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

Evaluating general purpose automated theorem proving systems

AbstractA key concern of ATP research is the development of more powerful systems, capable of solving more difficult problems within the same resource limits. In order to build more powerful systems, it is important to understand which systems, and hence which techniques, work well for what types of problems. This paper deals with the empirical evaluation of general purpose ATP systems, to determine which systems work well for what types of problems. This requires also dealing with the issues of assigning ATP problems into classes that are reasonably homogeneous with respect to the ATP systems that (attempt to) solve the problems, and assigning ratings to problems based on their difficulty

Deduction with XOR Constraints in Security API Modelling

We introduce XOR constraints, and show how they enable a theorem prover to reason effectively about security critical subsystems which employ bitwise XOR. Our primary case study is the API of the IBM 4758 hardware security module. We also show how our technique can be applied to standard security protocols

Automated deduction with built-in theories: completeness results and constraint solving techniques

Postprint (published version

A-ordered tableaux

In resolution proof procedures refinements based on A-orderings of literals have a long tradition and are well investigated. In tableau proof procedures such refinements were only recently introduced by the authors of the present paper. In this paper we prove the following results: we give a completeness proof of A-ordered ground clause tableaux which is a lot easier to follow than the previous one. The technique used in the proof is extended to the non-clausal case as well as to the non-ground case and we introduce an ordered version of Hintikka sets that shares the model existence property of standard Hintikks sets. We show that A-ordered tableaux are a proof confluent refinement of tableaux and that A-ordered tableaux together with the connection refinement yield an incomplete proof procedure. We introduce A-ordered first-order NNF tableaux, prove their completeness, and we briefly discuss implementation issues

Using a generalisation critic to find bisimulations for coinductive proofs

Coinduction is a method of growing importance in reasoning about functional languages, due to the increasing prominence of lazy data structures. Through the use of bisimulations and proofs that observational equivalence is a congruence in various domains it can be used to proof the congruence of two processes. Several proof tools have been developed to aid coinductive proofs but all require user interaction. Crucially they require the user to supply an appropriate relation which the system can then prove to be a bisimulation. A method is proposed which uses the idea of proof plans to make a heuristic guess at a suitable relation. If the proof fails for that relation the reasons for failure are analysed using a proof critic and a new relation is proposed to allow the proof to go through

Building and Combining Matching Algorithms

International audienceThe concept of matching is ubiquitous in declarative programming and in automated reasoning. For instance, it is a key mechanism to run rule-based programs and to simplify clauses generated by theorem provers. A matching problem can be seen as a particular conjunction of equations where each equation has a ground side. We give an overview of techniques that can be applied to build and combine matching algorithms. First, we survey mutation-based techniques as a way to build a generic matching algorithm for a large class of equational theories. Second, combination techniques are introduced to get combined matching algorithms for disjoint unions of theories. Then we show how these combination algorithms can be extended to handle non-disjoint unions of theories sharing only constructors. These extensions are possible if an appropriate notion of normal form is computable

A-ordered tableaux

In resolution proof procedures refinements based on A-orderings of literals have a long tradition and are well investigated. In tableau proof procedures such refinements were only recently introduced by the authors of the present paper. In this paper we prove the following results: we give a completeness proof of A-ordered ground clause tableaux which is a lot easier to follow than the previous one. The technique used in the proof is extended to the non-clausal case as well as to the non-ground case and we introduce an ordered version of Hintikka sets that shares the model existence property of standard Hintikks sets. We show that A-ordered tableaux are a proof confluent refinement of tableaux and that A-ordered tableaux together with the connection refinement yield an incomplete proof procedure. We introduce A-ordered first-order NNF tableaux, prove their completeness, and we briefly discuss implementation issues