Coalgebraic Semantics Research Articles

Probabilistic mediator: A coalgebraic perspective

Mediator is a component-based modeling language where components and systems can be specified separately and precisely. In this paper, we show how to analyze probabilistic Mediator, which supports both non-deterministic and probabilistic behavior of component-based systems, from a coalgebraic perspective. Probabilistic Mediator components are modeled as coalgebras, with the powerset monad P and the distribution monad D to capture their non-determinsitic and probabilistic behavior, respectively. A coalgebraic semantics of combinators on Mediator components is developed. Such a coalgebraic view induces a suitable notion of bisimulation for comparing probabilistic Mediator components and systems.

Semialgebras and Weak Distributive Laws

Motivated by recent work on weak distributive laws and their applications to coalgebraic semantics, we investigate the algebraic nature of semialgebras for a monad. These are algebras for the underlying functor of the monad subject to the associativity axiom alone-the unit axiom from the definition of an Eilenberg-Moore algebras is dropped. We prove that if the underlying category has coproducts, then semialgebras for a monad M are in fact the Eilenberg-Moore algebras for a suitable monad structure on the functor id + M , which we call the semifree monad M^s. We also provide concrete algebraic presentations for semialgebras for the maybe monad, the semigroup monad and the finite distribution monad. A second contribution is characterizing the weak distributive laws of the form M T => T M as strong distributive laws M^s T => T M^s subject to an additional condition.

A Unifying Coalgebraic Semantics Framework for Quantum Systems

As a quantum counterpart of labeled transition system (LTS), quantum labeled transition system (QLTS) is a powerful formalism for modeling quantum programs or protocols, and gives a categorical understanding for quantum computation. With the help of quantum branching monad, QLTS provides a framework extending some ideas in non-deterministic or probabilistic systems to quantum systems. On the other hand, quantum finite automata (QFA) emerged as a very elegant and simple model for resolving some quantum computational problems. In this paper, we propose the notion of reactive quantum system (RQS), a variant of QLTS capturing reactive system behavior, and develop a coalgebraic semantics for QLTS, RQS and QFA by an endofunctor on the category of convex sets, which has a final coalgebra. Such a coalgebraic semantics provides a unifying abstract interpretation for QLTS, RQS and QFA. The notions of bisimulation and simulation can be employed to compare the behavior of different types of quantum systems and judge whether a coalgebra can be behaviorally simulated by another.

Coalgebraic Semantics for Probabilistic Logic Programming

Probabilistic logic programming is increasingly important in artificial intelligence and related fields as a formalism to reason about uncertainty. It generalises logic programming with the possibility of annotating clauses with probabilities. This paper proposes a coalgebraic semantics on probabilistic logic programming. Programs are modelled as coalgebras for a certain functor F, and two semantics are given in terms of cofree coalgebras. First, the F-coalgebra yields a semantics in terms of derivation trees. Second, by embedding F into another type G, as cofree G-coalgebra we obtain a `possible worlds' interpretation of programs, from which one may recover the usual distribution semantics of probabilistic logic programming. Furthermore, we show that a similar approach can be used to provide a coalgebraic semantics to weighted logic programming.

Coalgebraic Operational Semantics for an Imperative Language

Operational semantics is a known and popular semantic method for describing the execution of programs in detail. The traditional definition of this method defines each step of a program as a transition relation. We present a new approach on how to define operational semantics as a coalgebra over a category of configurations. Our approach enables us to deal with a program that is written in a small but real imperative language containing also the common program constructs as input and output statements, and declarations. A coalgebra enables to define operational semantics in a uniform way and it describes the behavior of the programs. The state space of our coalgebra consists of the configurations modeling the actual states; the morphisms in a base category of the coalgebra are the functions defining particular steps during the program's executions. Polynomial endofunctor determines this type of systems. Another advantage of our approach is its easy implementation and graphical representation, which we illustrate on a simple program.

A compositional treatment of iterated open games

Compositional Game Theory is a new, recently introduced model of economic games based upon the computer science idea of compositionality. In it, complex and irregular games can be built up from smaller and simpler games, and the equilibria of these complex games can be defined recursively from the equilibria of their simpler subgames. This paper extends the model by providing a final coalgebra semantics for infinite games. In the course of this, we introduce a new operator on games to model the economic concept of subgame perfection.

Weak Completeness of Coalgebraic Dynamic Logics

We present a coalgebraic generalisation of Fischer and Ladner's Propositional Dynamic Logic (PDL) and Parikh's Game Logic (GL). In earlier work, we proved a generic strong completeness result for coalgebraic dynamic logics without iteration. The coalgebraic semantics of such programs is given by a monad T, and modalities are interpreted via a predicate lifting \^I whose transpose is a monad morphism from T to the neighbourhood monad. In this paper, we show that if the monad T carries a complete semilattice structure, then we can define an iteration construct, and suitable notions of diamond-likeness and box-likeness of predicate-liftings which allows for the definition of an axiomatisation parametric in T, \^I and a chosen set of pointwise program operations. As our main result, we show that if the pointwise operations are "negation-free" and Kleisli composition left-distributes over the induced join on Kleisli arrows, then this axiomatisation is weakly complete with respect to the class of standard models. As special instances, we recover the weak completeness of PDL and of dual-free Game Logic. As a modest new result we obtain completeness for dual-free GL extended with intersection (demonic choice) of games.

A coalgebraic semantics for causality in Petri nets

In this paper we revisit some pioneering efforts to equip Petri nets with compact operational models for expressing causality. The models we propose have a bisimilarity relation and a minimal representative for each equivalence class, and they can be fully explained as coalgebras on a presheaf category on an index category of partial orders. First, we provide a set-theoretic model in the form of a causal case graph, that is a labeled transition system where states and transitions represent markings and firings of the net, respectively, and are equipped with causal information. Most importantly, each state has a poset representing causal dependencies among past events. Our first result shows the correspondence with behavior structure semantics as proposed by Trakhtenbrot and Rabinovich. Causal case graphs may be infinitely-branching and have infinitely many states, but we show how they can be refined to get an equivalent finitely-branching model. In it, states only keep the most recent causes for each token, are up to isomorphism, and are equipped with a symmetry, i.e., a group of poset isomorphisms. Symmetries are essential for the existence of a minimal, often finite-state, model. This first part requires no knowledge of category theory. The next step is constructing a coalgebraic model. We exploit the fact that events can be represented as names, and event generation as name generation. Thus we can apply the Fiore–Turi framework, where the semantics of nominal calculi are modeled as coalgebras over presheaves. We model causal relations as a suitable category of posets with action labels, and generation of new events with causal dependencies as an endofunctor on this category. Presheaves indexed by labeled posets represent the functorial association between states and their causal information. Then we define a well-behaved category of coalgebras. Our coalgebraic model is still infinite-state, but we exploit the equivalence between coalgebras over a class of presheaves and History Dependent automata to derive a compact representation, which is equivalent to our set-theoretical compact model. Remarkably, state reduction is automatically performed along the equivalence.

Network-Conscious π-calculus – A Model of Pastry

A peer-to-peer (p2p) system provides the networking substrate for the execution of distributed applications. It is made of peers that interact over an overlay network. Overlay networks are highly dynamic, as peers can join and leave at any time. Traditional process calculi, such as π-calculus, CCS and others, seem inadequate to capture these kinds of networks, their routing mechanisms, and to verify their properties. In order to model network architecture in a more explicit way, in [Ugo Montanari and Matteo Sammartino. Network conscious π-calculus: A concurrent semantics. ENTCS, 286:291–306, 2012; Matteo Sammartino. A Network-Aware Process Calculus for Global Computing and its Categorical Framework. PhD thesis, University of Pisa, 2013. available at http://www.di.unipi.it/~sammarti/publications/thesis.pdf; Ugo Montanari and Matteo Sammartino. A network-conscious π-calculus and its coalgebraic semantics. Theor. Comput. Sci., 546:188–224, 2014] we have introduced the Network Conscious π-calculus (NCPi), an extension of the π-calculus with names representing network nodes and links. In [Ugo Montanari and Matteo Sammartino. A network–conscious π-calculus and its coalgebraic semantics. Theor. Comput. Sci., 546:188–224, 2014] (a simpler version of) NCPi has been equipped with a coalgebraic operational models, along the lines of Fiore-Turi presheaf-based approach [Marcelo P. Fiore and Daniele Turi. Semantics of name and value passing. In LICS 2001, pages 93–104. IEEE Computer Society, 2001], and with an equivalent History Dependent Automaton [Ugo Montanari and Marco Pistore. Structured coalgebras and minimal hd-automata for the π-calculus. Theor. Comput. Sci., 340(3):539–576, 2005], i.e., an (often) finite-state automaton suitable for verification. In this paper we first give a brief account of these results. Then, our contribution is the sketch of a NCPi representation of the p2p architecture Pastry. In particular, we give models of its overlay network and of a Distributed Hash Table built on top of it, and we give evidence of their correctness by proving convergence of routing mechanisms.

A new coalgebraic Lindström theorem

In a recent article, Alexander Kurz and Yde Venema establish a Lindstrom theorem for coalgebraic modal logic that is shown to imply a modal Lindstrom theorem by Maarten de Rijke. A later modal Lindstrom theorem has been established by Johan van Benthem, and this result still lacks a coalgebraic formulation. The main obstacle has so far been the lack of a suitable notion of ‘submodels’ in coalgebraic semantics, and the problem is left open by Kurz and Venema. In this article, we propose a solution to this problem and derive a general coalgebraic Lindstrom theorem along the lines of van Benthem's result. We provide several applications of the result.

Coalgebraic logic programming: from Semantics to Implementation

Coinductive definitions, such as that of an infinite stream, may often be described by elegant logic programs, but ones for which SLD-refutation is of no value as SLD-derivations fall into infinite loops. Such definitions give rise to questions of lazy corecursive derivations and parallelism, as execution of such logic programs can have both recursive and corecursive features at once. Observational and coalgebraic semantics have been used to study them abstractly. The programming developments have often occurred separately and have usually been implementation-led. Here, we give a coherent semantics-led account of the issues, starting with abstract category theoretic semantics, developing coalgebra to characterize naturally arising trees and proceeding towards implementation of a new dialect, CoALP, of logic programming, characterised by guarded lazy corecursion and parallelism.

A network-conscious π-calculus and its coalgebraic semantics

Traditional process calculi usually abstract away from network details, modeling only communication over shared channels. They, however, seem inadequate to describe new network architectures, such as Software Defined Networks, where programs are allowed to manipulate the infrastructure. In this paper we present the Network Conscious π-calculus ( NCPi), a proper extension of the π-calculus with an explicit notion of network: network links and nodes are represented as names, in full analogy with ordinary π-calculus names, and observations are routing paths through which data is transported. However, restricted links do not appear in the observations, which thus can possibly be as abstract as in the π-calculus. Then we construct a presheaf-based coalgebraic semantics for NCPi along the lines of Turi–Plotkin's approach, by indexing processes with the network resources they use: we give a model for observational equivalence in this context, and we prove that it admits an equivalent nominal automaton (HD-automaton), suitable for verification. Finally, we give a concurrent semantics for NCPi where observations are multisets of routing paths. We show that bisimilarity for this semantics is a congruence, and this property holds also for the concurrent version of the π-calculus.

Exploiting Parallelism in Coalgebraic Logic Programming

We present a parallel implementation of Coalgebraic Logic Programming (CoALP) in the programming language Go. CoALP was initially introduced to reflect coalgebraic semantics of logic programming, with coalgebraic derivation algorithm featuring both corecursion and parallelism. Here, we discuss how the coalgebraic semantics influenced our parallel implementation of logic programming.

Coalgebraic Characterizations of Context-Free Languages

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.

Coalgebras, Chu Spaces, and Representations of Physical Systems

We investigate the use of coalgebra to represent quantum systems, thus providing a basis for the use of coalgebraic methods in quantum information and computation. Coalgebras allow the dynamics of repeated measurement to be captured, and provide mathematical tools such as final coalgebras, bisimulation and coalgebraic logic. However, the standard coalgebraic framework does not accommodate contravariance, and is too rigid to allow physical symmetries to be represented. We introduce a fibrational structure on coalgebras in which contravariance is represented by indexing. We use this structure to give a universal semantics for quantum systems based on a final coalgebra construction. We characterize equality in this semantics as projective equivalence. We also define an analogous indexed structure for Chu spaces, and use this to obtain a novel categorical description of the category of Chu spaces. We use the indexed structures of Chu spaces and coalgebras over a common base to define a truncation functor from coalgebras to Chu spaces. This truncation functor is used to lift the full and faithful representation of the groupoid of physical symmetries on Hilbert spaces into Chu spaces, obtained in our previous work, to the coalgebraic semantics.

Coalgebraic semantics of modal logics: An overview

Coalgebras can be seen as a natural abstraction of Kripke frames. In the same sense, coalgebraic logics are generalised modal logics. In this paper, we give an overview of the basic tools, techniques and results that connect coalgebras and modal logic. We argue that coalgebras unify the semantics of a large range of different modal logics (such as probabilistic, graded, relational, conditional) and discuss unifying approaches to reasoning at this level of generality. We review languages defined in terms of the so-called cover modality, languages induced by predicate liftings as well as their common categorical abstraction, and present (abstract) results on completeness, expressiveness and complexity in these settings, both for basic languages as well as a number of extensions, such as hybrid languages and fixpoints.

Cut elimination in coalgebraic logics

We give two generic proofs for cut elimination in propositional modal logics, interpreted over coalgebras. We first investigate semantic coherence conditions between the axiomatisation of a particular logic and its coalgebraic semantics that guarantee that the cut-rule is admissible in the ensuing sequent calculus. We then independently isolate a purely syntactic property of the set of modal rules that guarantees cut elimination. Apart from the fact that cut elimination holds, our main result is that the syntactic and semantic assumptions are equivalent in case the logic is amenable to coalgebraic semantics. As applications we present a new proof of the (already known) interpolation property for coalition logic and newly establish the interpolation property for the conditional logics LCK and LCKID.

Optimizing Conditional Logic Reasoning within CoLoSS

The generic modal reasoner CoLoSS covers a wide variety of logics ranging from graded and probabilistic modal logic to coalition logic and conditional logics, being based on a broadly applicable coalgebraic semantics and an ensuing general treatment of modal sequent and tableau calculi. Here, we present research into optimisation of the reasoning strategies employed in CoLoSS. Specifically, we discuss strategies of memoisation and dynamic programming that are based on the observation that short sequents play a central role in many of the logics under study. These optimisations seem to be particularly useful for the case of conditional logics, for some of which dynamic programming even improves the theoretical complexity of the algorithm. These strategies have been implemented in CoLoSS; we give a detailed comparison of the different heuristics, observing that in the targeted domain of conditional logics, a substantial speed-up can be achieved.

An approach for machine-assisted verification of Timed CSP specifications

The real-time process calculus Timed CSP is capable of expressing properties such as deadlock-freedom and real-time constraints. It is therefore well-suited to model and verify embedded software. However, proofs about Timed CSP specifications are not ensured to be correct since comprehensive machine-assistance for Timed CSP is not yet available. In this paper, we present our formalization of Timed CSP in the Isabelle/HOL theorem prover, which we have formulated as an operational coalgebraic semantics together with bisimulation equivalences and coalgebraic invariants. This allows for semi-automated and mechanically checked proofs about Timed CSP specifications. Mechanically checked proofs enhance confidence in verification because corner cases cannot be overlooked. We additionally apply our formalization to an abstract specification with real-time constraints. This is the basis for our current work, in which we verify a simple real-time operating system deployed on a satellite. As this operating system has to cope with arbitrarily many threads, we use verification techniques from the area of parameterized systems for which we outline their formalization.

Bringing class diagrams to life

Research in formal methods emphasizes a fundamental interconnection between modeling, calculation and prototyping, made possible by a common unambiguous, mathematical semantics. This paper, building on a broader research agenda on coalgebraic semantics for Unified Modeling Language diagrams, concentrates on class diagrams and discusses how such a coalgebraic perspective can be of use not only for formalizing their specification, but also as a basis for prototyping.