 [32]mentioning

confidence: 99%

“…1-unambiguity is different from, though related with, unambiguity, as used to classify grammars in language theory, and studied for regular expressions by Book et al [1]. From [1]: "A regular expression is called unambiguous if every tape in the event can be generated from the expression in one way only". It follows almost directly from the definitions that the class of 1-unambiguous regular expressions is included in the class of unambiguous regular expressions.…”

Section: Definition 8 [32]mentioning

confidence: 99%

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The Inclusion Problem for Regular Expressions

Hovland

2010

Language and Automata Theory and Applications

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Abstract. This paper presents a new polynomial-time algorithm for the inclusion problem for certain pairs of regular expressions. The algorithm is not based on construction of finite automata, and can therefore be faster than the lower bound implied by the Myhill-Nerode theorem. The algorithm automatically discards unnecessary parts of the right-hand expression. In these cases the right-hand expression might even be 1-ambiguous. For example, if r is a regular expression such that any DFA recognizing r is very large, the algorithm can still, in time independent of r, decide that the language of ab is included in that of (a + r)b. The algorithm is based on a syntax-directed inference system. It takes arbitrary regular expressions as input, and if the 1-ambiguity of the right-hand expression becomes a problem, the algorithm will report this.

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Section: Definition 8 [32]mentioning

confidence: 99%

Section: Definition 8 [32]mentioning

confidence: 99%

The Inclusion Problem for Regular Expressions

Hovland

2010

Language and Automata Theory and Applications

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“…Thus the construction of [7] is optimal for large alphabets, i.e., if k = n Ω (1) . Since Theorem 1 is almost optimal for alphabets of fixed size, only improvements for alphabets of intermediate size, i.e., ω(1) = k = n o (1) , are still required.…”

Section: Theoremmentioning

confidence: 99%

“…Since Theorem 1 is almost optimal for alphabets of fixed size, only improvements for alphabets of intermediate size, i.e., ω(1) = k = n o (1) , are still required. In Section 2 we show how to construct small ε-free NFAs for a given regular expression R using ideas from [2,7].…”

Section: Theoremmentioning

confidence: 99%

“…All classical conversions [1,3,9,12] produce ε-free NFAs with worst-case size quadratic in the length of the given regular expression and for some time this was assumed to be optimal [10]. But then Hromkovic, Seibert and Wilke [7] constructed ε-free NFAs with surprisingly only O(n(log 2 n) 2 ) transitions for regular expressions of length n and this transformation can even be implemented to run in time O(n · log 2 n + m), where m is the size of the output [4].…”

Section: Introductionmentioning

confidence: 99%

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Regular Expressions and NFAs Without ε-Transitions

Schnitger

2006

Stacs 2006

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Abstract. We consider the problem of converting regular expressions into ε-free NFAs with as few transitions as possible. If the regular expression has length n and is defined over an alphabet of size k, then the previously best construction uses O(n·min{k, log 2 n}·log 2 n) transitions. We show that O(n · log 2 2k · log 2 n) transitions suffice. For small alphabets, for instance if k = O(log 2 log 2 n), we further improve the upper bound to O(k 1+log * n · n). In particular, O(2 log * 2 n · n) transitions and hence almost linear size suffice for the binary alphabet! Finally we show the lower bound Ω(n · log 2 2 2k) and as a consequence the upper bound O(n · log 2 2 n) of [7] for general alphabets is best possible. Thus the conversion problem is solved for large alphabets (k = n Ω(1) ) and almost solved for small alphabets (k = O(1)).

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One-Unambiguous Regular Languages

Brüggemann-KleinAnne¹,

WoodDerick²

1998

Information and Computation

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The ISO standard for the Standard Generalized Markup Language (SGML) provides a syntactic metalanguage for the de nition of textual markup systems. In the standard, the right-hand sides of productions are based on regular expressions; although only expressions that denote words unambiguously are allowed. In general, the fact that a word is denoted by an expression is witnessed by a sequence of occurrences of symbols in the expression that matches the word. In an unambiguous expression as de ned by Book, Even, Greibach, and Ott, each word has at most one witness. But the SGML standard also requires that a witness can be computed incrementally from the word with a one-symbol lookahead; we call such expressions 1-unambiguous. A regular language is 1-unambiguous if it is denoted by some 1-unambiguous expression. We give a Kleene theorem for 1-unambiguous languages and characterize them in terms of structural properties of the minimal deterministic automata that recognize them. This result enables us to decide whether a given regular expression denotes a 1-unambiguous language; if it does, then we can construct an equivalent 1-unambiguous expression in worst-case optimal time.

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Ambiguity in Graphs and Expressions

Cited by 70 publications

References 9 publications

The Inclusion Problem for Regular Expressions

The Inclusion Problem for Regular Expressions

Regular Expressions and NFAs Without ε-Transitions

One-Unambiguous Regular Languages

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