Strict General Setting for Building Decision Procedures into Theorem Provers

The efficient and flexible incorporating of decision procedures into theorem provers is very important for their successful use. There are several approaches for combining and augmenting of decision procedures; some of them support handling uninterpreted functions, congruence closure, lemma invoking etc. In this paper we present a variant of one general setting for building decision procedures into theorem provers (gs framework [18]). That setting is based on macro inference rules motivated by techniques used in different approaches. The general setting enables a simple describing of different combination/augmentation schemes. In this paper, we further develop and extend this setting by an imposed ordering on the macro inference rules. That ordering leads to a ”strict setting”. It makes implementing and using variants of well-known or new schemes within this framework a very easy task even for a non-expert user. Also, this setting enables easy comparison of different combination/augmentation schemes and combination of their ideas

Quantifier-Free Interpolation of a Theory of Arrays

The use of interpolants in model checking is becoming an enabling technology to allow fast and robust verification of hardware and software. The application of encodings based on the theory of arrays, however, is limited by the impossibility of deriving quantifier- free interpolants in general. In this paper, we show that it is possible to obtain quantifier-free interpolants for a Skolemized version of the extensional theory of arrays. We prove this in two ways: (1) non-constructively, by using the model theoretic notion of amalgamation, which is known to be equivalent to admit quantifier-free interpolation for universal theories; and (2) constructively, by designing an interpolating procedure, based on solving equations between array updates. (Interestingly, rewriting techniques are used in the key steps of the solver and its proof of correctness.) To the best of our knowledge, this is the first successful attempt of computing quantifier- free interpolants for a variant of the theory of arrays with extensionality

Conformal Gravity: Dark Matter and Dark Energy

This short review examines recent progress in understanding dark matter, dark energy, and galactic halos using theory that departs minimally from standard particle physics and cosmology. Strict conformal symmetry (local Weyl scaling covariance), postulated for all elementary massless fields, retains standard fermion and gauge boson theory but modifies Einstein-Hilbert general relativity and the Higgs scalar field model, with no new physical fields. Subgalactic phenomenology is retained. Without invoking dark matter, conformal gravity and a conformal Higgs model fit empirical data on galactic rotational velocities, galactic halos, and Hubble expansion including dark energy.Comment: 9 pp in revtex format. References added with minor text revision

Renforcement du noyau d un démonstrateur SMT (Conception et implantation de procédures de décisions efficaces)

Cette thèse s'intéresse à la démonstration automatique de la validité de formules mathématiques issues de la preuve de programmes. Elle se focalise tout particulièrement sur la Satisfiabilité Modulo Théories (SMT): un jeune domaine de recherche qui a connu de grands progrès durant la dernière décennie. Les démonstrateurs de cette famille ont des applications diverses dans la conception de microprocesseurs, la preuve de programmes, le model-checking, etc.Les démonstrateurs SMT offrent un bon compromis entre l'expressivité et l'efficacité. Ils reposent sur une coopération étroite d'un solveur SAT avec une combinaison de procédures de décision pour des théories spécifiques comme la théorie de l'égalité libre avec des symboles non interprétés, l'arithmétique linéaire sur les entiers et les rationnels, et la théorie des tableaux.L'objectif de cette thèse est d'améliorer l'efficacité et l'expressivité du démonstrateur SMT Alt-Ergo. Pour cela, nous proposons une nouvelle procédure de décision pour la théorie de l'arithmétique linéaire sur les entiers. Cette procédure est inspirée par la méthode de Fourier-Motzkin, mais elle utilise un simplexe sur les rationnels pour effectuer les calculs en pratique. Nous proposons également un nouveau mécanisme de combinaison, capable de raisonner dans l'union de la théorie de l'égalité libre, la théorie AC des symboles associatifs et commutatifs et une théorie arbitraire deShostak. Ce mécanisme est une extension modulaire et non intrusive de la procédure de completion close modulo AC avec la théorie de Shostak. Aussi, nous avons étendu Alt-Ergo avec des procédures de décision existantes pour y intégrer d'autres théories intéressantes comme la théorie de types de données énumérés et la théorie des tableaux. Enfin, nous avons exploré des techniques de simplification de formules en amont et l'amélioration de son solveur SAT.This thesis tackles the problem of automatically proving the validity of mathematical formulas generated by program verification tools. In particular, it focuses on Satisfiability Modulo Theories (SMT): a young research topic that has seen great advances during the last decade. The solvers of this family have various applications in hardware design, program verification, model checking, etc.SMT solvers offer a good compromise between expressiveness and efficiency. They rely on a tight cooperation between a SAT solver and a combination of decision procedures for specific theories, such as the free theory of equality with uninterpreted symbols, linear arithmetic over integers and rationals, or the theory of arrays.This thesis aims at improving the efficiency and the expressiveness of the Alt-Ergo SMT solver. For that, we designed a new decision procedure for the theory of linear integer arithmetic. This procedure is inspired by Fourier-Motzkin's method, but it uses a rational simplex to perform computations in practice. We have also designed a new combination framework, capable of reasoning in the union of the free theory of equality, the AC theory of associative and commutativesymbols, and an arbitrary signature-disjoint Shostak theory. This framework is a modular and non-intrusive extension of the ground AC completion procedure with the given Shostak theory. In addition, we have extended Alt-Ergo with existing decision procedures to integrate additional interesting theories, such as the theory of enumerated data types and the theory of arrays. Finally, we have explored preprocessing techniques for formulas simplification as well as the enhancement of Alt-Ergo's SAT solver.PARIS11-SCD-Bib. électronique (914719901) / SudocSudocFranceF

NATURAL DEDUCTION AS HIGHER-ORDER RESOLUTION

An interactive theorem prover, Isabelle, is under development. In LCF, each inference rule is represented by one function for forwards proof and another (a tactic) for backwards proof. In Isabelle, each inference rule is represented by a Horn clause. Resolution gives both forwards and backwards proof, supporting a large class of logics. Isabelle has been used to prove theorems in Martin-L\"of's Constructive Type Theory. Quantifiers pose several difficulties: substitution, bound variables, Skolemization. Isabelle's representation of logical syntax is the typed lambda-calculus, requiring higher- order unification. It may have potential for logic programming. Depth-first subgoaling along inference rules constitutes a higher-order Prolog

Nelson Oppen combination as a rewrite theory

Solving Satisfiability Modulo Theories (SMT) problems in a key piece in automating tedious mathematical proofs. It involves deciding satisfiability of formulas of a decidable theory, which can often be reduced to solving systems of equalities and disequalities, in a variety of theories such as linear and non-linear real and integer arithmetic, arrays, uninterpreted and Boolean algebra. While solvers exist for many such theories or their subsets, it is common for interesting SMT problems to span multiple theories. SMT solvers typically use refinements of the Nelson-Oppen combination method, an algorithm for producing a solver for the quantifier free fragment of the combination of a number of such theories via cooperation between solvers of those theories, for this case. Here, we present the Nelson-Oppen algorithm adapted for an order-sorted setting as a rewriting logic theory. We implement this algorithm in the Maude System and instantiate it with the theories of real and integer matrices to demonstrate its use in automated theorem proving, and with hereditarily finite sets with reals to show its use with non-convex theories. This is done using both SMT solvers written in Maude itself via reflection (Variant-based satisfiability) and using external solvers (CVC4 and Yices). This work can be considered a first step towards building a rich ecosystem of cooperating SMT solvers in Maude, that modeling and automated theorem proving tools typically written using the Maude System can leverage

Investigation, Development, and Evaluation of Performance Proving for Fault-tolerant Computers

A number of methodologies for verifying systems and computer based tools that assist users in verifying their systems were developed. These tools were applied to verify in part the SIFT ultrareliable aircraft computer. Topics covered included: STP theorem prover; design verification of SIFT; high level language code verification; assembly language level verification; numerical algorithm verification; verification of flight control programs; and verification of hardware logic

Galactic rotation curves of spiral galaxies and dark matter in f(R,T)f(\mathcal{R},T) gravity theory

Galactic rotation curve is a powerful indicator of the state of the gravitational field within a galaxy. Flatness of these curves implies the linear increase of mass to have constant velocity with the radial distance. In this paper, we focus on the possibility of explaining the flatness of observed rotation curves of spiral galaxies without postulating the existence of dark matter in the framework of $f(\mathcal{R},T)$ gravity where the gravitational Lagrangian is written by an arbitrary function of $\mathcal{R}$, the Ricci scalar and of $T$, the trace of stress-energy tensor $T_{\mu\nu}$. We derive the gravitational field equations in this gravity theory for the static spherically symmetric spacetime and solve the equations for metric coefficients using a specific model that has minimal coupling between matter and geometry. The orbital motion of a massive test particle moving in a stable circular orbit is considered and the behaviour of tangential velocity in the halo region with the help of the considered model is studied. The linear variation with distance of the interaction mass generated due to matter-geometry coupling successfully explains the galactic dynamics without the existence of dark matter at large distances from the galactic core.Comment: 13 pages, 4 figure