Probabilistic Boolean circuits have recently been proposed as a string-diagrammatic foundation for finite probabilistic programming. In this paper, we present a complete set of axioms for their semantics in terms of Markov kernels. Our approach is based on two intermediate results: completeness for partial Boolean circuits and completeness for probabilistic Boolean tapes, a diagrammatic language for rig categories.

Building on the recent axiomatisation of first-order bicategories, we develop a functorial semantics approach to the model theory of first-order logic. First-order theories 𝕋 are captured by free first-order bicategories ℱ_𝕋 and models of𝕋 are structure-preserving functors from ℱ_𝕋 to a first-order bicategory 𝐂. Elementary morphisms of models arise as lax natural transformations between such functors, and the classical Tarski-Vaught test and downward Löwenheim-Skolem theorem admit direct diagrammatic proofs. Our results instantiate classically when 𝐂 = Rel and hold uniformly for models valued in Rel(𝐃) over an arbitrary Boolean geometric category 𝐃 in which regular epis split.

The calculus of relations was introduced by De Morgan and Peirce during the second half of the 19th century, as an extension of Boole's algebra of classes. Later developments on quantification theory by Frege and Peirce himself, paved the way to what is known today as first-order logic, causing the calculus of relations to be long forgotten. This was until 1941, when Tarski raised the question on the existence of a complete axiomatisation for it. This question found only negative answers: there is no finite axiomatisation for the calculus of relations and many of its fragments, as shown later by several no-go theorems. In this paper we show that -- by moving from traditional syntax (cartesian) to a diagrammatic one (monoidal) -- it is possible to have complete axiomatisations for the full calculus. The no-go theorems are circumvented by the fact that our calculus, named the calculus of neo-Peircean relations, is more expressive than the calculus of relations and, actually, as expressive as first-order logic. The axioms are obtained by combining two well known categorical structures: cartesian and linear bicategories. arXiv admin note: substantial text overlap with arXiv:2401.07055

Tape diagrams provide a graphical representation for arrows of rig categories, namely categories equipped with two monoidal structures, ⊕ and ⊗, where ⊗ distributes over ⊕. However, their applicability is limited to categories where ⊕ is a biproduct, i.e., both a categorical product and a coproduct. In this work, we extend tape diagrams to deal with Kleisli categories of symmetric monoidal monads, presented by algebraic theories.

Effectful Mealy machines, which we introduce, are a generalization of Mealy machines with global effects determined by an effectful triple. We provide semantics of effectful Mealy machines in terms of both bisimilarity and traces: bisimilarity is characterized syntactically, via uniform feedback; traces are constructed coinductively in terms of streams. We prove that this framework characterizes standard causal processes and existing flavours of Mealy machine, bisimilarity, and trace equivalence. In the commutative case, we introduce a monoidal generalization of Raney's causal functions: monoidal causal processes.

Abstract Tape diagrams provide a convenient graphical notation for arrows of rig categories, i.e., categories equipped with two monoidal products, $$\oplus $$ ⊕ and $$\otimes $$ ⊗ . In this work, we introduce Kleene-Cartesian rig categories, namely rig categories where $$\otimes $$ ⊗ provides a Cartesian bicategory, while $$\oplus $$ ⊕ a Kleene bicategory.We show that the associated tape diagrams can conveniently deal with Hoare logic.

We introduce effectful Mealy machines - a general notion of Mealy machine with global effects - and give them semantics in terms of both bisimilarity and traces. Bisimilarity of effectful Mealy machines is characterized syntactically, via free uniform feedback. Traces of effectful Mealy machines are given a novel semantic coinductive universe in terms of effectful streams. We prove that this framework generalizes standard causal processes and captures existing flavours of Mealy machine, bisimilarity, and trace.

We introduce the calculus of neo-Peircean relations, a string diagrammatic extension of the calculus of binary relations that has the same expressivity as first order logic and comes with a complete axiomatisation. The axioms are obtained by combining two well known categorical structures: cartesian and linear bicategories.

Fo-bicategories are a categorification of Peirce's calculus of relations. Notably, their laws provide a proof system for first-order logic that is both purely equational and complete. This paper illustrates a correspondence between fo-bicategories and Lawvere's hyperdoctrines. To streamline our proof, we introduce peircean bicategories, which offer a more succinct characterization of fo-bicategories.

Abstract We formulate, in lattice-theoretic terms, two novel algorithms inspired by Bradley’s property directed reachability algorithm. For finding safe invariants or counterexamples, the first algorithm exploits over-approximations of both forward and backward transition relations, expressed abstractly by the notion of adjoints. In the absence of adjoints, one can use the second algorithm, which exploits lower sets and their principals. As a notable example of application, we consider quantitative reachability problems for Markov Decision Processes.

Abstract Up-to techniques are a well-known method for enhancing coinductive proofs of behavioural equivalences. We introduce up-to techniques for behavioural metrics between systems modelled as coalgebras, and we provide abstract results to prove their soundness in a compositional way. In order to obtain a general framework, we need a systematic way to lift functors: we show that the Wasserstein lifting of a functor, introduced in a previous work, corresponds to a change of base in a fibrational sense. This observation enables us to reuse existing results about soundness of up-to techniques in a fibrational setting. We focus on the fibrations of predicates and relations valued in a quantale. To illustrate our approach, we provide an example on distances between regular languages.

Rig categories with finite biproducts are categories with two monoidal products, where one is a biproduct and the other distributes over it. In this work we present tape diagrams, a sound and complete diagrammatic language for these categories, that can be intuitively thought as string diagrams of string diagrams. We test the effectiveness of our approach against the positive fragment of Tarski's calculus of relations.

String diagrams are a powerful and intuitive graphical syntax, originating in theoretical physics and later formalised in the context of symmetric monoidal categories. In recent years, they have found application in the modelling of various computational structures, in fields as diverse as Computer Science, Physics, Control Theory, Linguistics, and Biology. In several of these proposals, transformations of systems are modelled as rewrite rules of diagrams. These developments require a mathematical foundation for string diagram rewriting: whereas rewrite theory for terms is well-understood, the two-dimensional nature of string diagrams poses quite a few additional challenges. This work systematises and expands a series of recent conference papers, laying down such a foundation. As a first step, we focus on the case of rewrite systems for string diagrammatic theories that feature a Frobenius algebra. This common structure provides a more permissive notion of composition than the usual one available in monoidal categories, and has found many applications in areas such as concurrency, quantum theory, and electrical circuits. Notably, this structure provides an exact correspondence between the syntactic notion of string diagrams modulo Frobenius structure and the combinatorial structure of hypergraphs. Our work introduces a combinatorial interpretation of string diagram rewriting modulo Frobenius structures in terms of double-pushout hypergraph rewriting. We prove this interpretation to be sound and complete and we also show that the approach can be generalised to rewriting modulo multiple Frobenius structures. As a proof of concept, we show how to derive from these results a termination strategy for Interacting Bialgebras, an important rewrite theory in the study of quantum circuits and signal flow graphs.

We study the canonical weak distributive law $\delta$ of the powerset monad over the semimodule monad for a certain class of semirings containing, in particular, positive semifields. For this subclass we characterise $\delta$ as a convex closure in the free semimodule of a set. Using the abstract theory of weak distributive laws, we compose the powerset and the semimodule monads via $\delta$, obtaining the monad of convex subsets of the free semimodule.

This paper studies trace-based equivalences for systems combining nondeterministic and probabilistic choices. We show how trace semantics for such processes can be recovered by instantiating a coalgebraic construction known as the generalised powerset construction. We characterise and compare the resulting semantics to known definitions of trace equivalences appearing in the literature. Most of our results are based on the exciting interplay between monads and their presentations via algebraic theories.

Abstract Symmetric monoidal theories (SMTs) generalise algebraic theories in a way that make them suitable to express resource-sensitive systems, in which variables cannot be copied or discarded at will. In SMTs, traditional tree-like terms are replaced by string diagrams, topological entities that can be intuitively thought of as diagrams of wires and boxes. Recently, string diagrams have become increasingly popular as a graphical syntax to reason about computational models across diverse fields, including programming language semantics, circuit theory, quantum mechanics, linguistics, and control theory. In applications, it is often convenient to implement the equations appearing in SMTs as rewriting rules. This poses the challenge of extending the traditional theory of term rewriting, which has been developed for algebraic theories, to string diagrams. In this paper, we develop a mathematical theory of string diagram rewriting for SMTs. Our approach exploits the correspondence between string diagram rewriting and double pushout (DPO) rewriting of certain graphs, introduced in the first paper of this series. Such a correspondence is only sound when the SMT includes a Frobenius algebra structure. In the present work, we show how an analogous correspondence may be established for arbitrary SMTs, once an appropriate notion of DPO rewriting (which we call convex) is identified. As proof of concept, we use our approach to show termination of two SMTs of interest: Frobenius semi-algebras and bialgebras.

Abstract In this paper, we address the problem of proving confluence for string diagram rewriting, which was previously shown to be characterised combinatorially as double-pushout rewriting with interfaces (DPOI) on (labelled) hypergraphs. For standard DPO rewriting without interfaces, confluence for terminating rewriting systems is, in general, undecidable. Nevertheless, we show here that confluence for DPOI, and hence string diagram rewriting, is decidable. We apply this result to give effective procedures for deciding local confluence of symmetric monoidal theories with and without Frobenius structure by critical pair analysis. For the latter, we introduce the new notion of path joinability for critical pairs, which enables finitely many joins of a critical pair to be lifted to an arbitrary context in spite of the strong non-local constraints placed on rewriting in a generic symmetric monoidal theory.

Farkas' lemma is a celebrated result on the solutions of systems of linear inequalities, which finds application pervasively in mathematics and computer science. In this work we show how to formulate and prove Farkas' lemma in diagrammatic polyhedral algebra, a sound and complete graphical calculus for polyhedra. Furthermore, we show how linear programs can be modeled within the calculus and how some famous duality results can be proved.

We study the relationship between cartesian bicategories and a specialisation of Lawvere's hyperdoctrines, namely elementary existential doctrines. Both provide different ways of abstracting the structural properties of logical systems: the former in algebraic terms based on a string diagrammatic calculus, the latter in universal terms using the fundamental notion of adjoint functor. We prove that these two approaches are related by an adjunction, which can be strengthened to an equivalence by imposing further constraints on doctrines.

We prove that every finitely generated convex set of finitely supported probability distributions has a unique base. We apply this result to provide an alternative proof of a recent result: the algebraic theory of convex semilattices presents the monad of convex sets of probability distributions.

Abstract We describe the canonical weak distributive law $$\delta :\mathcal S\mathcal P\rightarrow \mathcal P\mathcal S$$ δ : S P → P S of the powerset monad $$\mathcal P$$ P over the S-left-semimodule monad $$\mathcal S$$ S , for a class of semirings S. We show that the composition of $$\mathcal P$$ P with $$\mathcal S$$ S by means of such $$\delta $$ δ yields almost the monad of convex subsets previously introduced by Jacobs: the only difference consists in the absence in Jacobs’s monad of the empty convex set. We provide a handy characterisation of the canonical weak lifting of $$\mathcal P$$ P to $$\mathbb {EM}(\mathcal S)$$ EM ( S ) as well as an algebraic theory for the resulting composed monad. Finally, we restrict the composed monad to finitely generated convex subsets and we show that it is presented by an algebraic theory combining semimodules and semilattices with bottom, which are the algebras for the finite powerset monad $$\mathcal P_f$$ P f .

Probabilistic automata (PA), also known as probabilistic nondeterministic labelled transition systems, combine probability and nondeterminism. They can be given different semantics, like strong bisimilarity, convex bisimilarity, or (more recently) distribution bisimilarity. The latter is based on the view of PA as transformers of probability distributions, also called belief states, and promotes distributions to first-class citizens. We give a coalgebraic account of distribution bisimilarity, and explain the genesis of the belief-state transformer from a PA. To do so, we make explicit the convex algebraic structure present in PA and identify belief-state transformers as transition systems with state space that carries a convex algebra. As a consequence of our abstract approach, we can give a sound proof technique which we call bisimulation up-to convex hull.

Abstract We extend the signal flow calculus—a compositional account of the classical signal flow graph model of computation—to encompass affine behaviour, and furnish it with a novel operational semantics. The increased expressive power allows us to define a canonical notion of contextual equivalence, which we show to coincide with denotational equality. Finally, we characterise the realisable fragment of the calculus: those terms that express the computations of (affine) signal flow graphs.

We argue that cartesian bicategories, often used as a general categorical algebra of relations, are also a natural setting for the study of the axiom of choice (AC). In this setting, AC manifests itself as an inequation asserting that every total relation contains a map. The generality of cartesian bicategories allows us to separate this formulation from other set-theoretically equivalent properties, for instance that epimorphisms split. Moreover, via a classification result, we show that cartesian bicategories satisfying choice tend to be those that arise from bicategories of spans.

Turi and Plotkin’s bialgebraic semantics is an abstract approach to specifying the operational semantics of a system, by means of a distributive law between its syntax (encoded as a monad) and its dynamics (an endofunctor). This setup is instrumental in showing that a semantic specification (a coalgebra) satisfies desirable properties: in particular, that it is compositional. In this work, we use the bialgebraic approach to derive well-behaved structural operational semantics of string diagrams, a graphical syntax that is increasingly used in the study of interacting systems across different disciplines. Our analysis relies on representing the two-dimensional operations underlying string diagrams in various categories as a monad, and their bialgebraic semantics in terms of a distributive law for that monad. As a proof of concept, we provide bialgebraic compositional semantics for a versatile string diagrammatic language which has been used to model both signal flow graphs (control theory) and Petri nets (concurrency theory). Moreover, our approach reveals a correspondence between two different interpretations of the Frobenius equations on string diagrams and two synchronisation mechanisms for processes, à la Hoare and à la Milner.

Graphical linear algebra is a diagrammatic language allowing to reason compositionally about different types of linear computing devices. In this paper, we extend this formalism with a connector for affine behaviour. The extension, which we call graphical affine algebra, is simple but remarkably powerful: it can model systems with richer patterns of behaviour such as mutual exclusion-with modules over the natural numbers as semantic domain-or non-passive electrical components-when considering modules over a certain field. Our main technical contribution is a complete axiomatisation for graphical affine algebra over these two interpretations. We also show, as case studies, how graphical affine algebra captures electrical circuits and the calculus of stateless connectors-a coordination language for distributed systems.

This paper studies trace-based equivalences for systems combining nondeterministic and probabilistic choices. We show how trace semantics for such processes can be recovered by instantiating a coalgebraic construction known as the generalised powerset construction. We characterise and compare the resulting semantics to known definitions of trace equivalences appearing in the literature. Most of our results are based on the exciting interplay between monads and their presentations via algebraic theories.

We introduce the resource calculus, a string diagrammatic language for concurrent systems. Significantly, it uses the same syntax and operational semantics as the signal flow calculus --- an algebraic formalism for signal flow graphs, which is a combinatorial model of computation of interest in control theory. Indeed, our approach stems from the simple but fruitful observation that, by replacing real numbers (modelling signals) with natural numbers (modelling resources) in the operational semantics, concurrent behaviour patterns emerge. The resource calculus is canonical: we equip it and its stateful extension with equational theories that characterise the underlying space of definable behaviours---a convex algebraic universe of additive relations---via isomorphisms of categories. Finally, we demonstrate that our calculus is sufficiently expressive to capture behaviour definable by classical Petri nets.

Up-to techniques are a well-known method for enhancing coinductive proofs of behavioural equivalences. We introduce up-to techniques for behavioural metrics between systems modelled as coalgebras and we provide abstract results to prove their soundness in a compositional way. In order to obtain a general framework, we need a systematic way to lift functors: we show that the Wasserstein lifting of a functor, introduced in a previous work, corresponds to a change of base in a fibrational sense. This observation enables us to reuse existing results about soundness of up-to techniques in a fibrational setting. We focus on the fibrations of predicates and relations valued in a quantale, for which pseudo-metric spaces are an example. To illustrate our approach we provide an example on distances between regular languages.

The Calculus of Conjunctive Queries (CCQ) has foundational status in database theory. A celebrated theorem of Chandra and Merlin states that CCQ query inclusion is decidable. Its proof transforms logical formulas to graphs: each query has a natural model - a kind of graph - and query inclusion reduces to the existence of a graph homomorphism between natural models. We introduce the diagrammatic language Graphical Conjunctive Queries (GCQ) and show that it has the same expressivity as CCQ. GCQ terms are string diagrams, and their algebraic structure allows us to derive a sound and complete axiomatisation of query inclusion, which turns out to be exactly Carboni and Walters' notion of cartesian bicategory of relations. Our completeness proof exploits the combinatorial nature of string diagrams as (certain cospans of) hypergraphs: Chandra and Merlin's insights inspire a theorem that relates such cospans with spans. Completeness and decidability of the (in)equational theory of GCQ follow as a corollary. Categorically speaking, our contribution is a model-theoretic completeness theorem of free cartesian bicategories (on a relational signature) for the category of sets and relations.

Abstract interpretation is a method to automatically find invariants of programs or pieces of code whose semantics is given via least fixed-points. Up-to techniques have been introduced as enhancements of coinduction, an abstract principle to prove properties expressed via greatest fixed-points.

Symmetric monoidal categories have become ubiquitous as a formal environment for the analysis of compound systems in a compositional, resource-sensitive manner using the graphical syntax of string diagrams. Recently, reasoning with string diagrams has been implemented concretely via double-pushout (DPO) hypergraph rewriting. The hypergraph representation has the twin advantages of being convenient for mechanisation and of completely absorbing the structural laws of symmetric monoidal categories, leaving just the domain-specific equations explicit in the rewriting system.

We study different behavioral metrics, such as those arising from both branching and linear-time semantics, in a coalgebraic setting. Given a coalgebra $\alpha\colon X \to HX$ for a functor $H \colon \mathrm{Set}\to \mathrm{Set}$, we define a framework for deriving pseudometrics on $X$ which measure the behavioral distance of states. A crucial step is the lifting of the functor $H$ on $\mathrm{Set}$ to a functor $\overline{H}$ on the category $\mathrm{PMet}$ of pseudometric spaces. We present two different approaches which can be viewed as generalizations of the Kantorovich and Wasserstein pseudometrics for probability measures. We show that the pseudometrics provided by the two approaches coincide on several natural examples, but in general they differ. If $H$ has a final coalgebra, every lifting $\overline{H}$ yields in a canonical way a behavioral distance which is usually branching-time, i.e., it generalizes bisimilarity. In order to model linear-time metrics (generalizing trace equivalences), we show sufficient conditions for lifting distributive laws and monads. These results enable us to employ the generalized powerset construction.

Probabilistic automata (PA) combine probability and nondeterminism. They can be given different semantics, like strong bisimilarity, convex bisimilarity, or (more recently) distribution bisimilarity. The latter is based on the view of PA as transformers of probability distributions, also called belief states, and promotes distributions to first-class citizens. We give a coalgebraic account of the latter semantics, and explain the genesis of the belief-state transformer from a PA. To do so, we make explicit the convex algebraic structure present in PA and identify belief-state transformers as transition systems with state space that carries a convex algebra. As a consequence of our abstract approach, we can give a sound proof technique which we call bisimulation up-to convex hull.

Herein we develop category-theoretic tools for understanding network-style diagrammatic languages. The archetypal network-style diagrammatic language is that of electric circuits; other examples include signal flow graphs, Markov processes, automata, Petri nets, chemical reaction networks, and so on. The key feature is that the language is comprised of a number of components with multiple (input/output) terminals, each possibly labelled with some type, that may then be connected together along these terminals to form a larger network. The components form hyperedges between labelled vertices, and so a diagram in this language forms a hypergraph. We formalise the compositional structure by introducing the notion of a hypergraph category. Network-style diagrammatic languages and their semantics thus form hypergraph categories, and semantic interpretation gives a hypergraph functor. The first part of this thesis develops the theory of hypergraph categories. In particular, we introduce the tools of decorated cospans and corelations. Decorated cospans allow straightforward construction of hypergraph categories from diagrammatic languages: the inputs, outputs, and their composition are modelled by the cospans, while the 'decorations' specify the components themselves. Not all hypergraph categories can be constructed, however, through decorated cospans. Decorated corelations are a more powerful version that permits construction of all hypergraph categories and hypergraph functors. These are often useful for constructing the semantic categories of diagrammatic languages and functors from diagrams to the semantics. To illustrate these principles, the second part of this thesis details applications to linear time-invariant dynamical systems and passive linear networks.

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.

String diagrams are a powerful and intuitive graphical syntax for terms of symmetric monoidal categories (SMCs). They find many applications in computer science and are becoming increasingly relevant in other fields such as physics and control theory.

OASIS TOSCA aims at solving the problem of managing complex applications across heterogeneous clouds by providing a standard, vendor-agnostic language to describe them. TOSCA permits defining a cloud application as an orchestration of typed components, which can be instantiated by matching other TOSCA applications. In this paper we first present two types of behaviour-aware matching of applications, based on a notion of simulation. We then relax this notion by permitting to match an operation with a sequence of available operations, and present a coinductive procedure to compute such relaxed simulation.

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 investigate the possibility of deriving metric trace semantics in a coalgebraic framework. First, we generalize a technique for systematically lifting functors from the category Set of sets to the category PMet of pseudometric spaces, by identifying conditions under which also natural transformations, monads and distributive laws can be lifted. By exploiting some recent work on an abstract determinization, these results enable the derivation of trace metrics starting from coalgebras in Set. More precisely, for a coalgebra in Set we determinize it, thus obtaining a coalgebra in the Eilenberg-Moore category of a monad. When the monad can be lifted to PMet, we can equip the final coalgebra with a behavioral distance. The trace distance between two states of the original coalgebra is the distance between their images in the determinized coalgebra through the unit of the monad. We show how our framework applies to nondeterministic automata and probabilistic automata.

Up-to techniques are useful tools for optimising proofs of behavioural equivalence of processes. Bisimulations up-to context can be safely used in any language specified by GSOS rules. We showed this result in a previous paper by exploiting the well-known observation by Turi and Plotkin that such languages form bialgebras. In this paper, we prove the soundness of up-to contextual closure for weak bisimulations of systems specified by cool rule formats, as defined by Bloom to ensure congruence of weak bisimilarity. However, the weak transition systems obtained from such cool rules give rise to lax bialgebras, rather than to bialgebras. Hence, to reach our goal, we extend our previously developed categorical framework to an ordered setting.

Network theory uses the string diagrammatic language of monoidal categories to study graphical structures formally, eschewing specialised translations into intermediate formalisms. Recently, there has been a concerted research focus on developing a network theoretic approach to signal flow graphs, which are classical structures in control theory, signal processing and a cornerstone in the study of feedback. In this approach, signal flow graphs are given a relational denotational semantics in terms of formal power series.

The paper is devoted to an analysis of the concurrent features of asynchronous systems. A preliminary step is represented by the introduction of a non-interleaving extension of barbed equivalence. This notion is then exploited in order to prove thatconcurrency cannot be observedthrough asynchronous interactions, i.e., that the interleaving and concurrent versions of a suitable asynchronous weak equivalence actually coincide. The theory is validated on some case studies, related to nominal calculi (π-calculus) and visual specification formalisms (Petri nets). Additionally, we prove that a class of systems which is deemed (output-buffered) asynchronous, according to a characterization that was previously proposed in the literature, falls into our theory.

We study behavioral metrics in an abstract coalgebraic setting. Given a coalgebra α: X → FX in Set, where the functor F specifies the branching type, we define a framework for deriving pseudometrics on X which measure the behavioral distance of states. A first crucial step is the lifting of the functor F on Set to a functor F in the category PMet of pseudometric spaces. We present two different approaches which can be viewed as generalizations of the Kantorovich and Wasserstein pseudometrics for probability measures. We show that the pseudometrics provided by the two approaches coincide on several natural examples, but in general they differ. Then a final coalgebra for F in Set can be endowed with a behavioral distance resulting as the smallest solution of a fixed-point equation, yielding the final F-coalgebra in PMet. The same technique, applied to an arbitrary coalgebra α: X → FX in Set, provides the behavioral distance on X. Under some constraints we can prove that two states are at distance 0 if and only if they are behaviorally equivalent.

In this paper we focus on the synthesis of labelled transition systems (LTSs) for process calculi using Mobile Ambients (MAs) as a testbed. Our proposal is based on a graphical encoding: a process is mapped into a graph equipped with interfaces such that the denotation is fully abstract with respect to the standard structural congruence. Graphs with interfaces are amenable to the synthesis mechanism based on borrowed contexts (BCs), which is an instance of relative pushouts (RPOs). The BC mechanism allows the effective construction of an LTS that has graphs with interfaces as states and labels, and such that the associated bisimilarity is a congruence. We focus here on the analysis of an LTS over processes as graphs with interfaces: we use the LTS on graphs to recover an LTS directly defined over the structure of MA processes and define a set of SOS inference rules capturing the same operational semantics.

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.

Barbed bisimilarity is a widely used behavioral equivalence for interactive systems: given a set of predicates (denoted “barbs” and representing basic observations on states) and a set of contexts (representing the possible execution environments), two systems are deemed to be equivalent if they verify the same barbs whenever inserted inside any of the chosen contexts. Despite its flexibility and expressiveness, this definition of equivalence is unsatisfactory because often the quantification is over an infinite set of contexts, thus making barbed bisimilarity very hard to be verified. Should a labeled operational semantics be available, more efficient observational equivalences might be adopted. To this end, a series of techniques has been proposed to derive labeled transition systems (LTSs) from unlabeled ones, the main example being Leifer and Milner’s theory of reactive systems. The underlying intuition is that labels should be the “minimal” contexts that allow for a reduction step to be performed. However, minimality is difficult to asses, whereas the set of “intuitively” correct labels is often easily devised by the ingenuity of the researcher. This article introduces a framework that characterizes (weak) barbed bisimilarity via LTSs whose labels are (not necessarily minimal) contexts. Differently from previous proposals, our theory does not depend on the way the labeled transitions are built but instead relies on a simple set-theoretical presentation for identifying those properties such an LTS should verify to (1) capture the barbed bisimilarities of the underlying system and (2) ensure that such bisimilarities are congruences. Furthermore, we adopt suitable proof techniques to make feasible the verification of such properties. To provide a test-bed for our formalism, we instantiate it by addressing the semantics of the Mobile Ambients calculus, recasting its barbed bisimilarities via label-based behavioral equivalences.

We introduce bisimulation up to congruence as a technique for proving language equivalence of non-deterministic finite automata. Exploiting this technique, we devise an optimisation of the classical algorithm by Hopcroft and Karp. We compare our approach to the recently introduced antichain algorithms, by analysing and relating the two underlying coinductive proof methods. We give concrete examples where we exponentially improve over antichains; experimental results moreover show non negligible improvements.

Concurrent Constraint Programming (ccp) is a well-established declarative framework from concurrency theory. Its foundations and principles e.g., semantics, proof systems, axiomatizations, have been thoroughly studied for over the last two decades. In contrast, the development of algorithms and automatic verification procedures for ccp have hitherto been far too little considered. To the best of our knowledge there is only one existing verification algorithm for the standard notion of ccp program (observational) equivalence. In this paper we first show that this verification algorithm has an exponential-time complexity even for programs from a representative sub-language of ccp; the summation-free fragment (ccp\+). We then significantly improve on the complexity of this algorithm by providing two alternative polynomial-time decision procedures for ccp\+ program equivalence. Each of these two procedures has an advantage over the other. One has a better time complexity. The other can be easily adapted for the full language of ccp to produce significant state space reductions. The relevance of both procedures derives from the importance of ccp\+. This fragment, which has been the subject of many theoretical studies, has strong ties to first-order logic and an elegant denotational semantics, and it can be used to model real-world situations. Its most distinctive feature is that of confluence, a property we exploit to obtain our polynomial procedures.

This special issue of Mathematical Structures in Computer Science contains a selection of papers presented at three satellite events of CONCUR'09, which was held between 31 August and 5 September 2009 in Bologna (Italy). Specifically, it contains three papers from the 16th International Workshop on Expressiveness in Concurrency (EXPRESS'09), one paper from the 2nd Interaction and Concurrency Experience (ICE'09) and two papers from the 6th Workshop on Structural Operational Semantics (SOS'09).

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.

Web services represent a promising technology for the development of distributed heterogeneous software systems. In this setting, a major issue is to establish whether two services can be used interchangeably in any context. To this aim, our paper first briefly reviews the results contained in a recent article by the same authors, where a suitable notion of behavioural equivalence for Web services was introduced. Our work then extends those results, in order to account for ontologybased service specifications. Next, a concrete example scenario – a car rental system – is presented, and it is then used to illustrate how the equivalence between services can be fruitfully employed for correctly addressing two prominent, modularity-related problems: the publication of correct service specifications and the replaceability of (sub)services.

The semantics of process calculi has traditionally been specified by labelled transition systems (LTS), but with the development of name calculi it turned out that reaction rules (i.e., unlabelled transition rules) are often more natural. This leads to the question of how behavioural equivalences (bisimilarity, trace equivalence, etc.) defined for LTS can be transferred to unlabelled transition systems. Recently, in order to answer this question, several proposals have been made with the aim of automatically deriving an LTS from reaction rules in such a way that the resulting equivalences are congruences. Furthermore these equivalences should agree with the standard semantics, whenever one exists. In this paper we propose saturated semantics, based on a weaker notion of observation and orthogonal to all the previous proposals, and we demonstrate the appropriateness of our semantics by means of two examples: logic programming and a subset of the open ð-calculus. Indeed, we prove that our equivalences are congruences and that they coincide with logical equivalence and open bisimilarity respectively, while equivalences studied in previous works are strictly finer.