Ricardo Caferra

A method for simultaneous search for refutations and models by equational constraint solving

A method is proposed to systematize the simultaneous search for a refutation and Herbrand models of a given conjecture. It is based on an extension of resolution using equational problems and the inference system included in the method is proved to be sound and refutationally complete. For some classes of formulas the method is indeed a decision procedure. In particular it is a decision procedure for the Bernays-Schonfinkel class (a class for which no resolution term ordering strategy is known to be a decision procedure). Some examples of model construction - including one for which other resolution based decision procedures fail to detect satisfiability - are developed in detail. The method is also useful in cases in which model construction is not required. The search space in resolution based deductions can be greatly decreased. This is shown in solving a question-answering problem, considered to be hard. Models are built by constructing relations on Herbrand universe. The relationship between these models and finite ones is established. The class of these constructible relations is precisely characterized. Some of the rules introduced, in order to extend resolution, are essentially new. It is proved that they are necessary for enlarging the class of models the method is able to build. A brief comparison with existing methods which bear similarity with ours, either in the use of constraints or in the search of a model, shows the originality of our proposal. Some hints about directions for extending the class of formulas for which models can be constructed are given.

Extending resolution for model construction

A method is proposed to systematize the simultaneous search for a refutation and Herbrand models of a given conjecture. It is based on an extension of resolution using equational problems and the inference system included in the method is proved to be sound and refutational complete. For some classes of formulae the method is indeed a decision procedure. Some examples of model construction — including one for which other resolution based decision procedures fail to detect satisfiability — are developed in detail.

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.

Model Building and Interactive Theory Discovery

\bigwedge_{i=1}^n P_i

Rewriting term-graphs with priority

, 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.

An application of many-valued logic to decide propositional S5 formulae: a strategy designed for a parameterized tableaux-based Theorem Prover

We show how our former approach to model building can be extended into a unified approach to model building and model checking, able to guide the discovery of theories in which a model can be built for a given formula. In contrast with other enumeration approaches used to decide some classes of first-order formulae, our approach automatically discovers (in the best case) or strongly guides (in the worst case) discovery of such theories. For practical reasons, the method has been developed in resolution style and implemented as an extension of a resolution-based theorem prover, but the same principles can be applied to the connexion method, tableaux ⋯ and of course the models built by our method can be used by theorem provers based on other calculi. Detailed examples are given for the new notions.

Linear Temporal Logic and Propositional Schemata, Back and Forth

We define a new class of rewrite systems operating over term-graphs. Our aim is twofold. First we propose to extend classical first-order rewrite rules in order to process easily data-structures with pointers (e.g., circular lists, doubly linked lists etc). For that, our rules provide specific features such as pointer (edges) redirections, relabeling of existing nodes etc. Unfortunately, such features are very often source of non confluence. Our second aim is then to ensure confluence of the considered rewrite systems in the new class. We introduce the notion of term-graphs with priority and show that orthogonal rewrite systems are confluent in our setting

Towards systematic analysis of theorem provers search spaces: first steps

It is well known that for deciding on the validity of a S5 formula it is possible to bound the maximum number of worlds (say N) on which the formula must be tested to decide. In doing modal logic with many-valued logic the number of truth-values to consider depends on N. The idea of using a theorem prover parameterized by the set of truth-values is imperative. We profit of possibilities offered by a parameterized theorem prover in order to use the information of a failure for a n-valued logic in the validity test for a 2xn-valued logic (corresponding to a change from m to m+1 worlds). This feature allows us to establish the main result of the paper: a strategy for doing modal logic with many-valued logics. To do so, we simulate by a many-valued logic the truth values set of a modal formula in the different worlds. Though the idea of doing modal logic with many-valued logic exists in the bibliography (see [HuC 68], [Res 69]) it has been used only in obtaining theoretical results. We propose in this work a strategy allowing to mechanize effectively modal logic with many-valued logic. Moreover this strategy furnish a decision procedure for S5 not needing transformation into a normal form (as in [HuC 68]). Our aim is not to compete with other ways of doing modal logic (i.e. with resolution or connexion method - work of Farinas del Cerro, Ohlbach, Wallen and others) but to use a parameterized many-valued tableaux based system to do also modal logic. A m-valued logic theorem prover (parameterized by m) based on these ideas and result has been implemented on a SUN-3 workstation. Some running examples are shown.

Combining enumeration and deductive techniques in order to increase the class of constructible infinite models

This 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.