IA Scholar Query: Kripke-Galois Frames and their Logics.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgWed, 21 Sep 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440The Axiomatic Approach to Non-Classical Model Theory
https://scholar.archive.org/work/eis66kn4lvfj7ewxu3xnqlxzwy
Institution theory represents the fully axiomatic approach to model theory in which all components of logical systems are treated fully abstractly by reliance on category theory. Here, we survey some developments over the last decade or so concerning the institution theoretic approach to non-classical aspects of model theory. Our focus will be on many-valued truth and on models with states, which are addressed by the two extensions of ordinary institution theory known as L-institutions and stratified institutions, respectively. The discussion will include relevant concepts, techniques, and results from these two areas.Răzvan Diaconescuwork_eis66kn4lvfj7ewxu3xnqlxzwyWed, 21 Sep 2022 00:00:00 GMTDiffusion of Information on Networked Lattices by Gossip
https://scholar.archive.org/work/3262ryt2tzg2jjuhrqh2xs6c7u
We study time-dependent dynamics on a network of order lattices, where structure-preserving lattice maps are used to fuse lattice-valued data over vertices and edges. The principal contribution is a novel asynchronous Laplacian, generalizing the usual graph Laplacian, adapted to a network of heterogeneous lattices. The resulting gossip algorithm is shown to converge asymptotically to stable "harmonic" distributions of lattice data. This general theorem is applicable to several general problems, including lattice-valued consensus, Kripke semantics, and threat detection, all using asynchronous local update rules.Hans Riess, Robert Ghristwork_3262ryt2tzg2jjuhrqh2xs6c7uMon, 19 Sep 2022 00:00:00 GMTRethinking the notion of oracle
https://scholar.archive.org/work/426m2pxnjrasbawctyc2hxvh4u
We present three different perspectives of oracle. First, an oracle is a blackbox; second, an oracle is an endofunctor on the category of represented spaces; and third, an oracle is an operation on the object of truth-values. These three perspectives create a link between the three fields, computability theory, synthetic descriptive set theory, and effective topos theory.Takayuki Kiharawork_426m2pxnjrasbawctyc2hxvh4uMon, 19 Sep 2022 00:00:00 GMTQuantitative Hennessy-Milner Theorems via Notions of Density
https://scholar.archive.org/work/e6pqdvuqafe5vhro27jrv6it7i
The classical Hennessy-Milner theorem is an important tool in the analysis of concurrent processes; it guarantees that any two non-bisimilar states in finitely branching labelled transition systems can be distinguished by a modal formula. Numerous variants of this theorem have since been established for a wide range of logics and system types, including quantitative versions where lower bounds on behavioural distance (e.g.~in weighted, metric, or probabilistic transition systems) are witnessed by quantitative modal formulas. Both the qualitative and the quantitative versions have been accommodated within the framework of coalgebraic logic, with distances taking values in quantales, subject to certain restrictions, such as being so-called value quantales. While previous quantitative coalgebraic Hennessy-Milner theorems apply only to liftings of set functors to (pseudo-)metric spaces, in the present work we provide a quantitative coalgebraic Hennessy-Milner theorem that applies more widely to functors native to metric spaces; notably, we thus cover, for the first time, the well-known Hennessy-Milner theorem for continuous probabilistic transition systems, where transitions are given by Borel measures on metric spaces, as an instance. In the process, we also relax the restrictions imposed on the quantale, and additionally parametrize the technical account over notions of closure and, hence, density, providing associated variants of the Stone-Weierstrass theorem; this allows us to cover, for instance, behavioural ultrametrics.Jonas Forster, Sergey Goncharov, Dirk Hofmann, Pedro Nora, Lutz Schröder, Paul Wildwork_e6pqdvuqafe5vhro27jrv6it7iTue, 30 Aug 2022 00:00:00 GMTTopological duality for distributive lattices, and applications
https://scholar.archive.org/work/xnnm6lpovzeg7euoh4xupsxexm
This material will be published by Cambridge University Press as "Topological Duality for Distributive Lattices: Theory and Applications" by Mai Gehrke and Sam van Gool. This pre-publication is free to view and download for personal use only. Not for re-distribution, re-sale, or use in derivative works. \copyright Mai Gehrke and Sam van Gool This book is a course in Stone-Priestley duality theory, with applications to logic and theoretical computer science. Our target audience are graduate students and researchers in mathematics and computer science. Our aim is to get in a fairly full palette of duality tools as directly and quickly as possible, then to illustrate and further elaborate these tools within the setting of three emblematic applications: semantics of propositional logics, domain theory in logical form, and the theory of profinite monoids for the study of regular languages and automata. This pre-publication contains the first part of the book, a graduate level 'crash course' in duality theory as it is practiced now, and a chapter on applications to domain theory.Mai Gehrke, Sam van Goolwork_xnnm6lpovzeg7euoh4xupsxexmFri, 26 Aug 2022 00:00:00 GMTThe Lattice-Theoretic Essence of Property Directed Reachability Analysis
https://scholar.archive.org/work/56clzxebhfdllnmkfw3ozchi2a
We present LT-PDR, a lattice-theoretic generalization of Bradley's property directed reachability analysis (PDR) algorithm. LT-PDR identifies the essence of PDR to be an ingenious combination of verification and refutation attempts based on the Knaster-Tarski and Kleene theorems. We introduce four concrete instances of LT-PDR, derive their implementation from a generic Haskell implementation of LT-PDR, and experimentally evaluate them. We also present a categorical structural theory that derives these instances.Mayuko Kori, Natsuki Urabe, Shin-ya Katsumata, Kohei Suenaga, Ichiro Hasuowork_56clzxebhfdllnmkfw3ozchi2aSat, 13 Aug 2022 00:00:00 GMTA non-distributive logic for semiconcepts of a context and its modal extension with semantics based on Kripke contexts
https://scholar.archive.org/work/egdntiaa5bds5ij4y4fxis6k3m
A non-distributive two-sorted hypersequent calculus PDBL and its modal extension MPDBL are proposed for the classes of pure double Boolean algebras and pure double Boolean algebras with operators respectively. A relational semantics for PDBL is next proposed, where any formula is interpreted as a semiconcept of a context. For MPDBL, the relational semantics is based on Kripke contexts, and a formula is interpreted as a semiconcept of the underlying context. The systems are shown to be sound and complete with respect to the relational semantics. Adding appropriate sequents to MPDBL results in logics with semantics based on reflexive, symmetric or transitive Kripke contexts. One of these systems is a logic for topological pure double Boolean algebras. It is demonstrated that, using PDBL, the basic notions and relations of conceptual knowledge can be expressed and inferences involving negations can be obtained. Further, drawing a connection with rough set theory, lower and upper approximations of semiconcepts of a context are defined. It is then shown that, using the formulae and sequents involving modal operators in MPDBL, these approximation operators and their properties can be captured.Prosenjit Howlader, Mohua Banerjeework_egdntiaa5bds5ij4y4fxis6k3mFri, 22 Jul 2022 00:00:00 GMTAlgebraic coherent confluence and higher globular Kleene algebras
https://scholar.archive.org/work/koh4cvnwdvb4bajkvcyomqtcaa
We extend the formalisation of confluence results in Kleene algebras to a formalisation of coherent confluence proofs. For this, we introduce the structure of higher globular Kleene algebra, a higher-dimensional generalisation of modal and concurrent Kleene algebra. We calculate a coherent Church-Rosser theorem and a coherent Newman's lemma in higher Kleene algebras by equational reasoning. We instantiate these results in the context of higher rewriting systems modelled by polygraphs.Cameron Calk, Eric Goubault, Philippe Malbos, Georg Struthwork_koh4cvnwdvb4bajkvcyomqtcaaTue, 19 Jul 2022 00:00:00 GMTFormal Methods for Quantum Programs: A Survey
https://scholar.archive.org/work/7wdnffr4mnho5izjec4llrcehi
While recent progress in quantum hardware open the door for significant speedup in certain key areas (cryptography, biology, chemistry, optimization, machine learning, etc), quantum algorithms are still hard to implement right, and the validation of such quantum programs is achallenge. Moreover, importing the testing and debugging practices at use in classical programming is extremely difficult in the quantum case, due to the destructive aspect of quantum measurement. As an alternative strategy, formal methods are prone to play a decisive role in the emerging field of quantum software. Recent works initiate solutions for problems occurring at every stage of the development process: high-level program design, implementation, compilation, etc. We review the induced challenges for an efficient use of formal methods in quantum computing and the current most promising research directions.Christophe Chareton, Sébastien Bardin, Dongho Lee, Benoît Valiron, Renaud Vilmart, Zhaowei Xuwork_7wdnffr4mnho5izjec4llrcehiFri, 08 Apr 2022 00:00:00 GMTModelling Value-oriented Legal Reasoning in LogiKEy
https://scholar.archive.org/work/67imjzrx6zbc3ek4r2ern4nywu
The logico-pluralist LogiKEy knowledge engineering methodology and framework is applied to the modelling of a theory of legal balancing in which legal knowledge (cases and laws) is encoded by utilising context-dependent value preferences. The theory obtained is then used to formalise, automatically evaluate, and reconstruct illustrative property law cases (involving appropriation of wild animals) within the Isabelle proof assistant system, illustrating how LogiKEy can harness interactive and automated theorem proving technology to provide a testbed for the development and formal verification of legal domain-specific languages and theories. Modelling value-oriented legal reasoning in that framework, we establish novel bridges between latest research in knowledge representation and reasoning in non-classical logics, automated theorem proving, and applications in legal reasoning.Christoph Benzmüller and David Fuenmayor and Bertram Lomfeldwork_67imjzrx6zbc3ek4r2ern4nywuWed, 30 Mar 2022 00:00:00 GMTAlgebraic verification of probabilistic and concurrent systems
https://scholar.archive.org/work/nttgh6gvkbaaxlb2dwmervdeo4
This thesis provides an algebraic modelling and verification of probabilistic concurrent systems in the style of Kleene algebra. Without concurrency, it is shown that the equational theory of continuous probabilistic Kleene algebra is completewith respect to an automata model under standard simulation equivalence. This yields a minimisation-based decision procedure for the algebra. Without probability, an event structure model of Hoare et al.'s concurrent Kleene algebra is constructed. These two algebras are then \merged" to provide probabilistic concurrent Kleene algebra which is used to discover and prove development rules for probabilistic concurrent systems (e.g. rely/guarantee calculus). Soundness of thenew algebra is ensured by models based on probabilistic automata (interleaving) and probabilistic bundle event structures (true concurrency) quotiented with the respective simulation equivalences. Lastly, event structures with implicit probabilitiesare constructed to provide a state based model for the soundness of the probabilistic rely/guarantee rules.Mananjanahary Tahiry Rabehajawork_nttgh6gvkbaaxlb2dwmervdeo4Mon, 28 Mar 2022 00:00:00 GMTUnified inverse correspondence for DLE-Logics
https://scholar.archive.org/work/gquhoaiojjdtpajii3g4noik3a
By exploiting the algebraic and order theoretic mechanisms behind Sahlqvist correspondence, the theory of unified correspondence provides powerful tools for correspondence and canonicity across different semantics and signatures, covering all the logics whose algebraic semantics are given by normal (distributive) lattice expansions (referred to as (D)LEs). In particular, the algorithm ALBA, parametric in each (D)LE, effectively computes the first order correspondents of (D)LE-inductive formulas. We present an algorithm that makes use of ALBA's rules and algebraic language to invert its steps in the DLE setting; therefore effectively computing an inductive formula starting from its first order correspondent.Willem Conradie, Andrea De Domenico, Giuseppe Greco, Alessandra Palmigiano, Mattia Panettiere, Apostolos Tzimouliswork_gquhoaiojjdtpajii3g4noik3aThu, 17 Mar 2022 00:00:00 GMTModal reduction principles across relational semantics
https://scholar.archive.org/work/fkw6s7iaifey5m4jux3kenykyu
The present paper establishes systematic connections among the first-order correspondents of Sahlqvist modal reduction principles in various relational semantic settings which include crisp and many-valued Kripke frames, and crisp and many-valued polarity-based frames (aka enriched formal contexts). Building on unified correspondence theory, we aim at introducing a theoretical environment which makes it possible to: (a) compare and inter-relate the various frame correspondents (in different relational settings) of any given Sahlqvist modal reduction principle; (b) recognize when first-order sentences in the frame-correspondence languages of different types of relational structures encode the same "modal content"; (c) meaningfully transfer and represent well known relational properties such as reflexivity, transitivity, symmetry, seriality, confluence, density, across different semantic contexts. These results can be understood as a first step in a research program aimed at making correspondence theory not just (methodologically) unified, but also (effectively) parametric.Willem Conradie, Andrea De Domenico, Krishna Manoorkar, Alessandra Palmigiano, Mattia Panettiere, Daira Pinto Prieto, Apostolos Tzimouliswork_fkw6s7iaifey5m4jux3kenykyuTue, 01 Feb 2022 00:00:00 GMTSemantic Cut Elimination for the Logic of Bunched Implications, Formalized in Coq
https://scholar.archive.org/work/eluouarchvcuhpuewz5xbkknlu
The logic of bunched implications (BI) is a substructural logic that forms the backbone of separation logic, the much studied logic for reasoning about heap-manipulating programs. Although the proof theory and metatheory of BI are mathematically involved, the formalization of important metatheoretical results is still incipient. In this paper we present a self-contained formalized, in the Coq proof assistant, proof of a central metatheoretical property of BI: cut elimination for its sequent calculus. The presented proof is *semantic*, in the sense that is obtained by interpreting sequents in a particular "universal" model. This results in a more modular and elegant proof than a standard Gentzen-style cut elimination argument, which can be subtle and error-prone in manual proofs for BI. In particular, our semantic approach avoids unnecessary inversions on proof derivations, or the uses of cut reductions and the multi-cut rule. Besides modular, our approach is also robust: we demonstrate how our method scales, with minor modifications, to (i) an extension of BI with an arbitrary set of simple structural rules, and (ii) an extension with an S4-like modality.Dan Fruminwork_eluouarchvcuhpuewz5xbkknluFri, 10 Dec 2021 00:00:00 GMTA Presheaf Semantics for Quantified Temporal Logics
https://scholar.archive.org/work/q66raeklkvam3i4adfac243s6i
Temporal logics stands for a widely adopted family of formalisms for the verification of computational devices, enriching propositional logics by operators predicating on the step-wise behaviour of a system. Its quantified extensions allow to reason on the properties of the individual components of the system at hand. The expressiveness of the resulting logics poses problems in correctly identifying a semantics that exploit its features without resorting to the imposition of restrictions on the acceptable behaviours. In this paper we address this issue by means of counterpart models and relational presheaves.Fabio Gadducci, Davide Trottawork_q66raeklkvam3i4adfac243s6iMon, 01 Nov 2021 00:00:00 GMTA Characterization Result for Non-Distributive Logics
https://scholar.archive.org/work/7bcbzubpajgilalcw6fch4ugdm
Recent published work has addressed the Shalqvist correspondence problem for non-distributive logics. The natural question that arises is to identify the fragment of first-order logic that corresponds to logics without distribution, lifting van Benthem's characterization result for modal logic to this new setting. Carrying out this project is the contribution of the present article. The article is intended as a demonstration and application of a project of reduction of non-distributive logics to (sorted) residuated modal logics. The reduction is an application of recent representation results by this author for normal lattice expansions and a generalization of a canonical and fully abstract translation of the language of substructural logics into the language of their companion sorted, residuated modal logics. The reduction of non-distributive logics to sorted modal logics makes the proof of a van Benthem characterization of non-distributive logics nearly effortless, by adapting and reusing existing results, demonstrating the usefulness and suitability of this approach in studying logics that may lack distribution.Chrysafis Hartonaswork_7bcbzubpajgilalcw6fch4ugdmThu, 14 Oct 2021 00:00:00 GMTDuality for Normal Lattice Expansions and Sorted, Residuated Frames with Relations
https://scholar.archive.org/work/h5umfugxbfgjllwkjhh4rmsbqq
We revisit the problem of Stone duality for lattices with various quasioperators, first studied in [14], presenting a fresh duality result. The new result is an improvement over that of [14] in two important respects. First, the axiomatization of frames in [14] was rather cumbersome and it is now simplified, partly by incorporating Gehrke's proposal [8] of section stability for relations. Second, morphisms are redefined so as to preserve Galois stable (and co-stable) sets and we rely for this, partly again, on Goldblatt's [11] recently proposed definition of bounded morphisms for polarities, though we need to strengthen the definition in order to get a Stone duality result. In studying the dual algebraic structures associated to polarities with relations we demonstrate that stable/co-stable set operators result as the Galois closure of the restriction of classical (though sorted) image operators generated by the frame relations to Galois stable/co-stable sets. This provides a proof, at the representation level, that non-distributive logics can be viewed as fragments of sorted, residuated (poly)modal logics, a research direction initiated in [16,17].Chrysafis Hartonaswork_h5umfugxbfgjllwkjhh4rmsbqqWed, 13 Oct 2021 00:00:00 GMT