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Robert A. Reckhow coauthored the 1979 paper establishing a general computational framework for propositional proof systems with Stephen Cook. Their formulation represents a proof system by a polynomial-time computable surjection onto the logical tautologies, separating verification cost from the length of the shortest proof.^[ref]

The paper compares systems using efficient translations of proofs. If proofs in one system can be converted to proofs in another by a polynomial-time procedure, the translated proofs can grow only polynomially in length. Such a comparison is stronger than observing that both systems eventually prove the same formulas: completeness alone places no useful bound on the cost of converting evidence. This gives proof complexity a way to compare proof formats as computational representations.^[ref]

Stephen Cook is a computer scientist whose 1971 paper established a foundational connection between the class NP and Boolean satisfiability. The Cook-Levin theorem is expressed in modern terminology by saying that Boolean satisfiability is NP-complete.^[ref]

The proof encodes the possible steps of a polynomially bounded Nondeterministic Turing machine as a Boolean formula. A satisfying assignment describes an accepting computation, and the formula has polynomial size. This makes polynomial-time reduction a way to compare apparently different computational tasks: an efficient general solver for the encoded problem would solve every problem in NP. Cook's broader work also connects mathematical logic with proof complexity.^[ref]

By Stephen Cook and Robert A. Reckhow. Journal of Symbolic Logic 44(1), 36–50, March 1979.

Author-hosted original: www.cs.toronto.edu/~sacook/homepage/cook_reckhow.pdf

Locator: §1, Definition 1.3 and pp.38–39 describe the polynomial-time surjective proof function and encoding invalid proofs as a fixed tautology.