Section: New Results

Inductive Reasoning

Participant : Sorin Stratulat.

Automated inductive reasoning for term rewriting has been extensively studied in the literature. Classes of equations and term rewriting systems (TRSs) with decidable inductive validity have been identified and used to automatize the inductive reasoning. In [9] , we give procedures for deciding the inductive validity of equations in some standard TRSs on natural numbers and lists. Contrary to previous decidability results, our procedures can automatically decide without involving induction reasoning the inductive validity of arbitrary equations for these TRSs, that is, without imposing any syntactical restrictions on the form of equations. We also report on the complexity of our decision procedures. These decision procedures are implemented in our automated provers for inductive theorems of TRSs and experiments are reported.

In [30] , we detail the integration in SPIKE, an implicit induction theorem prover, of two reasoning modules operating over naturals combined with interpreted symbols. The first integration schema is à la Boyer-Moore, based on the combination of a congruence closure procedure with a decision procedure for linear arithmetic over rationals/reals. The second follows a ‘black-box’ approach and is based on external SMT solvers. It is shown that the two extensions significantly increase the power of SPIKE; their performances are compared when proving a non-trivial application.

In the setting of classical first-order logic with inductive predicates, two kinds of sequent-based induction reasoning are distinguished: cyclic and structural. Proving their equivalence is of great theoretical and practical interest for the automated reasoning community. Previously, it has been shown how to transform any structural proof developed with the LKID system into a cyclic proof using the CLKID${}^{\omega }$ system. However, the inverse transformation was only conjectured. In [29] , we provide a simple procedure that performs the inverse transformation for an extension of LKID with explicit induction rules issued from the structural analysis of CLKID${}^{\omega }$ proofs, then establish the equivalence of the two systems. This result is further refined for an extension of LKID with Noetherian induction rules. We show that Noetherian induction subsumes the two kinds of reasoning. This opens the perspective for building new effective induction proof methods and validation techniques supported by (higher-order) certification systems integrating the Noetherian induction principle, like Coq.