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

The Use of Proof Planning for Cooperative Theorem Proving

AbstractWe describebarnacle: a co-operative interface to theclaminductive theorem proving system. For the foreseeable future, there will be theorems which cannot be proved completely automatically, so the ability to allow human intervention is desirable; for this intervention to be productive the problem of orienting the user in the proof attempt must be overcome. There are many semi-automatic theorem provers: we call our style of theorem provingco-operative, in that the skills of both human and automaton are used each to their best advantage, and used together may find a proof where other methods fail. The co-operative nature of thebarnacleinterface is made possible by the proof planning technique underpinningclam. Our claim is that proof planning makes new kinds of user interaction possible.Proof planning is a technique for guiding the search for a proof in automatic theorem proving. Common patterns of reasoning in proofs are identified and represented computationally as proof plans, which can then be used to guide the search for proofs of new conjectures. We have harnessed the explanatory power of proof planning to enable the user to understand where the automatic prover got to and why it is stuck. A user can analyse the failed proof in terms ofclam's specification language, and hence override the prover to force or prevent the application of a tactic, or discover a proof patch. This patch might be to apply further rules or tactics to bridge the gap between the effects of previous tactics and the preconditions needed by a currently inapplicable tactic

STRICT: a language and tool set for the design of very large scale integrated circuits

PhD ThesisAn essential requirement for the design of large VLSI circuits is a design methodology which would allow the designer to overcome the complexity and correctness issues associated with the building of such circuits. We propose that many of the problems of the design of large circuits can be solved by using a formal design notation based upon the functional programming paradigm, that embodies design concepts that have been used extensively as the framework for software construction. The design notation should permit parallel, sequential, and recursive decompositions of a design into smaller components, and it should allow large circuits to be constructed from simpler circuits that can be embedded in a design in a modular fashion. Consistency checking should be provided as early as possible in a design. Such a methodology would structure the design of a circuit in much the same way that procedures, classes, and control structures may be used to structure large software systems. However, such a design notation must be supported by tools which automatically check the consistency of the design, if the methodology is to be practical. In principle, the methodology should impose constraints upon circuit design to reduce errors and provide' correctness by construction' . It should be possible to generate efficient and correct circuits, by providing a route to a large variety of design tools commonly found in design systems: simulators, automatic placement and routing tools, module generators, schematic capture tools, and formal verification and synthesis tools