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17
Presenting distributive laws
 In CALCO
, 2013
"... Abstract. Distributive laws of a monad T over a functor F are categorical tools for specifying algebracoalgebra interaction. They proved to be important for solving systems of corecursive equations, for the specification of wellbehaved structural operational semantics and, more recently, also fo ..."
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Abstract. Distributive laws of a monad T over a functor F are categorical tools for specifying algebracoalgebra interaction. They proved to be important for solving systems of corecursive equations, for the specification of wellbehaved 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 transformations. 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 contextfree languages. 1
Enhanced Coalgebraic Bisimulation
, 2013
"... We present a systematic study of bisimulationupto 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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We present a systematic study of bisimulationupto 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 bisimulationupto, 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 bisimulationupto, 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 bisimulationupto is a sound proof technique. 1
Coalgebraic characterizations of contextfree languages
 Logical Methods in Computer Science
"... Abstract. In this article, we provide three coalgebraic characterizations of the class of contextfree languages, each based on the idea of adding coalgebraic structure to an existing algebraic structure by specifying outputderivative 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 contextfree languages, each based on the idea of adding coalgebraic structure to an existing algebraic structure by specifying outputderivative 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 termalgebra; 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 upto: beyond firstorder transition systems
"... Abstract. The bisimulation proof method can be enhanced by employing ‘bisimulations upto ’ techniques. A comprehensive theory of such enhancements has been developed for firstorder (i.e., CCSlike) 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 upto ’ techniques. A comprehensive theory of such enhancements has been developed for firstorder (i.e., CCSlike) labelled transition systems (LTSs) and bisimilarity, based on the notion of compatible function for fixedpoint theory. We transport this theory onto languages whose bisimilarity and LTS go beyond those of firstorder models. The approach consists in exhibiting fully abstract translations of the more sophisticated LTSs and bisimilarities onto the firstorder ones. This allows us to reuse directly the large corpus of upto techniques that are available on firstorder LTSs. The only ingredient that has to be manually supplied is the compatibility of basic upto techniques that are specific to the new languages. We investigate the method on the picalculus, the λcalculus, and a (callbyvalue) λcalculus with references. 1
Contextfree coalgebras
, 2013
"... In this article, we provide a coalgebraic account of parts of the mathematical theory underlying contextfree languages. We characterize contextfree languages, and power series and streams generalizing or corresponding to the contextfree 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 underlying contextfree languages. We characterize contextfree languages, and power series and streams generalizing or corresponding to the contextfree 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 contextfree stream. 1
Foundational Extensible Corecursion
, 2014
"... This paper presents a theoretical framework for defining corecursive functions safely in a total setting, based on corecursion upto and relational parametricity. The end product is a general corecursor that allows corecursive (and even recursive) calls under wellbehaved operations, including con ..."
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This paper presents a theoretical framework for defining corecursive functions safely in a total setting, based on corecursion upto and relational parametricity. The end product is a general corecursor that allows corecursive (and even recursive) calls under wellbehaved operations, including constructors. Corecursive functions that are well behaved can be registered as such, thereby increasing the corecursor’s expressiveness. To the extensible corecursor corresponds 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 extensions 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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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.