101 research outputs found

    Fragments and frame classes:Towards a uniform proof theory for modal fixed point logics

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    This thesis studies the proof theory of modal fixed point logics. In particular, we construct proof systems for various fragments of the modal mu-calculus, interpreted over various classes of frames. With an emphasis on uniform constructions and general results, we aim to bring the relatively underdeveloped proof theory of modal fixed point logics closer to the well-established proof theory of basic modal logic. We employ two main approaches. First, we seek to generalise existing methods for basic modal logic to accommodate fragments of the modal mu-calculus. We use this approach for obtaining Hilbert-style proof systems. Secondly, we adapt existing proof systems for the modal mu-calculus to various classes of frames. This approach yields proof systems which are non-well-founded, or cyclic.The thesis starts with an introduction and some mathematical preliminaries. In Chapter 3 we give hypersequent calculi for modal logic with the master modality, building on work by Ori Lahav. This is followed by an Intermezzo, where we present an abstract framework for cyclic proofs, in which we give sufficient conditions for establishing the bounded proof property. In Chapter 4 we generalise existing work on Hilbert-style proof systems for PDL to the level of the continuous modal mu-calculus. Chapter 5 contains a novel cyclic proof system for the alternation-free two-way modal mu-calculus. Finally, in Chapter 6, we present a cyclic proof system for Guarded Kleene Algebra with Tests and take a first step towards using it to establish the completeness of an algebraic counterpart

    Categorical Modelling of Logic Programming: Coalgebra, Functorial Semantics, String Diagrams

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    Logic programming (LP) is driven by the idea that logic subsumes computation. Over the past 50 years, along with the emergence of numerous logic systems, LP has also grown into a large family, the members of which are designed to deal with various computation scenarios. Among them, we focus on two of the most influential quantitative variants are probabilistic logic programming (PLP) and weighted logic programming (WLP). In this thesis, we investigate a uniform understanding of logic programming and its quan- titative variants from the perspective of category theory. In particular, we explore both a coalgebraic and an algebraic understanding of LP, PLP and WLP. On the coalgebraic side, we propose a goal-directed strategy for calculating the probabilities and weights of atoms in PLP and WLP programs, respectively. We then develop a coalgebraic semantics for PLP and WLP, built on existing coalgebraic semantics for LP. By choosing the appropriate functors representing probabilistic and weighted computation, such coalgeraic semantics characterise exactly the goal-directed behaviour of PLP and WLP programs. On the algebraic side, we define a functorial semantics of LP, PLP, and WLP, such that they three share the same syntactic categories of string diagrams, and differ regarding to the semantic categories according to their data/computation type. This allows for a uniform diagrammatic expression for certain semantic constructs. Moreover, based on similar approaches to Bayesian networks, this provides a framework to formalise the connection between PLP and Bayesian networks. Furthermore, we prove a sound and complete aximatization of the semantic category for LP, in terms of string diagrams. Together with the diagrammatic presentation of the fixed point semantics, one obtain a decidable calculus for proving the equivalence between propositional definite logic programs

    Tools and Algorithms for the Construction and Analysis of Systems

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    This open access book constitutes the proceedings of the 28th International Conference on Tools and Algorithms for the Construction and Analysis of Systems, TACAS 2022, which was held during April 2-7, 2022, in Munich, Germany, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2022. The 46 full papers and 4 short papers presented in this volume were carefully reviewed and selected from 159 submissions. The proceedings also contain 16 tool papers of the affiliated competition SV-Comp and 1 paper consisting of the competition report. TACAS is a forum for researchers, developers, and users interested in rigorously based tools and algorithms for the construction and analysis of systems. The conference aims to bridge the gaps between different communities with this common interest and to support them in their quest to improve the utility, reliability, exibility, and efficiency of tools and algorithms for building computer-controlled systems

    Foundations of Software Science and Computation Structures

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    This open access book constitutes the proceedings of the 25th International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2022, which was held during April 4-6, 2022, in Munich, Germany, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2022. The 23 regular papers presented in this volume were carefully reviewed and selected from 77 submissions. They deal with research on theories and methods to support the analysis, integration, synthesis, transformation, and verification of programs and software systems

    From enhanced coinduction towards enhanced induction

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    International audienceThere exist a rich and well-developed theory of enhancements of the coinduction proof method, widely used on behavioural relations such as bisimilarity. We study how to develop an analogous theory for inductive behaviour relations, i.e., relations defined from inductive observables. Similarly to the coinductive setting, our theory makes use of (semi)-progressions of the form R->F(R), where R is a relation on processes and F is a function on relations, meaning that there is an appropriate match on the transitions that the processes in R can perform in which the process derivatives are in F(R). For a given preorder, an enhancement corresponds to a sound function, i.e., one for which R->F(R) implies that R is contained in the preorder; and similarly for equivalences. We introduce weights on the observables of an inductive relation, and a weight-preserving condition on functions that guarantees soundness. We show that the class of functions contains non-trivial functions and enjoys closure properties with respect to desirable function constructors, so to be able to derive sophisticated sound functions (and hence sophisticated proof techniques) from simpler ones. We consider both strong semantics (in which all actions are treated equally) and weak semantics (in which one abstracts from internal transitions). We test our enhancements on a few non-trivial examples

    Tools and Algorithms for the Construction and Analysis of Systems

    Get PDF
    This open access book constitutes the proceedings of the 28th International Conference on Tools and Algorithms for the Construction and Analysis of Systems, TACAS 2022, which was held during April 2-7, 2022, in Munich, Germany, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2022. The 46 full papers and 4 short papers presented in this volume were carefully reviewed and selected from 159 submissions. The proceedings also contain 16 tool papers of the affiliated competition SV-Comp and 1 paper consisting of the competition report. TACAS is a forum for researchers, developers, and users interested in rigorously based tools and algorithms for the construction and analysis of systems. The conference aims to bridge the gaps between different communities with this common interest and to support them in their quest to improve the utility, reliability, exibility, and efficiency of tools and algorithms for building computer-controlled systems

    Automated Deduction – CADE 28

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    This open access book constitutes the proceeding of the 28th International Conference on Automated Deduction, CADE 28, held virtually in July 2021. The 29 full papers and 7 system descriptions presented together with 2 invited papers were carefully reviewed and selected from 76 submissions. CADE is the major forum for the presentation of research in all aspects of automated deduction, including foundations, applications, implementations, and practical experience. The papers are organized in the following topics: Logical foundations; theory and principles; implementation and application; ATP and AI; and system descriptions

    Coalgebraic semantics for probabilistic logic programming

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    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 cofree 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

    Functorial semantics as a unifying perspective on logic programming

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    Logic programming and its variations are widely used for formal reasoning in various areas of Computer Science, most notably Artificial Intelligence. In this paper we develop a systematic and unifying perspective for (ground) classical, probabilistic, weighted logic programs, based on categorical algebra. Our departure point is a formal distinction between the syntax and the semantics of programs, now regarded as separate categories. Then, we are able to characterise the various variants of logic program as different models for the same syntax category, i.e. structure-preserving functors in the spirit of Lawvere’s functorial semantics. As a first consequence of our approach, we showcase a series of semantic constructs for logic programming pictorially as certain string diagrams in the syntax category. Secondly, we describe the correspondence between probabilistic logic programs and Bayesian networks in terms of the associated models. Our analysis reveals that the correspondence can be phrased in purely syntactical terms, without resorting to the probabilistic domain of interpretation

    Abstract Congruence Criteria for Weak Bisimilarity

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    We introduce three general compositionality criteria over operational semantics and prove that, when all three are satisfied together, they guarantee weak bisimulation being a congruence. Our work is founded upon Turi and Plotkin's mathematical operational semantics and the coalgebraic approach to weak bisimulation by Brengos. We demonstrate each criterion with various examples of success and failure and establish a formal connection with the simply WB cool rule format of Bloom and van Glabbeek. In addition, we show that the three criteria induce lax models in the sense of Bonchi et al
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