Cook–Reckhow proof system

Cook–Reckhow proof system is a formal framework introduced to study the efficiency of propositional proof systems in theoretical computer science. It was proposed by Stephen Cook and Robert Reckhow to formalize notions of proof verification related to NP and co-NP and to connect proof length with computational resources in settings influenced by Alan Turing, Kurt Gödel, and the P versus NP problem. The framework underlies research across institutions such as MIT, Princeton University, and Carnegie Mellon University and informs investigations at conferences like STOC and FOCS.

Definition and formalism

A Cook–Reckhow proof system is defined as a polynomial-time computable function f with domain Σ* and range consisting of all tautologies in propositional logic; for this definition see work by Stephen Cook and Robert Reckhow and expositions by researchers at University of California, Berkeley and Stanford University. The formalism requires that for every tautology φ there exists a string π such that f(π)=φ, reflecting ideas originating in Gödel's completeness theorem and the Cook–Levin theorem; verification of f(π)=φ must be feasible in time bounded by a polynomial in the length of φ, an approach that connects to the NP-completeness framework developed by Richard Karp and Michael Garey. The system frames proof length as the size of witnesses π, linking to classical results by Ladner and to later structural inquiries by researchers at University of Illinois Urbana–Champaign and University of Toronto.

Examples and instantiations

Concrete instantiations include the Frege system and the Extended Frege system, each studied by groups at Princeton University and Harvard University; other examples are Resolution, studied in work by Iliano Cervesato and research groups at Microsoft Research. Systems such as Cutting planes and Nullstellensatz provide algebraic instances explored in collaborations with scholars at Columbia University and ETH Zurich, while bounded-depth variants like AC0-Frege are examined by teams at IBM Research and University of Chicago. The Extended Frege system relates to conjectures about efficient proof search discussed by researchers affiliated with Yale University and Cornell University.

Complexity and simulations

The Cook–Reckhow framework formalizes simulations between systems: a system P p-simulates Q if proofs in Q can be transformed into P with polynomial overhead, a notion analyzed in seminars at Massachusetts Institute of Technology and conferences such as ICALP. Simulation results compare systems like Resolution, Frege system, and Extended Frege system; complexity-theoretic implications draw on work by László Babai, Manindra Agrawal, and scholars from University of Toronto and University of Cambridge. The relationship between proof systems and complexity classes such as NP, co-NP, and PSPACE is central, with reductions and completeness proofs echoing techniques developed by Richard Lipton, Noam Nisan, and teams at University of Waterloo.

Upper and lower bounds

Upper bounds in the Cook–Reckhow setting include polynomial-size proofs for certain tautology families in Extended Frege, results pursued by researchers at Princeton University and University of California, San Diego; lower bounds, notably superpolynomial or exponential separations, are major open problems with partial results by groups including Alexander Razborov, Sanjeev Arora, and researchers at DIMACS. Notable lower-bound techniques involve feasible interpolation and communication complexity arguments developed with contributions from Eyal Kushilevitz and Noam Nisan and pursued at Tel Aviv University and Weizmann Institute of Science. Hardness results for Resolution and bounded-depth Frege have been established in work by Jan Krajíček and Avi Wigderson, with connections to circuit lower bounds explored by Ryan Williams at Stanford University.

Variants and extensions

Extensions of the Cook–Reckhow notion include non-uniform or advice-augmented proof systems studied at University of California, Los Angeles and oracle-based variants influenced by Bennett and Gill-style oracle results; cryptographic flavors relate to conjectures investigated by researchers at ETH Zurich and Tel Aviv University. Other variants introduce size-measure refinements, automatizability considerations, and average-case analyses pursued by teams at Microsoft Research and Google Research, while connections to bounded arithmetic theories such as S_1^2 and T_2 are developed by scholars at University of Chicago and Rutgers University. Cross-disciplinary extensions touch on proof complexity applications in formal verification at Carnegie Mellon University and satisfiability-solving heuristics explored at University of Oxford.

Category:Proof complexity