Propositional proof complexity is the study of the sizes of propositional proofs, and more generally, the resources necessary to certify propositional tautologies. Questions about proof sizes have connections with computational complexity, theories of arithmetic, and satisfiability algorithms. This is article includes a broad survey of the field, and a technical exposition of some recently developed techniques for proving lower bounds on proof sizes.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S1079898600001931

The complexity of propositional proofs

§1. Introduction. The classical propositional calculus has an undeserved reputation among logicians as being essentially trivial. I hope to convince the reader that it presents some of the most challenging and intriguing problems in modern logic. Although the problem of the complexity of propositional proofs is very natural, it has been investigated systematically only since the late 1960s. Interest in the problem arose from two fields connected with computers, automated theorem proving and computational complexity theory. The earliest paper in the subject is a ground-breaking article by Tseitin [62], the published version of a talk given in 1966 at a Leningrad seminar. In the three decades since that talk, substantial progress has been made in determining the relative complexity of proof systems, and in proving strong lower bounds for some restricted proof systems. However, major problems remain to challenge researchers. The present paper provides a survey of the field, and of some of the techniques that have proved successful in deriving lower bounds on the complexity of proofs. A major area only touched upon here is the proof theory of bounded arithmetic and its relation to the complexity of propositional proofs. The reader is referred to the book by Buss [10] for background in bounded arithmetic. The forthcoming book by Krajicek [40] also gives a good introduction to bounded arithmetic, as well as covering most of the basic results in complexity of propositional proofs.

We give the first nontrivial model-independent time?space tradeoffs for satisfiability. Namely, we show that SAT cannot be solved in n1+o(1) time and n1?? space for any ?>0 general random-access nondeterministic Turing machines. In particular, SAT cannot be solved deterministically by a Turing machine using quasilinear time and n space. We also give lower bounds for log-space uniform NC1 circuits and branching programs. Our proof uses two basic ideas. First we show that if SAT can be solved nondeterministically with a small amount of time then we can collapse a nonconstant number of levels of the polynomial-time hierarchy. We combine this work with a result of Nepomnja??ii that shows that a nondeterministic computation of superlinear time and sublinear space can be simulated in alternating linear time. A simple diagonalization yields our main result. We discuss how these bounds lead to a new approach to separating the complexity classes NL and NP. We give some possibilities and limitations of this approach.

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Time—Space Tradeoffs for Satisfiability

Complexity of Boolean algebras

Model theoretic aspects of computational complexity

A class of languages RUD derived from the class of rudimentary relations is studied. Two characterizations of RUD are established, one using linear-time relative computation and the other using language-theoretic operations. Also, some connections between RUD and classes of languages defined by resource-bounded Turing machines are given.

Celia Wrathall

Papers

Rudimentary Predicates and Relative Computation