An Approach to Abductive Reasoning in Equational Logic

http://ijcai.org/papers13/contents.php - Posters: Constraints, Satisfiability, and Search (ijcai13.org)International audienceAbduction has been extensively studied in propositional logic because of its many applications in artificial intelligence. However, its intrinsic complexity has been a limitation to the implementation of abductive reasoning tools in more expressive logics. We have devised such a tool in ground flat equational logic, in which literals are equations or disequations between constants. Our tool is based on the computation of prime implicates. It uses a relaxed paramodulation calculus, designed to generate all prime implicates of a formula, together with a carefully defined data structure storing the implicates and able to efficiently detect, and remove, redundancies. In addition to a detailed description of this method, we present an analysis of some experimental results

A Generic Framework for Implicate Generation Modulo Theories

International audienceThe clausal logical consequences of a formula are called its implicates. The generation of these implicates has several applications, such as the identification of missing hypotheses in a logical specification. We present a procedure that generates the implicates of a quantifier-free formula modulo a theory. No assumption is made on the considered theory, other than the existence of a decision procedure. The algorithm has been implemented (using the solvers MiniSAT, CVC4 and Z3) and experimental results show evidence of the practical relevance of the proposed approach

Improved False Causal Loop Detection in Polychronous Specificationof Embedded Software

As opposed to single clocked synchronous programming paradigms, polychronous formalism allows specification of concurrent data flow computation on signals such that various data flows can evolve asynchronous with respect to each other. Explicit constraints and constraints implied by the syntactic structures impart certain intrinsic properties to models specified polychronously. One of the major steps in designing a synthesis engine for polychronous specifications is the characterization of specified models into categories such as inherently sequential or inherently multi-threaded. In this paper, we are concerned with sequentially implementable polychronous specification where computation is divided into a totally ordered sequence of logical instants. Data flow computation within an instant happens based on the implied data flow order. This order or data dependency often varies from one instant to another. Thus determining if there is an instant at which the data flow order forms a causal cycle is an important problem. In the current polychronous compilers, such as SIGNAL compiler and EmCodeSyn, this is solved without due effort, by rejecting any program which has a buffer-free structural cycle. However, a clocked dependency graph can be used to construct logical constraints representing the instants with a possible causal loop. The satisfiability of such constraints would imply that such a loop is realizable and hence the specification has a possible deadlock. The reachability of this instant with a given set of initial conditions would verify if the program should be rejected. In the past, the work on such constraints and their satisfiability has not been implemented even though for pure Boolean signals and clocks this could have been done using a satisfiability solver. With the advent to SAT modulo theory (SMT) solvers, this can now be extended to a more general class of specifications. Moreover, model checking on an abstraction of the specification can provide more information about the reachability of instants at which cyclic data dependency is realized. This paper presents an improved polychronous synthesis tool accepting a much larger class of specifications than could be done before. In our experimental results, we demonstrate the capabilities of our causality analysis methods and show that our synthesis tool performs better than previous strategies, including our own past work

Prime implicate generation in equational logic

The work presented in this memoir deals with the generation of prime implicates in ground equational logic, i.e., of the most general consequences of formulae containing equations and disequations between ground terms.It is divided in three parts. First, two calculi that generate implicates are defined. Their deductive-completeness is proved, meaning they can both generate all the implicates up to redundancy of equational formulae.Second, a tree data structure to store the generated implicates is proposed along with algorithms to detect redundancies and prune the branches of the tree accordingly. This data structure is adapted to the different kinds of clauses (with and without function symbols, with and without constraints) and to the various formal definitions of redundancy used in the calculi since each calculus uses different -- although similar -- redundancy criteria. Termination and correction proofs are provided with each algorithm. Finally, an experimental evaluation of the different prime implicate generation methods based on research prototypes written in Ocaml is conducted including a comparison with state-of-the-art prime implicate generation tools. This experimental study is used to identify the most efficient variants of the proposed algorithms. These show promising results overstepping the state of the art.Ce mémoire présente le résultat de mon travail de thèse sur la génération d'impliqués premiers en logique équationnelle fermée, i.e., la génération des conséquences les plus générales de formules logiques contenants des équations et des disequations entre termes sans variables. Ce mémoire est divisé en trois parties. Tout d'abord, deux calculs de génération d'impliqués sont définis. Leur complétude pour la déduction est prouvée, ce qui signifie qu'ils sont tous deux capables de générer l'ensemble des impliqués modulo redondance d'une formule équationnelle fermée. Dans une deuxième partie, une structure de données arborescente est proposée pour stocker les impliqués générés, accompagnée d'algorithmes pour déceler les redondances et couper les branches de l'arbre lorsque c'est nécessaire. Cette structure de données est adaptée aux différents types de clauses (avec et sans symboles de fonctions, avec et sans contraintes) ainsi qu'aux différentes notions de redondance utilisées dans les calculs. En effet, chaque calcul utilise un critère de redondance légèrement différent des autres. Les preuves de correction et de terminaison des algorithmes sont fournies pour chaque algorithme. Enfin, une évaluation expérimentale des différentes méthodes de génération d'impliqués premiers est réalisée. Pour cela, un prototype de ces méthodes, écrit en Ocaml est comparé à des outils de génération d'impliqués premiers récents.Les résultats de ces expériences sont utilisés pour identifier les variantes les plus efficaces des algorithmes proposés. Les résultats sont prometteurs et dans la plupart des cas, meilleurs que ceux de l'état de l'art

Prime implicate generation in equational logic

The work presented in this memoir deals with the generation of prime implicates in ground equational logic, i.e., of the most general consequences of formulae containing equations and disequations between ground terms.It is divided in three parts. First, two calculi that generate implicates are defined. Their deductive-completeness is proved, meaning they can both generate all the implicates up to redundancy of equational formulae.Second, a tree data structure to store the generated implicates is proposed along with algorithms to detect redundancies and prune the branches of the tree accordingly. This data structure is adapted to the different kinds of clauses (with and without function symbols, with and without constraints) and to the various formal definitions of redundancy used in the calculi since each calculus uses different -- although similar -- redundancy criteria. Termination and correction proofs are provided with each algorithm. Finally, an experimental evaluation of the different prime implicate generation methods based on research prototypes written in Ocaml is conducted including a comparison with state-of-the-art prime implicate generation tools. This experimental study is used to identify the most efficient variants of the proposed algorithms. These show promising results overstepping the state of the art.Ce mémoire présente le résultat de mon travail de thèse sur la génération d'impliqués premiers en logique équationnelle fermée, i.e., la génération des conséquences les plus générales de formules logiques contenants des équations et des disequations entre termes sans variables. Ce mémoire est divisé en trois parties. Tout d'abord, deux calculs de génération d'impliqués sont définis. Leur complétude pour la déduction est prouvée, ce qui signifie qu'ils sont tous deux capables de générer l'ensemble des impliqués modulo redondance d'une formule équationnelle fermée. Dans une deuxième partie, une structure de données arborescente est proposée pour stocker les impliqués générés, accompagnée d'algorithmes pour déceler les redondances et couper les branches de l'arbre lorsque c'est nécessaire. Cette structure de données est adaptée aux différents types de clauses (avec et sans symboles de fonctions, avec et sans contraintes) ainsi qu'aux différentes notions de redondance utilisées dans les calculs. En effet, chaque calcul utilise un critère de redondance légèrement différent des autres. Les preuves de correction et de terminaison des algorithmes sont fournies pour chaque algorithme. Enfin, une évaluation expérimentale des différentes méthodes de génération d'impliqués premiers est réalisée. Pour cela, un prototype de ces méthodes, écrit en Ocaml est comparé à des outils de génération d'impliqués premiers récents.Les résultats de ces expériences sont utilisés pour identifier les variantes les plus efficaces des algorithmes proposés. Les résultats sont prometteurs et dans la plupart des cas, meilleurs que ceux de l'état de l'art