SATCHMOREBID: SATCHMO(RE) with BIDirectional relevancy

SATCHMORE was introduced as a mechanism to integrate relevancy testing with the model-generation theorem prover SATCHMO. This made it possible to avoid invoking some clauses that appear in no refutation, which was a major drawback of the SATCHMO approach. SATCHMORE relevancy, however, is driven by the entire set of negative clauses and no distinction is accorded to the query negation. Under unfavorable circumstances, such as in the presence of large amounts of negative data, this can reduce the efficiency of SATCHMORE. In this paper we introduce a further refinement of SATCHMO called SATCHMOREBID: SATCHMORE with BIDirectional relevancy. SATCHMOREBID uses only the negation of the query for relevancy determination at the start. Other negative clauses are introduced on demand and only if a refutation is not possible using the current set of negative clauses. The search for the relevant negative clauses is performed in a forward chaining mode as opposed to relevancy propagation in SATCHMORE which is based on backward chaining. SATCHMOREBID is shown to be refutationally sound and complete. Experiments on a prototype SATCHMOREBID implementation point to its potential to enhance the efficiency of the query answering process in disjunctive databases. Donald Loveland, Ph.D.: He is Emeritus Professor of Computer Science at Duke University. He received his Ph.D. in mathematics from New York University and taught at NYU and CMU prior to joining Duke in 1973. His research in automated deduction includes defining the model elimination proof procedure and the notion of linear resolution. He is author of one book and editor/co-editor of two other books on automated theorem proving. He has done research in the areas of algorithms, complexity, expert systems and logic programming. He is an AAAI Fellow, ACM Fellow and winner of the Herbrand Award in Automated Reasoning. Adnan H. Yahya, Ph.D.: He is an associate professor at the Department of Electrical Engineering, Birzeit University, Palestine. He received his Diploma and PhD degrees from St. Petersburg Electrotechnical University and Nothwestern University in 1979 and 1984 respectively. His research interests are in Artificial Intelligence in general and in the areas of Deductive Databases, Logic Programming and Nonmonotonic Reasoning in particular. He had several visiting appointments at universities and research labs in the US, Germany, France and the UK. Adnan Yahya is a member of the ACM, IEEE and IEEE Computer Society.     

I—SATCHMORE：An Improvement of A—SATCHMORE

This paper presents an improvement of A-SATCHMORE (SATCHMORE with Availability).A-SATCHMORE incorporates relevancy testing and availability checking into SATCHMO to prune away irrelevant forward chaining.However ,considering every consequent atom of those non-Horn clauses being derivable,A-SATCHMORE may suffer from a potential explosion of the search space when some of such consequent atoms are actually underivable.This paper introduces a solution for this problem and shows its correctness.     

Non-Horn clause logic programming without contrapositives

We present an extension of Prolog-style Horn clause logic programming to full first order logic. This extension has some advantages over other such extensions that have been proposed. We compare this method with the model elimination strategy which Stickel has recently implemented very efficiently, and with Loveland's extension of Prolog to near-Horn clauses. This new method is based on the author's simplified problem reduction format but permits a better control of the splitting rule than does the simplified problem reduction format. In contrast to model elimination, this new method does not require the use of contrapositives of clauses, permitting a better control of the search. This method has been implemented in C Prolog and has turned out to be a respectable and surprisingly compact first-order theorem prover. This implementation uses depth-first iterative deepening and caching of answers to avoid repeated solution of the same subgoal. We show that the time and space used by this method are polynomial functions of certain natural parameters of the search space, unlike other known methods. We discuss the relation of these upper bounds to Savitch's theorem relating nondeterministic time to deterministic space.This research was supported in part by the National Science Foundation under grant DCR-8516243.     

Inapproximability Results for Set Splitting and Satisfiability Problems with No Mixed Clauses

We prove hardness results for approximating set splitting problems and also instances of satisfiability problems which have no mixed clauses, i.e., every clause has either all its literals unnegated or all of them negated. Results of Håstad imply tight hardness results for set splitting when all sets have size exactly $k \ge 4$ elements and also for non-mixed satisfiability problems with exactly $k$ literals in each clause for $k \ge 4$. We consider the case $k=3$. For the MAX E3-SET SPLITTING, problem in which all sets have size exactly 3, we prove an NP-hardness result for approximating within any factor better than ${\frac{19}{20}}$. This result holds even for satisfiable instances of MAX E3-SET SPLITTING, and is based on a PCP construction due to Håstad. For non-mixed MAX 3SAT, we give a PCP construction which is a slight variant of the one given by Håstad for MAX 3SAT, and use it to prove the NP-hardness of approximating within a factor better than ${\frac{11}{12}}$.     

Toward leaner binary-clause reasoning in a satisfiability solver

Binary-clause reasoning has been shown to reduce the size of the search space on many satisfiability problems, but has often been so expensive that run-time was higher than that of a simpler procedure that explored a larger space. The method of Sharir for detecting strongly connected components in a directed graph can be adapted to performing lean resolution on a set of binary clauses. Beyond simply detecting unsatisfiability, the goal is to find implied equivalent literals, implied unit clauses, and implied binary clauses.     

Toward leaner binary-clause reasoning in a satisfiability solver

Binary-clause reasoning has been shown to reduce the size of the search space on many satisfiability problems, but has often been so expensive that run-time was higher than that of a simpler procedure that explored a larger space. The method of Sharir for detecting strongly connected components in a directed graph can be adapted to performing lean resolution on a set of binary clauses. Beyond simply detecting unsatisfiability, the goal is to find implied equivalent literals, implied unit clauses, and implied binary clauses.     

基于目标演绎距离的一阶逻辑子句集预处理方法

一阶逻辑定理证明是人工智能的核心基础,研究一阶逻辑自动定理证明器的相关理论和高效的算法实现具有重要的学术意义。当前一阶逻辑自动定理证明器首先通过子句集预处理约简子句集规模,然后通过演绎方法对定理进行判定。现有的应用于证明器中的子句集预处理方法普遍只从与目标子句项符号相关性角度出发,不能很好地从文字的互补对关系中体现子句间的演绎。为了在子句集预处理时从演绎的角度刻画子句间的关系,定义了目标演绎距离的概念并给出了计算方法,提出了一种基于目标演绎距离的一阶逻辑子句集预处理方法。首先对原始子句集进行包含冗余子句约简并应用纯文字删除规则,然后根据目标子句计算剩余子句集中的文字目标演绎距离、子句目标演绎距离,并最终通过设定子句演绎距离阈值来实现对子句集的进一步预处理。将该预处理方法应用于顶尖证明器Vampire,以2017年国际一阶逻辑自动定理证明器标准一阶逻辑问题组竞赛例为测试对象,在标准的300 s内,加入提出的子句集预处理方法的Vampire4.1相比原始的Vampire4.1多证明4个定理,能证明10个Vampire4.1未证明的定理,占其未证明定理总数的13.5%;在证明的定理中,提出的子句集预处理方法能对77.2%的子句集进行约简,最大子句集约简规模达到51.7%。实验结果表明,提出的一阶逻辑子句集预处理方法是一种有效的方法,能有效地约简一阶逻辑子句集的规模,提高一阶逻辑自动定理证明器的证明能力。     

CLIN-S - A Semantically Guided First-Order Theorem Prover

CLIN-S is an instance-based, clause-form first-order theorem prover. CLIN-S employs three inference procedures: semantic hyper-linking, which uses semantics to guide the proof search and performs well on non-Horn parts of the proofs involving small literals, rough resolution, which removes large literals in the proofs, and UR resolution, which proves the Horn parts of the proofs. A semantic structure for the input clauses is given as input. During the search for the proof, ground instances of the input clauses are generated and new semantic structures are built based on the input semantics and a model of the ground clause set. A proof is found if the ground clause set is unsatisfiable. In this article, we describe the system architecture and major inference rules used in CLIN-S.     

I-SATCHMO: An Improvement of SATCHMO

We introduce a method for reducing the redundant search space for SATCHMO's model generation approach by means of intelligent backtracking. During the reasoning, we mark an asserted consequent atom as useful whenever it has been used as an antecedent atom for forward chaining. We show that a splitting of the consequence of a non-Horn clause is unnecessary if one of its consequent atoms is found not to be useful at the time it is retracted from the database on backtracking, and therefore the remaining splitting over the clause's consequence can be immediately abandoned. In this way, much of the redundant search space can be eliminated. Our method is simple in principle, easy to implement in Prolog, independent of other refinements, and effective for model generation theorem proving.     

A Disjunctive Positive Refinement of Model Elimination and its Application to Subsumption Deletion

The Model Elimination (ME) calculus is a refutationally complete,goal-oriented calculus for first-order clause logic. In this article, weintroduce a new variant called disjunctive positive ME (DPME); it improveson Plaisteds positive refinement of ME in that reduction steps areallowed only with positive literals stemming from clauses having at leasttwo positive literals (so-called disjunctive clauses). DPME is motivated byits application to various kinds of subsumption deletion: in order to applysubsumption deletion in ME equally successful as in resolution, it iscrucial to employ a version of ME that minimizes ancestor context (i.e., thenecessary A-literals to find a refutation). DPME meets this demand. Wedescribe several variants of ME with subsumption, the most important onesbeing ME with backward and forward subsumption and theT*-Context Check. We compare their pruning power, also takinginto consideration the well-known regularity restriction. All proofs aresupplied. The practicability of our approach is demonstrated with experiments.