Towards a framework to integrate proof search paradigms (original) (raw)

Integrating Automated and Interactive Theorem Proving

Applied Logic Series, 1998

This paper highlights a project to integrate interactive and automated theorem proving in Software Veri cation. Its aim is to combine the advantages of the two paradigms. We report on the integration concepts, and on the experimental results with a prototype implementation.

An Interactive Driver for Goal-directed Proof Strategies

Electronic Notes in Theoretical Computer Science, 2009

Interactive Theorem Provers (ITPs) are tools meant to assist the user during the formal development of mathematics. Automatic proof searching procedures are a desirable aid, and most ITPs supply the user with an extensive set of facilities to improve automation. However, the black-box nature of most automatic procedure conflicts with the interactive nature of these tools: a newcomer running an automatic procedure learns nothing by its execution (especially in case of failure), and a trained user has no opportunities to interactively guide the procedure towards the solution, e.g. pruning wrong or not promising branches of the search tree. In this paper we discuss the implementation of the resolution based automatic procedure of the Matita ITP, explicitly conceived to be interactively driven by the user through a suitable, simple graphical interface.

Problem-oriented applications of automated theorem proving

Lecture Notes in Computer Science, 1996

This paper provides an overall view of an approach to developing a coherent ATP-system which can deal with a variety of logics and with different applications in a tailored way. The paper also summarizes research results achieved in the course of this development.

Learning from experts to aid the automation of proof search

2009

Most formal methods give rise to proof obligations which are putative lemmas that need proof. Discharging these POs can become a bottleneck in the use of formal methods in practical applications. Some techniques for reducing this bottleneck are known—it is our aim to increase the repertoire of techniques by tackling learning from proof attempts. Even after obvious fixed heuristics are used, there remains the problem of what to do with POs that are not discharged automatically.

Progress in Automated Theorem Proving, 1997-2001

Despite some impressive individual achievements, the ex- treme difficulty of Automated Theorem Proving (ATP) means that progress in ATP is slow relative to, e.g., some aspects of commercial information technology. The (relatively) slow progress has two distinct disadvantages. First, for the researchers, it is difficult to determine if a direction of in- vestigation is making a meaningful contribution. Second, for unaware observers, a lack of progress leads to a loss of interest and confidence in the field. In this context it is important that progress in ATP be measured, monitored, and recognized. This paper presents quantitative measures that show progress in ATP, from mid-1997 to mid-2001. The measures are based on collected performance data from ATP systems.

The search efficiency of theorem proving strategies: an analytical comparison

1994

We analyze the search e ciency of a number of common refutational theorem proving strategies for rst-order logic. Search e ciency is concerned with the total number of proofs and partial proofs generated, rather than with the sizes of the proofs. We s h o w that most common strategies produce search spaces of exponential size even on simple sets of clauses, or else are not sensitive to the goal. However, clause linking, which uses a reduction to propositional calculus, has behavior that is more favorable in some respects, a property that it shares with methods that cache subgoals. A strategy which is of interest for term-rewriting based theorem proving is the A-ordering strategy, a n d we discuss it in some detail. We show some advantages of A-ordering over other strategies, which m a y help to explain its e ciency in practice. We also point out some of its combinatorial ine ciencies, especially in relation to goal-sensitivity and irrelevant clauses. In addition, SLD-resolution, which is of importance for Prolog implementation, has combinatorial ine ciencies this may suggest basing Prolog implementations on a di erent theorem proving strategy.

Increasing the versatility of heuristic based theorem provers

Lecture Notes in Computer Science, 1993

Heuristic based theorem proving systems typically impose a fixed ordering on the strategies which they embody. The ordering reflects the general experience of the system designer. As a consequence, there will exist a variety of specific instances where the fixed ordering breaks down. We present an approach which liberates such systems by introducing a more versatile framework for organising proof strategies.

The search efficiency of theorem proving strategies

Springer eBooks, 1994

We analyze the search e ciency of a number of common refutational theorem proving strategies for rst-order logic. Search e ciency is concerned with the total number of proofs and partial proofs generated, rather than with the sizes of the proofs. We s h o w that most common strategies produce search spaces of exponential size even on simple sets of clauses, or else are not sensitive to the goal. However, clause linking, which uses a reduction to propositional calculus, has behavior that is more favorable in some respects, a property that it shares with methods that cache subgoals. A strategy which is of interest for term-rewriting based theorem proving is the A-ordering strategy, a n d we discuss it in some detail. We show some advantages of A-ordering over other strategies, which m a y help to explain its e ciency in practice. We also point out some of its combinatorial ine ciencies, especially in relation to goal-sensitivity and irrelevant clauses. In addition, SLD-resolution, which is of importance for Prolog implementation, has combinatorial ine ciencies this may suggest basing Prolog implementations on a di erent theorem proving strategy.

Proceedings of Logic Programming and Automated Reasoning '94

1994

Theorem proving is the systematic derivation of a mathcm-aticM proof from a set of axioms by the use of rules of inference. We ~re interested in a related but far less explored problem: the analysis and correction of false conjectures, especiMly where that correction involves finding a collection of antecedents that, together with a set of axioms, transform non-theorems into theorems. Most failed search trees are huge, and special care is to be taken in order to tackle the combinatorial explosion phenoraenom Fortunately, the planning search space generated by proof plans, see [1], are moderately small. We have explored the possibility of using this technique in the implementation of an abduction mechanism to correct non-theorems.

Progress in Automated Theorem Proving, 1997-1999

2000

Despite some impressive individual achievements, the extreme difficulty of Automated Theorem Proving (ATP) means that progress in ATP is slow relative to, e.g., some aspects of commercial information technology. The (relatively) slow progress has two distinct disadvantages. First, for the researchers, it is difficult to determine if a direction of investigation is making a meaningful contribution. Second, for unaware observers, a lack of progress leads to a loss of interest and confidence in the field. A serious outcome of this loss of interest and confidence has been the withdrawal of significant funding for ATP research. In this context of slow progress, it is important that progress in ATP be measured, monitored, and recognized. This paper presents quantitative measures that show progress in ATP, from mid-1997 to the end of 1999. The measures are based on collected performance data from ATP systems.