Abstract
First-order logic is an important part of mathematical logic, and automated theorem proving is an interdisciplinary field of mathematics and computer science. The paper presents an automated theorem prover for first-order logic, called C S E _ E 1.0, which is a combination of two provers contradiction separation extension (CSE) and E, where CSE is based on the recently-introduced multi-clause standard contradiction separation (S-CS) calculus for first-order logic and E is the well-known equational theorem prover for first-order logic based on superposition and rewriting. The motivation of the combined prover C S E _ E 1.0 is to (1) evaluate the capability, applicability and generality of C S E _ E , and (2) take advantage of novel multi-clause S-CS dynamic deduction of CSE and mature equality handling of E to solve more and harder problems. In contrast to other improvements of E, C S E _ E 1.0 optimizes E mainly from the inference mechanism aspect. The focus of the present work is given to the description of C S E _ E including its S-CS rule, heuristic strategies, and the S-CS dynamic deduction algorithm for implementation. In terms of combination, in order not to lose the capability of E and use C S E _ E to solve some hard problems which are unsolved by E, C S E _ E 1.0 schedules the running of the two provers in time. It runs plain E first, and if E does not find a proof, it runs plain CSE, then if it does not find a proof, some clauses inferred in the CSE run as lemmas are added to the original clause set and the combined clause set handed back to E for further proof search. C S E _ E 1.0 is evaluated through benchmarks, e.g., CASC-26 (2017) and CASC-J9 (2018) competition problems (FOFdivision). Experimental results show that C S E _ E 1.0 indeed enhances the performance of E to a certain extent.
Highlights
Automated theorem proving (ATP) is an important branch of artificial intelligence, whose research involves mathematics and computer science, and has been successfully applied in many application areas [1,2,3]
Instead of treating a contradiction as a complementary pair, can we extend it into a contradiction coming from two or more than two clauses? Can the inference rule and techniques go beyond binary resolution to enhance the efficiency and versatility of contemporary ATP systems?
This strategy is implemented mainly based on the following unit clause attributes: (1) deduction weight: the deduction weight of a unit clause is the number of times that the clause has participated in the standard contradiction separation (S-CS) dynamic deduction; (2) symbol count: this is used to characterize the statistics of symbols in a unit clause
Summary
Automated theorem proving (ATP) is an important branch of artificial intelligence, whose research involves mathematics and computer science, and has been successfully applied in many application areas [1,2,3]. (for clause/literal selection, axioms selection and proof search) have been studied from both the analytical and empirical perspectives These methods have effectively improved the capability of ATP systems. Their improvements are mainly from the perspectives of the effective selection of axioms and the effective use of strategies for the improvement of E It is an open question whether E can be improved from the inference mechanism point of view and the S-CS rule can be applied effectively to E. A multi-clause dynamic deduction algorithm based on the S-CS rule and the lemmas’ filtration method are proposed in Sections 4 and 5 respectively. Multi-clause S-CS-based dynamic deduction theory for first-order logic was introduced in [21], which gives the theoretical foundation for the CSE prover. We provide a review of the basic concepts of the S-CS rule and gives some theoretical analysis
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