Deduction by Combining Semantic Tableaux and Integer Programming

In this paper we propose to extend the current capabilities of automated reasoning systems by making use of techniques from integer programming We describe the architecture of an automated reasoning system based on a Herbrand procedure enumeration of formula instan ces on clauses The input are arbitrary sentences of rst order logic The translation into clauses is done incrementally and is controlled by a semantic tableau procedure using uni cation This amounts to an incremental polynomial CNF transformation which at the same time encodes part of the tableau structure and therefore tableau speci c re nements that reduce the search space Checking propositional unsatis ability of the resulting sequence of clauses can either be done with a symbolic inference system such as the Davis Putnam procedure or it can be done using integer programming If the latter is used a number of advantages become apparent Introduction In this paper we propose to extend the current capabilities of automated reaso ning AR systems by combining the inference procedure semantic tableaux with integer program IP solvers We show that the resulting system has properties which are interesting for such applications as formal program veri cation In Sec tion we summarize some facts on semantic tableaux in order to make the paper reasonably self contained In Section we give a tableau based polynomial time translation from propositional logic into IPs This translation will be lifted to full rst order logic in Section With an extended example we illustrate how the system is supposed to work Section and in Section we summarize the possible synergy e ects from marrying AR and OR in the way suggested Finally we mention related and ongoing work We had to omit all proofs due to limited space Semantic Tableaux First we state some standard notions of computational logic that will be used in the following consult Fitting for details Let us x a rst order language whose terms and formulae are built up from countable sets of predicate symbols function symbols constant symbols and object variables in the usual manner for each arity there are countably many function and predicate symbols We use the logical connectives conjunction disjunction implication and negation This research was supported by Deutsche Forschungsgemeinschaft within the Schwerpunkt programm Deduktion Bernhard Beckert and Reiner H ahnle and the quanti er symbols and An atom is a formula of the form p t tn where p is a predicate symbol and t tn are terms Atoms and their negations are called literals A clause is a disjunction of literals A formula is in conjunctive normal form CNF if it is a conjunction of clauses A variable is free if it is not bound by a quanti er or A sentence is a formula not containing any free variables We use the standard notions of satis ability and model A sentence is called a tautology if it is true in all models i e if its negation is unsatis able Substitutions are mappings from variables to terms and are extended to formulae as usual We denote a substitution by fx t xn tng where fx xng are the variables that occur in the term it is applied to The application of to a term t is denoted by t Semantic or analytic tableaux are a sound and complete calculus for doing logical inferences in full rst order logic They were developed in the s from Gentzen systems For an introduction which covers the material needed here see Fitting Following Fitting we divide the set of formulae of into four classes for formulae of conjunctive type for formulae of disjunctive type for quanti ed formulae of universal type and nally for quanti ed formulae of existential type This is called uniform notation it simpli es presentation and proofs considerably The classi cation is motivated by the tableau expansion rules which are associated with each formula The rules characterize the assertion of a truth value to a formula by means of asserting truth values to its direct subfor mulae For example holds if and only if and hold In the upper part of Table I the rule schemata for the various formula types are given Premises and conclusions are separated by a horizontal bar while vertical bars in the conclusion denote di erent extensions which are to be thought as disjunctions In the lower part of Table I the correspondence between formulae and formula types is shown