Extensional Crisis and Proving Identity
Extensionality axioms are common when reasoning about data collections, such as arrays and functions in program analysis, or sets in mathematics. An extensionality axiom asserts that two collections are equal if they consist of the same elements at the same indices. Using extensionality is often required to show that two collections are equal. A typical example is the set theory theorem ( ∀ x)( ∀ y)x ∪ y = y ∪ x.Interestingly, while humans have no problem with proving such set identities using extensionality, they are very hard for superposition theorem provers because of the calculi they use.In this paper we show how addition of a new inference rule, called extensionality resolution, allows first-order theorem provers to easily solve problems no modern first-order theorem prover can solve. We illustrate this by running the Vampire theorem prover with extensionality resolution on a number of set theory and array problems. Extensionality resolution helps Vampire to solve problems from the TPTP library of first-order problems that were never solved before by any prover.
KeywordsInference Rule Theorem Prover Collection Type Extensionality Axiom Saturation Algorithm
Unable to display preview. Download preview PDF.
- 1.Experimental results of this paper, http://vprover.org/extres
- 2.Armando, A., Bonacina, M.P., Ranise, S., Schulz, S.: New Results on Rewrite-Based Satisfiability Procedures. ACM Trans. Comput. Log. 10(1) (2009)Google Scholar
- 3.Bachmair, L., Ganzinger, H.: Resolution Theorem Proving. In: Handbook of Automated Reasoning, pp. 19–99. Elsevier and MIT Press (2001)Google Scholar
- 6.Barrett, C., Stump, A., Tinelli, C.: The Satisfiability Modulo Theories Library (SMT-LIB) (2010), www.SMT-LIB.org
- 8.Bledsoe, W.W., Boyer, R.S.: Computer Proofs of Limit Theorems. In: Proc. of IJCAI, pp. 586–600 (1971)Google Scholar
- 11.de Moura, L., Bjørner, N.: Generalized, efficient array decision procedures. In: FMCAD, pp. 45–52. IEEE (2009)Google Scholar
- 16.Nieuwenhuis, R., Rubio, A.: Paramodulation-Based Theorem Proving. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning, vol. I, ch. 7, pp. 371–443. Elsevier Science (2001)Google Scholar
- 17.Prevosto, V., Waldmann, U.: SPASS+T. In: Proc. of ESCoR, pp. 18–33 (2006)Google Scholar