We present a comparative study of four product operators on weighted languages: (i) the convolution, (ii) the shuffle, (iii) the infiltration and (iv) the Hadamard product. Exploiting the fact that the set of weighted languages is a final coalgebra, we use coinduction to prove that an operator of the classical difference calculus, the Newton transform, generalises from infinite sequences to weighted languages. We show that the Newton transform is an isomorphism of rings that transforms the Hadamard product of two weighted languages into their infiltration product, and we develop various representations for the Newton transform of a language, together with concrete calculation rules for computing them.

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.

In the concurrency theory, various semantic equivalences on transition systems are based on traces decorated with some additional observations, generally referred to as decorated traces. Using the generalized powerset construction, recently introduced by a subset of the authors (Silva et al.2010 FSTTCS. LIPIcs8 272–283), we give a coalgebraic presentation of decorated trace semantics. The latter include ready, failure, (complete) trace, possible futures, ready trace and failure trace semantics for labelled transition systems, and ready, (maximal) failure and (maximal) trace semantics for generative probabilistic systems. This yields a uniform notion of minimal representatives for the various decorated trace equivalences, in terms of final Moore automata. As a consequence, proofs of decorated trace equivalence can be given by coinduction, using different types of (Moore-) bisimulation (up-to context).

We give a new presentation of Brzozowski's algorithm to minimize finite automata using elementary facts from universal algebra and coalgebra and building on earlier work by Arbib and Manes on a categorical presentation of Kalman duality between reachability and observability. This leads to a simple proof of its correctness and opens the door to further generalizations. Notably, we derive algorithms to obtain minimal language equivalent automata from Moore nondeterministic and weighted automata.

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Coalgebra is an abstract framework for the uniform study of different kinds of dynamical systems. An endofunctor $F$ determines both the type of systems ($F$-coalgebras) and a notion of behavioral equivalence ($\sim_F$) amongst them. Many types of transition systems and their equivalences can be captured by a functor $F$. For example, for deterministic automata the derived equivalence is language equivalence, while for non-deterministic automata it is ordinary bisimilarity. The powerset construction is a standard method for converting a nondeterministic automaton into an equivalent deterministic one as far as language is concerned. In this paper, we lift the powerset construction on automata to the more general framework of coalgebras with structured state spaces. Examples of applications include partial Mealy machines, (structured) Moore automata, and Rabin probabilistic automata.

Several dynamical systems, such as deterministic automata and labelled transition systems, can be described as coalgebras of so-called Kripke polynomial functors, built up from constants and identities, using product, coproduct and powerset. Locally finite Kripke polynomial coalgebras can be characterized up to bisimulation by a specification language that generalizes Kleene's regular expressions for finite automata. In this paper we equip this specification language with an axiomatization and prove it sound and complete with respect to bisimulation, using a purely coalgebraic argument. We demonstrate the usefulness of our framework by providing a finite equational system for (non-)deterministic finite automata, labelled transition systems with explicit termination and automata on guarded strings. © 2009 IEEE.

The complex interactions appearing in service-oriented computing make coordination a key concern in service-oriented systems. In this paper, we present a fault-based method to generate test cases for component connectors from specifications. For connectors, faults are caused by possible errors during the development process, such as wrongly used channels, missing or redundant subcircuits, or circuits with wrongly constructed topology. We give test cases and connectors a unifying formal semantics by using the notion of design, and generate test cases by solving constraints obtained from the specification and faulty connectors. A prototype symbolic test case generator serves to demonstrate the automatizing of the approach.

We study rational streams (over a field) from a coalgebraic perspective. Exploiting the finality of the set of streams, we present an elementary and uniform proof of the equivalence of four notions of representability of rational streams: by finite dimensional linear systems; by finite stream circuits; by finite weighted stream automata; and by finite dimensional subsystems of the set of streams.

We develop a coinductive calculus of streams based on the presence of a final coalgebra structure on the set of streams (infinite sequences of real numbers). The main ingredient is the notion of stream derivative, which can be used to formulate both coinductive proofs and definitions. In close analogy to classical analysis, the latter are presented as behavioural differential equations. A number of applications of the calculus are presented, including difference equations, analytical differential equations, continued fractions, and some problems from discrete mathematics and combinatorics.

Abstract We develop a denotational semantics for POOL, a parallel object-oriented programming language. The main contribution of this semantics is an accurate mathematical model of the most important concept in object-oriented programming: the object. This is achieved by structuring the semantics in layers working at three different levels: for statements, objects and programs. For each of these levels we define a specialized mathematical domain of processes, which we use to assign a meaning to each language construct. This is done in the mathematical framework of complete metric spaces. We also define operators that translate between these domains. At the program level we give a precise definition of the observable input/output behaviour of a particular program, which could be used at a later stage to decide the issue of full abstractness. We illustrate our semantic techniques by first applying them to a toy language similar to CSP.

A compositional semantics characterizing bisimulation equivalence is derived from transition system specifications in the SOS style, satisfying certain syntactic syntactic conditions. We use Aczel's nonstandard set theory for solving a recursive equation for a domain fo processes. It contains non-well-founded elements modelling possibly infinite behaviour. Semantic interpretations of syntactic operators are obtained by defining the operational semantics for terms consisting of both syntactic and semantic (processes)entities. Finally, we return to standard set theory by observing that a similar, though less general, result can be obtained with the use of complete metric spaces.

Different semantic models are studied for a language called POOL: parallel object-oriented language. It is a simplified version of POOL-T, a language that is actually used to write programs for a parallel machine. The most important aspect of this language is that it describes a system as a collection of communicating objects that all have internal activities which are executed in parallel. For POOL, operational and denotational semantics have been developed previously. The former aims at the intuitive operational meaning of the language, whereas the main characteristic of the latter is compositionality. In this paper, the author relates both models, which are quite different, and proves the semantic correctness of the denotational semantics with respect to the operational semantics. These semantic investigations take place in the mathematical framework of complete metric spaces. For the operational semantics a simple space of functions from states to compact sets of streams (which are sequences of states) is used; for the denotational semantics, a domain of processes is used, which is the solution of a reflexive domain equation over a category of complete metric spaces. The main mathematical tool we use is Banach’s theorem, which states that contractions on complete metric spaces have unique fixed points. Both the operational and the denotational semantics are reformulated and are presented, as well as many operators on the semantic domains, as the fixed point of a suitably defined contraction. In this way, a formal equivalence between both models is established. For this purpose, an intermediate domain, which is first compared to the operational model by means of an abstraction operator, is introduced. This function takes processes, which are treelike structures, as arguments and yields sets of streams as results. Next, it is shown that both intermediate and the denotational model are fixed points of the same contraction, from which their equality follows. From both facts, the main result of this study follows: The operational meaning of a POOL program is equal to the denotational meaning to which the abstraction operator is applied. In this manner, the correctness of the denotational semantics with respect to the operational semantics is established.

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