Linear Temporal Logic and Propositional Schemata, Back and Forth (extended version)

This paper relates the well-known Linear Temporal Logic with the logic of propositional schemata introduced by the authors. We prove that LTL is equivalent to a class of schemata in the sense that polynomial-time reductions exist from one logic to the other. Some consequences about complexity are given. We report about first experiments and the consequences about possible improvements in existing implementations are analyzed.Comment: Extended version of a paper submitted at TIME 2011: contains proofs, additional examples & figures, additional comparison between classical LTL/schemata algorithms up to the provided translations, and an example of how to do model checking with schemata; 36 pages, 8 figure

A Decidable Class of Nested Iterated Schemata (extended version)

Many problems can be specified by patterns of propositional formulae depending on a parameter, e.g. the specification of a circuit usually depends on the number of bits of its input. We define a logic whose formulae, called "iterated schemata", allow to express such patterns. Schemata extend propositional logic with indexed propositions, e.g. P_i, P_i+1, P_1, and with generalized connectives, e.g. /\i=1..n or i=1..n (called "iterations") where n is an (unbound) integer variable called a "parameter". The expressive power of iterated schemata is strictly greater than propositional logic: it is even out of the scope of first-order logic. We define a proof procedure, called DPLL*, that can prove that a schema is satisfiable for at least one value of its parameter, in the spirit of the DPLL procedure. However the converse problem, i.e. proving that a schema is unsatisfiable for every value of the parameter, is undecidable so DPLL* does not terminate in general. Still, we prove that it terminates for schemata of a syntactic subclass called "regularly nested". This is the first non trivial class for which DPLL* is proved to terminate. Furthermore the class of regularly nested schemata is the first decidable class to allow nesting of iterations, i.e. to allow schemata of the form /\i=1..n (/\j=1..n ...).Comment: 43 pages, extended version of "A Decidable Class of Nested Iterated Schemata", submitted to IJCAR 200

Combining Enumeration and Deductive Techniques in order to Increase the Class of Constructible Infinite Models

AbstractA new method for building infinite models for first-order formulae is presented. The method combines enumeration techniques with existing deductive (in a broad sense) ones. Its soundness and completeness w.r.t. the class of models that can be represented by equational constraints are proven. This shows that the use of enumeration techniques strictly increases the power of existing methods for building Herbrand models that are not complete in this sense. Some strategies are proposed to reduce the search space. We give examples and show how to use this approach for building interactively a model of a formula introduced by Goldfarb in his proof of the undecidability of the Gödel class with identity. This formula is satisfiable but has no finite model

Linear Temporal Logic and Propositional Schemata, Back and Forth

Session: p-Automata and Obligation Games - http://www.isp.uni-luebeck.de/time11/International audienceThis paper relates the well-known Linear Temporal Logic with the logic of propositional schemata introduced in elsewhere by the authors. We prove that LTL is equivalent to a class of schemata in the sense that polynomial-time reductions exist from one logic to the other. Some consequences about complexity are given. We report about first experiments and the consequences about possible improvements in existing implementations are analyzed

Decidability and Undecidability Results for Propositional Schemata

International audienceWe define a logic of propositional formula schemata adding to the syntax of propositional logic indexed propositions and iterated connectives ranging over intervals parameterized by arithmetic variables. The satisfiability problem is shown to be undecidable for this new logic, but we introduce a very general class of schemata, called bound-linear, for which this problem becomes decidable. This result is obtained by reduction to a particular class of schemata called regular, for which we provide a sound and complete terminating proof procedure. This schemata calculus allows one to capture proof patterns corresponding to a large class of problems specified in propositional logic. We also show that the satisfiability problem becomes again undecidable for slight extensions of this class, thus demonstrating that bound-linear schemata represent a good compromise between expressivity and decidability

Baghera Assessment Project, designing an hybrid and emergent educational society

Edited by Sophie Soury-Lavergne ; Available at: http://www-leibniz.imag.fr/LesCahiers/2003/Cahier81/BAP_CahiersLaboLeibniz.PDFResearch reportThe Baghera Assessment Project (BAP) has the objective to ex plore a new avenue for the design of e-Learning environments. The key features of BAP's approach are: (i) the concept of emergence in multi-agents systems as modelling framework, (ii) the shaping of a new theoretic al framework for modelling student knowledge, namely the cK¢ model. This new model has been constructed, based on the current research in cognitive science and education, to bridge research on education and research on the design of learning environments

Abstraction, partage de structure et retour arrière non aveugle dans la méthode de réduction matricielle en démonstration automatique de théorèmes

Université : Université scientifique et médicale de GrenobleUne technique d'abstraction et de partage de structure pour une methode de démonstration automatique non expérimentée jusqu'à présent est proposée. On donne aussi une généralisation de la règle d'inférence pour le cas propositionnel. Une preuve est séparée en plan + validation, ce qui correspond à séparer la partie purement déductive de l'algorithme d'unification. Cette séparation est utilisée pour détecter les ensembles responsables des échecs de validation pour un retour arrière non aveugl

A Significant Extension of Logic Programming By Adapting Model Building Rules

. A method by the authors for automated model building is extended and specialized in a natural way in order to increase the possibilities of logic programming. A rather complete, though reasonably short, description of the ideas and technicalities of the former method is given in order to make the paper self-contained. Specialization of several key rules permits to obtain three main theoretical results concerning extensions of logic programming: non-ground negative facts as well as inductive consequences can be deduced from programs. Goals containing negations, quantifications and logical connectives are allowed. It is proven that the proposed extension is strictly more powerful than SLDNF. Several non-trivial running examples show evidence of the interest of our approach. Last but not least, a nice side effect exploits the model building capabilities of the approach: it is shown on one representative example how the method can be used to detect (and to correct) errors in logic progra..