Khaled Bsaïes

Static Analysis for the Synthesis of Eureka Properties for Transforming Logic programs

Much research has been devoted to logic program transformation using Unfold/Fold transformations. However there has been relatively little work on the use and the synthesis of so-called Eureka properties. Experience has shown that in many cases Unfold/Fold transformations are not sufficient to produce efficient logic programs and we have to consider the application of suitable properties. This paper deals with the way to discover these properties. First we introduce notions of a well-behaved clause and definite program. We prove that under certain conditions the application of the Unfold/Fold transformations preserves the notion of well-behavedness. We introduce a strategy based on static program analysis for the synthesis of Eureka properties for a class of logic programs called candidate programs. Finally we illustrate this strategy by a complete example.

A methodology for constructing logic programs

A methodology for constructing definite programs is presented. This presentation consists of two parts. The first part is the writing of an initial specification in Horn clauses form. In this part some paradigms of construction are outlined. The specification obtained is moded and typed; moreover it has some important specific qualities: structure, clearness, correctness and executability. However the initial specification is generally not efficient. For this reason the second part of the presentation is the stage of transformation. In this part some strategies based on folding, unfolding and generation of schemata are presented.

Appropriate lemmae discovery

An inductive proof attempt may fail as the available induction hypotheses cannot be applied to simplify the conclusion. One of the major problems which arise when performing inductive proofs is to transform the conclusion of an induction step in order to make the hypothesis applicable. Often, to overcome this problem, several additional lemmae are needed. However, most inductive theorem provers rely upon user intervention in supplying the required lemmae. In contrast, we present a method for generating automatically lemmae. Generation of lemmae is motivated by attempts to find appropriate instantiations for non-induction variables in the inductive step. We consider implicative formulae of the form ∀x-yΓ(x-,y-) ← Δ(x-), where Γ and Δ are conjunction of atoms and x- and y- are vectors of universal and of existential variables respectively.

Predicate Synthesis from Inductive Proof Attempt of Faulty Conjectures

We present a method for patching faulty conjectures in automatic theorem proving. The method is based on well-known folding /unfolding deduction rules. The conjectures we are interested in here are implicative formulas that are of the following form: \(\forall \overline{x} \phi( \overline{ x})=\forall \overline{x} \exists \overline{Y} \Gamma(\overline{x},\overline{Y}) \leftarrow \Delta(\overline{x})\). A faulty conjecture is a statement \(\forall \overline{x} \phi( \overline{x})\), which is not provable in some given program \({\cal T}\), defining all the predicates occurring in φ, i.e, \({\cal M(T)}\not \models \forall \overline{x} \phi( \overline{ x})\), where \({\cal M(T)}\) means the least Herbrand model of \({\cal T}\), but it would be if enough conditions, say P, were assumed to hold, i.e., \({\cal M(T\cup P)}\models \forall \overline{x} \phi( \overline{x})\leftarrow P\), where \({\cal P}\) is the definition of P. The missing hypothesis P is called a corrective predicate for φ. To construct P, we use the abduction mechanism that is the process of hypothesis formation. In this paper, we use the logic based approach because it is suitable for the application of deductive rules.

On using Eureka properties for transforming generate and test logic programs

Three transformation techniques to generate and test logic programs are presented. The key idea in transforming such programs is that the tester should prune incorrect search trees by interleaving testers into generators immediately after the testers become active. For this purpose, mode input-output analysis makes it possible to avoid useless computations. The basic idea is to replace the test made on output parameters by a test of input parameters and to rearrange the atoms in the body of clauses. >