Reseña: Efficient, mechanically-verified validation of satisfiability solvers

PorJosé A. Alonso

Se ha publicado una tesis doctoral de razonamiento formalizado en ACL2 Efficient, mechanically-verified validation of satisfiability solvers.

Su autor es Nathan Wetzler (de The Mechanized Theorem Proving Group en The University of Texas at Austin).

Los directores de la tesis son Warren A. Hunt, Jr. y Marijn J. H. Heule.

Su resumen es

Satisfiability (SAT) solvers are commonly used for a variety of applications, including hardware verification, software verification, theorem proving, debugging, and hard combinatorial problems. These applications rely on the efficiency and correctness of SAT solvers. When a problem is determined to be unsatisfiable, how can one be confident that a SAT solver has fully exhausted the search space? Traditionally, unsatisfiability results have been expressed using resolution or clausal proof systems. Resolution-based proofs contain perfect reconstruction information, but these proofs are extremely large and difficult to emit from a solver. Clausal proofs rely on rediscovery of inferences using a limited number of techniques, which typically takes several orders of magnitude longer than the solving time. Moreover, neither of these proof systems has been able to express contemporary solving techniques such as bounded variable addition. This combination of issues has left SAT solver authors unmotivated to produce proofs of unsatisfiability.

The work from this dissertation focuses on validating satisfiability solver output in the unsatisfiability case. We developed a new clausal proof format called DRAT that facilitates compact proofs that are easier to emit and capable of expressing all contemporary solving and preprocessing techniques. Furthermore, we implemented a validation utility called DRAT-trim that is able to validate proofs in a time similar to that of the discovery time. The DRAT format has seen widespread adoption in the SAT community and the DRAT-trim utility was used to validate the results of the 2014 SAT Competition.

DRAT-trim uses many advanced techniques to realize its performance gains, so why should the results of DRAT-trim be trusted? Mechanical verification enables users to model programs and algorithms and then prove their correctness with a proof assistant, such as ACL2. We designed a new modeling technique for ACL2 that combines efficient model execution with an agile and convenient theory. Finally, we used this new technique to construct a fast, mechanically-verified validation tool for proofs of unsatisfiability. This research allows SAT solver authors and users to have greater confidence in their results and applications by ensuring the validity of unsatisfiability results.

Reseña: TryLogic tutorial (An approach to learning Logic by proving and refuting)

PorJosé A. Alonso

Se ha publicado un artículo de aplicación del razonamiento asistido por ordenador a la enseñanza titulado TryLogic tutorial: An approach to learning Logic by proving and refuting

Sus autores son Patrick Terrematte y João Marcos (del Group for Logic, Language, Information, Theory and Applications (LoLITA) en la Federal University of Rio Grande do Norte (UFRN) de Brasil).

Su resumen es

Aiming to offer a framework for blended learning to the teaching of proof theory, the present paper describes an interactive tutorial, called TryLogic, teaching how to solve logical conjectures either by proofs or refutations. The paper also describes the integration of our infrastructure with the Virtual Learning Environment Moodle through the IMS Learning Tools Interoperability specification, and evaluates the tool we have developed.

El trabajo se ha presentado en TTL2015 (Fourth International Conference on Tools for Teaching Logic)

Reseña: Formal study of functional orbits in finite domains

PorJosé A. Alonso

Se ha publicado un artículo de razonamiento formalizado en Coq titulado Formal study of functional orbits in finite domains.

Su autor es Jean-François Dufourd (del grupo Computer Science in Geometry and Graphics (IGG) en la Univ. de Estrasburgo, Francia).

Su resumen es

In computer science, functional orbits − i.e tracks left by iterating a function − in a finite domain naturally appear at a high or a low level. This paper introduces a Coq logical orbit framework, the purpose of which is to help rigorously developing software systems with some complex data structures from specification to implementation.

The result is a Coq library of orbit concepts formalized as definitions, lemmas and theorems. Most of them are inspired by our previous work in geometric modelling, where combinatorial hypermaps were used to describe surface subdivisions appearing in computational geometry problems, e.g. building convex hulls or Delaunay diagrams. Now, this domain remains a reference for us, but our results are drastically extended and usable in other computer science areas. The proof of Floyd’s cycle-detection algorithm, known as “the tortoise and the hare”, confirms that point.

The library contains operations to observe, traverse and update orbits − addition, deletion, mutation, transposition − with proofs of their behavior. It focuses on the important case where the involved function is a partial injection. In this case, it defines a connectivity relationship and evaluates the variation of the number of connected components during updates.

El trabajo se ha publicado en Theoretical Computer Science.

Reseña: Representations of finite groups

PorJosé A. Alonso

Se ha publicado un artículo de razonamiento formalizado en Isabelle/HOL sobre álgebra titulado Representations of finite groups.

Su autor es Jeremy Sylvestre (de la Univ. de Alberta en Canadá).

Su resumen es

We provide a formal framework for the theory of representations of finite groups, as modules over the group ring. Along the way, we develop the general theory of groups (relying on the group_add class for the basics), modules, and vector spaces, to the extent required for theory of group representations. We then provide formal proofs of several important introductory theorems in the subject, including Maschke’s theorem, Schur’s lemma, and Frobenius reciprocity. We also prove that every irreducible representation is isomorphic to a submodule of the group ring, leading to the fact that for a finite group there are only finitely many isomorphism classes of irreducible representations. In all of this, no restriction is made on the characteristic of the ring or field of scalars until the definition of a group representation, and then the only restriction made is that the characteristic must not divide the order of the group.

El trabajo se ha publicado el 12 de agosto en AFP (The Archive of Formal Proofs).

El código de las correspondientes teorías en Isabelle/HOL se encuentra aquí.

Reseña: A symbolic approach to abstract algebra in HOL Light

PorJosé A. Alonso

Se ha publicado un artículo de razonamiento formalizado en HOL Light sobre álgebra titulado A symbolic approach to abstract algebra in HOL Light.

Su autor es Marco Maggesi (de la Univ. de Florencia en Italia).

Su resumen es

Formalising algebraic structures (groups, rings, fields, vector spaces, lattices, …) is known to be challenging task which is often undertaken by exploiting various kind of extra-logical mechanisms (axiomatic classes, modules, locales, coercions, …) provided by most modern theorem provers.

We want to explore an alternative strategy, where algebraic structures are implemented via a deep embedding of mathematical formulas and are managed as first class objects in the HOL theory. Along the way, we provide a mechanism of generalised conversions, which extends Paulson’s conversions by allowing to compute with equivalence relations. We developed generalised conversions to support rewriting in our system, but they can be used independently and may have an interest in its own.

El trabajo se ha presentado en CICM 2015 (the 8th Conference on Intelligent Computer Mathematics).

El código de las correspondientes teorías en HOL Light se encuentra aquí.