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The Promise Constraint Satisfaction Problem (PCSP) is a generalization of the Constraint Satisfaction Problem (CSP) that includes approximation variants of satisfiability and graph coloring problems. Barto [LICS '19] has shown that a specific PCSP, the problem to find a valid Not-All-Equal solution to a 1-in-3-SAT instance, is not finitely tractable in that it can be solved by a trivial reduction to a tractable CSP, but such a CSP is necessarily over an infinite domain (unless P=NP). We initiate a systematic study of this phenomenon by giving a general necessary condition for finite tractability and characterizing finite tractability within a class of templates - the "basic" tractable cases in the dichotomy theorem for symmetric Boolean PCSPs allowing negations by Brakensiek and Guruswami [SODA'18]., LIPIcs, Vol. 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021), pages 11:1-11:16

The non-commutative sequoid operator $\oslash$ on games was introduced to capture algebraically the presence of state in history-sensitive strategies in game semantics, by imposing a causality relation on the tensor product of games. Coalgebras for the functor $A \oslash \_$ - i.e. morphisms from $S$ to $A \oslash S$ - may be viewed as state transformers: if $A \oslash \_$ has a final coalgebra, $!A$, then the anamorphism of such a state transformer encapsulates its explicit state, so that it is shared only between successive invocations. We study the conditions under which a final coalgebra $!A$ for $A \oslash \_$ is the carrier of a cofree commutative comonoid on $A$. That is, it is a model of the exponential of linear logic in which we can construct imperative objects such as reference cells coalgebraically, in a game semantics setting. We show that if the tensor decomposes into the sequoid, the final coalgebra $!A$ may be endowed with the structure of the cofree commutative comonoid if there is a natural isomorphism from $!(A \times B)$ to $!A \otimes !B$. This condition is always satisfied if $!A$ is the bifree algebra for $A \oslash \_$, but in general it is necessary to impose it, as we establish by giving an example of a sequoidally decomposable category of games in which plays will be allowed to have transfinite length. In this category, the final coalgebra for the functor $A \oslash \_$ is not the cofree commutative comonoid over A: we illustrate this by explicitly contrasting the final sequence for the functor $A \oslash \_$ with the chain of symmetric tensor powers used in the construction of the cofree commutative comonoid as a limit by Melli\'es, Tabareau and Tasson., Comment: Accepted for publication in the proceedings of CALCO 2017, published in the Dagstuhl LIPIcs series. 15pp + 2pp bibliography + 12 pp Appendix (the appendix is not part of the conference version)

International audience; We introduce IH, a sound and complete graphical theory of vector subspaces over the field of polynomial fractions, with relational composition. The theory is constructed in modular fashion, using Lack's approach to composing PROPs with distributive laws. We then view string diagrams of IH as generalised stream circuits by using a formal Laurent series semantics. We characterize the subtheory where circuits adhere to the classical notion of signal flow graphs, and illustrate the use of the graphical calculus on several examples.

A series of recent papers introduces a coalgebraic semantics for logic programming, where the behavior of a goal is represented by a parallel model of computation called coinductive tree. This semantics fails to be compositional, in the sense that the coalgebra formalizing such behavior does not commute with the substitutions that may apply to a goal. We suggest that this is an instance of a more general phenomenon, occurring in the setting of interactive systems (in particular, nominal process calculi), when one tries to model their semantics with coalgebrae on presheaves. In those cases, compositionality can be obtained through saturation. We apply the same approach to logic programming: the resulting semantics is compositional and enjoys an elegant formulation in terms of coalgebrae on presheaves and their right Kan extensions.

Name passing calculi are nowadays one of the preferred formalisms for the specification of concurrent and distributed systems with a dynamically evolving topology. Despite their widespread adoption as a theoretical tool, though, they still face some unresolved semantic issues, since the standard operational, denotational and logical methods often proved inadequate to reason about these formalisms. A domain which has been successfully employed for languages with asymmetric communication, like the ?-calculus, are presheaf categories based on (injective) relabellings, such as $Set^{\mathbb{I}}$ . Calculi with symmetric binding, in the spirit of the fusion calculus, give rise to novel research challenges. In this work we examine the explicit fusion calculus, and propose to model its syntax and semantics using the presheaf category $Set^\mathbb{E}$ , where $\mathbb{E}$ is the category of equivalence relations and equivalence preserving morphisms.

International audience; Barbed bisimilarity is a widely-used behavioural 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, this definition of equivalence is unsatisfactory, since often the quantification is over an infinite set of contexts, thus making barbed bisimilarity very hard to be verified. Should a labelled operational semantics be available for our system, more efficient observational equivalences might be adopted. To this end, a series of techniques have been proposed to derive labelled transition systems (LTSs) from unlabelled ones, the main example being Leifer and Milner's reactive systems. The underlying intuition is that labels are the "minimal" contexts that allow for a reduction to be performed. We introduce a framework that characterizes (weak) barbed bisimilarity via transition systems whose labels are (possibly minimal) contexts. Differently from other proposals, our theory is not dependent on the way LTSs are built, and it relies on a simple set-theoretical presentation. To provide a test-bed for our formalism, we instantiate it by addressing the semantics of mobile ambients and HoCore, recasting the (weak) barbed bisimilarities of these calculi via label-based behavioural equivalences.

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.

International audience; It is a known fact that the subobjects of an object in an adhesive category form a distributive lattice. Building on this observation, in the paper we show how the representation theorem for finite distributive lattices applies to subobject lattices. In particular, we introduce a notion of irreducible object in an adhesive category, and we prove that any finite object of an adhesive category can be obtained as the colimit of its irreducible subobjects. Furthermore we show that every arrow between finite objects in an adhesive category can be interpreted as a lattice homomorphism between subobject lattices and, conversely, we characterize those homomorphisms between subobject lattices which can be seen as arrows.

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.