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  • Certified Quantum Computation in Isabelle/HOL
    J. Autom. Reason. (IF 1.431) Pub Date : 2020-12-24
    Anthony Bordg, Hanna Lachnitt, Yijun He

    In this article we present an ongoing effort to formalise quantum algorithms and results in quantum information theory using the proof assistant Isabelle/HOL. Formal methods being critical for the safety and security of algorithms and protocols, we foresee their widespread use for quantum computing in the future. We have developed a large library for quantum computing in Isabelle based on a matrix

  • Schematic Refutations of Formula Schemata
    J. Autom. Reason. (IF 1.431) Pub Date : 2020-11-19
    David M. Cerna, Alexander Leitsch, Anela Lolic

    Proof schemata are infinite sequences of proofs which are defined inductively. In this paper we present a general framework for schemata of terms, formulas and unifiers and define a resolution calculus for schemata of quantifier-free formulas. The new calculus generalizes and improves former approaches to schematic deduction. As an application of the method we present a schematic refutation formalizing

  • A Decidable Class of Security Protocols for Both Reachability and Equivalence Properties
    J. Autom. Reason. (IF 1.431) Pub Date : 2020-10-21
    Véronique Cortier, Stéphanie Delaune, Vaishnavi Sundararajan

    We identify a new decidable class of security protocols, both for reachability and equivalence properties. Our result holds for an unbounded number of sessions and for protocols with nonces. It covers all standard cryptographic primitives. Our class sets up three main assumptions. (i) Protocols need to be “simple”, meaning that an attacker can precisely identify from which participant and which session

  • HO $$\pi $$ π in Coq
    J. Autom. Reason. (IF 1.431) Pub Date : 2020-09-14
    Guillaume Ambal, Sergueï Lenglet, Alan Schmitt

    We present a formalization of HO\(\pi \) in Coq, a process calculus where messages carry processes. Such a higher-order calculus features two very different kinds of binder: process input, similar to \(\lambda \)-abstraction, and name restriction, whose scope can be expanded by communication. For the latter, we compare four approaches to represent binders: locally nameless, de Bruijn indices, nominal

  • Formalising $$\varSigma $$ Σ -Protocols and Commitment Schemes Using CryptHOL
    J. Autom. Reason. (IF 1.431) Pub Date : 2020-09-09
    D. Butler, A. Lochbihler, D. Aspinall, A. Gascón

    Machine-checked proofs of security are important to increase the rigour of provable security. In this work we present a formalised theory of two fundamental two party cryptographic primitives: \(\varSigma \)-protocols and Commitment Schemes. \(\varSigma \)-protocols allow a prover to convince a verifier that they possess some knowledge without leaking information about the knowledge. Commitment schemes

  • Machine Learning Guidance for Connection Tableaux
    J. Autom. Reason. (IF 1.431) Pub Date : 2020-09-05
    Michael Färber, Cezary Kaliszyk, Josef Urban

    Connection calculi allow for very compact implementations of goal-directed proof search. We give an overview of our work related to connection tableaux calculi: first, we show optimised functional implementations of connection tableaux proof search, including a consistent Skolemisation procedure for machine learning. Then, we show two guidance methods based on machine learning, namely reordering of

  • Mechanisation of the AKS Algorithm
    J. Autom. Reason. (IF 1.431) Pub Date : 2020-09-02
    Hing Lun Chan, Michael Norrish

    The AKS algorithm (by Agrawal, Kayal and Saxena) is a significant theoretical result, establishing “PRIMES in P” by a brilliant application of ideas from finite fields. This paper describes an implementation of the AKS algorithm in our theorem prover HOL4, together with a proof of its correctness and its computational complexity. The complexity analysis is based on a conservative computation model

  • Synthesizing Precise and Useful Commutativity Conditions
    J. Autom. Reason. (IF 1.431) Pub Date : 2020-08-29
    Kshitij Bansal, Eric Koskinen, Omer Tripp

    Reasoning about commutativity between data-structure operations is an important problem with many applications. In the sequential setting, commutativity can be used to reason about the correctness of refactoring, compiler transformations, and identify instances of non-determinism. In parallel contexts, commutativity dates back to the database (Weihl in IEEE Trans Comput 37(12):1488–1505, 1988) and

  • Higher-Order Quantifier Elimination, Counter Simulations and Fault-Tolerant Systems
    J. Autom. Reason. (IF 1.431) Pub Date : 2020-08-29
    Silvio Ghilardi, Elena Pagani

    We develop quantifier elimination procedures for fragments of higher order logic arising from the formalization of distributed systems (especially of fault-tolerant ones). Such procedures can be used in symbolic manipulations like the computation of pre/post images and of projections. We show in particular that our procedures are quite effective in producing counter abstractions that can be model-checked

  • TacticToe: Learning to Prove with Tactics
    J. Autom. Reason. (IF 1.431) Pub Date : 2020-08-20
    Thibault Gauthier, Cezary Kaliszyk, Josef Urban, Ramana Kumar, Michael Norrish

    We implement an automated tactical prover TacticToe on top of the HOL4 interactive theorem prover. TacticToe learns from human proofs which mathematical technique is suitable in each proof situation. This knowledge is then used in a Monte Carlo tree search algorithm to explore promising tactic-level proof paths. On a single CPU, with a time limit of 60 s, TacticToe proves 66.4% of the 7164 theorems

  • An Isabelle/HOL Formalisation of the SPARC Instruction Set Architecture and the TSO Memory Model
    J. Autom. Reason. (IF 1.431) Pub Date : 2020-08-14
    Zhé Hóu, David Sanan, Alwen Tiu, Yang Liu, Koh Chuen Hoa, Jin Song Dong

    The SPARC instruction set architecture (ISA) has been used in various processors in workstations, embedded systems, and in mission-critical industries such as aviation and space engineering. Hence, it is important to provide formal frameworks that facilitate the verification of hardware and software that run on or interface with these processors. In this work, we give the first formal model for multi-core

  • Natural Projection as Partial Model Checking
    J. Autom. Reason. (IF 1.431) Pub Date : 2020-08-13
    Gabriele Costa, Letterio Galletta, Pierpaolo Degano, David Basin, Chiara Bodei

    Verifying the correctness of a system as a whole requires establishing that it satisfies a global specification. When it does not, it would be helpful to determine which modules are incorrect. As a consequence, specification decomposition is a relevant problem from both a theoretical and practical point of view. Until now, specification decomposition has been independently addressed by the control

  • ICE-Based Refinement Type Discovery for Higher-Order Functional Programs
    J. Autom. Reason. (IF 1.431) Pub Date : 2020-08-01
    Adrien Champion, Tomoya Chiba, Naoki Kobayashi, Ryosuke Sato

    We propose a method for automatically finding refinement types of higher-order function programs. Our method is an extension of the Ice framework of Garg et al. for finding invariants. In addition to the usual positive and negative samples in machine learning, their Ice framework uses implication constraints, which consist of pairs (x, y) such that if x satisfies an invariant, so does y. From these

  • Simulating Strong Practical Proof Systems with Extended Resolution
    J. Autom. Reason. (IF 1.431) Pub Date : 2020-07-31
    Benjamin Kiesl, Adrián Rebola-Pardo, Marijn J. H. Heule, Armin Biere

    Proof systems for propositional logic provide the basis for decision procedures that determine the satisfiability status of logical formulas. While the well-known proof system of extended resolution—introduced by Tseitin in the sixties—allows for the compact representation of proofs, modern SAT solvers (i.e., tools for deciding propositional logic) are based on different proof systems that capture

  • Multi-cost Bounded Tradeoff Analysis in MDP
    J. Autom. Reason. (IF 1.431) Pub Date : 2020-07-28
    Arnd Hartmanns, Sebastian Junges, Joost-Pieter Katoen, Tim Quatmann

    We provide a memory-efficient algorithm for multi-objective model checking problems on Markov decision processes (MDPs) with multiple cost structures. The key problem at hand is to check whether there exists a scheduler for a given MDP such that all objectives over cost vectors are fulfilled. We cover multi-objective reachability and expected cost objectives, and combinations thereof. We further transfer

  • Automated Proof of Bell–LaPadula Security Properties
    J. Autom. Reason. (IF 1.431) Pub Date : 2020-07-27
    Maximiliano Cristiá, Gianfranco Rossi

    Almost 50 years ago, D. E. Bell and L. LaPadula published the first formal model of a secure system, known today as the Bell–LaPadula (BLP) model. BLP is described as a state machine by means of first-order logic and set theory. The authors also formalize two state invariants known as security condition and *-property. Bell and LaPadula prove that all the state transitions preserve these invariants

  • Chain Reduction for Binary and Zero-Suppressed Decision Diagrams
    J. Autom. Reason. (IF 1.431) Pub Date : 2020-07-21
    Randal E. Bryant

    Chain reduction enables reduced ordered binary decision diagrams (BDDs) and zero-suppressed binary decision diagrams (ZDDs) to each take advantage of the other’s ability to symbolically represent Boolean functions in compact form. For any Boolean function, its chain-reduced ZDD (CZDD) representation will be no larger than its ZDD representation, and at most twice the size of its BDD representation

  • Decidable $${\exists }^*{\forall }^*$$ ∃ ∗ ∀ ∗ First-Order Fragments of Linear Rational Arithmetic with Uninterpreted Predicates
    J. Autom. Reason. (IF 1.431) Pub Date : 2020-07-18
    Marco Voigt

    First-order linear rational arithmetic enriched with uninterpreted predicates yields an interesting and very expressive modeling language. However, already the presence of a single uninterpreted predicate symbol of arity one or greater renders the associated satisfiability problem undecidable. We identify two decidable fragments, both based on the Bernays–Schönfinkel–Ramsey prefix class. Due to the

  • CoCon: A Conference Management System with Formally Verified Document Confidentiality
    J. Autom. Reason. (IF 1.431) Pub Date : 2020-07-16
    Andrei Popescu, Peter Lammich, Ping Hou

    We present a case study in formally verified security for realistic systems: the information flow security verification of the functional kernel of a web application, the CoCon conference management system. We use the Isabelle theorem prover to specify and verify fine-grained confidentiality properties, as well as complementary safety and “traceback” properties. The challenges posed by this development

  • Fine-Grained Complexity of Safety Verification
    J. Autom. Reason. (IF 1.431) Pub Date : 2020-07-14
    Peter Chini, Roland Meyer, Prakash Saivasan

    We study the fine-grained complexity of Leader Contributor Reachability (\({\textsf {LCR}} \)) and Bounded-Stage Reachability (\({\textsf {BSR}} \)), two variants of the safety verification problem for shared memory concurrent programs. For both problems, the memory is a single variable over a finite data domain. Our contributions are new verification algorithms and lower bounds. The latter are based

  • A Learning-Based Approach to Synthesizing Invariants for Incomplete Verification Engines
    J. Autom. Reason. (IF 1.431) Pub Date : 2020-07-13
    Daniel Neider, P. Madhusudan, Shambwaditya Saha, Pranav Garg, Daejun Park

    We propose a framework for synthesizing inductive invariants for incomplete verification engines, which soundly reduce logical problems in undecidable theories to decidable theories. Our framework is based on the counterexample guided inductive synthesis principle and allows verification engines to communicate non-provability information to guide invariant synthesis. We show precisely how the verification

  • Parameterized Model Checking on the TSO Weak Memory Model
    J. Autom. Reason. (IF 1.431) Pub Date : 2020-06-27
    Sylvain Conchon, David Declerck, Fatiha Zaïdi

    We present an extended version of the model checking modulo theories framework for verifying parameterized systems under the TSO weak memory model. Our extension relies on three main ingredients: (1) an axiomatic theory of the TSO memory model based on relations over (read, write) events, (2) a TSO-specific backward reachability algorithm and (3) an SMT solver for reasoning about TSO formulas. One

  • Constructive Decision via Redundancy-Free Proof-Search
    J. Autom. Reason. (IF 1.431) Pub Date : 2020-06-24
    Dominique Larchey-Wendling

    We give a constructive account of Kripke–Curry’s method which was used to establish the decidability of implicational relevance logic (\(\mathbf{R}_{{\rightarrow }}\)). To sustain our approach, we mechanize this method in axiom-free Coq, abstracting away from the specific features of \(\mathbf{R}_{{\rightarrow }}\) to keep only the essential ingredients of the technique. In particular we show how to

  • Formalizing Bachmair and Ganzinger’s Ordered Resolution Prover
    J. Autom. Reason. (IF 1.431) Pub Date : 2020-06-17
    Anders Schlichtkrull, Jasmin Blanchette, Dmitriy Traytel, Uwe Waldmann

    We present an Isabelle/HOL formalization of the first half of Bachmair and Ganzinger’s chapter on resolution theorem proving, culminating with a refutationally complete first-order prover based on ordered resolution with literal selection. We developed general infrastructure and methodology that can form the basis of completeness proofs for related calculi, including superposition. Our work clarifies

  • Formalizing the LLL Basis Reduction Algorithm and the LLL Factorization Algorithm in Isabelle/HOL.
    J. Autom. Reason. (IF 1.431) Pub Date : 2020-06-09
    René Thiemann,Ralph Bottesch,Jose Divasón,Max W Haslbeck,Sebastiaan J C Joosten,Akihisa Yamada

    The LLL basis reduction algorithm was the first polynomial-time algorithm to compute a reduced basis of a given lattice, and hence also a short vector in the lattice. It approximates an NP-hard problem where the approximation quality solely depends on the dimension of the lattice, but not the lattice itself. The algorithm has applications in number theory, computer algebra and cryptography. In this

  • Probably Partially True: Satisfiability for Łukasiewicz Infinitely-Valued Probabilistic Logic and Related Topics
    J. Autom. Reason. (IF 1.431) Pub Date : 2020-06-06
    Marcelo Finger, Sandro Preto

    We study probabilistic-logic reasoning in a context that allows for “partial truths”, focusing on computational and algorithmic properties of non-classical Łukasiewicz Infinitely-valued Probabilistic Logic. In particular, we study the satisfiability of joint probabilistic assignments, which we call ŁIPSAT. Although the search space is initially infinite, we provide linear algebraic methods that guarantee

  • Proof-Producing Synthesis of CakeML from Monadic HOL Functions
    J. Autom. Reason. (IF 1.431) Pub Date : 2020-06-06
    Oskar Abrahamsson, Son Ho, Hrutvik Kanabar, Ramana Kumar, Magnus O. Myreen, Michael Norrish, Yong Kiam Tan

    We introduce an automatic method for producing stateful ML programs together with proofs of correctness from monadic functions in HOL. Our mechanism supports references, exceptions, and I/O operations, and can generate functions manipulating local state, which can then be encapsulated for use in a pure context. We apply this approach to several non-trivial examples, including the instruction encoder

  • Unbounded-Time Safety Verification of Guarded LTI Models with Inputs by Abstract Acceleration
    J. Autom. Reason. (IF 1.431) 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 1.431) 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 1.431) 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 1.431) 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 1.431) 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 1.431) 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 1.431) 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

  • The MetaCoq Project
    J. Autom. Reason. (IF 1.431) Pub Date : 2020-02-18
    Matthieu Sozeau; Abhishek Anand; Simon Boulier; Cyril Cohen; Yannick Forster; Fabian Kunze; Gregory Malecha; Nicolas Tabareau; Théo Winterhalter

    The MetaCoq project aims to provide a certified meta-programming environment in Coq. It builds on Template-Coq, a plugin for Coq originally implemented by Malecha (Extensible proof engineering in intensional type theory, Harvard University, http://gmalecha.github.io/publication/2015/02/01/extensible-proof-engineering-in-intensional-type-theory.html, 2014), which provided a reifier for Coq terms and

  • System-Level Non-interference of Constant-Time Cryptography. Part II: Verified Static Analysis and Stealth Memory
    J. Autom. Reason. (IF 1.431) Pub Date : 2020-02-17
    Gilles Barthe, Gustavo Betarte, Juan Diego Campo, Carlos Luna, David Pichardie

    This paper constitutes the second part of a paper published in Barthe et al. (J Autom Reason, 2017. https://doi.org/10.1007/s10817-017-9441-5). Cache-based attacks are a class of side-channel attacks that are particularly effective in virtualized or cloud-based environments, where they have been used to recover secret keys from cryptographic implementations. One common approach to thwart cache-based

  • Loop-Type Sequent Calculi for Temporal Logic
    J. Autom. Reason. (IF 1.431) Pub Date : 2020-02-13
    R. Alonderis, R. Pliuškevičius, A. Pliuškevičienė, H. Giedra

    Various types of calculi (Hilbert, Gentzen sequent, resolution calculi, tableaux) for propositional linear temporal logic (PLTL) have been invented. In this paper, a sound and complete loop-type sequent calculus \(\mathbf{G} _\text {L}{} \mathbf{T} \) for PLTL with the temporal operators “next” and “henceforth always” (\({\mathbf{PLTL}}^{n,a}\)) is considered at first. We prove that all rules of \(\mathbf{G}

  • Verified Analysis of Random Binary Tree Structures
    J. Autom. Reason. (IF 1.431) Pub Date : 2020-02-08
    Manuel Eberl; Max W. Haslbeck; Tobias Nipkow

    This work is a case study of the formal verification and complexity analysis of some famous probabilistic algorithms and data structures in the proof assistant Isabelle/HOL. In particular, we consider the expected number of comparisons in randomised quicksort, the relationship between randomised quicksort and average-case deterministic quicksort, the expected shape of an unbalanced random Binary Search

  • Graph Theory in Coq: Minors, Treewidth, and Isomorphisms
    J. Autom. Reason. (IF 1.431) Pub Date : 2020-01-31
    Christian Doczkal; Damien Pous

    We present a library for graph theory in Coq/Ssreflect. This library covers various notions on simple graphs, directed graphs, and multigraphs. We use it to formalize several results from the literature: Menger’s theorem, the excluded-minor characterization of treewidth-two graphs, and a correspondence between multigraphs of treewidth at most two and terms of certain algebras.

  • Formal Reasoning Under Cached Address Translation
    J. Autom. Reason. (IF 1.431) Pub Date : 2020-01-18
    Hira Taqdees Syeda; Gerwin Klein

    Operating system (OS) kernels achieve isolation between user-level processes using hardware features such as multi-level page tables and translation lookaside buffers (TLBs). The TLB caches address translation, and therefore correctly controlling the TLB is a fundamental security property of OS kernels—yet all large-scale formal OS verification projects we are aware of leave the correct functionality

  • The 2D Dependency Pair Framework for Conditional Rewrite Systems—Part II: Advanced Processors and Implementation Techniques
    J. Autom. Reason. (IF 1.431) Pub Date : 2020-01-16
    Salvador Lucas, José Meseguer, Raúl Gutiérrez

    Proving termination of programs in ‘real-life’ rewriting-based languages like CafeOBJ, Haskell, Maude, etc., is an important subject of research. To advance this goal, faithfully capturing the impact in the termination behavior of the main language features (e.g., conditions in program rules) is essential. In Part I of this work, we have introduced a 2D Dependency Pair Framework for automatically proving

  • A Library for Formalization of Linear Error-Correcting Codes
    J. Autom. Reason. (IF 1.431) Pub Date : 2020-01-06
    Reynald Affeldt; Jacques Garrigue; Takafumi Saikawa

    Error-correcting codes add redundancy to transmitted data to ensure reliable communication over noisy channels. Since they form the foundations of digital communication, their correctness is a matter of concern. To enable trustful verification of linear error-correcting codes, we have been carrying out a systematic formalization in the Coq proof-assistant. This formalization includes the material that

  • A Relaxation of Üresin and Dubois’ Asynchronous Fixed-Point Theory in Agda
    J. Autom. Reason. (IF 1.431) Pub Date : 2019-12-10
    Matthew L. Daggitt; Ran Zmigrod; Timothy G. Griffin

    Üresin and Dubois’ paper “Parallel Asynchronous Algorithms for Discrete Data” shows how a class of synchronous iterative algorithms may be transformed into asynchronous iterative algorithms. They then prove that the correctness of the resulting asynchronous algorithm can be guaranteed by reasoning about the synchronous algorithm alone. These results have been used to prove the correctness of various

  • Exploring the Structure of an Algebra Text with Locales
    J. Autom. Reason. (IF 1.431) Pub Date : 2019-11-30
    Clemens Ballarin

    Locales, the module system of the theorem prover Isabelle, were designed so that developments in abstract algebra could be represented faithfully and concisely. Whether these goals were met is assessed through a case study. Parts of an algebra textbook, Jacobson’s Basic Algebra, that are challenging structurally were formalised. Key parts of the formalisation are presented in greater detail. An analysis

  • Efficient Strategies for CEGAR-Based Model Checking
    J. Autom. Reason. (IF 1.431) Pub Date : 2019-11-11
    Ákos Hajdu; Zoltán Micskei

    Automated formal verification is often based on the Counterexample-Guided Abstraction Refinement (CEGAR) approach. Many variants of CEGAR have been developed over the years as different problem domains usually require different strategies for efficient verification. This has lead to generic and configurable CEGAR frameworks, which can incorporate various algorithms. In our paper we propose six novel

  • An Assertional Proof of Red–Black Trees Using Dafny
    J. Autom. Reason. (IF 1.431) 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

  • First-Order Automated Reasoning with Theories: When Deduction Modulo Theory Meets Practice
    J. Autom. Reason. (IF 1.431) Pub Date : 2019-09-23
    Guillaume Burel; Guillaume Bury; Raphaël Cauderlier; David Delahaye; Pierre Halmagrand; Olivier Hermant

    We discuss the practical results obtained by the first generation of automated theorem provers based on Deduction modulo theory. In particular, we demonstrate the concrete improvements such a framework can bring to first-order theorem provers with the introduction of a rewrite feature. Deduction modulo theory is an extension of predicate calculus with rewriting both on terms and propositions. It is

  • Automatically Verifying Temporal Properties of Pointer Programs with Cyclic Proof
    J. Autom. Reason. (IF 1.431) 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

  • Implicit Computational Complexity of Subrecursive Definitions and Applications to Cryptographic Proofs
    J. Autom. Reason. (IF 1.431) Pub Date : 2019-07-31
    Patrick Baillot; Gilles Barthe; Ugo Dal Lago

    We define a call-by-value variant of Gödel’s system \(\textsf {T} \) with references, and equip it with a linear dependent type and effect system, called \(\textsf {d}\ell \textsf {T} \), that can estimate the time complexity of programs, as a function of the size of their inputs. We prove that the type system is intentionally sound, in the sense that it over-approximates the complexity of executing

  • Formalizing the Cox–Ross–Rubinstein Pricing of European Derivatives in Isabelle/HOL
    J. Autom. Reason. (IF 1.431) 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

  • Formalization of Metatheory of the Quipper Quantum Programming Language in a Linear Logic
    J. Autom. Reason. (IF 1.431) Pub Date : 2019-06-22
    Mohamed Yousri Mahmoud; Amy P. Felty

    We develop a linear logical framework within the Hybrid system and use it to reason about the type system of a quantum lambda calculus. In particular, we consider a practical version of the calculus called Proto-Quipper, which contains the core of Quipper. Quipper is a quantum programming language under active development and recently has gained much popularity among the quantum computing communities

  • A Verified Implementation of the Berlekamp–Zassenhaus Factorization Algorithm
    J. Autom. Reason. (IF 1.431) 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 1.431) 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 1.431) 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 1.431) 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 1.431) 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

  • Solving Quantifier-Free First-Order Constraints Over Finite Sets and Binary Relations
    J. Autom. Reason. (IF 1.431) Pub Date : 2019-04-08
    Maximiliano Cristiá; Gianfranco Rossi

    In this paper we present a solver for a first-order logic language where sets and binary relations can be freely and naturally combined. The language can express, at least, any full set relation algebra on finite sets. It provides untyped, hereditarily finite sets, whose elements can be variables, and basically all the classic set and relational operators used in formal languages such as B and Z. Sets

  • Evaluating Winding Numbers and Counting Complex Roots Through Cauchy Indices in Isabelle/HOL
    J. Autom. Reason. (IF 1.431) Pub Date : 2019-04-03
    Wenda Li; Lawrence C. Paulson

    In complex analysis, the winding number measures the number of times a path (counter-clockwise) winds around a point, while the Cauchy index can approximate how the path winds. We formalise this approximation in the Isabelle theorem prover, and provide a tactic to evaluate winding numbers through Cauchy indices. By further combining this approximation with the argument principle, we are able to make

  • Combining Induction and Saturation-Based Theorem Proving
    J. Autom. Reason. (IF 1.431) Pub Date : 2019-03-28
    M. Echenim; N. Peltier

    A method is devised to integrate reasoning by mathematical induction into saturation-based proof procedures based on resolution or superposition. The obtained calculi are capable of handling formulas in which some of the quantified variables range over inductively defined domains (which, as is well-known, cannot be expressed in first-order logic). The procedure is defined as a set of inference rules

  • QPCF: Higher-Order Languages and Quantum Circuits
    J. Autom. Reason. (IF 1.431) Pub Date : 2019-03-13
    Luca Paolini; Mauro Piccolo; Margherita Zorzi

    qPCF is a paradigmatic quantum programming language that extends PCF with quantum circuits and a quantum co-processor. Quantum circuits are treated as classical data that can be duplicated and manipulated in flexible ways by means of a dependent type system. The co-processor is essentially a standard QRAM device, albeit we avoid to store permanently quantum states in between two co-processor’s calls

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