Nicolas Peltier

A Method for Building Models Automatically. Experiments with an Extension of OTTER

A previous work on Herbrand model construction is extended in two ways. The first extension increases the capabilities of the method, by extending one of its key rules. The second, more important one, defines a new method for simultaneous search of refutations and models for set of equational clauses. The essential properties of the new method are given. The main theoretical result of the paper is the characterization of conditions assuring that models can be built. Both methods (for equational and non equational clauses) have been implemented as an extension of OTTER. Several running examples are given, in particular a new automatic solution of the ternary algebra problem first solved by Winker.

A Schemata Calculus for Propositional Logic

We define a notion of formula schema handling arithmetic parameters, indexed propositional variables (e.g. P i ) and iterated conjunctions/disjunctions (e.g.

A New Method for Automated Finite Model Building Exploiting Failures and Symmetries

\bigwedge_{i=1}^n P_i

Decidability and undecidability results for propositional schemata

, where n is a parameter ). Iterated conjunctions or disjunctions are part of their syntax. We define a sound and complete (w.r.t. satisfiability) tableaux-based proof procedure for this language. This schemata calculus (called stab ) allows one to capture proof patterns corresponding to a large class of problems specified in propositional logic. Although the satisfiability problem is undecidable for unrestricted schemata, we identify a class of them for which stab always terminates. An example shows evidence that the approach is applicable to non-trivial practical problems. We give some precise technical hints to pursue the present work.

A decidable class of nested iterated schemata

A method for building nite models is proposed. It combines enumeration of the set of interpretations on a nite domain with strategies in order to prune signiicantly the search space. The main new ideas underlying our method are to beneet from symmetries and from the information extracted from the structure of the problem and from failures of model veriication tests. The algorithms formalizing the approach are given and the standard properties (termination, completeness, and soundness) are proven. The method can deal with rst-order logic with equality. In contrast to existing ones, it does not require to transform the initial problem into a normal form and can be easily extended to other logics. Experimental results and comparisons with related works are reported.

A Resolution Calculus for First-order Schemata

We define a logic of propositional formula schemata adding to the syntax of propositional logic indexed propositions (e.g., pi) and iterated connectives ∨ or ∧ ranging over intervals parameterized by arithmetic variables (e.g., ∧i-1n pi, where n is a parameter). 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 (called STAB) 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

An Instantiation Scheme for Satisfiability Modulo Theories

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.Pi, Pi+1, P1 or Pn, and with generalized connectives, e.g.

Simplifying and Generalizing Formulae in Tableaux. Pruning the Search Space and Building Models

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Tree Automata and Automated Model Building

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Semantic Generalizations for Proving and Disproving Conjectures by Analogy

\bigvee_{\rm i = 1}^n