Automated mathematical induction

Projet EURECAProofs 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

Automated mathematical induction

This is a new version of Technical Report 1663, INRIA, 1992.Proofs by induction are important in many computer science and artificial intelligence applications, in particular, in program verification and specification systems. We present a new method to prove (and disprove) automatically inductive properties. Given a set of axioms, a well-suited induction scheme is construted automatically. We call such an induction scheme a test set. Then, for proving a property, we just instantiate it with terms from the test set and apply pure algebraic simplification 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

Strategic Issues, Problems and Challenges in Inductive Theorem Proving

Abstract(Automated) Inductive Theorem Proving (ITP) is a challenging field in automated reasoning and theorem proving. Typically, (Automated) Theorem Proving (TP) refers to methods, techniques and tools for automatically proving general (most often first-order) theorems. Nowadays, the field of TP has reached a certain degree of maturity and powerful TP systems are widely available and used. The situation with ITP is strikingly different, in the sense that proving inductive theorems in an essentially automatic way still is a very challenging task, even for the most advanced existing ITP systems. Both in general TP and in ITP, strategies for guiding the proof search process are of fundamental importance, in automated as well as in interactive or mixed settings. In the paper we will analyze and discuss the most important strategic and proof search issues in ITP, compare ITP with TP, and argue why ITP is in a sense much more challenging. More generally, we will systematically isolate, investigate and classify the main problems and challenges in ITP w.r.t. automation, on different levels and from different points of views. Finally, based on this analysis we will present some theses about the state of the art in the field, possible criteria for what could be considered as substantial progress, and promising lines of research for the future, towards (more) automated ITP

Automated Certification of Implicit Induction Proofs

International audienceTheorem proving is crucial for the formal validation of properties about user specifications. With the help of the Coq proof assistant, we show how to certify properties about conditional specifications that are proved using automated proof techniques like those employed by the Spike prover, a rewrite-based implicit induction proof system. The certification methodology is based on 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. We propose improvements of the certification process and show that the certification time is reasonable even for industrial-size applications. As a case study, we automatically prove and certify more than 40% of the lemmas needed for the validation of a conformance algorithm for the ABR protocol

多項式解釈による順序付けを用いた書き換え帰納法

Tohoku University外山芳

Test-sets und Termersetzungen für die Generierung rekursiv definierter Algorithmen aus Existenzaussagen

In dieser Arbeit wurde ein Verfahren vorgestellt, mit dem man rekursiv definierte Algorithmen aus Gueltigkeitsbeweisen von Existenzformeln extrahieren kann.Das Verfahren beschränkt sich auf einen einfachen Formalismus und basiert auf Test-sets und einem Vereinfachungsmechanismus.Termersetzungen und logische Simplifikationen bilden den Kern dieses Vereinfachungsmechanismus, waehrend Test-sets eine Beschreibung des initialen Modells einer Axiommenge darstellen.In this thesis we presented a method for extracting recursive defined algorithms from existentially quantified formulas, being based on a simple formalism, test sets and a simplification strategy.Term rewriting and logical simplification represent the core of that simplification strategy and test sets the description of the initial model of a set of axioms

Test-sets und Termersetzungen für die Generierung rekursiv definierter Algorithmen aus Existenzaussagen

In dieser Arbeit wurde ein Verfahren vorgestellt, mit dem man rekursiv definierte Algorithmen aus Gueltigkeitsbeweisen von Existenzformeln extrahieren kann.Das Verfahren beschränkt sich auf einen einfachen Formalismus und basiert auf Test-sets und einem Vereinfachungsmechanismus.Termersetzungen und logische Simplifikationen bilden den Kern dieses Vereinfachungsmechanismus, waehrend Test-sets eine Beschreibung des initialen Modells einer Axiommenge darstellen.In this thesis we presented a method for extracting recursive defined algorithms from existentially quantified formulas, being based on a simple formalism, test sets and a simplification strategy.Term rewriting and logical simplification represent the core of that simplification strategy and test sets the description of the initial model of a set of axioms