An Improvement of Herbrand's Theorem and Its Application to Model Generation Theorem Proving

An Improvement on Sub-Herbrand Universe Computation

“…Similar as in [3] The above algorithm for computing SHUs in a give clause set S is not efficient. According to Algorithm 1, to computer an SHU corresponding to an argument i , it processes all clauses in S once, and at that time, it matches i with every argument in S. For this reason, when the number of arguments in S and the number of SHUs in S are large, it will take too long time to finish the computation.…”

“…Addressing to this problem, He et al [3] proposed a method for computing a sub-universe of the Herbrand universe, denoted a sub-Herbrand universe, for each argument of predicates or functions in a given clause set S, and they proved that S is unsatisfiable if and only if there is a finite unsatisfiable set of ground instances of clauses of S derived by instantiating each variable, which appears as an argument of predicate symbols or function symbols, in S over its corresponding sub-Herbrand universes. Because such subuniverses are usually smaller (sometimes considerably so) than the Herbrand universe of S, the number of ground instances that need to be considered for reasoning can be reduced in many cases.…”

“…However, the algorithm proposed in [3] is inefficient. In order to compute a sub-Herbrand universe corresponding to an argument in a given clause set S, it has to processes all *Address correspondence to this author at the Graduate School of Information Science, Aichi Prefectural University, Nagakute-cho, Aichi-gun, Aichi 480-1198, Japan; Tel: +81-561-64-1111, Ext.…”

An Improvement on Sub-Herbrand Universe Computation

“…The rest of the paper is organized as follows: We review the previous algorithm proposed in [3] in the next section, then introduce our efficient algorithm in Section 3. Section 4 shows the correctness of our approach, and Section 5 reports the experimental results on benchmarks.…”