Specifying and Proving a Sorting Algorithm

Rapport de recherche LIFCThis work investigates the question of automaticity of algorithm proofs, through the typical example of a sorting algorithm. The first part introduces two specification languages for Java programs. In the second part one of them is used to specify a sorting algorithm by selection. The suggested specifications are enhanced until obtaining a complete solution by the current automated theorem provers. This report is a part of Elena Tushkanova's diploma project (equivalent to a master thesis) entitled “Modular Specification of Object Oriented Programs” from the Yaroslavl State University, Russia, translated from Russian into English

Automatic Decidability for Theories Modulo Integer Offsets

Many verification problems can be reduced to a satisfiability problem modulo theories. For building satisfiability procedures the rewriting-based approach uses a general calculus for equational reasoning named superposition. Schematic superposition, in turn, provides a mean to reason on the derivations computed by superposition. Until now, schematic superposition was only studied for standard superposition. We present a schematic superposition calculus modulo a fragment of arithmetics, namely the theory of Integer Offsets. This new schematic calculus is used to prove the decidability of the satisfiability problem for some theories extending Integer Offsets. We illustrate our theoretical contribution on theories representing extensions of classical data structures, e.g., lists and records. An implementation in the rewriting-based Maude system constitutes a practical contribution. It enables automatic decidability proofs for theories of practical use

Automatic Decidability for Theories with Counting Operators

International audienceThe notion of schematic paramodulation has been introduced to reason on properties of (standard) paramodulation. We present a schematic paramodulation calculus modulo a fragment of arithmetics, namely the theory of Integer Offsets. This new schematic calculus is used to prove the decidability of the satisfiability problem for some theories equipped with counting operators. We illustrate our theoretical contribution on theories representing extensions of classical data structures, e.g., lists and records

Specifying Generic Java Programs: two case studies

International audienceThis work investigates the question of modular specification of generic Java classes and methods. We propose extensions to the Krakatoa Modeling Language, a part of the Why platform for proving that a Java or C program is a correct implementation of some specification. The new constructs we propose for the specification of generic Java programs are presented through two significant examples: the specification of the generic method for sorting arrays which comes from the java.util.Arrays class in the Java API, and the specification of the java.util.HashMap class defining a generic hash map and its use for memoization. The main ingredient is the notion of theories and the instantiation relation between them. We discuss soundness conditions and their verification

Modular Specification of Java Programs

This work investigates the question of modular specification of generic Java classes and methods. The first part introduces a specification language for Java programs. In the second part the language is used to specify an array sorting algorithm by selection. The third and the fourth parts define a syntax proposal for the specification a generic Java programs, through two examples. The former is the specification of the generic method for sorting arrays which comes in the java.util.Arrays class of the Java API. The latter is the specification of the java.util.HashMap class and its use for memoization

Specifying Generic Java Programs: two case studies

International audienceThis work investigates the question of modular specification of generic Java classes and methods. We propose extensions to the Krakatoa Modeling Language, a part of the Why platform for proving that a Java or C program is a correct implementation of some specification. The new constructs we propose for the specification of generic Java programs are presented through two significant examples: the specification of the generic method for sorting arrays which comes from the java.util.Arrays class in the Java API, and the specification of the java.util.HashMap class defining a generic hash map and its use for memoization. The main ingredient is the notion of theories and the instantiation relation between them. We discuss soundness conditions and their verification

Calculs schématiques pour l'analyse de procédures de décision

Calculs schématiques pour l'analyse de procédures de décisionIn this thesis we address problems related to the verification of software-based systems. We aremostly interested in the (safe) design of decision procedures used in verification. In addition, we alsoconsider a modularity problem for a modeling language used in the Why verification platform.Many verification problems can be reduced to a satisfiability problem modulo theories (SMT). In orderto build satisfiability procedures Armando et al. have proposed in 2001 an approach based on rewriting.This approach uses a general calculus for equational reasoning named paramodulation. In general, afair and exhaustive application of the rules of paramodulation calculus (PC) leads to a semi-decisionprocedure that halts on unsatisfiable inputs (the empty clause is then generated) but may diverge onsatisfiable ones. Fortunately, it may also terminate for some theories of interest in verification, and thusit becomes a decision procedure. To reason on the paramodulation calculus, a schematic paramodulationcalculus (SPC) has been studied, notably to automatically prove decidability of single theories and oftheir combinations. The advantage of SPC is that if it halts for one given abstract input, then PC haltsfor all the corresponding concrete inputs. More generally, SPC is an automated tool to check propertiesof PC like termination, stable infiniteness and deduction completeness.A major contribution of this thesis is a prototyping environment for designing and verifying decisionprocedures. This environment, based on the theoretical studies, is the first implementation of theschematic paramodulation calculus. It has been implemented from scratch on the firm basis provided bythe Maude system based on rewriting logic. We show that this prototype is very useful to derive decidabilityand combinability of theories of practical interest in verification. It helps testing new saturationstrategies and experimenting new extensions of the original (schematic) paramodulation calculus.This environment has been applied for the design of a schematic paramodulation calculus dedicated tothe theory of Integer Offsets. This contribution is the first extension of the notion of schematic paramodulationto a built-in theory. This study has led to new automatic proof techniques that are different fromthose performed manually in the literature. The assumptions to apply our proof techniques are easyto satisfy for equational theories with counting operators. We illustrate our theoretical contribution ontheories representing extensions of classical data structures such as lists and records.We have also addressed the problem of modular specification of generic Java classes and methods.We propose extensions to the Krakatoa Modeling Language, a part of the Why platform for provingthat a Java or C program is a correct implementation of some specification. The key features arethe introduction of parametricity both for types and for theories and an instantiation relation betweentheories. The proposed extensions are illustrated on two significant examples: the specification of thegeneric method for sorting arrays and for generic hash map.Both problems considered in this thesis are related to SMT solvers. Firstly, decision procedures areat the core of SMT solvers. Secondly, the Why platform extracts verification conditions from a sourceprogram annotated by specifications, and then transmits them to SMT solvers or proof assistants to checkthe program correctness.BESANCON-Bib. Electronique (250560099) / SudocSudocFranceF