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

Nominal Logic Programming

Nominal logic is an extension of first-order logic which provides a simple foundation for formalizing and reasoning about abstract syntax modulo consistent renaming of bound names (that is, alpha-equivalence). This article investigates logic programming based on nominal logic. We describe some typical nominal logic programs, and develop the model-theoretic, proof-theoretic, and operational semantics of such programs. Besides being of interest for ensuring the correct behavior of implementations, these results provide a rigorous foundation for techniques for analysis and reasoning about nominal logic programs, as we illustrate via examples.Comment: 46 pages; 19 page appendix; 13 figures. Revised journal submission as of July 23, 200

Fast subsumption checks using anti-links

The concept of "anti-link" is defined, and useful equivalence-preserving operations based on anti-links are introduced.These operations eliminate a potentially large number of subsumed paths in a negation normal form formula.Those anti-links that directly indicate the presence of subsumed paths are characterized. The operations have linear time complexity in the size of that part of the formula containing the anti-link. The problem of removing all subsumed paths in an NNF formula is shown to be NP-hard, even though such formulas may be small relative to the size of their path sets. The general problem of determining whether there exists a pair of subsumed paths associated with an arbitrary anti-link is shown to be NP-complete. Additional techniques based on "strictly pure full blocks" are introduced and are also shown to eliminate redundant subsumption checks. The effectiveness of these techniques is examined with respect to some benchmark examples from the literature

Translating logic programs into conditional rewriting systems

In this paper a translation from a subclass of logic programs consisting of the simply moded logic programs into rewriting systems is defined. In these rewriting systems conditions and explicit substitutions may be present. We argue that our translation is more natural than previously studied ones and establish a result showing its correctness

Applicative Bisimilarities for Call-by-Name and Call-by-Value λμ-Calculus

International audienceWe propose the first sound and complete bisimilarities for the call-by-name and call-by-value untyped λµ-calculus, defined in the applicative style. We give equivalence examples to illustrate how our relations can be used; in particular, we prove David and Py's counter-example, which cannot be proved with Lassen's preexisting normal form bisimilarities for the λµ-calculus

Confluence of Conditional Rewriting in Logic Form

We characterize conditional rewriting as satisfiability in a Herbrand-like model of terms where variables are also included as fresh constant symbols extending the original signature. Confluence of conditional rewriting and joinability of conditional critical pairs is characterized similarly. Joinability of critical pairs is then translated into combinations of (in)feasibility problems which can be efficiently handled by a number of automatic tools. This permits a more efficient use of standard results for proving confluence of conditional term rewriting systems, most of them relying on auxiliary proofs of joinability of conditional critical pairs, perhaps with additional syntactical and (operational) termination requirements on the system. Our approach has been implemented in a new system: CONFident . Its ability to (dis)prove confluence of conditional term rewriting systems is witnessed by means of some benchmarks comparing our tool with existing tools for similar purposes

Confluence of Conditional Term Rewrite Systems via Transformations

Conditional term rewriting is an intuitive yet complex extension of term rewriting. In order to benefit from the simpler framework of unconditional rewriting, transformations have been defined to eliminate the conditions of conditional term rewrite systems. Recent results provide confluence criteria for conditional term rewrite systems via transformations, yet they are restricted to CTRSs with certain syntactic properties like weak left-linearity. These syntactic properties imply that the transformations are sound for the given CTRS. This paper shows how to use transformations to prove confluence of operationally terminating, right-stable deterministic conditional term rewrite systems without the necessity of soundness restrictions. For this purpose, it is shown that certain rewrite strategies, in particular almost U-eagerness and innermost rewriting, always imply soundness

Decision procedures for equality logic with uninterpreted functions

In dit proefschrift presenteren we een aantal technieken om vervulbaarheid (satisfiability) vast te stellen binnen beslisbare delen van de eerste orde logica met gelijkheid. Het doel van dit proefschrift is voornamelijk het ontwikkelen van nieuwe technieken in plaats van het ontwikkelen van een effici¨ente implementatie om vervulbaarheid vast te stellen. Als algemeen logisch raamwerk gebruiken we de eerste orde predikaten logica zonder kwantoren. We beschrijven enkele basisprocedures om vervulbaarheid van propositionele formules vast te stellen: de DP procedure, de DPLL procedure, en een techniek gebaseerd op BDDs. Deze technieken zijn eigenlijk families van algoritmen in plaats van losse algoritmen. Hun gedrag wordt bepaald door een aantal keuzen die ze maken gedurende de uitvoering. We geven een formele beschrijving van resolutie, en we analyseren gedetailleerd de relatie tussen resolutie en DPLL. Het is bekend dat een DPLL bewijs van onvervulbaarheid (refutation) rechtstreeks kan worden getransformeerd naar een resolutie bewijs van onvervulbaarheid met een vergelijkbare lengte. In dit proefschrift wordt een transformatie ge¨introduceerd van zo’n DPLL bewijs naar een resolutie bewijs dat de kortst mogelijke lengte heeft. We presenteren GDPLL, een generalisatie van de DPLL procedure. Deze is bruikbaar voor het vervulbaarheidsprobleem voor beslisbare delen van de eerste orde logica zonder kwantoren. Voldoende eigenschappen worden ge¨identificeerd om de correctheid, de be¨eindiging en de volledigheid van GDPLL te bewijzen. We beschrijven manieren om vervulbaarheid vast te stellen binnen de logica met gelijkheid en niet-ge¨interpreteerde functies (EUF). Dit soort logica is voorgesteld om abstracte hardware ontwerpen te verifi¨eren. Het snel kunnen vaststellen van vervulbaarheid binnen deze logica is belangrijk om dergelijke verificaties te laten slagen. In de afgelopen jaren zijn er verschillende procedures voorgesteld om de vervulbaarheid van dergelijke formules vast te stellen. Wij beschrijven een nieuwe aanpak om vervulbaarheid vast te stellen van formules uit de logica met gelijkheid die in de conjunctieve normaal vorm zijn gegeven. Centraal in deze aanpak staat ´e´en enkele bewijsregel genaamd gelijkheidsresolutie. Voor deze ene regel bewijzen wij correctheid en volledigheid. Op grond van deze regel stellen we een volledige procedure voor om vervulbaarheid van dit soort formules vast te stellen, en we bewijzen de correctheid ervan. Daarnaast presenteren we nog een nieuwe procedure om vervulbaarheid vast te stellen van EUF-formules, gebaseerd op de GDPLL methode. Tot slot breiden we BDDs voor propositionele logica uit naar logica met gelijkheid. We bewijzen dat alle paden in deze uitgebreide BDDs vervulbaar zijn. In een constante hoeveelheid tijd kan vastgesteld worden of de formule een tautologie is, een tegenspraak is, of slechts vervulbaar is