We propose a method for combining the clause linking theorem proving method with theorem proving methods based on orderings. This may be useful for incorporating term-rewriting based approaches into clause linking. In this way, some of the propositional inefficiencies of ordering-based approaches may be overcome, while at the same time incorporating the advantages of ordering methods into clause linking. The combination also provides a natural way to combine resolution on non-ground clauses, with the clause linking method, which is essentially a ground method. We describe the method, prove completeness, and show that the enumeration part of clause linking with semantics can be reduced to polynomial time in certain cases. We analyze the complexity of the proposed method, and also give some plausibility arguments concerning its expected performance. 1 Introduction There are at least two basic approaches to the study of automated deduction. One approach concentrates on solving...

We first show that ground term-rewriting systems can be completed in a polynomial number of rewriting steps, if the appropriate data structure for terms is used. We then apply this result to study the lengths of critical pair proofs in non-ground systems, and obtain bounds on the lengths of critical pair proofs in the non-ground case. We show how these bounds depend on the types of inference steps that are allowed in the proofs. 1 Introduction We are interested in developing theoretical techniques for evaluating the efficiency of automated inference methods. This includes bounding proof sizes, as well as bounding the size of the total search space generated. Such investigations can provide insights into the comparative strengths of various inference systems, insights that might otherwise be missed. This can also aid in the development of new methods and new inference rules, as we will show. We first consider equational deduction for systems of ground equations. We note that in general...

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. We present a new automated proof method for first-order classical logic, aimed at limiting the combinatorial explosion of the search. It is non-clausal, based on BDDs (binary decision diagrams) and on new strategies that control the size and traversal of the search space by controlling the amount of information, in Shannon's sense, gained at each step of the proof. Our prover does not search blindly for a proof, but thinks a lot to decide of a course of action. Practical results show that this pays off. 1 Introduction We present a complete refutation method for first-order classical logic that aims at controlling the growth and at guiding the traversal of the search space intelligently. Our starting point is [10], which proves that finding whether a given proposition in this logic is obvious, for several different reasonable definitions of non-obviousness, is \Sigma p 2 -complete. This not only means that proving is hard, but also that any complete proof method is built on, or hid...

ic programming and all applications of deduction. The idea of "strategy analysis" is new. Most of the work on search in artificial intelligence concentrates on the design of heuristics (e.g., [5]). Most of the research in complexity related to theorem proving studies the complexity of propositional proofs as part of the quest for NP 6= co\GammaN P (e.g., see [10] for a survey), or works with complexity measures based on the Herbrand theorem to determine lower bounds for sets of clauses, not upper bounds for strategies (e.g., [2, 4, 7]). In resolution theorem proving, the classical source for the modelling of search is [3], which was not concerned with evaluating the complexity of the strategies. The primary objective of strategy analysis is to study the complexity of searching for a proof. An approach to this problem was proposed in [6]. It applies classical techniques from algorithm analysis to derive worst-case upper

Mathematical logic formalizes the process of mathematical reasoning. For centuries, it has been a dream of mathematicians to do mathematical reasoning mechanically. In the TPTP library, one finds thousands of problems from various domains of mathematics such as group theory, number theory, set theory, etc. Many of these problems can now be solved with state of the art automated theorem provers. Theorem proving also has applications in artificial intelligence and formal verification. As a formal method, theorem proving has been used to verify the correctness of various hardware and software designs. In this thesis, we propose a novel first-order theorem proving strategy -- ordered semantic hyper linking (OSHL). OSHL is an instance-based theorem proving strategy. It proves first-order unsatisfiability by generating instances of first-order clauses and proving the set of instances to be propositionally unsatisfiable. OSHL can use semantics, i.e. domain information, to guide its search. OS...