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  • Unbounded-Time Safety Verification of Guarded LTI Models with Inputs by Abstract Acceleration
    J. Autom. Reason. (IF 1.172) Pub Date : 2020-05-29
    Dario Cattaruzza, Alessandro Abate, Peter Schrammel, Daniel Kroening

    Reachability analysis of dynamical models is a relevant problem that has seen much progress in the last decades, however with clear limitations pertaining to the nature of the dynamics and the soundness of the results. This article focuses on sound safety verification of unbounded-time (infinite-horizon) linear time-invariant (LTI) models with inputs using reachability analysis. We achieve this using

  • Building Strategies into QBF Proofs
    J. Autom. Reason. (IF 1.172) Pub Date : 2020-05-22
    Olaf Beyersdorff, Joshua Blinkhorn, Meena Mahajan

    Strategy extraction is of great importance for quantified Boolean formulas (QBF), both in solving and proof complexity. So far in the QBF literature, strategy extraction has been algorithmically performed from proofs. Here we devise the first QBF system where (partial) strategies are built into the proof and are piecewise constructed by simple operations along with the derivation. This has several

  • From QBFs to MALL and Back via Focussing
    J. Autom. Reason. (IF 1.172) Pub Date : 2020-05-22
    Anupam Das

    In this work we investigate how to extract alternating time bounds from ‘focussed’ proof systems. Our main result is the obtention of fragments of \(\mathsf {MALL} {\mathsf {w} }\) (\(\mathsf {MALL} \) with weakening) complete for each level of the polynomial hierarchy. In one direction we encode QBF satisfiability and in the other we encode focussed proof search, and we show that the composition of

  • Formalization of the Poincaré Disc Model of Hyperbolic Geometry
    J. Autom. Reason. (IF 1.172) Pub Date : 2020-04-30
    Danijela Simić, Filip Marić, Pierre Boutry

    We describe formalization of the Poincaré disc model of hyperbolic geometry within the Isabelle/HOL proof assistant. The model is defined within the complex projective line \(\mathbb {C}{}P^1\)and is shown to satisfy Tarski’s axioms except for Euclid’s axiom—it is shown to satisfy it’s negation, and, moreover, to satisfy the existence of limiting parallels axiom.

  • Formalization of Euler–Lagrange Equation Set Based on Variational Calculus in HOL Light
    J. Autom. Reason. (IF 1.172) Pub Date : 2020-03-06
    Yong Guan, Jingzhi Zhang, Guohui Wang, Ximeng Li, Zhiping Shi, Yongdong Li

    As the theoretical foundation of Lagrangian mechanics, Euler–Lagrange equation sets are widely applied in building mathematical models of physical systems, especially in solving dynamics problems. However, their preconditions are often not fully satisfied in practice. Therefore, it is necessary to verify their applications. The purpose of the present work is to conduct such verification by establishing

  • SPASS-AR: A First-Order Theorem Prover Based on Approximation-Refinement into the Monadic Shallow Linear Fragment
    J. Autom. Reason. (IF 1.172) Pub Date : 2020-02-25
    Andreas Teucke, Christoph Weidenbach

    We introduce FO-AR, an approximation-refinement approach for first-order theorem proving based on counterexample-guided abstraction refinement. A given first-order clause set N is transformed into an over-approximation \(N^{\prime }\) in a decidable first-order fragment. That means if \(N^{\prime }\) is satisfiable so is N. However, if \(N^{\prime }\) is unsatisfiable, then the approximation provides

  • Theorem Proving for Pointwise Metric Temporal Logic Over the Naturals via Translations
    J. Autom. Reason. (IF 1.172) Pub Date : 2020-02-19
    Ullrich Hustadt, Ana Ozaki, Clare Dixon

    We study translations from metric temporal logic (MTL) over the natural numbers to linear temporal logic (LTL). In particular, we present two approaches for translating from MTL to LTL which preserve the ExpSpace complexity of the satisfiability problem for MTL. In each of these approaches we consider the case where the mapping between states and time points is given by (i) a strict monotonic function

  • Hammer for Coq: Automation for Dependent Type Theory.
    J. Autom. Reason. Pub Date : 2018-08-03
    Łukasz Czajka,Cezary Kaliszyk

    Hammers provide most powerful general purpose automation for proof assistants based on HOL and set theory today. Despite the gaining popularity of the more advanced versions of type theory, such as those based on the Calculus of Inductive Constructions, the construction of hammers for such foundations has been hindered so far by the lack of translation and reconstruction components. In this paper,

  • A Verified SAT Solver Framework with Learn, Forget, Restart, and Incrementality.
    J. Autom. Reason. Pub Date : 2018-08-03
    Jasmin Christian Blanchette,Mathias Fleury,Peter Lammich,Christoph Weidenbach

    We developed a formal framework for conflict-driven clause learning (CDCL) using the Isabelle/HOL proof assistant. Through a chain of refinements, an abstract CDCL calculus is connected first to a more concrete calculus, then to a SAT solver expressed in a functional programming language, and finally to a SAT solver in an imperative language, with total correctness guarantees. The framework offers

  • Verified iptables Firewall Analysis and Verification.
    J. Autom. Reason. Pub Date : 2018-08-03
    Cornelius Diekmann,Lars Hupel,Julius Michaelis,Maximilian Haslbeck,Georg Carle

    This article summarizes our efforts around the formally verified static analysis of iptables rulesets using Isabelle/HOL. We build our work around a formal semantics of the behavior of iptables firewalls. This semantics is tailored to the specifics of the filter table and supports arbitrary match expressions, even new ones that may be added in the future. Around that, we organize a set of simplification

  • A Verified ODE Solver and the Lorenz Attractor.
    J. Autom. Reason. Pub Date : 2018-08-03
    Fabian Immler

    A rigorous numerical algorithm, formally verified with Isabelle/HOL, is used to certify the computations that Tucker used to prove chaos for the Lorenz attractor. The verification is based on a formalization of a diverse variety of mathematics and algorithms. Formalized mathematics include ordinary differential equations and Poincaré maps. Algorithms include low level approximation schemes based on

  • The Role of the Mizar Mathematical Library for Interactive Proof Development in Mizar.
    J. Autom. Reason. Pub Date : 2018-08-03
    Grzegorz Bancerek,Czesław Byliński,Adam Grabowski,Artur Korniłowicz,Roman Matuszewski,Adam Naumowicz,Karol Pąk

    The Mizar system is one of the pioneering systems aimed at supporting mathematical proof development on a computer that have laid the groundwork for and eventually have evolved into modern interactive proof assistants. We claim that an important milestone in the development of these systems was the creation of organized libraries accumulating all previously available formalized knowledge in such a

  • OWL Reasoning: Subsumption Test Hardness and Modularity.
    J. Autom. Reason. Pub Date : 2018-08-03
    Nicolas Matentzoglu,Bijan Parsia,Uli Sattler

    Reasoning with SROIQ(D) , the logic that underpins the popular Web Ontology Language (OWL), has a high worst case complexity (N2Exptime). Decomposing the ontology into modules prior to classification, and then classifying the composites one-by-one, has been suggested as a way to mitigate this complexity in practice. Modular reasoning is currently motivated by the potential for reducing the hardness

  • Genetic Programming + Proof Search = Automatic Improvement.
    J. Autom. Reason. Pub Date : 2018-08-03
    Zoltan A Kocsis,Jerry Swan

    Search Based Software Engineering techniques are emerging as important tools for software maintenance. Foremost among these is Genetic Improvement, which has historically applied the stochastic techniques of Genetic Programming to optimize pre-existing program code. Previous work in this area has not generally preserved program semantics and this article describes an alternative to the traditional

  • Relative Termination via Dependency Pairs.
    J. Autom. Reason. Pub Date : 2017-01-01
    José Iborra,Naoki Nishida,Germán Vidal,Akihisa Yamada

    A term rewrite system is terminating when no infinite reduction sequences are possible. Relative termination generalizes termination by permitting infinite reductions as long as some distinguished rules are not applied infinitely many times. Relative termination is thus a fundamental notion that has been used in a number of different contexts, like analyzing the confluence of rewrite systems or the

  • Higher-Order Pattern Anti-Unification in Linear Time.
    J. Autom. Reason. Pub Date : 2017-01-01
    Alexander Baumgartner,Temur Kutsia,Jordi Levy,Mateu Villaret

    We present a rule-based Huet's style anti-unification algorithm for simply typed lambda-terms, which computes a least general higher-order pattern generalization. For a pair of arbitrary terms of the same type, such a generalization always exists and is unique modulo α -equivalence and variable renaming. With a minor modification, the algorithm works for untyped lambda-terms as well. The time complexity

  • A Fully Automatic Theorem Prover with Human-Style Output.
    J. Autom. Reason. Pub Date : 2017-01-01
    M Ganesalingam,W T Gowers

    This paper describes a program that solves elementary mathematical problems, mostly in metric space theory, and presents solutions that are hard to distinguish from solutions that might be written by human mathematicians.

  • An Approximation Framework for Solvers and Decision Procedures.
    J. Autom. Reason. Pub Date : 2017-01-01
    Aleksandar Zeljić,Christoph M Wintersteiger,Philipp Rümmer

    We consider the problem of automatically and efficiently computing models of constraints, in the presence of complex background theories such as floating-point arithmetic. Constructing models, or proving that a constraint is unsatisfiable, has various applications, for instance for automatic generation of test inputs. It is well-known that a naïve encoding of constraints into simpler theories (for

  • The OWL Reasoner Evaluation (ORE) 2015 Competition Report.
    J. Autom. Reason. Pub Date : 2017-01-01
    Bijan Parsia,Nicolas Matentzoglu,Rafael S Gonçalves,Birte Glimm,Andreas Steigmiller

    The OWL Reasoner Evaluation competition is an annual competition (with an associated workshop) that pits OWL 2 compliant reasoners against each other on various standard reasoning tasks over naturally occurring problems. The 2015 competition was the third of its sort and had 14 reasoners competing in six tracks comprising three tasks (consistency, classification, and realisation) over two profiles

  • Complexity and Resource Bound Analysis of Imperative Programs Using Difference Constraints.
    J. Autom. Reason. Pub Date : 2017-01-01
    Moritz Sinn,Florian Zuleger,Helmut Veith

    Difference constraints have been used for termination analysis in the literature, where they denote relational inequalities of the form x'≤y+c , and describe that the value of x in the current state is at most the value of y in the previous state plus some constant c∈Z . We believe that difference constraints are also a good choice for complexity and resource bound analysis because the complexity of

  • A Formal C Memory Model for Separation Logic.
    J. Autom. Reason. Pub Date : 2016-01-01
    Robbert Krebbers

    The core of a formal semantics of an imperative programming language is a memory model that describes the behavior of operations on the memory. Defining a memory model that matches the description of C in the C11 standard is challenging because C allows both high-level (by means of typed expressions) and low-level (by means of bit manipulation) memory accesses. The C11 standard has restricted the interaction

  • Labelled Interpolation Systems for Hyper-Resolution, Clausal, and Local Proofs.
    J. Autom. Reason. Pub Date : 2016-01-01
    Matthias Schlaipfer,Georg Weissenbacher

    Craig's interpolation theorem has numerous applications in model checking, automated reasoning, and synthesis. There is a variety of interpolation systems which derive interpolants from refutation proofs; these systems are ad-hoc and rigid in the sense that they provide exactly one interpolant for a given proof. In previous work, we introduced a parametrised interpolation system which subsumes existing

  • On Definitions of Constants and Types in HOL.
    J. Autom. Reason. Pub Date : 2016-01-01
    Rob Arthan

    This paper reports on a simpler and more powerful replacement for the principles for defining new constants that were previously provided in the various HOL implementations. We discuss the problems that the new principle is intended to solve and sketch the proofs that it is conservative and that it subsumes the earlier definitional principles. The new definitional principle for constants has been implemented

  • The Higher-Order Prover Leo-II.
    J. Autom. Reason. Pub Date : 2015-01-01
    Christoph Benzmüller,Nik Sultana,Lawrence C Paulson,Frank Theiß

    Leo-II is an automated theorem prover for classical higher-order logic. The prover has pioneered cooperative higher-order-first-order proof automation, it has influenced the development of the TPTP THF infrastructure for higher-order logic, and it has been applied in a wide array of problems. Leo-II may also be called in proof assistants as an external aid tool to save user effort. For this it is crucial

  • Reinterpreting Dependency Schemes: Soundness Meets Incompleteness in DQBF.
    J. Autom. Reason. Pub Date : null
    Olaf Beyersdorff,Joshua Blinkhorn,Leroy Chew,Renate Schmidt,Martin Suda

    Dependency quantified Boolean formulas (DQBF) and QBF dependency schemes have been treated separately in the literature, even though both treatments extend QBF by replacing the linear order of the quantifier prefix with a partial order. We propose to merge the two, by reinterpreting a dependency scheme as a mapping from QBF into DQBF. Our approach offers a fresh insight on the nature of soundness in

  • Long-Distance Q-Resolution with Dependency Schemes.
    J. Autom. Reason. Pub Date : null
    Tomáš Peitl,Friedrich Slivovsky,Stefan Szeider

    Resolution proof systems for quantified Boolean formulas (QBFs) provide a formal model for studying the limitations of state-of-the-art search-based QBF solvers that use these systems to generate proofs. We study a combination of two proof systems supported by the solver DepQBF: Q-resolution with generalized universal reduction according to a dependency scheme and long distance Q-resolution. We show

  • Verifying OpenJDK's Sort Method for Generic Collections.
    J. Autom. Reason. Pub Date : null
    Stijn de Gouw,Frank S de Boer,Richard Bubel,Reiner Hähnle,Jurriaan Rot,Dominic Steinhöfel

    TimSort is the main sorting algorithm provided by the Java standard library and many other programming frameworks. Our original goal was functional verification of TimSort with mechanical proofs. However, during our verification attempt we discovered a bug which causes the implementation to crash by an uncaught exception. In this paper, we identify conditions under which the bug occurs, and from this

  • Extraction of Expansion Trees.
    J. Autom. Reason. Pub Date : null
    Alexander Leitsch,Anela Lolic

    We define a new method for proof mining by CERES (cut-elimination by resolution) that is concerned with the extraction of expansion trees in first-order logic (see Miller in Stud Log 46(4):347-370, 1987) with equality. In the original CERES method expansion trees can be extracted from proofs in normal form (proofs without quantified cuts) as a post-processing of cut-elimination. More precisely they

  • An Assertional Proof of Red–Black Trees Using Dafny
    J. Autom. Reason. (IF 1.172) Pub Date : 2019-10-03
    Ricardo Peña

    Red–black trees are convenient data structures for inserting, searching, and deleting keys with logarithmic costs. However, keeping them balanced requires careful programming, and sometimes to deal with a high number of cases. In this paper, we present a functional version of a red–black tree variant called left-leaning, due to R. Sedgewick, which reduces the number of cases to be dealt with to a few

  • Automatically Verifying Temporal Properties of Pointer Programs with Cyclic Proof
    J. Autom. Reason. (IF 1.172) Pub Date : 2019-08-09
    Gadi Tellez, James Brotherston

    In this article, we investigate the automated verification of temporal properties of heap-aware programs. We propose a deductive reasoning approach based on cyclic proof. Judgements in our proof system assert that a program has a certain temporal property over memory state assertions, written in separation logic with user-defined inductive predicates, while the proof rules of the system unfold temporal

  • Formalizing the Cox–Ross–Rubinstein Pricing of European Derivatives in Isabelle/HOL
    J. Autom. Reason. (IF 1.172) Pub Date : 2019-07-04
    Mnacho Echenim, Hervé Guiol, Nicolas Peltier

    We formalize in the proof assistant Isabelle essential basic notions and results in financial mathematics. We provide generic formal definitions of concepts such as markets, portfolios, derivative products, arbitrages or fair prices, and we show that, under the usual no-arbitrage condition, the existence of a replicating portfolio for a derivative implies that the latter admits a unique fair price

  • A Verified Implementation of the Berlekamp–Zassenhaus Factorization Algorithm
    J. Autom. Reason. (IF 1.172) Pub Date : 2019-06-17
    Jose Divasón, Sebastiaan J. C. Joosten, René Thiemann, Akihisa Yamada

    We formally verify the Berlekamp–Zassenhaus algorithm for factoring square-free integer polynomials in Isabelle/HOL. We further adapt an existing formalization of Yun’s square-free factorization algorithm to integer polynomials, and thus provide an efficient and certified factorization algorithm for arbitrary univariate polynomials. The algorithm first performs factorization in the prime field \(\mathrm

  • Efficient Verified (UN)SAT Certificate Checking
    J. Autom. Reason. (IF 1.172) Pub Date : 2019-06-04
    Peter Lammich

    SAT solvers decide the satisfiability of Boolean formulas in conjunctive normal form. They are commonly used for software and hardware verification. Modern SAT solvers are highly complex and optimized programs. As a single bug in the solver may invalidate the verification of many systems, SAT solvers output certificates for their answer, which are then checked independently. However, even certificate

  • Automated Reasoning with Power Maps
    J. Autom. Reason. (IF 1.172) Pub Date : 2019-05-07
    G. I. Moghaddam, R. Padmanabhan, Yang Zhang

    In this paper, we employ automated deduction techniques to prove and generalize some well-known theorems in group theory that involve power maps \( x^n\). The difficulty lies in the fact that the term \(x^n\) cannot be expressed in the syntax of first-order logic when n is an integer variable. Here we employ a new concept of “power-like functions” by extracting relevant equational properties valid

  • Homogeneous Length Functions on Groups: Intertwined Computer and Human Proofs
    J. Autom. Reason. (IF 1.172) Pub Date : 2019-04-16
    Siddhartha Gadgil

    We describe a case of an interplay between human and computer proving which played a role in the discovery of an interesting mathematical result (Fritz et al. in Algebra Number Theory 12:1773–1786, 2018). The unusual feature of the use of computers here was that a computer generated but human readable proof was read, understood, generalized and abstracted by mathematicians to obtain the key lemma in

  • A Formalized General Theory of Syntax with Bindings: Extended Version
    J. Autom. Reason. (IF 1.172) Pub Date : 2019-04-16
    Lorenzo Gheri, Andrei Popescu

    We present the formalization of a theory of syntax with bindings that has been developed and refined over the last decade to support several large formalization efforts. Terms are defined for an arbitrary number of constructors of varying numbers of inputs, quotiented to alpha-equivalence and sorted according to a binding signature. The theory contains a rich collection of properties of the standard

  • Strong Extension-Free Proof Systems
    J. Autom. Reason. (IF 1.172) Pub Date : 2019-02-22
    Marijn J. H. Heule, Benjamin Kiesl, Armin Biere

    We introduce proof systems for propositional logic that admit short proofs of hard formulas as well as the succinct expression of most techniques used by modern SAT solvers. Our proof systems allow the derivation of clauses that are not necessarily implied, but which are redundant in the sense that their addition preserves satisfiability. To guarantee that these added clauses are redundant, we consider

  • Scalable Fine-Grained Proofs for Formula Processing
    J. Autom. Reason. (IF 1.172) Pub Date : 2019-01-04
    Haniel Barbosa, Jasmin Christian Blanchette, Mathias Fleury, Pascal Fontaine

    We present a framework for processing formulas in automatic theorem provers, with generation of detailed proofs. The main components are a generic contextual recursion algorithm and an extensible set of inference rules. Clausification, skolemization, theory-specific simplifications, and expansion of ‘let’ expressions are instances of this framework. With suitable data structures, proof generation adds

  • Conflict-Driven Satisfiability for Theory Combination: Transition System and Completeness
    J. Autom. Reason. (IF 1.172) Pub Date : 2019-01-04
    Maria Paola Bonacina, Stéphane Graham-Lengrand, Natarajan Shankar

    Many applications depend on solving the satisfiability of formulæ involving propositional logic and first-order theories, a problem known as Satisfiability Modulo Theory. This article presents a new method for satisfiability modulo a combination of theories, named CDSAT, for Conflict-Driven SATisfiability. CDSAT also solves Satisfiability Modulo Assignment problems that may include assignments to first-order

  • A Resolution-Based Theorem Prover for $${\textsf {K}}_{n}^{}$$Kn : Architecture, Refinements, Strategies and Experiments
    J. Autom. Reason. (IF 1.172) Pub Date : 2018-12-17
    Cláudia Nalon, Ullrich Hustadt, Clare Dixon

    In this paper we describe the implementation of , a resolution-based prover for the basic multimodal logic \({\textsf {K}}_{n}^{}\). The prover implements a resolution-based calculus for both local and global reasoning. The user can choose different normal forms, refinements of the basic resolution calculus, and strategies. We describe these options in detail and discuss their implications. We provide

  • Limited Second-Order Functionality in a First-Order Setting
    J. Autom. Reason. (IF 1.172) Pub Date : 2018-12-17
    Matt Kaufmann, J Strother Moore

    We describe how we have defined in ACL2 a weak version of the Common Lisp functional apply, which takes a function and list of actuals and applies the function to the actuals. Our version, called apply$, does not operate on functions but on ordinary objects—symbols and lists representing lambda expressions—some of which are interpreted as functions. We define a syntactic notion of “tameness” to identify

  • OptiMathSAT : A Tool for Optimization Modulo Theories
    J. Autom. Reason. (IF 1.172) Pub Date : 2018-12-15
    Roberto Sebastiani, Patrick Trentin

    Optimization Modulo Theories (\(\text {OMT}\)) is an extension of SMT which allows for finding models that optimize given objectives. OptiMathSAT is an OMT solver which allows for solving a list of optimization problems on SMT formulas with linear objective functions—on the Boolean, the rational and the integer domains, and on their combination thereof—including (partial weighted) MaxSMT . Multiple

  • A Verified Implementation of Algebraic Numbers in Isabelle/HOL
    J. Autom. Reason. (IF 1.172) Pub Date : 2018-12-09
    Sebastiaan J. C. Joosten, René Thiemann, Akihisa Yamada

    We formalize algebraic numbers in Isabelle/HOL. Our development serves as a verified implementation of algebraic operations on real and complex numbers. We moreover provide algorithms that can identify all the real or complex roots of rational polynomials, and two implementations to display algebraic numbers, an approximative version and an injective precise one. We obtain verified Haskell code for

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