How Many Times Do We Need an Assumption to Prove a Tautology in Minimal Logic: An Example on the Compression Power of Classical Reasoning
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چکیده
In this article we present a class of formulas φn, n ∈ Nat, that need at least 2 assumption occurrences to be proved in a normal proof in Natural Deduction for purely implicational minimal propositional logic. In purely implicational classical propositional logic, with Peirce’s rule, each φn is proved with only one assumption occurrence in Natural Deduction in a normal proof. Besides that, the formulas φn have exponentially sized proofs in cut-free Sequent Calculus. In fact 2 is the lower-bound for normal proofs in ND and cut-free Sequent proofs. We briefly discuss the consequences of the existence of this class of formulas for designing automatic proof-procedures based on these deductive systems.
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تاریخ انتشار 2014