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The Undecidability of kProvability
 Annals of Pure and Applied Logic
, 1989
"... The kprovability problem is, given a first order formula φ and an integer k, to determine if φ has a proof consisting of k or fewer lines (i.e., formulas or sequents). This paper shows that the kprovability problem for the sequent calculus is undecidable. Indeed, for every r.e. srt X... ..."
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Cited by 27 (0 self)
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The kprovability problem is, given a first order formula φ and an integer k, to determine if φ has a proof consisting of k or fewer lines (i.e., formulas or sequents). This paper shows that the kprovability problem for the sequent calculus is undecidable. Indeed, for every r.e. srt X...
ON THE NUMBER OF STEPS IN PROOFS
, 1989
"... In this paper we prove some results about the complexity of proofs. We consider proofs in Hilbertstyle formal systems such as in [17J. Thus a proof is a sequence of formulas satisfying certain conditions. We caD view the formulas as being strings of symbols; hence the whole proof is a string too. W ..."
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Cited by 17 (2 self)
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In this paper we prove some results about the complexity of proofs. We consider proofs in Hilbertstyle formal systems such as in [17J. Thus a proof is a sequence of formulas satisfying certain conditions. We caD view the formulas as being strings of symbols; hence the whole proof is a string too. We consider the following measures of complexity of proofs: length ( = the number of symbols in the proof), depth ( = the maximal depth of a formula in the proof) and number o! steps ( = the number of formulas in the proof). For a particular formaI system and a given formula A we consider the shortest length of a proof of A, the minimal depth ofa proof of A and the minimal number of steps in a proof of A. The main results are the following: (1) a bound on the depth in terms of the number of steps: Theorem 2.2, (2) a bound on the depth in terms of the length: Theorem 2.3, (3) a bound on the length in terms of the number of steps for restricted systems: Theorem 3.1. These results are applied to obtain several corollaries. In particular we show: (1) a bound on the number of steps in a cutfree proof, (2) some speedup results, (3) bounds on the number of steps in proofs of ParisHarrington sentences. Some paper
Upper Bound on the Height of Terms in Proofs with Cuts
, 1998
"... We describe an upper bound on the heights of terms occurring in a most general unifier of a system of pairs of terms that contains unknowns of two types. An unknown belongs to the first type if all occurrences of this unknown have the same depth; we call such unknown an unknown of the cut type. ..."
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We describe an upper bound on the heights of terms occurring in a most general unifier of a system of pairs of terms that contains unknowns of two types. An unknown belongs to the first type if all occurrences of this unknown have the same depth; we call such unknown an unknown of the cut type. Unknowns of the second type (unknowns of not the cut type) are unknowns that have arbitrary occurrences. We bound from above the heights of terms occurring in a most general unifier in terms of the number of unknowns of not the cut type and of the height of the system. This bound yields an upper bound on the sizes of proofs in the Gentzen sequent calculus LK. Namely, we show that one can transform a proof D in LK by substituting some free terms in places of variables in such a way that the heights of terms occurring in the proof may be bounded from above by ar [D] h 1 \Delta q \Gamma [D] \Delta h 0 , where ar [D] is the maximal arity of function symbols occurring in D, h 1 is the...
Linear lower bounds and simulations in Frege systems with substitutions
, 1997
"... . We investigate the complexity of proofs in Frege (F), Substitution Frege (sF) and Renaming Frege (rF) systems. Starting from a recent work of Urquhart and using Kolmogorov Complexity we give a more general framework to obtain superlogarithmic lower bounds for the number of lines in both treelike ..."
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. We investigate the complexity of proofs in Frege (F), Substitution Frege (sF) and Renaming Frege (rF) systems. Starting from a recent work of Urquhart and using Kolmogorov Complexity we give a more general framework to obtain superlogarithmic lower bounds for the number of lines in both treelike and daglike sF. We show the previous known lower bound, extend it to the treelike case and, for another class of tautologies, we give new lower bounds that in the daglike case slightly improve the previous one. Also we show optimality of Urquhart's lower bounds giving optimal proofs. Finally we give the following two simulation results: (1) treelike sF psimulates daglike sF; (2) Treelike F psimulates treelike rF . 1 Introduction Since the work of Cook and Reckhow [CR], the study of complexity of proofs in propositional logic is viewed as related to main questions like NP 6= coNP in Complexity Theory. The main open problem is whether for all propositional proof systems th...