We propose a novel formalism, called Frame Logic (abbr., F-logic), that accounts in a clean and declarative fashion for most of the structural aspects of object-oriented and frame-based languages. These features include object identity, complex objects, inheritance, polymorphic types, query methods, encapsulation, and others. In a sense, F-logic stands in the same relationship to the objectoriented paradigm as classical predicate calculus stands to relational programming. F-logic has a model-theoretic semantics and a sound and complete resolution-based proof theory. A small number of fundamental concepts that come from object-oriented programming have direct representation in F-logic; other, secondary aspects of this paradigm are easily modeled as well. The paper also discusses semantic issues pertaining to programming with a deductive object-oriented language based on a subset of F-logic.

Theory resolution constitutes a set of complete procedures for incorporating theories into a resolution theorem-proving program, thereby making it unnecessary to resolve directly upon axioms of the theory. This can greatly reduce the length of proofs and the size of the search space. Theory resolution effects a beneficial division of labor, improving the performance of the theorem prover and increasing the applicability of the specialized reasoning procedures. Total theory resolution utilizes a decision procedure that is capable of determining unsatisfiability of any set of clauses using predicates in the theory. Partial theory resolution employs a weaker decision procedure that can determine potential unsatisfiability of sets of literals. Applications include the building in of both mathematical and special decision procedures, e.g., for the taxonomic information furnished by a knowledge representation system. Theory resolution is a generalization of numerous previously known resolution refinements. Its power is demonstrated by comparing solutions of "Schubert's Steamroller" challenge problem with and without building in axioms through theory resolution. 1 1

In this paper we provide algorithms for reasoning with partitions of related logical axioms in propositional and first-order logic (FOL). We also provide a greedy algorithm that automatically decomposes a set of logical axioms into partitions. Our motivation is two-fold. First, we are concerned with how to reason e#ectively with multiple knowledge bases that have overlap in content. Second, we are concerned with improving the e#ciency of reasoning over a set of logical axioms by partitioning the set with respect to some detectable structure, and reasoning over individual partitions. Many of the reasoning procedures we present are based on the idea of passing messages between partitions. We present algorithms for reasoning using forward message-passing and using backward message-passing with partitions of logical axioms. Associated with each partition is a reasoning procedure. We characterize a class of reasoning procedures that ensures completeness and soundness of our message-passing ...

A general framework for translating logical formulae from one logic into another logic is presented. The framework is instantiated with two different approaches to translating modal logic formulae into predicate logic. The first one, the well known ‘relational’ translation makes the modal logic’s possible worlds structure explicit by introducing a distinguished predicate symbol to represent the accessibility relation. In the second approach, the ‘functional ’ translation method, paths in the possible worlds structure are represented by compositions of functions which map worlds to accessible worlds. On the syntactic level this means that every flexible symbol is parametrized with particular terms denoting whole paths from the initial world to the actual world. The ‘target logic’ for the translation is a first-order many-sorted logic with built in equality. Therefore the ‘source logic’ may also be first-order many-sorted with built in equality. Furthermore flexible function symbols are allowed. The modal operators may be parametrized with arbitrary terms and particular properties of the accessibility relation may be specified within the

Even though it is not very often admitted, partial functions do play a significant role in many practical applications of deduction systems. Kleene has already given a semantic account of partial functions using three-valued logic decades ago, but there has not been a satisfactory mechanization. Recent years have seen a thorough investigation of the framework of many-valued truth-functional logics. However, strong Kleene logic, where quantification is restricted and therefore not truth-functional, does not fit the framework directly. We solve this problem by applying recent methods from sorted logics. This paper presents a resolution calculus that combines the proper treatment of partial functions with the efficiency of sorted calculi.

. The purpose of this paper is to make some practically relevant results in automated theorem proving available to many-valued logics with suitable modifications. We are working with a notion of many-valued first-order clauses which any finitely-valued logic formula can be translated into and that has been used several times in the literature, but in an ad hoc way. We give a many-valued version of polarity which in turn leads to natural many-valued counterparts of Horn formulas, hyperresolution, and a Davis-Putnam procedure. We show that the many-valued generalizations share many of the desirable properties of the classical versions. Our results justify and generalize several earlier results on theorem proving in many-valued logics. KEYWORDS: many-valued logic, polarity, Horn formula, direct products of structures, resolution, Davis-Putnam procedure Introduction The purpose of this paper is to make some practically relevant results in automated theorem proving available to many-value...

Abstract. In this paper we present an ordered theory resolution calculus and prove its completeness. Theory reasoning means to relieve a calculus from explicitly drawing inferences in a given theory by special purpose inference rules (e.g. E-resolution for equality reasoning). We take advantage of orderings (e.g. simplification orderings) by disallowing to resolve upon clauses which violate certain maximality constraints; stated positively, a resolvent may only be built if all the selected literals are maximal in their clauses. By this technique the search space is drastically pruned. As an instantiation for theory reasoning we show that equality can be built in by rigid E-unification. Keywords: Automated Theorem Proving, Theory Resolution. 1.

The aim of this chapter is twofold: first, introducing the basic concepts of analytic tableaux and, secondly, presenting state-of-the-art techniques for using non-clausal tableaux in automated deduction.

this paper, we will show how to extend model elimination with theory reasoning. Technically, theory reasoning means to relieve a calculus from explicit reasoning in some domain (e.g. equality, partial orders) by taking apart the domain knowledge and treating it by special inference rules. In an implementation, this results in a universal "foreground" reasoner that calls a specialized "background" reasoner for theory reasoning. Theory reasoning comes in two variants (Sti85) : total and

We present modifications of model elimination which do not necessitate the use of contrapositives. These restart model elimination calculi are proven sound and complete. The corresponding proof procedures are evaluated by a number of runtime experiments and they are compared to other well known provers. Finally we relate our results to other calculi, namely the connection method, modified problem reduction format and Near-Horn Prolog.