# Linear lower bounds and simulations in frege systems with substitutions

## Abstract

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 tree-like and dag-like *sF*. We show the previous known lower bound, extend it to the tree-like case and, for another class of tautologies, we give new lower bounds that in the dag-like 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) tree-like *sF p*-simulates dag-like *sF*; (2) Tree-like *F p*-simulates tree-like *rF*.

## Keywords

Proof System Binary String Propositional Variable Kolmogorov Complexity Random String## Preview

Unable to display preview. Download preview PDF.

## References

- [B-YGN]R. A. Baeza-Yates, R. Gavaldá, G. Navarro. Bounding the Expected Length of Longest Common Subsequences and Forest. Invited talk at the
*Third South American Workshop on String Processing*(WSP'96), Recife (Brazil), Aug. 8–9 1996.Google Scholar - [Bol]M. Bonet. Number of symbols in Frege proofs with and without the deduction rule. In
*Arithmetic, proof theory and computational complexity*. Oxford University Press Eds. P. Clote and J. Krajiček-1992.Google Scholar - [BB]M. Bonet, S. Buss. The deduction rule and linear and near-linear proof simulations.
*Journal of Symbol Logic*,**58**(1993) pp. 688–709.Google Scholar - [Bul]S. Buss. Some remarks on length of propositional proofs.
*Archive for Mathematical Logic*.**34**(1995) pp. 377–394.CrossRefGoogle Scholar - [Bu2]S. Buss,
*Lectures on proof theory*. TR SOCS-96.1 School of C.S.-Mc Gill University 1996.Google Scholar - [CR]S. Cook, R. Reckhow. The relative efficiency of propositional proof systems.
*Journal of Symbolic Logic*,**44**(1979) pp. 36–50.Google Scholar - [G]J. H. Gallier.
*Logic for Computer Science-Foundations of Automatic Theorem Proving*. J. Wiley & Sons. 1987Google Scholar - [HM]J. R. Hindley, D. Meredith. Principal type schemes and condensed detachment.
*Journal of Symbolic Logic*, 55 (1990) pp. 90–105.Google Scholar - [IP]K. Iwana, T. Pitassi. Exponential Lower bounds for the tree-like Hajós Calculus.
*Manuscript*1997.Google Scholar - [Krl]J. Krajiček. Speed-up for propositional Frege systems via generalizations of proofs.
*Commentationes Mathernaticae Universiatatis Carolinae*,**30**(1989) pp. 137–140.Google Scholar - [Kr2]J. Krajiček. On the number of steps in proof.
*Annals of Pure and Applied Logic*, 41 (1989) pp. 153–178.CrossRefGoogle Scholar - [KP]J. Krajiček, P. Pudlák. Propositional proof systems, the consistency of first order theories and the complexity of computation.
*Journal of Symbolic Logic*,**54**(1989) pp. 1063–1079ia.Google Scholar - [LV]M.Li, P. Vitanyi.
*An Introduction to Kolmogorov Complexity and its Application*. Springer-Verlag 1993.Google Scholar - [Or]V.P. Orevkov. Reconstruction of a proof from its scheme.
*Soviet Mathematics Doklady***35**(1987) pp. 326–329.Google Scholar - [Orl]V.P. Orevkov. On lower bounds on the lengths of proofs in propositional logic. (In Russian),
*in Proc, of All Union Conf. Metody Matem. v Problemach iskusstvennogo intellekta i sistematicheskoje programmirovanie*, Vilnius, Vol I. 1980. pp. 142–144.Google Scholar - [P]R. Parikh Some results on the length of proofs.
*Trans. A.M.S.*, 177 (1973) pp. 29–36.Google Scholar - [Pr]A. N. Prior.
*Formal Logic*. Oxford, second edition, 1960.Google Scholar - [PB]P. Pudlak, S. Buss. How to he without being (easily) convicted and the length of proofs in propositional calculus.
*8th Workshop on CSL, Kazimierz, Poland, September 1994*, Springer Verlag LNCS n.995, 1995, pp. 151–162.Google Scholar - [Ur]A. Urquhart. The number of lines in Frege proof with substitution.
*Archive for Mathematical Logic*. To appear.Google Scholar