I. P. de Guzmán

Reductions for non-clausal theorem proving

This paper presents the TAS methodology as a new framework for generating non-clausal Automated Theorem Provers. We present a complete description of the ATP for Classical Propositional Logic, named TAS-D, but the ideas, which make use of implicants and implicates can be extended in a natural manner to first-order logic, and non-classical logics. The method is based on the application of a number of reduction strategies on subformulas, in a rewrite-system style, in order to reduce the complexity of the formula as much as possible before branching. Specifically, we introduce the concept of complete reduction, and extensions of the pure literal rule and ofthe collapsibility theorems; these strategies allow to limit the size ofthe search space. In addition, TAS-D is a syntactical countermodel construction. As an example of the power of TAS-D we study a class of formulas which has linear proofs (in the number of branchings) when either resolution or dissolution with factoring is applied. When applying our method to these formulas we get proofs without branching. In addition, some experimental results are reported. Copyright 2001 Elsevier Science B.V.

A New Algebraic Tool for Automatic Theorem Provers

The concepts of implicates and implicants are widely used in several fields of “Automated Reasoning”. Particularly, our research group has developed several techniques that allow us to reduce efficiently the size of the input, and therefore the complexity of the problem. These techniques are based on obtaining and using implicit information that is collected in terms of unitary implicates and implicants. Thus, we require efficient algorithms to calculate them. In classical propositional logic it is easy to obtain efficient algorithms to calculate the set of unitary implicants and implicates of a formula. In temporal logics, contrary to what we see in classical propositional logic, these sets may contain infinitely many members. Thus, in order to calculate them in an efficient way, we have to base the calculation on the theoretical study of how these sets behave. Such a study reveals the need to make a generalization of Lattice Theory, which is very important in “Computational Algebra”. In this paper we introduce the multisemilattice structure as a generalization of the semilattice structure. Such a structure is proposed as a particular type of poset. Subsequently, we offer an equivalent algebraic characterization based on non-deterministic operators and with a weakly associative property. We also show that from the structure of multisemilattice we can obtain an algebraic characterization of the multilattice structure. This paper concludes by showing the relevance of the multisemilattice structure in the design of algorithms aimed at calculating unitary implicates and implicants in temporal logics. Concretely, we show that it is possible to design efficient algorithms to calculate the unitary implicants/implicates only if the unitary formulae set has the multisemilattice structure.

Reducing signed propositional formulas

Abstract New strategies of reduction for finite-valued propositional logics are introduced in the framework of the TAS1 methodology developed by the authors [1]. A new data structure, the Δ^-sets, is introduced to store information about the formula being analysed, and its usefulness is shown by developing efficient strategies to decrease the size of signed propositional formulas, viz., new criteria to detect the validity or unsatisfiability of subformulas, and a strong generalisation of the pure literal rule.

Generalizations of lattices via non-deterministic operators

Benado (Cehoslovak. Mat. Z. 79(4) (1954) 105-129) and later Hansen (Discrete Math. 33(1) (1981) 99-101) have offered an algebraic characterization of multilattice (i.e., a poset where every pair of elements satisfies that any upper bound is greater than or equal to a minimal upper bound, and also satisfies the dual property). To that end, they introduce two algebraic operators that are a generalization of the operators @? and @? in a lattice. However, in Martinez et al. (Math. Comput. Sci. Eng. (2001) 238-248), we give the only algebraic characterization of the multisemilattice structure that exists in the literature. Moreover, this characterization allows us to give a more adequate characterization of the multilattice structure. The main advantage of our algebraic characterizations is that they are natural generalizations of the semilattice and lattice structures. It is well-known that in the lattice theory we can use indistinctly pairs of elements or finite subsets to characterize them. However, this is not true when we work with multilattices. For this reason in this paper we introduce two new structures from the ordered point of view, called universal multisemilattice and universal multilattice, and we propose an equivalent algebraic characterization for them. These new structures are generalizations, on one hand, of semilattice and lattice and, on the other hand, of multisemilattice and multilattice, respectively. The algebraic characterizations have the same advantages as the two introduced by us in Martinez et al. The most important purpose of this paper is to deepen the theoretical study of universal multisemilattices and universal multilattices.

Ideal non-deterministic operators as a formal framework to reduce the key finding problem

A formal development in the framework of the lattice theory for functional dependencies and minimal keys is presented. Beyond this theoretical study, a technique to prune the key finding problem, named scheme pruning transformation, is proposed in this work. This transformation is founded on theoretical results and has linear cost in the worst case. Moreover, this approach has provided a better size reduction than the usual techniques existing in the literature.

Non-deterministic ideal operators: An adequate tool for formalization in Data Bases

In this paper, we propose the application of formal methods to Software Engineering. The most used data model is the relational model and we present, within the general framework of lattice theory, this analysis of functional dependencies. For this reason, we characterize the concept of f-family by means of a new concept which we call non-deterministic ideal operator (nd.ideal-o). The study of nd.ideal-o.s allows us to obtain results about functional dependencies as trivial particularizations, to clarify the semantics of the functional dependencies and to progress in their efficient use, and to extend the concept of schema. Moreover, the algebraic characterization of the concept of Key of a schema allows us to propose new formal definitions in the lattice framework for classical normal forms in relation schemata. We give a formal definition of the normal forms for functional dependencies more frequently used in the bibliography: the second normal form (2FN), the third normal form(3FN) and Boyce-Codds normal form (FNBC).

Satisfiability Testing for Boolean Formulas Using Δ-trees

The tree-based data structure of Δ-tree for propositional formulas is introduced in an improved and optimised form. The Δ-trees allow a compact representation for negation normal forms as well as for a number of reduction strategies in order to consider only those occurrences of literals which are relevant for the satisfiability of the input formula. These reduction strategies are divided into two subsets (meaning- and satisfiability-preserving transformations) and can be used to decrease the size of a negation normal form A at (at most) quadratic cost. The reduction strategies are aimed at decreasing the number of required branchings and, therefore, these strategies allow to limit the size of the search space for the SAT problem.

Bases for closed sets of implicants and implicates in temporal logic

Abstract. Prime implicates and implicants are used in several areas of Artificial Intelligence. However, their calculation is not always an easy task. Nevertheless, it is important to remark the distinction between (i) computing the prime implicates and implicants and (ii) using the information they contain.In this paper, we present a way in which (ii) can be done without actually doing (i) by limiting prime implicants and implicates management to unitary implicants and implicates. Besides, we outline how the use of this technique is particularly relevant in the field of automated deduction in temporal logics. The information contained in temporal implicates and implicants can be used to design transformations of temporal formulae able to increase the power of automated deduction techniques for temporal logics. Particularly, we have developed a theory for unitary temporal implicates and implicants that can be more efficiently computed than prime implicants, while still providing the information needed to design this kind of transformations.The theory we have developed in this paper is easily extensible to cover different types of temporal logics, and is integrable in different automated deduction methods for these temporal logics.

Implicates and reduction techniques for temporal logics

Reduction strategies are introduced for the future fragment of a temporal propositional logic on linear discrete time, named FNext. These reductions are based on the information collected from the syntactic structure of the formula, which allows the development of efficient strategies to decrease the size of temporal propositional formulas, viz. new criteria to detect the validity or unsatisfiability of subformulas, and a strong generalisation of the pure literal rule. These results, used as an inner processing step, allow to improve the performance of any automated theorem prover.

Generalization of some properties of relations in the context of functional temporal×modal logic

In this paper, we generalize the definitions of transitivity, reflexivity, symmetry, Euclidean and serial properties of relations in the context of a functional approach for temporal×modal logic. The main result is the proof of definability of these definitions which is obtained by using algebraic characterizations. As a consequence, we will have in our temporal×modal context the generalizations of modal logics T, S4, S5, KD45, etc. These new logics will allow us to establish connections among time flows in very different ways, which enables us to carry out different relations among asynchronous systems. Our further research is focused on the construction of logics with these properties and the design of theorem provers for these logics.