Can a Higher-Order and a First-Order Theorem Prover Cooperate?
State-of-the-art first-order automated theorem proving systems have reached considerable strength over recent years. However, in many areas of mathematics they are still a long way from reliably proving theorems that would be considered relatively simple by humans. For example, when reasoning about sets, relations, or functions, first-order systems still exhibit serious weaknesses. While it has been shown in the past that higher-order reasoning systems can solve problems of this kind automatically, the complexity inherent in their calculi and their inefficiency in dealing with large numbers of clauses prevent these systems from solving a whole range of problems.
We present a solution to this challenge by combining a higher-order and a first-order automated theorem prover, both based on the resolution principle, in a flexible and distributed environment. By this we can exploit concise problem formulations without forgoing efficient reasoning on first-order subproblems. We demonstrate the effectiveness of our approach on a set of problems still considered non-trivial for many first-order theorem provers.
KeywordsInference Rule Theorem Prover Open Goal Primitive Equality Natural Deduction
Unable to display preview. Download preview PDF.
- 2.Benzmüller, C.: Equality and Extensionality in Higher-Order Theorem Proving. PhD thesis, Universität des Saarlandes, Germany (1999)Google Scholar
- 8.Benzmüller, C., Sorge, V.: Oants – An open approach at combining Interactive and Automated Theorem Proving. In: Proc. of Calculemus-2000, AK Peters (2001)Google Scholar
- 10.Brown, C.E.: Set Comprehension in Church’s Type Theory. PhD thesis, Dept. of Mathematical Sciences, Carnegie Mellon University, USA (2004)Google Scholar
- 11.de Nivelle, H.: The Bliksem Theorem Prover, Version 1.12. Max-Planck-Institut, Saarbrücken, Germany (1999), http://www.mpi-sb.mpg.de/bliksem/manual.ps
- 12.Denzinger, J., Fuchs, D.: Cooperation of Heterogeneous Provers. In: Proc. IJCAI-16, pp. 10–15. Morgan Kaufmann, San Francisco (1999)Google Scholar
- 13.Fisher, M., Ireland, A.: Multi-agent proof-planning. In: CADE-15 Workshop “Using AI methods in Deduction” (1998)Google Scholar
- 15.Hurd, J.: An LCF-style interface between HOL and first-order logic. In: Voronkov, A. (ed.) CADE 2002. LNCS (LNAI), vol. 2392, pp. 134–138. Springer, Heidelberg (2002)Google Scholar
- 16.Kerber, M.: On the Representation of Mathematical Concepts and their Translation into First Order Logic. PhD thesis, Universität Kaiserslautern, Germany (1992)Google Scholar
- 22.Sorge, V.: OANTS: A Blackboard Architecture for the Integration of Reasoning Techniques into Proof Planning. PhD thesis, Universität des Saarlandes, Germany (2001)Google Scholar