SRASS - a Semantic Relevance Axiom Selection System Geoff Sutcliffe ⋆ ⋆ University of Miami, USA geoff@cs.miami.edu Abstract This paper describes the design, implementation, and testing of a system for selecting necessary axioms from a large set also containing superfluous axioms, to obtain a proof of a conjecture. The selection is determined by semantics of the axioms and conjecture, ordered heuristically by a syntactic relevance measure. The system is able to solve many problems that cannot be solved alone by the underlying conventional automated reasoning system. 1 Introduction In recent years the ability of systems to reason over large theories – theories in which there are many functors and pred- icates, many axioms of which typically only a few are required for the proof of a theorem, and many theorems to be proved from the same set of axioms – has become more important. Large theory problems are becoming more prevalent as large knowledge bases, e.g., ontologies and large mathematical knowledge bases, are translated into forms suitable for automated reasoning, and mechanical generation of automated reasoning problems becomes more common. This work addresses the issue of selecting necessary axioms, from a large set also containing superfluous axioms, to obtain a proof of a conjecture. It is based on the idea in Petr Pudlak’s PhD research, which he attributes to Jiˇ r´ ı Vyskoˇ cil. In contrast to existing work that selects axioms based on syntactic characteristics, this work uses semantics to guide the selection. 2 Semantic Relevance Axiom Selection M 0 C A 1 M 1 M 2 M 0 A 2 A 3 C A 1 M 1 M 0 C A 1 M 1 M 2 M 0 A 2 C The selection starts with an empty set of selected axioms. At each iteration the process looks for a model of the selected axioms and the negation of the conjec- ture. If no such model exists then the conjecture is a logical consequence of the selected axioms. If such a model exists then an unselected axiom that is false in the model is moved to the set of selected axioms. The newly selected axiom excludes the model (and possibly other models) from the models of the selected axioms and negated conjecture, eventually leading to the situation where there are no models of the selected axioms and the negated conjecture. The figure shows the idea. The plane represents the space of interpretations, the rectan- gle encompasses the models of the conjecture C, and an oval encompasses the models of the corresponding axiom A i . In the first iteration, when the set of selected axioms is empty, the model M 0 of the negation of the conjecture, ¬C, is found. That leads to the selection of the axiom A 1 , which is false in the model. Iteratively, the model M 1 of {A 1 , ¬C} is found, leading to the selection of A 2 , the model M 2 of {A 1 ,A 2 , ¬C} is found, leading to the selection of A 3 , at which point there is no model of {A 1 ,A 2 ,A 3 , ¬C}, proving that C is a logical consequence of {A 1 ,A 2 ,A 3 }. In the last part of the figure this is seen by the intersection of the axiom ovals lying within the conjecture rectangle. Example: Consider the simple propositional problem, to prove the conjecture C = b from the axioms E 1 = a | b, E 2 = b ⇒ a, E 3 =(¬a &(b | c)) | (a & ¬b & ¬c). and E 4 = b | (a ⇔ c). The conjecture C can be proved from E 3 and E 4 , i.e., E 1 and E 2 are superfluous. The following table shows a possible sequence of models and selected axioms. Note how E 1 , E 2 , and E 3 are true in the first model, so only E 4 can be selected. E 1 and E 3 are false in the second model, but E 1 is found first. E 2 is true in every model of ¬C, and thus can never be selected. If the model {a, ¬b, c} had been used in the second iteration, then E 3 would have been selected, leading to immediate success. Selected set Model Axiom 1 {} {a, ¬b, ¬c} E 4 = b | (a ⇔ c) 2 {E 4 } {¬a, ¬b, ¬c} E 1 = a | b 3 {E 1 ,E 4 } {a, ¬b, c} E 3 =(¬a &(b | c)) | (a & ¬b & ¬c) 4 {E 1 ,E 3 ,E 4 } - -