Results 1 - 10 of 21

Coalgebraic Bisimulation-Up-To

by Jurriaan Rot , Marcello Bonsangue , Jan Rutten , 2013
"... Bisimulation-up-to enhances the bisimulation proof method for process equivalence. We present its generalization from labelled transition systems to arbitrary coalgebras, and show that for a large class of systems, enhancements such as bisimulation up to bisimilarity, up to equivalence and up to con ..."
Abstract - Cited by 17 (7 self) - Add to MetaCart
Bisimulation-up-to enhances the bisimulation proof method for process equivalence. We present its generalization from labelled transition systems to arbitrary coalgebras, and show that for a large class of systems, enhancements such as bisimulation up to bisimilarity, up to equivalence and up to context are sound proof techniques. This allows for simplified bisimulation proofs for many different types of state-based systems.

Presenting distributive laws

by Marcello M. Bonsangue, Helle H. Hansen, Er Kurz, Jurriaan Rot - 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 ..."
Abstract - Cited by 8 (4 self) - Add to MetaCart
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
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Nature

by Matias Lee , 1993
"... We present a format for the specification of probabilistic transition systems that guarantees that bisimulation equivalence is a congruence for any operator defined in this format. In this sense, the format is somehow comparable to the ntyft/ntyxt format in a non-probabilistic setting. We also study ..."
Abstract - Cited by 4 (0 self) - Add to MetaCart
We present a format for the specification of probabilistic transition systems that guarantees that bisimulation equivalence is a congruence for any operator defined in this format. In this sense, the format is somehow comparable to the ntyft/ntyxt format in a non-probabilistic setting. We also study the modular construction of probabilistic transition systems specifi-cations and prove that some standard conservative extension theorems also hold in our setting. Finally, we show that the trace congruence for image-finite processes induced by our format is precisely bisimulation on probabilistic systems. Note:
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Enhanced Coalgebraic Bisimulation

by J. Rot, F. Bonchi, M. Bonsangue, D. Pous, J. Rutten, A. Silva , 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 ..."
Abstract - Cited by 4 (3 self) - Add to MetaCart
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.
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Coalgebraic characterizations of context-free languages

by Joost Winter, Marcello M. Bonsangue, Jan Rutten - 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.
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Structural Operational Semantics for Stochastic and Weighted Transition Systems

by Bartek Klin, Vladimiro Sassone , 2013
"... We introduce weighted GSOS, a general syntactic framework to specify well-behaved transition systems where transitions are equipped with weights coming from a commutative monoid. We prove that weighted bisimilarity is a congruence on systems defined by weighted GSOS specifications. We illustrate the ..."
Abstract - Cited by 2 (0 self) - Add to MetaCart
We introduce weighted GSOS, a general syntactic framework to specify well-behaved transition systems where transitions are equipped with weights coming from a commutative monoid. We prove that weighted bisimilarity is a congruence on systems defined by weighted GSOS specifications. We illustrate the flexibility of the framework by instantiating it to handle some special cases, most notably that of stochastic transition systems. Through examples we provide weighted-GSOS definitions for common stochastic operators in the literature.
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Stream Differential Equations: Specification Formats and Solution Methods

by Helle Hvid Hansen, Clemens Kupke, Jan Rutten , 2014
"... Streams, or infinite sequences, are infinite objects of a very simple type, yet they have a rich theory partly due to their ubiquity in mathematics and computer science. Stream differential equations are a coinductive method for specifying streams and stream operations, and their theory has been dev ..."
Abstract - Cited by 2 (2 self) - Add to MetaCart
Streams, or infinite sequences, are infinite objects of a very simple type, yet they have a rich theory partly due to their ubiquity in mathematics and computer science. Stream differential equations are a coinductive method for specifying streams and stream operations, and their theory has been developed in many papers over the past two decades. In this paper we present a survey of the many results in this area. Our focus is on the classification of different formats of stream differential equations, their solution methods, and the classes of streams they can define. Moreover, we describe in detail the connection between the so-called syntactic solution method and abstract GSOS.

Coalgebraic up-to techniques

by Damien Pous
"... A simple algorithm for checking language equivalence of finite automata con-sists in trying to compute a bisimulation that relates them. This is possible because language equivalence can be characterised coinductively, as the largest bisimulation. More precisely, consider an automaton 〈S, t, o〉, whe ..."
Abstract - Cited by 2 (0 self) - Add to MetaCart
A simple algorithm for checking language equivalence of finite automata con-sists in trying to compute a bisimulation that relates them. This is possible because language equivalence can be characterised coinductively, as the largest bisimulation. More precisely, consider an automaton 〈S, t, o〉, where S is a (finite) set of states, t: S → P(S)A is a non-deterministic transition function, and o: S → 2 is the characteristic function of the set of accepting states. Such an automation gives rise to a determinised automaton 〈P(S), t], o]〉, where t] : P(S) → P(S)A and o] : P(S) → 2 are the natural extensions of t and o to sets. A bisimulation is a relation R between sets of states such that for all sets of states X,Y, X R Y entails: 1. o](X) = o](Y), and 2. for all letter a, t]a(X) R t a(Y). The coinductive characterisation is the following one: two sets of states recognise the same language if and only if they are related by some bisimulation. Taking inspiration from concurrency theory [4,5], one can improve this proof technique by weakening the second item in the definition of bisimulation: given a function f on binary relations, a bisimulation up to f is a relation R between states such that for all sets X,Y, X R Y entails: 1. o](X) = o](Y), and 2. for all letter a, t]a(X) f(R) t a(Y). For well-chosen functions f, bisimulations up to f are contained in a bisimula-tion, so that the improvement is sound. So is the function mapping each relation to its equivalence closure. In this particular case, one recover the standard al-gorithm by Hopcroft and Karp [2]: two sets can be skipped whenever they can already be related by a sequence of pairwise related states. One can actually do more, by considering the function c mapping each rela-tion to its congruence closure: the smallest equivalence relation which contains
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GSOS for non-deterministic processes with quantitative aspects

by Marino Miculan, Marco Peressotti
"... Recently,some general frameworks have been proposed as unifying theories for processes combining non-determinism with quantitative aspects (such as probabilistic or stochastically timed executions), aiming to provide general results and tools. This paper provides two contributions in this respect. F ..."
Abstract - Cited by 1 (1 self) - Add to MetaCart
Recently,some general frameworks have been proposed as unifying theories for processes combining non-determinism with quantitative aspects (such as probabilistic or stochastically timed executions), aiming to provide general results and tools. This paper provides two contributions in this respect. First, we present a general GSOS specification format (and a corresponding notion of bisimulation) for non-deterministic processes with quantitative aspects. These specifications define labelled transition systems according to the ULTraS model, an extension of the usual LTSs where the transition relation associates any source state and transition label with state reachability weight functions (like, e.g., probability distributions). This format, hence called Weight Function SOS (WFSOS), covers many known systems and their bisimulations (e.g. PEPA, TIPP, PCSP) and GSOS formats (e.g. GSOS, Weighted GSOS, Segala-GSOS, among others). The second contribution is a characterization of these systems as coalgebras of a class of functors, parametric on the weight structure. This result allows us to prove soundness of the WFSOS specification format, and that bisimilarities induced by these specifications are always congruences.

Foundational Extensible Corecursion

by Jasmin Christian Blanchette , Andrei Popescu , Dmitriy Traytel , 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 ..."
Abstract - Cited by 1 (1 self) - Add to MetaCart
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.