Automata-Driven Automated Induction

Abstract

This work investigates inductive theorem proving techniques for first-order functions whose meaning and domains can be specified by Horn Clauses built up from the equality and finitely many unary membership predicates. In contrast with other works in the area, constructors are not assumed to be free. Techniques originating from tree automata are used to describe ground constructor terms in normal form, on which the induction proofs are built up. Validity of (free) constructor clauses is checked by an original technique relying on the recent discovery of a complete axiomatisation of finite trees and their rational subsets. Validity of clauses with defined symbols or non-free constructor terms is reduced to the latter case by appropriate inference rules using a notion of ground reducibility for these symbols. We show how to check this property by generating proof obligations which can be passed over to the inductive prover.

Mechanically certifying formula-based Noetherian induction reasoning

“…SPIKE was initially designed to deal with free constructors for which no equality relation can be established between any two different constructor symbols. Several extensions have been operated on SPIKE since (Bouhoula et al, 1992) in order to deal with: i) non-free constructors (Bouhoula and Jouannaud, 2001), ii) parameterised specifications (Bouhoula, 1994(Bouhoula, , 1996, iii) associative-commutative theories (Berregeb et al, 1996), iv) observational proofs (Berregeb et al, 1998;Bouhoula and Rusinowitch, 2002), and v) simultaneous check of the completeness and ground convergence properties of a specification (Bouhoula, 2009). Most of them led to distinct proof systems that are no longer maintained in spite of their theoretical and practical interests.…”

Automated Induction with Constrained Tree Automata

“…The idea of using such formalism for induction theorem proving is also in e.g. [4,10], because it is known that they can generate the languages of normal-forms for arbitrary term rewriting systems.…”

“…Some progress has been done e.g. in [4] and [5] in the direction of handling specification with non-free constructors, with severe restrictions (see related work below).…”

“…Our method subsumes former test set induction procedures like [6,1,4], by reusing former theoretical works on tree automata with constraints. It is sound and refutationally complete (any conjecture that is not valid in the initial model will be disproved) when R is sufficiently complete and the constructor subsystem R C is terminating.…”

“…The first author and Jouannaud [4] have used tree automata techniques to generalize test set induction to specifications with non-free constructors. This work has been generalized in [5] for membership equational logic.…”