74 research outputs found

    Left-handed completeness

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    We give a new proof of the completeness of the left-handed star rule of Kleene algebra. The proof is significantly shorter than previous proofs and exposes the rich interaction of algebra and coalgebra in the theory of Kleene algebra

    Cyclic Proofs and Jumping Automata

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    We consider a fragment of a cyclic sequent proof system for Kleene algebra, and we see it as a computational device for recognising languages of words. The starting proof system is linear and we show that it captures precisely the regular languages. When adding the standard contraction rule, the expressivity raises significantly; we characterise the corresponding class of languages using a new notion of multi-head finite automata, where heads can jump

    An Elementary Proof of the FMP for Kleene Algebra

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    Kleene Algebra (KA) is a useful tool for proving that two programs are equivalent by reasoning equationally. Because it abstracts from the meaning of primitive programs, KA's equational theory is decidable, so it integrates well with interactive theorem provers. This raises the question: which equations can we (not) prove using the laws of KA? Moreover, which models of KA are complete, in the sense that they satisfy exactly the provable equations? Kozen (1994) answered these questions by characterizing KA in terms of its language model. Concretely, equivalences provable in KA are exactly those that hold for regular expressions. Pratt (1980) observed that KA is complete w.r.t. relational models, i.e., that its provable equations are those that hold for any relational interpretation. A less known result due to Palka (2005) says that finite models are complete for KA, i.e., that provable equivalences coincide with equations satisfied by all finite KAs. Phrased contrapositively, the latter is a finite model property (FMP): any unprovable equation is falsified by a finite KA. These results can be argued using Kozen's theorem, but the implication is mutual: given that KA is complete w.r.t. finite (resp. relational) models, Palka's (resp. Pratt's) arguments show that it is complete w.r.t. the language model. We embark on a study of the different complete models of KA, and the connections between them. This yields a fourth result subsuming those of Palka and Pratt, namely that KA is complete w.r.t. finite relational models. Next, we put an algebraic spin on Palka's techniques, which yield an elementary proof of the finite model property, and by extension, of Kozen's and Pratt's theorems. In contrast with earlier approaches, this proof relies not on minimality or bisimilarity of automata, but rather on representing the regular expressions involved in terms of transformation automata

    On Tools for Completeness of Kleene Algebra with Hypotheses

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    In the literature on Kleene algebra, a number of variants have been proposed which impose additional structure specified by a theory, such as Kleene algebra with tests (KAT) and the recent Kleene algebra with observations (KAO), or make specific assumptions about certain constants, as for instance in NetKAT. Many of these variants fit within the unifying perspective offered by Kleene algebra with hypotheses, which comes with a canonical language model constructed from a given set of hypotheses. For the case of KAT, this model corresponds to the familiar interpretation of expressions as languages of guarded strings. A relevant question therefore is whether Kleene algebra together with a given set of hypotheses is complete with respect to its canonical language model. In this paper, we revisit, combine and extend existing results on this question to obtain tools for proving completeness in a modular way. We showcase these tools by giving new and modular proofs of completeness for KAT, KAO and NetKAT, and we prove completeness for new variants of KAT: KAT extended with a constant for the full relation, KAT extended with a converse operation, and a version of KAT where the collection of tests only forms a distributive lattice

    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

    On the Logical Strength of Confluence and Normalisation for Cyclic Proofs

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    Kleene Algebra with Hypotheses

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    International audienceWe study the Horn theories of Kleene algebras and star continuous Kleene algebras, from the complexity point of view. While their equational theories coincide and are PSpace-complete, their Horn theories differ and are undecidable. We characterise the Horn theory of star continuous Kleene algebras in terms of downward closed languages and we show that when restricting the shape of allowed hypotheses, the problems lie in various levels of the arithmetical or analytical hierarchy. We also answer a question posed by Cohen about hypotheses of the form 1 = S where S is a sum of letters: we show that it is decidable

    A Complete Proof System for 1-Free Regular Expressions Modulo Bisimilarity

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    Robin Milner (1984) gave a sound proof system for bisimilarity of regular expressions interpreted as processes: Basic Process Algebra with unary Kleene star iteration, deadlock 0, successful termination 1, and a fixed-point rule. He asked whether this system is complete. Despite intensive research over the last 35 years, the problem is still open. This paper gives a partial positive answer to Milner's problem. We prove that the adaptation of Milner's system over the subclass of regular expressions that arises by dropping the constant 1, and by changing to binary Kleene star iteration is complete. The crucial tool we use is a graph structure property that guarantees expressibility of a process graph by a regular expression, and is preserved by going over from a process graph to its bisimulation collapse

    Foundations of Software Science and Computation Structures

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    This open access book constitutes the proceedings of the 22nd International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2019, which took place in Prague, Czech Republic, in April 2019, held as part of the European Joint Conference on Theory and Practice of Software, ETAPS 2019. The 29 papers presented in this volume were carefully reviewed and selected from 85 submissions. They deal with foundational research with a clear significance for software science

    A hierarchy of languages, logics, and mathematical theories

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    We present mathematics from a foundational perspective as a hierarchy in which each tier consists of a language, a logic, and a mathematical theory. Each tier in the hierarchy subsumes all preceding tiers in the sense that its language, logic, and mathematical theory generalize all preceding languages, logics, and mathematical theories. Starting from the root tier, the mathematical theories in this hierarchy are: combinatory logic restricted to the identity I, combinatory logic, ZFC set theory, constructive type theory, and category theory. The languages of the first four tiers correspond to the languages of the Chomsky hierarchy: in combinatory logic Ix = x gives rise to a regular language; the language generated by S, K in combinatory logic is context-free; first-order logic is context-sensitive; and the typed lambda calculus of type theory is recursively enumerable. The logic of each tier can be characterized in terms of the cardinality of the set of its truth values: combinatory logic restricted to I has 0 truth values, while combinatory logic has 1, first-order logic 2, constructive type theory 3, and categeory theory omega_0. We conjecture that the cardinality of objects whose existence can be established in each tier is bounded; for example, combinatory logic is bounded in this sense by omega_0 and ZFC set theory by the least inaccessible cardinal. We also show that classical recursion theory presents a framework for generating the above hierarchy in terms of the initial functions zero, projection, and successor followed by composition and m-recursion, starting with the zero function I in combinatory logic This paper begins with a theory of glossogenesis, i.e. a theory of the origin of language, since this theory shows that natural language has deep connections to category theory and since it was through these connections that the last tier and ultimately the whole hierarchy were discovered. The discussion covers implications of the hierarchy for mathematics, physics, cosmology, theology, linguistics, extraterrestrial communication, and artificial intelligence
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