Linear Lower Bounds and Simulations in Frege Systems with Substitutions


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.

DOI: 10.1007/BFb0028010

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@inproceedings{Bonet1997LinearLB, title={Linear Lower Bounds and Simulations in Frege Systems with Substitutions}, author={Maria Luisa Bonet and Nicola Galesi}, booktitle={CSL}, year={1997} }