Incorporating Decision Procedures in Implicit Induction

Abstract

In this paper we present an approach to integrating reasoning specialists into cover set induction based on constraint contextual rewriting. The approach has been successfully used to incorporate decision procedures into the SPIKE prover. Computer experiments on non-trivial verification problems illustrating the effectiveness of the proposed technique are given. The generality of the approach allows for the integration of computer algebra algorithms and techniques into induction theorem provers. To illustrate this, we discuss the integration of the Buchberger algorithm into our framework.

Rewriting Induction + Linear Arithmetic = Decision Procedure

“…The integration of a decision procedures for the theory of linear integer arithmetic into inductive reasoning has been previously considered in [10,28,3], with the main focus on contextual rewriting which integrates rewriting with decision procedures. The proof method developed in this paper is in general incomparable to these methods, though, since instead of the more complex contextual rewriting without constraints, regular constrained rewriting is employed.…”

'Descente Infinie' Induction-Based Saturation Procedures

“…This result opens the perspective of designing hybrid inference systems that would be able to perform both implicit induction and saturation-based proofs with minor changes, as explained in [11]. It also means that reasoning techniques can be shared: for example, we expect that the approach for integrating Computer Algebra algorithms into implicit induction systems [1] be easily adaptable for saturation-based systems.…”

Automated Certification of Implicit Induction Proofs

“…Concerning the ABR proofs, they have been performed by a restricted version of the Spike inference system and their certification was established within seconds for most of them. Compared to the Spike proofs from [17], the arithmetic reasoning was simulated with lemmas instead of using the complex decision procedures integrated into Spike [1]. Generating complete translations, i.e.…”

“…The bottom-line is that these improvements allowed us to certify big proof scripts in a reasonable time, 20 times faster than in [23]. 1 The structure of the paper is as follows: after introducing the basic notions and notations in Section 2, we present in Section 3 the restricted inference system and a new representation of the implicit induction proofs for which the underlying induction principle is an instance of Noetherian induction governed by an induction ordering over equalities. The conformance algorithm and its Spike specification are discussed in Section 4.…”