Algebraically complete semirings and Greibach normal form

Ésik, Zoltán; Leiß, Hans

doi:10.1016/j.apal.2004.10.008

Cited by 19 publications

(31 citation statements)

References 19 publications

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“…The additional expressive power obtained by the context-free expressions presented here is due to an explicit inclusion of a concatenation operator. 2 This provides an additional perspective on the treatment given here, in which 'context-freeness' is obtained by the addition of a new operator to a calculus of regular expressions 3 , and may pave the way for an investigation of (1) extending this approach to other coinductively defined operators, and (2) extending this approach to a generalized notion of context-freeness for other functors.…”

Section: Context-free Expressionsmentioning

confidence: 99%

“…Note that, because language equivalence of context-free languages is not semi-decidable, there cannot be any complete finitary axiomatization of behavioural equivalence. [2] As an illustration of context-free expressions, it is easy to see that the expression μx. (axb+1) will be mapped onto the language {a n b n }.…”

Section: T∼ μY(t[y/x]) If Y Is Not Free In T μXt ∼ T[μxt/x]mentioning

confidence: 99%

“…We take a similar approach, but coalgebraic in nature and substituting the Kleene star with a unique fixed point operator. Whereas [9,2] are interested in providing solutions to systems of equations of the form x = t using least fixed points, we look at systems of behavioural differential equations and give a semantic solution in terms of context-free languages (in Sect. 4) and syntactic solutions in terms of regular expressions with unique fixed points (in Sect.…”

Section: Introductionmentioning

confidence: 99%

“…In [9,2], Kleene algebras have been extended with a least fixed point operator to axiomatize fragments of the theory of context-free languages. We take a similar approach, but coalgebraic in nature and substituting the Kleene star with a unique fixed point operator.…”

Section: Introductionmentioning

confidence: 99%

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Context-Free Languages, Coalgebraically

Winter

Bonsangue

Rütten

2011

Algebra and Coalgebra in Computer Science

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Abstract. We give a coalgebraic account of context-free languages using the functor D(X) = 2 × X A for deterministic automata over an alphabet A, in three different but equivalent ways: (i) by viewing context-free grammars as D-coalgebras; (ii) by defining a format for behavioural differential equations (w.r.t. D) for which the unique solutions are precisely the context-free languages; and (iii) as the D-coalgebra of generalized regular expressions in which the Kleene star is replaced by a unique fixed point operator. In all cases, semantics is defined by the unique homomorphism into the final coalgebra of all languages, paving the way for coinductive proofs of context-free language equivalence. Furthermore, the three characterizations can serve as the basis for the definition of a general coalgebraic notion of context-freeness, which we see as the ultimate long-term goal of the present study.

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Section: Context-free Expressionsmentioning

confidence: 99%

Section: T∼ μY(t[y/x]) If Y Is Not Free In T μXt ∼ T[μxt/x]mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Context-Free Languages, Coalgebraically

Winter

Bonsangue

Rütten

2011

Algebra and Coalgebra in Computer Science

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“…The characterization of the context-free languages as least solutions of algebraic inequalities involving µ goes back to a 1971 paper of Gruska [4]. More recently, several researchers have given equational axioms for semirings with µ and have developed fragments of the equational theory of context-free languages [3,5,6,7,8,9].…”

Section: Introductionmentioning

confidence: 99%