Proofs and Games Course Material

Proof complexity is a vivid field in the intersection of logic and complexity that investigates proofs towards the main aim of understanding the complexity of theorem proving procedures. Most research in proof complexity concerns proving lower bounds to the lengths of proofs. The aim of this course is to present an up-to-date introduction to propositional proof complexity with emphasis on game-theoretic techniques, connections to parameterized complexity and DPLL algorithms. 1 Propositional Proof Complexity One of the starting points of propositional proof complexity is the seminal paper of Cook and Reckhow [17] where they formalized propositional proof systems as polynomial-time computable functions which have as their range the set of all propositional tautologies. In that paper, Cook and Reckhow also observed a fundamental connection between lengths of proofs and the separation of complexity classes: they showed that there exists a propositional proof system which has polynomial-size proofs for all tautologies (a polynomially bounded proof system) if and only if the class NP is closed under complementation. From this observation the so called Cook-Reckhow programme was derived which serves as one of the major motivations for propositional proof complexity: to separate NP from coNP (and hence P from NP) it suffices to show super-polynomial lower bounds to the size of proofs in all propositional proof systems. Although the first super-polynomial lower bound to the lengths of proofs had already been shown by Tseitin in the late 60’s for a sub-system of resolution [44], the first major achievement in this programme was made by Haken in 1985 when he showed an exponential lower bound to the proof size in Resolution for a sequence of propositional formulas describing the pigeonhole principle [25]. In the last two decades these lower bounds were extended to a number of further propositional systems such as the Nullstellensatz system [5], Cutting Planes [14, 35], Polynomial Calculus [16, 39], or bounded-depth Frege systems [1, 4, 6, 32]. For all these proof systems we know exponential lower bounds to the lengths of proofs for concrete sequences of tautologies arising mostly from natural propositional encodings of combinatorial statements. For proving these lower bounds, a number of generic approaches and general techniques have been developed. Most notably, there is the method of feasible interpolation developed by Kraj́ıček [28], the size-width trade-off introduced ★ Part of this ESSLLI course material is based on the surveys [8, 9] and the research papers [10–12]. Part of this work was done while the first author was visiting Sapienza University of Rome under support of grant N. 20517 by the John Templeton Foundation. by Ben-Sasson and Wigderson [7], and the use of pseudorandom generators in proof complexity [2, 29,30]. Despite this enormous success many questions still remain open. In particular Frege systems currently form a strong barrier [13], and all current lower bound methods seem to be insufficient for these strong systems. A detailed survey of recent advances in propositional proof complexity is contained in [43]. Let us mention that the separation of complexity classes is not the only motivation for studying lengths of proofs. In particular for strong systems like Frege and its extensions there is a fruitful connection to bounded arithmetic which adds insight to both subjects (cf. [27]). Further, understanding weak systems as Resolution is vital to applications as the design of efficient SAT solvers (see e. g. [34] for a more elaborate argument). Last not least, propositional proof complexity has over the years grown into a mature field and many researchers believe that understanding propositional proofs and proving lower bounds— arguably the hardest task in complexity—is a very important and beautiful field of logic which is justified in its own right. 2 Concepts from Classical Proof Complexity We start by introducing some of the relevant concepts and proof systems. Propositional proof systems were defined in a very general way by Cook and Reckhow in [17] as polynomial time functions P which have as its range the set of all tautologies. Definition 1 (Cook, Reckhow [17]). A propositional proof system is a polynomial time computable function P with rng(P ) = TAUT. A string with P ( ) = ' is called a P -proof of the tautology '. The intuition behind this definition is that given a proof it should be easy to determine which formula is actually proven and to verify the correctness of the proof. Nevertheless it might be difficult to generate proofs for a given formula and proofs might be very long compared to the size of the formula proven. Probably the simplest proof system is the truth-table system that proves formulas by checking all propositional assignments. In the sense of Definition 1 proofs in the truth-table system consist of the proven formula ' together with a string 12 ∣Var(')∣ . As most formulas require exactly exponential proof size in this system it is neither very interesting from the application oriented nor from the proof complexity perspective. But also all the usually studied proof systems are captured by the above definition. Let us illustrate this by an example. One of the most widely used proof systems is the resolution calculus and its variants introduced by Davis and Putnam [20] and Robinson [40]. Resolution is a refutation system that operates with clauses which are finite sets of negated or unnegated variables called literals. A clause is associated with the disjunction of the literals it contains and a set of clauses is associated with the conjunction of its clauses. Therefore finite sets of clauses correspond to propositional formulas in conjunctive normal form. A clause is satisfied by a propositional assignment if at least one literal of the clause is satisfied by the assignment. Therefore by definition the empty clause