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17
Presenting distributive laws
- In CALCO
, 2013
"... Abstract. Distributive laws of a monad T over a functor F are categor-ical tools for specifying algebra-coalgebra interaction. They proved to be important for solving systems of corecursive equations, for the specifica-tion of well-behaved structural operational semantics and, more recently, also fo ..."
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Abstract. Distributive laws of a monad T over a functor F are categor-ical tools for specifying algebra-coalgebra interaction. They proved to be important for solving systems of corecursive equations, for the specifica-tion of well-behaved structural operational semantics and, more recently, also for enhancements of the bisimulation proof method. If T is a free monad, then such distributive laws correspond to simple natural trans-formations. However, when T is not free it can be rather difficult to prove the defining axioms of a distributive law. In this paper we describe how to obtain a distributive law for a monad with an equational presentation from a distributive law for the underlying free monad. We apply this result to show the equivalence between two different representations of context-free languages. 1
Enhanced Coalgebraic Bisimulation
, 2013
"... We present a systematic study of bisimulation-up-to techniques for coalgebras. This enhances the bisimulation proof method for a large class of state based systems, including labelled transition systems but also stream systems and weighted automata. Our approach allows for compositional reasoning ab ..."
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Cited by 4 (3 self)
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We present a systematic study of bisimulation-up-to techniques for coalgebras. This enhances the bisimulation proof method for a large class of state based systems, including labelled transition systems but also stream systems and weighted automata. Our approach allows for compositional reasoning about the soundness of enhancements. Applications include the soundness of bisimulation up to bisimilarity, up to equivalence and up to congruence. All in all, this gives a powerful and modular framework for simplified coinductive proofs of equivalence. 1.
Coinductive Proof Techniques for Language Equivalence
"... Abstract. Language equivalence can be checked coinductively by establishing a bisimulation on suitable deterministic automata. We improve and extend this technique with bisimulation-up-to, which is an enhancement of the bisimulation proof method. First, we focus on the regular operations of union, c ..."
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Abstract. Language equivalence can be checked coinductively by establishing a bisimulation on suitable deterministic automata. We improve and extend this technique with bisimulation-up-to, which is an enhancement of the bisimulation proof method. First, we focus on the regular operations of union, concatenation and Kleene star, and illustrate our method with new proofs of classical results such as Arden’s rule. Then we extend our enhanced proof method to incorporate language complement and intersection. Finally we define a general format of behavioural differential equations, in which one can define operations on languages for which bisimulation-up-to is a sound proof technique. 1
Coalgebraic characterizations of context-free languages
- Logical Methods in Computer Science
"... Abstract. In this article, we provide three coalgebraic characterizations of the class of context-free languages, each based on the idea of adding coalgebraic structure to an existing algebraic structure by specifying output-derivative pairs. Final coalgebra semantics then gives an interpretation fu ..."
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Abstract. In this article, we provide three coalgebraic characterizations of the class of context-free languages, each based on the idea of adding coalgebraic structure to an existing algebraic structure by specifying output-derivative pairs. Final coalgebra semantics then gives an interpretation function into the final coalgebra of all languages with the usual output and derivative operations. The first characterization is based on systems, where each derivative is given as a finite language over the set of nonterminals; the second characterization on systems where derivatives are given as elements of a term-algebra; and the third characterization is based on adding coalgebraic structure to a class of closed (unique) fixed point expressions. We prove equivalences between these characterizations, discuss the generalization from languages to formal power series, as well as the relationship to the generalized powerset construction. 1.
Bisimulations up-to: beyond first-order transition systems
"... Abstract. The bisimulation proof method can be enhanced by employing ‘bisimulations up-to ’ techniques. A comprehensive theory of such enhance-ments has been developed for first-order (i.e., CCS-like) labelled transition systems (LTSs) and bisimilarity, based on the notion of compatible function for ..."
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Abstract. The bisimulation proof method can be enhanced by employing ‘bisimulations up-to ’ techniques. A comprehensive theory of such enhance-ments has been developed for first-order (i.e., CCS-like) labelled transition systems (LTSs) and bisimilarity, based on the notion of compatible function for fixed-point theory. We transport this theory onto languages whose bisimilarity and LTS go beyond those of first-order models. The approach consists in exhibiting fully abstract translations of the more sophisticated LTSs and bisimilarities onto the first-order ones. This allows us to reuse directly the large corpus of up-to techniques that are available on first-order LTSs. The only ingredient that has to be manually supplied is the compatibility of basic up-to techniques that are specific to the new languages. We investigate the method on the pi-calculus, the λ-calculus, and a (call-by-value) λ-calculus with references. 1
Context-free coalgebras
, 2013
"... In this article, we provide a coalgebraic account of parts of the mathematical theory un-derlying context-free languages. We characterize context-free languages, and power series and streams generalizing or corresponding to the context-free languages, by means of systems of behavioural differential ..."
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In this article, we provide a coalgebraic account of parts of the mathematical theory un-derlying context-free languages. We characterize context-free languages, and power series and streams generalizing or corresponding to the context-free languages, by means of systems of behavioural differential equations; and prove a number of results, some of which are new, and some of which are new proofs of existing theorems, using the techniques of bisimulation and bisimulation up to linear combinations. Furthermore, we establish a link between automatic sequences and these systems of equations, allowing us to, given an automaton generating an automatic sequence, easily construct a system of behavioural differential equations yielding this sequence as a context-free stream. 1
Foundational Extensible Corecursion
, 2014
"... This paper presents a theoretical framework for defining corecur-sive functions safely in a total setting, based on corecursion up-to and relational parametricity. The end product is a general corecur-sor that allows corecursive (and even recursive) calls under well-behaved operations, including con ..."
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Cited by 1 (1 self)
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This paper presents a theoretical framework for defining corecur-sive functions safely in a total setting, based on corecursion up-to and relational parametricity. The end product is a general corecur-sor that allows corecursive (and even recursive) calls under well-behaved operations, including constructors. Corecursive functions that are well behaved can be registered as such, thereby increasing the corecursor’s expressiveness. To the extensible corecursor cor-responds an equally flexible coinduction principle. The metatheory is formalized in the Isabelle proof assistant and forms the core of a prototype tool. The approach is foundational: The corecursor is derived from first principles, without requiring new axioms or ex-tensions of the logic. This ensures that no inconsistencies can be introduced by omissions in a termination or productivity check.
Kleene Algebra with Equations
"... We identify sufficient conditions for the construction of free language models for systems of Kleene algebra with additional equations. The construction applies to a broad class of extensions of KA and provides a uniform approach to deductive completeness and coalgebraic decision procedures. ..."
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Cited by 1 (1 self)
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We identify sufficient conditions for the construction of free language models for systems of Kleene algebra with additional equations. The construction applies to a broad class of extensions of KA and provides a uniform approach to deductive completeness and coalgebraic decision procedures.
