86 research outputs found

    Generators and Bases for Monadic Closures

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    It is well-known that every regular language admits a unique minimal deterministic acceptor. Establishing an analogous result for non-deterministic acceptors is significantly more difficult, but nonetheless of great practical importance. To tackle this issue, a number of sub-classes of non-deterministic automata have been identified, all admitting canonical minimal representatives. In previous work, we have shown that such representatives can be recovered categorically in two steps. First, one constructs the minimal bialgebra accepting a given regular language, by closing the minimal coalgebra with additional algebraic structure over a monad. Second, one identifies canonical generators for the algebraic part of the bialgebra, to derive an equivalent coalgebra with side effects in a monad. In this paper, we further develop the general theory underlying these two steps. On the one hand, we show that deriving a minimal bialgebra from a minimal coalgebra can be realized by applying a monad on an appropriate category of subobjects. On the other hand, we explore the abstract theory of generators and bases for algebras over a monad

    Canonical Algebraic Generators in Automata Learning

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    Many methods for the verification of complex computer systems require the existence of a tractable mathematical abstraction of the system, often in the form of an automaton. In reality, however, such a model is hard to come up with, in particular manually. Automata learning is a technique that can automatically infer an automaton model from a system -- by observing its behaviour. The majority of automata learning algorithms is based on the so-called L* algorithm. The acceptor learned by L* has an important property: it is canonical, in the sense that, it is, up to isomorphism, the unique deterministic finite automaton of minimal size accepting a given regular language. Establishing a similar result for other classes of acceptors, often with side-effects, is of great practical importance. Non-deterministic finite automata, for instance, can be exponentially more succinct than deterministic ones, allowing verification to scale. Unfortunately, identifying a canonical size-minimal non-deterministic acceptor of a given regular language is in general not possible: it can happen that a regular language is accepted by two non-isomorphic non-deterministic finite automata of minimal size. In particular, it thus is unclear which one of the automata should be targeted by a learning algorithm. In this thesis, we further explore the issue and identify (sub-)classes of acceptors that admit canonical size-minimal representatives. In more detail, the contributions of this thesis are three-fold. First, we expand the automata (learning) theory of Guarded Kleene Algebra with Tests (GKAT), an efficiently decidable logic expressive enough to model simple imperative programs. In particular, we present GL*, an algorithm that learns the unique size-minimal GKAT automaton for a given deterministic language, and prove that GL* is more efficient than an existing variation of L*. We implement both algorithms in OCaml, and compare them on example programs. Second, we present a category-theoretical framework based on generators, bialgebras, and distributive laws, which identifies, for a wide class of automata with side-effects in a monad, canonical target models for automata learning. Apart from recovering examples from the literature, we discover a new canonical acceptor of regular languages, and present a unifying minimality result. Finally, we show that the construction underlying our framework is an instance of a more general theory. First, we see that deriving a minimal bialgebra from a minimal coalgebra can be realized by applying a monad on a category of subobjects with respect to an epi-mono factorisation system. Second, we explore the abstract theory of generators and bases for algebras over a monad: we discuss bases for bialgebras, the product of bases, generalise the representation theory of linear maps, and compare our ideas to a coalgebra-based approach

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    Canonical Algebraic Generators in Automata Learning

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    Many methods for the verification of complex computer systems require the existence of a tractable mathematical abstraction of the system, often in the form of an automaton. In reality, however, such a model is hard to come up with, in particular manually. Automata learning is a technique that can automatically infer an automaton model from a system -- by observing its behaviour. The majority of automata learning algorithms is based on the so-called L* algorithm. The acceptor learned by L* has an important property: it is canonical, in the sense that, it is, up to isomorphism, the unique deterministic finite automaton of minimal size accepting a given regular language. Establishing a similar result for other classes of acceptors, often with side-effects, is of great practical importance. Non-deterministic finite automata, for instance, can be exponentially more succinct than deterministic ones, allowing verification to scale. Unfortunately, identifying a canonical size-minimal non-deterministic acceptor of a given regular language is in general not possible: it can happen that a regular language is accepted by two non-isomorphic non-deterministic finite automata of minimal size. In particular, it thus is unclear which one of the automata should be targeted by a learning algorithm. In this thesis, we further explore the issue and identify (sub-)classes of acceptors that admit canonical size-minimal representatives.Comment: PhD thesi

    Generators and Bases for Monadic Closures

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    It is well-known that every regular language admits a unique minimal deterministic acceptor. Establishing an analogous result for non-deterministic acceptors is significantly more difficult, but nonetheless of great practical importance. To tackle this issue, a number of sub-classes of nondeterministic automata have been identified, all admitting canonical minimal representatives. In previous work, we have shown that such representatives can be recovered categorically in two steps. First, one constructs the minimal bialgebra accepting a given regular language, by closing the minimal coalgebra with additional algebraic structure over a monad. Second, one identifies canonical generators for the algebraic part of the bialgebra, to derive an equivalent coalgebra with side effects in a monad. In this paper, we further develop the general theory underlying these two steps. On the one hand, we show that deriving a minimal bialgebra from a minimal coalgebra can be realized by applying a monad on an appropriate category of subobjects. On the other hand, we explore the abstract theory of generators and bases for algebras over a monad

    Proof-theoretic Semantics for Intuitionistic Multiplicative Linear Logic

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    This work is the first exploration of proof-theoretic semantics for a substructural logic. It focuses on the base-extension semantics (B-eS) for intuitionistic multiplicative linear logic (IMLL). The starting point is a review of Sandqvist’s B-eS for intuitionistic propositional logic (IPL), for which we propose an alternative treatment of conjunction that takes the form of the generalized elimination rule for the connective. The resulting semantics is shown to be sound and complete. This motivates our main contribution, a B-eS for IMLL , in which the definitions of the logical constants all take the form of their elimination rule and for which soundness and completeness are established

    Preservation and reflection of bisimilarity via invertible steps

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    In the theory of coalgebras, distributive laws give a general perspective on determinisation and other automata constructions. This perspective has recently been extended to include so-called weak distributive laws, covering several constructions on state-based systems that are not captured by regular distributive laws, such as the construction of a belief-state transformer from a probabilistic automaton, and ultrafilter extensions of Kripke frames. In this paper we first observe that weak distributive laws give rise to the more general notion of what we call an invertible step: a pair of natural transformations that allows to move coalgebras along an adjunction. Our main result is that part of the construction induced by an invertible step preserves and reflects bisimilarity. This covers results that have previously been shown by hand for the instances of ultrafilter extensions and belief-state transformers

    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

    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
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