The complexity of theorem-proving procedures

  title={The complexity of theorem-proving procedures},
  author={Stephen A. Cook},
  journal={Proceedings of the third annual ACM symposium on Theory of computing},
  • S. Cook
  • Published 3 May 1971
  • Mathematics, Computer Science
  • Proceedings of the third annual ACM symposium on Theory of computing
It is shown that any recognition problem solved by a polynomial time-bounded nondeterministic Turing machine can be “reduced” to the problem of determining whether a given propositional formula is a tautology. Here “reduced” means, roughly speaking, that the first problem can be solved deterministically in polynomial time provided an oracle is available for solving the second. From this notion of reducible, polynomial degrees of difficulty are defined, and it is shown that the problem of… 
Complete problems for deterministic polynomial time
Several problems which are complete for P, the class of languages recognizable in deterministic polynomial time, are introduced, to serve to differentiate those sets in P which are not recognizable in logarithmic space from those which are, providing such differentiation is possible.
P = NP
It is proved that any algorithm that solves the SUBSET-SUM problem must perform a super-polynomial number of computations for some input of size, O(n), where n is the size of the set in the problem.
Complexity of solvable cases of the decision problem for the predicate calculus
  • H. R. Lewis
  • Mathematics, Computer Science
    19th Annual Symposium on Foundations of Computer Science (sfcs 1978)
  • 1978
We analyze the computational complexity of determining whether F is satisfiable when F is a formula of the classical predicate calculus which obeys certain syntactic restrictions. For example, for
The complexity of satisfiability problems
An infinite class of satisfiability problems is considered which contains these two particular problems as special cases, and it is shown that every member of this class is either polynomial-time decidable or NP-complete.
A Solution of the P versus NP Problem
The relationship between the complexity classes P and NP is a question that has not yet been answered by the Theory of Computation. The existence of a language in NP, proven not to belong to P, is
New problems complete for nondeterministic log space
Several problems are shown equivalent to reachability in undirected graphs, including bipartiteness and satisfiability of formulas with the “exclusive or” connective.
The Complexity of Propositional Proofs
  • A. Urquhart
  • Computer Science, Mathematics
    Bulletin of Symbolic Logic
  • 1995
A survey of the field, and of some of the techniques that have proved successful in deriving lower bounds on the complexity of proofs, of the classical propositional calculus and its relation to bounded arithmetic.
The Relative Efficiency of Propositional Proof Systems
All standard Hilbert type systems and natural deduction systems are equivalent, up to application of a polynomial, as far as minimum proof length goes, and extended Frege systems are introduced, which allow introduction of abbreviations for formulas.
A Mechanical Proof of the Cook-Levin Theorem
This paper presents a formal proof of the correctness of the translation of the Cook-Levin theorem, and the proof is verified with the theorem prover ACL2.


A Computing Procedure for Quantification Theory
In the present paper, a uniform proof procedure for quantification theory is given which is feasible for use with some rather complicated formulas and which does not ordinarily lead to exponentiation.
Formal languages and their relation to automata
  • J. Hopcroft, J. Ullman
  • Computer Science
    Addison-Wesley series in computer science and information processing
  • 1969
The theory of formal languages as a coherent theory is presented and its relationship to automata theory is made explicit, including the Turing machine and certain advanced topics in language theory.
Characterizations of Pushdown Machines in Terms of Time-Bounded Computers
A class of machines called auxiliary pushdown machines is introduced, characterized in terms of time-bounded Turing machines, and corollaries are derived which answer some open questions in the field.
An Efficient Algorithm for Graph Isomorphism
It is shown that the re ordered graphs form a sufficiency condition for isomorphism; namely, if the reordered graphs are identical, then the given graphs are isomorphic.
Bennett: On Spectra
  • Doctoral Dissertation, Princeton University,
  • 1962
Proc. of the 1964 International Congress for Logic
  • Proc. of the 1964 International Congress for Logic
Predictably Computable Functionals and Definitions by Recursion. Zeitschrift fHr math
  • Logik und Grundlagen der Math
  • 1964
Predictably Computable Functionals and Definitions by Recursion. Zeitschrift für math
  • Logik und Grundlagen der Math
  • 1964