Emmanuel Kounalis

Automatic proofs by induction in theories without constructors

Abstract Inductionless induction consists of using pure equational reasoning for proving the validity of an equation in the initial algebra of a set of equational axioms, which would normally require some kind of induction. Under given hypotheses, the equation is valid iff adding it to the set of axioms does not result in an inconsistency. This inconsistency can be found by the Knuth-Bendix completion algorithm, provided that the signature of the algebra is split into free constructors and defined symbols, which must be completely defined in terms of constructors. This is the base of the so-called inductive completion algorithm of Huet and Hullot. Two key concepts, inductive reducibility and inductive co-reducibility , allow us to extend these techniques in various directions: incomplete specifications, nonfree constructors, no constructors specified, equational term rewriting systems. The method is adapted for proving the consistency property of an enrichment of a specification by new operators and new equations. In addition, we get also a simple algorithm to exhibit a set of constructors of a specification. Finally, inductive co-reducibility is reduced to inductive reducibility and an algorithm for deciding inductive reducibility is given for left linear term rewriting systems.

Automated Mathematical Induction

Proofs by induction are important in many computer science and artifical intelligence applications, in particular, in program verification and specification systems. We present a new method to prove (and disprove) automatically inductives properties. Given a set of axioms, a well-suited induction scheme is constructed automatically. We call such and induction scheme a test set. Then, for proving a property, we just instantiate it with terms from the test set and apply pure algebraic simplifications to the result. This method needs no completion and explicit induction. However it retains their positive features, namely, the completeness of the former and the robustness of the latter. It has been implemented in the theorem-prover SPIKE.

Reasoning with conditional axioms

We present methods for automatically proving theorems in theories axiomatized by a set of Horn clauses. These methods address both deductive and inductive reasoning. They are based on the concept of simplification and require minimal human interaction.

On word problems in Horn theories

We interpret Horn clauses as conditional rewrite rules. Then we give sufficient conditions so that the word problem can be decided by conditional normalization in some Horn theory. We also show how to prove theorems in the initial models of Horn theories.

A LOGICAL FRAMEWORK FOR CONCEPT LEARNING WITH BACKGROUND KNOWLEDGE

A central process in many kinds of learning is the process of generalization or concept learning from a set of training instances (a set of examples and counterexamples) in the presence of some Background Knowledge. Given a set of examples and counterexamples of a concept, the learner induces a general concept that describes all of the positive examples and none of the counterexamples, and is consistent with the Background Knowledge. We present here a logical framework to induce concept descriptions from a given set of examples and counterexamples in the presence of Background Knowledge described by a set of Horn clauses. In particular, we provide: 1) a definition of what is meant by “a learnable concept” from examples and counterexamples in the presence of Background Knowledge described by a set H of Horn clauses”. Broadly speaking, in our framework, a learnable concept C is an atom which is true (valid) in the least Herbrand model of H but false in some other models of H and, 2) a methodology to induce general concept descriptions from concept examples and counterexamples in Horn theories. We give an automatic method, based on a well-founded ordering on the elements of H, which is the basis for checking validity in the least Herbrand model of H. This work generalizes, unifies, and provides a logical framework of previous studies on the concept-learning-from-examples paradigm.