Deciding Definability by Deterministic Regular Expressions

Czerwiński, Wojciech; David, Claire; Losemann, Katja; Martens, Wim

doi:10.1007/978-3-642-37075-5_19

Cited by 8 publications

(6 citation statements)

References 28 publications

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“…The important question is then whether a regular language is DRE definable. This problem has been shown to be PSpace-complete [12]. Since the language of the expression (a + b) * b(a + b) is not DRE definable [7], but it can be easily expressed by a poNFA, DRE definability is nontrivial for poNFAs.…”

Section: Deterministic Regular Expressions and Partially Ordered Nfasmentioning

confidence: 99%

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Complexity of universality and related problems for partially ordered NFAs

Krötzsch¹,

Masopust²,

Thomazo³

2017

Information and Computation

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Partially ordered nondeterministic finite automata (poNFAs) are NFAs whose transition relation induces a partial order on states, that is, for which cycles occur only in the form of self-loops on a single state. A poNFA is universal if it accepts all words over its input alphabet. Deciding universality is PSpace-complete for poNFAs, and we show that this remains true even when restricting to a fixed alphabet. This is nontrivial since standard encodings of alphabet symbols in, e.g., binary can turn self-loops into longer cycles. A lower coNP-complete complexity bound can be obtained if we require that all self-loops in the poNFA are deterministic, in the sense that the symbol read in the loop cannot occur in any other transition from that state. We find that such restricted poNFAs (rpoNFAs) characterize the class of R-trivial languages, and we establish the complexity of deciding if the language of an NFA is R-trivial. Nevertheless, the limitation to fixed alphabets turns out to be essential even in the restricted case: deciding universality of rpoNFAs with unbounded alphabets is PSpace-complete. Based on a close relation between universality and the problems of inclusion and equivalence, we also obtain the complexity results for these two problems. Finally, we show that the languages of rpoNFAs are definable by deterministic (one-unambiguous) regular expressions, which makes them interesting in schema languages for XML data.

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Section: Deterministic Regular Expressions and Partially Ordered Nfasmentioning

confidence: 99%

“…Finally, note that the converse of Theorem 34 does not hold. The expression b * a(b * a) * is deterministic [12] and it can be easily verified that its minimal DFA is not partially ordered. Therefore, the expression defines a language that is not R-trivial.…”

Section: Deterministic Regular Expressions and Partially Ordered Nfasmentioning

confidence: 99%

Complexity of universality and related problems for partially ordered NFAs

Krötzsch¹,

Masopust²,

Thomazo³

2017

Information and Computation

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“…As shown in [9], L(DREG) ⊂ L(REG) (also see [16,39], or Lemma 5 below). Like for determinism of regular expressions, the key idea behind our definition of deterministic regex is that a matcher for the expression treats terminals (and variable references) as states.…”

Section: Deterministic Regexmentioning

confidence: 74%

“…Aspects include computing the Glushkov automaton and deciding the membership problem (e. g. [8,31,44]), static analysis (cf. [40]), deciding whether a regular language is deterministic (e. g. [16,31,39]), closure properties and descriptional complexity [37], and learning (e. g. [5]). One noteworthy extension are counter operators (e. g. [29,31,36]), which we briefly address in Section 8.…”

Section: Deterministic Regular Expressionsmentioning

confidence: 99%

“…Then L is generated by the non-deterministic regular expression (ababab) * (ε ∨(aba)), and one can show that L is not a deterministic regular language by using the BKW-algorithm [9] (also [16,39] Figure 2: Illustration of the unary DFA in the proof of Theorem 6. Note that here, we do not distinguish between accepting and non-accepting states…”

Section: Expressive Powermentioning

confidence: 99%

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Deterministic regular expressions with back-references

Freydenberger

Schmid

2019

Journal of Computer and System Sciences

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Most modern libraries for regular expression matching allow back-references (i. e., repetition operators) that substantially increase expressive power, but also lead to intractability. In order to find a better balance between expressiveness and tractability, we combine these with the notion of determinism for regular expressions used in XML DTDs and XML Schema. This includes the definition of a suitable automaton model, and a generalization of the Glushkov construction. We demonstrate that, compared to their non-deterministic superclass, these deterministic regular expressions with back-references have desirable algorithmic properties (i. e., efficiently solvable membership problem and some decidable problems in static analysis), while, at the same time, their expressive power exceeds that of deterministic regular expressions without back-references. IntroductionRegular expressions were introduced in 1956 by Kleene [34] and quickly found wide use in both theoretical and applied computer science, including applications in bioinformatics [41], programming languages [49], model checking [48], and XML schema languages [47]. While the theoretical interpretation of regular expressions remains mostly unchanged (as expressions that describe exactly the class of regular languages), modern applications use variants that vary greatly in expressive power and algorithmic properties. This paper tries to find common ground between two of these variants with opposing approaches to the balance between expressive power and tractability. REGEXThe first variant that we consider are regex, regular expressions that are extended with a backreference operator. This operator is used in almost all modern programming languages (like e. g. Java, PERL, and .NET). For example, the regex x : (a ∨ b) * · &x defines {ww | w ∈ {a, b} * }, as (a ∨ b) * can create a w ∈ {a, b} * , which is then stored in the variable x and repeated with the reference &x. Hence, back-references allow to define non-regular languages; but with the side effect that the membership problem is NP-complete (cf. Aho [2]).Regex were first examined from a theoretical point of view by Aho [2], but without fully defining the semantics. There were various proposals for semantics, of which we mention the first by Câmpeanu, Salomaa, Yu [10], and the recent one by Schmid [46], which is the basis for this paper. Apart from defining the semantics, there was work on the expressive power [10,11,25], the static analysis [11,23,24], and the tractability of the membership problem (investigated in terms of a strongly restricted subclass of regex) [21,22]. They have also been compared to related models in database theory, e. g. graph databases [4,26] and information extraction [20,24]. * This work represents an extended version of the paper "Deterministic Regular Expressions with Back-References" presented at STACS 2017 and published in LIPICS (http://dx.doi.org/10.4230/LIPIcs.STACS.2017.33 ).3. The intersection-emptiness problem for DRX is undecidable, but in PSPACE for variablestar-free...

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A Note on Decidable Separability by Piecewise Testable Languages

Czerwiński

Martens

Rooijen

et al. 2015

Fundamentals of Computation Theory

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Piecewise testable languages form the first level of the Straubing-Thérien hierarchy. The membership problem for this level is decidable and testing if the language of a DFA is piecewise testable is NL-complete. The question has not yet been addressed for NFAs. We fill in this gap by showing that it is PSpace-complete. The main result is then the lower-bound complexity of separability of regular languages by piecewise testable languages. Two regular languages are separable by a piecewise testable language if the piecewise testable language includes one of them and is disjoint from the other. For languages represented by NFAs, separa-bility by piecewise testable languages is known to be decidable in PTime. We show that it is PTime-hard and that it remains PTime-hard even for minimal DFAs.

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Deciding Definability by Deterministic Regular Expressions

Cited by 8 publications

References 28 publications

Complexity of universality and related problems for partially ordered NFAs

Complexity of universality and related problems for partially ordered NFAs

Deterministic regular expressions with back-references

A Note on Decidable Separability by Piecewise Testable Languages

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