Space Complexity in Propositional Calculus

We study space complexity in the framework of propositional proofs. We consider a natural model analogous to Turing machines with a read-only input tape and such popular propositional proof systems as resolution, polynomial calculus, and Frege systems. We propose two different space measures, corresponding to the maximal number of bits, and clauses/monomials that need to be kept in the memory simultaneously. We prove a number of lower and upper bounds in these models, as well as some structural results concerning the clause space for resolution and Frege systems.

[1]  Maria Luisa Bonet,et al.  A study of proof search algorithms for resolution and polynomial calculus , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[2]  Jacobo Torán.  Lower Bounds for Space in Resolution , 1999, CSL.

[3]  G. S. Tseitin.  On the Complexity of Derivation in Propositional Calculus , 1983 .

[4]  Jacobo Torán,et al.  Space Bounds for Resolution , 1999, STACS.

[5]  Nathan Linial,et al.  Minimal non-two-colorable hypergraphs and minimal unsatisfiable formulas , 1986, J. Comb. Theory, Ser. A.

[6]  Jan Krajícek,et al.  Interpolation theorems, lower bounds for proof systems, and independence results for bounded arithmetic , 1997, Journal of Symbolic Logic.

[7]  Alexander A. Razborov,et al.  Read-once branching programs, rectangular proofs of the pigeonhole principle and the transversal calculus , 1997, STOC '97.

[8]  Stephen A. Cook,et al.  The Relative Efficiency of Propositional Proof Systems , 1979, Journal of Symbolic Logic.

[9]  Jan Krajícek,et al.  Bounded arithmetic, propositional logic, and complexity theory , 1995, Encyclopedia of mathematics and its applications.

[10]  Russell Impagliazzo,et al.  Lower bounds for the polynomial calculus and the Gröbner basis algorithm , 1999, computational complexity.

[11]  Dexter Kozen,et al.  Lower bounds for natural proof systems , 1977, 18th Annual Symposium on Foundations of Computer Science (sfcs 1977).

[12]  Alasdair Urquhart,et al.  The Complexity of Propositional Proofs , 1995, Bulletin of Symbolic Logic.

[13]  Stephen Cook,et al.  Corrections for "On the lengths of proofs in the propositional calculus preliminary version" , 1974, SIGA.

[14]  Balakrishnan Krishnamurthy.  Short proofs for tricky formulas , 2004, Acta Informatica.

[15]  Toniann Pitassi,et al.  Propositional Proof Complexity: Past, Present and Future , 2001, Bull. EATCS.

[16]  Russell Impagliazzo,et al.  Using the Groebner basis algorithm to find proofs of unsatisfiability , 1996, STOC '96.