CSE_E 1.0: An Integrated Automated Theorem Prover for First-Order Logic

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.

CSE - A Automated Theorem Prover Based on Standard Contradiction Separation Dynamic Deduction

“…(4) Analyze and store the static attributes of clauses, literals, terms, and the dynamic attributes are dynamically adjusted during the S-CS deduction process, such as acceptable deduction weight and unacceptable deduction weight [10].…”

“…CSE designs related heuristic strategies based on the characteristics of S-CS dynamic deduction, for example, how to select input clauses and literals from the input clauses to construct standard contradictions, and how to control the dynamic deduction in order to improve the deduction efficiency. The heuristic strategies in CSE includes clause selection strategy, literal selection strategy, which are generated by describing the attributes of clause and literal, and some of attributes can refer to our published paper [10][11].…”

A multi-clause dynamic deduction algorithm based on standard contradiction separation rule

“…(1) Acceptable deduction (AD) weight of a clause [22,32]. This measure is based on two concepts: deduction weight of a clause and acceptable deduction.…”

“…(2) Unacceptable deduction (UD) weight of a clause [22,32]. UD weight is the number of times that a clause has participated in an unacceptable S-CS dynamic deduction during a theorem proving.…”

“…The resolvent of binary resolution usually contains many literals, but the S-CS rule can eliminate multiple literals by separating a standard contradiction (a literal set), so the generated S-CSC usually contains less literals. The number of literals in an input clause can reflect the number of literals in the generated S-CSC to certain extent [22], which is used to control the number of literals in an S-CSC, thus improving the deduction efficiency.…”