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logic Beyond Formulas: A Proof System on Graphs 20

Logic Beyond Formulas: A Proof System on Graphs

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35th annual acm/ieee symposium on logic in computer science (LICS)

作者： Acclavio, Matteo Horne, Ross Strassburger, Lutz Univ Luxembourg Comp Sci Esch Sur Alzette Luxembourg INRIA Equipe Partout Ecole Polytech LIX UMR 7161 Rocquencourt France

ISBN: (纸本)9781450371049

In this paper we present a proof system that operates on graphs instead of formulas. We begin our quest with the well-known correspondence between formulas and cographs, which are undirected graphs that do not have P-4 (the four-vertex path) as vertex-induced subgraph;and then we drop that condition and look at arbitrary (undirected) graphs. the consequence is that we lose the tree structure of the formulas corresponding to the cographs. therefore we cannot use standard proof theoretical methods that depend on that tree structure. In order to overcome this difficulty, we use a modular decomposition of graphs and some techniques from deep inference where inference rules do not rely on the main connective of a formula. For our proof system we show the admissibility of cut and a generalization of the splitting property. Finally, we show that our system is a conservative extension of multiplicative linear logic (MLL) with mix, meaning that if a graph is a cograph and provable in our system, then it is also provable in MLL+mix.

关键词： Proof theory cographs graph modules prime graphs cut elimination deep inference splitting analycity

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Modal logics with Composition on Finite Forests: Expressivity and Complexity 20

Modal Logics with Composition on Finite Forests: Expressivit...

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35th annual acm/ieee symposium on logic in computer science (LICS)

作者： Bednarczyk, Bartosz Demri, Stephane Fervari, Raul Mansutti, Alessio Tech Univ Dresden Dresden Germany Univ Wroclaw Wroclaw Poland Univ Paris Saclay ENS Paris Saclay CNRS LSV Gif Sur Yvette France Univ Nacl Cordoba FAMAF Cordoba Argentina Consejo Nacl Invest Cient & Tecn Buenos Aires DF Argentina

ISBN: (纸本)9781450371049

We study the expressivity and complexity of two modal logics interpreted on finite forests and equipped with standard modalities to reason on submodels. the logic ML(vertical bar) extends the modal logic K with the composition operator vertical bar from ambient logic, whereas ML(*) features the separating conjunction * from separation logic. Both operators are second-order in nature. We show that ML(vertical bar) is as expressive as the graded modal logic GML (on trees) whereas ML(*) is strictly less expressive than GML. Moreover, we establish that the satisfiability problem is TOWER-complete for ML(*), whereas it is (only) AEXP(POL)-complete for ML(vertical bar), a result which is surprising given their relative expressivity. As by-products, we solve open problems related to sister logics such as static ambient logic and modal separation logic.

关键词： modal logic on trees separation logic static ambient logic graded modal logic expressive power complexity

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Consuming and Persistent Types for Classical logic 20

Consuming and Persistent Types for Classical Logic

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35th annual acm/ieee symposium on logic in computer science (LICS)

作者： Kesner, Delia Vial, Pierre Univ Paris Paris France Inst Univ France Paris France Inria LS2N CNRS Paris France

ISBN: (纸本)9781450371049

We prove that type systems are able to capture exact measures related to dynamic properties of functional programs with control operators, which allow implementing intricate continuations and backtracking. Our type systems give the number of evaluation steps to normal form as well as the size of this normal form without any evaluation being needed. We focus on Parigot's lambda mu-calculus, a computational interpretation of classical natural deduction, and our type systems are based upon non-idempotent intersection and union types. We introduce two kinds of arrows: consuming ones, which type function applications that are evaluated/destroyed during reduction, and persistent ones, which type those that are never consumed. these two forms of arrows are the essential tool of our typing systems to deal with control operators, as we are going to show. the main contribution of this paper is a type framework capturing exact measures within the lambda mu-calculus for 3 different evaluation strategies, namely, head, leftmost-outermost and maximal, associated to the well-known notions of head, weak and strong normalization respectively. Moreover, this is done in a parametric way: we do not provide one type system for each reduction strategy, as it is usually done in the literature, but we give a unique typing system parametrizing key concepts and factorizing common proofs.

关键词： classical logic lambda-mu calculus intersection and union types exact measures natural deduction

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Block products for algebras over countable words and applications to logic 19

Block products for algebras over countable words and applica...

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proceedings of the 34th annual acm/ieee symposium on logic in computer science

作者： Bharat Adsul Saptarshi Sarkar A. V. Sreejith IIT Bombay India IIT Goa India

We propose a seamless integration of the block product operation to the recently developed algebraic framework for regular languages of countable words. A simple but subtle accompanying block product principle has been established. Building on this, we generalize the well-known algebraic characterizations of first-order logic (resp. first-order logic with two variables) in terms of strongly (resp. weakly) iterated block products. We use this to arrive at a complete analogue of Schützenberger-McNaughton-Papert theorem for countable words. We also explicate the role of block products for linear temporal logic by formulating a novel algebraic characterization of a natural fragment.

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An Efficient Normalisation Procedure for Linear Temporal logic and Very Weak Alternating Automata 20

An Efficient Normalisation Procedure for Linear Temporal Log...

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35th annual acm/ieee symposium on logic in computer science (LICS)

作者： Sickert, Salomon Esparza, Javier Tech Univ Munich Munich Germany

ISBN: (纸本)9781450371049

In the mid 80s, Lichtenstein, Pnueli, and Zuck proved a classical theorem stating that every formula of Past LTL (the extension of LTL with past operators) is equivalent to a formula of the form Lambda(n)(i=)(1) GF(phi i) VFG(psi i), where phi(i), and psi(i) contain only past operators. Some years later, Chang, Manna, and Pnueli built on this result to derive a similar normal form for LTL. Both normalisation procedures have a non-elementary worst-case blow-up, and follow an involved path from formulas to counter-free automata to star-free regular expressions and back to formulas. We improve on both points. We present a direct and purely syntactic normalisation procedure for LTL yielding a normal form, comparable to the one by Chang, Manna, and Pnueli, that has only a single exponential blow-up. As an application, we derive a simple algorithm to translate LTL into deterministic Rabin automata. the algorithm normalises the formula, translates it into a special very weak alternating automaton, and applies a simple determinisation procedure, valid only for these special automata.

关键词： Linear Temporal logic Normal Form Weak Alternating Automata Deterministic Automata

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Resource approximation for the λμ-calculus 22

Resource approximation for the λμ-calculus

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proceedings of the 37th annual acm/ieee symposium on logic in computer science

作者： Davide Barbarossa Dipartimento di Informatica - Scienza e Tecnologia Università di Bologna Italy

ISBN: (纸本)9781450393515

the λμ-calculus plays a central role in the theory of programming languages as it extends the Curry-Howard correspondence to classical logic. A major drawback is that it does not satisfy Böhm’s theorem and it lacks the corresponding notion of approximation. On the contrary, we show that Ehrhard and Regnier’s Taylor expansion can be easily adapted, thus providing a resource conscious approximation theory. this produces a sensible λμ-theory with which we prove some advanced properties of the λμ-calculus, such as Stability and Perpendicular Lines Property, from which the impossibility of parallel computations follows.

关键词： λ-calculus λμ-calculus Taylor expansion linear logic resource approximation

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Walk refinement, walk logic, and the iteration number of the Weisfeiler-Leman algorithm 19

Walk refinement, walk logic, and the iteration number of the...

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proceedings of the 34th annual acm/ieee symposium on logic in computer science

作者： Moritz Lichter Ilia Ponomarenko Pascal Schweitzer TU Kaiserslautern Institute of the Russian Academy of Sciences

We show that the 2-dimensional Weisfeiler-Leman algorithm stabilizes n-vertex graphs after at most O(n log n) iterations. this implies that if such graphs are distinguishable in 3-variable first order logic with counting, then they can also be distinguished in this logic by a formula of quantifier depth at most O(n log n).For this we exploit a new refinement based on counting walks and argue that its iteration number differs from the classic Weisfeiler-Leman refinement by at most a logarithmic factor. We then prove matching linear upper and lower bounds on the number of iterations of the walk refinement. this is achieved with an algebraic approach by exploiting properties of semisimple matrix algebras. We also define a walk logic and a bijective walk pebble game that precisely correspond to the new walk refinement.

关键词： Weisfeiler-Leman algorithm

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Matching μ-logic 19

Matching μ-logic

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proceedings of the 34th annual acm/ieee symposium on logic in computer science

作者： Xiaohong Chen Grigore Roşu University of Illinois at Urbana-Champaign

Matching logic is a logic for specifying and reasoning about structure by means of patterns and pattern matching. this paper makes two contributions. First, it proposes a sound and complete proof system for matching logic in its full generality. Previously, sound and complete deduction for matching logic was known only for particular theories providing equality and membership. Second, it proposes matching μ-logic, an extension of matching logic with a least fixpoint μ-binder. It is shown that matching μ-logic captures as special instances many important logics in mathematics and computer science, including first-order logic with least fixpoints, modal μ-logic as well as dynamic logic and various temporal logics such as infinite/finite-trace linear temporal logic and computation tree logic, and notably reachability logic, the underlying logic of the K framework for programming language semantics and formal analysis. Matching μ-logic therefore serves as a unifying foundation for specifying and reasoning about fixpoints and induction, programming languages and program specification and verification.

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Forbidden Induced Subgraphs and the Loś-Tarski theorem

Forbidden Induced Subgraphs and the Loś-Tarski Theorem

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36th annual acm/ieee symposium on logic in computer science, LICS 2021

作者： Chen, Yijia Flum, Jorg Shanghai Jiao Tong University Department of Computer Science China Mathematisches Institut Albert-Ludwigs-Universität Freiburg Germany

ISBN: (纸本)9781665448956

Let \mathcal C be a class of finite and infinite graphs that is closed under induced subgraphs. the well-known Loś-Tarski theorem from classical model theory implies that \mathcal C is definable in first-order logic (FO) by a sentence if and only if \mathcal C has a finite set of forbidden induced finite subgraphs. It provides a powerful tool to show nontrivial characterizations of graphs of small vertex cover, of bounded tree-depth, of bounded shrub-depth, etc. in terms of forbidden induced finite subgraphs. Furthermore, by the Completeness theorem, we can compute from the corresponding forbidden induced subgraphs. Our results (a) and (b) show that this machinery fails on finite graphs.(a)there is a class of finite graphs that is definable in FO and closed under induced subgraphs but has no finite set of forbidden induced subgraphs.(b)Even if we only consider classes \mathcal C of finite graphs that can be characterized by a finite set of forbidden induced subgraphs such a characterization cannot be computed from an FO-sentence that defines \mathcal C and the size of the characterization cannot be bounded by f(||) for any computable function *** their importance in graph theory, our results also significantly strengthen similar known theorems for arbitrary structures. © 2021 ieee.

关键词： Machinery

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Resolving finite indeterminacy A definitive constructive universal prime ideal theorem 20

Resolving finite indeterminacy A definitive constructive uni...

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35th annual acm/ieee symposium on logic in computer science (LICS)

作者： Schuster, Peter Wessel, Daniel Univ Verona Dipartimento Informat Verona Italy

ISBN: (纸本)9781450371049

Dynamical methods were designed to eliminate the ideal objects abstract algebra abounds with. Typically granted by an incarnation of Zorn's Lemma, those ideal objects often serve for proving the semantic conservation of additional non-deterministic sequents, that is, with finite but not necessarily singleton succedents. Eliminating ideal objects dynamically was possible also because (finitary) coherent or geometric logic predominates in that area: the use of a nondeterministic axiom can be captured by a finite branching of the proof tree. Incidentally, a paradigmatic case has widely been neglected in dynamical algebra: Krull's Lemma for prime ideals. Digging deeper just about that case, which we have dealt with only recently (with Yengui), has now brought us to unearth the general phenomenon underlying dynamical algebra: Given a claim of computational nature that usually is proved by the semantic conservation of certain additional non-deterministic axioms, there is a finite labelled tree belonging to a suitable inductively generated class which tree encodes the desired computation. Our characterisation works in the fairly universal setting of a consequence relation enriched with non-deterministic axioms;uniformises many of the known instances of the dynamical method;generalises the proof-theoretic conservation criterion we have offered before (with Rinaldi);and last but not least links the syntactical with the semantic approach: every ideal object used for the customary proof of a concrete claim can be approximated by one of the corresponding tree's branches.

关键词： dynamical proof non-deterministic axiom geometric logic proof-theoretic conservation finite tree computational content inductive generation Krull's Lemma prime ideals

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