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Comprehension and Description in Tableaux
, 1997
"... Various approaches have been invented for enabling an automated theorem proving program to find proofs in set theory. The present approach is completely automatic and quite successful on many problems which are showcased as challenge problems for provers in set theory. In fact, this procedure fi ..."
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Various approaches have been invented for enabling an automated theorem proving program to find proofs in set theory. The present approach is completely automatic and quite successful on many problems which are showcased as challenge problems for provers in set theory. In fact, this procedure finds proofs of several of these examples without search. We implement the comprehension schema by means of tableau reduction and expansion rules. We also discuss the implementation of the definite descriptor in tableaux and special rules for handling equality effectively and in a tractable way in set theory. 1 Introduction An inference rule that "builds in" set theory at the inference level is the objective of Research Problem 8. More precisely, just as the employment of paramodulation permits one to avoid using any equality axioms other than reflexivity, the soughtafter inference rule for set theory would permit one to avoid using a number of the axioms in Godel's approach. L...
Discoveries and Experiments in the Automation of Mathematical Reasoning
, 2002
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NonElementary SpeedUps in Proof Length by Different Variants of Classical Analytic Calculi
"... . In this paper, different variants of classical analytic calculi for firstorder logic are compared with respect to the length of proofs possible in such calculi. A cutfree sequent calculus is used as a prototype for different other analytic calculi like analytic tableau or various connection calc ..."
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. In this paper, different variants of classical analytic calculi for firstorder logic are compared with respect to the length of proofs possible in such calculi. A cutfree sequent calculus is used as a prototype for different other analytic calculi like analytic tableau or various connection calculi. With modified branching rules (firules), nonelementary shorter minimal proofs can be obtained for a class of formulae. Moreover, by a simple translation technique and a standard sequent calculus, analytic cuts, i.e., cuts where the cut formulae occur as subformulae in the input formula, can be polynomially simulated. 1 Introduction Analytic firstorder calculi like (freevariable) analytic tableaux [4, 15, 23] or various connection calculi [5] are well suited for implementing automated deduction on a computer. In order to search for a proof in such calculi, only subformulae of the input formula have to be considered. This property, often referred to as the subformula property, makes...