Tel Aviv University the Raymond and Beverly Sackler Faculty of Exact Sciences School of Computer Science Studies in Algebraic and Propositional Proof Complexity

The field of proof complexity aims at characterizing which statements have short proofs in a given formal proof system. This thesis is a contribution to proof complexity broadly construed as the field that studies the sizes of structured or symbolic proofs. Our focus will be on the development and complexity-theoretic study of new frameworks, mainly of an algebraic nature, for providing, among other things, proofs of propositional tautologies. We further link and motivate the proof systems we explore with certain questions, mainly from algebraic complexity. The main results of this thesis can be divided into four parts, as follows. MULTILINEAR PROOFS: We introduce an algebraic proof system that operates with multilinear arithmetic formulas. We show that this proof system is fairly strong, even when restricted to multilinear arithmetic formulas of a very small depth. Specifically, we show that algebraic proofs manipulating depth-2 multilinear arithmetic formulas polynomially simulate resolution, Polynomial Calculus (PC) and Polynomial Calculus with Resolution (PCR) proofs. We provide polynomial size proofs manipulating depth-3 multilinear arithmetic formulas for the pigeonhole principle tautologies and the Tseitin’s graph tautologies. By known lower bounds, this demonstrates that algebraic proof systems manipulating depth-3 multilinear formulas are strictly stronger than resolution, PC and PCR, and have an exponential gap over bounded-depth Frege for both the pigeonhole principle and Tseitin’s graph tautologies. We illustrate a connection between lower bounds on multilinear proofs and lower bounds on multilinear circuits. In particular, we show that (an explicit) super-polynomial size separation between proofs manipulating general arithmetic circuits and proofs manipulating multilinear circuits implies a super-polynomial size lower bound on multilinear circuits for an explicit family of polynomials. The short multilinear proofs for hard tautologies are established via a connection between depth-3 multilinear proofs and extensions of resolution, described as follows: RESOLUTION OVER LINEAR EQUATIONS WITH APPLICATIONS TO MULTILINEAR PROOFS: We develop and study the complexity of propositional proof systems of varying strength extending resolution by allowing it to operate with disjunctions of linear equations instead of clauses. We demonstrate polynomial-size refutations for hard tautologies like the pigeonhole principle, Tseitin graph tautologies and the clique-coloring tautologies in these proof systems. Using (monotone) interpolation we establish an exponential-size lower bound on refutations in a certain, strong, fragment of resolution over linear equations, as well as a general polynomial upper bound on (non-monotone) interpolants in this fragment. We show that proofs operating with depth-3 multilinear formulas polynomially simulate a certain, strong, fragment of resolution over linear equations (by which the aforementioned upper bounds on multilinear proofs follow). We then connect resolution over linear equations with extensions of the cutting planes proof system. SYMBOLIC PROOFS OF POLYNOMIAL IDENTITIES: To transform algebraic propositional proof systems operating with arithmetic formulas into formal proof systems one usually augments the system with an “auxiliary” proof system capable of manipulating arithmetic formulas by means of the polynomial-ring axioms. We investigate basic structural and complexity characterizations of the latter proof system and its fragments. Specifically, a symbolic proof for establishing that a given arithmetic formula Φ computes the zero polynomial (or equivalently, that two given arithmetic formulas compute the same polynomial) is a sequence of formulas, starting with Φ and deriving the formula 0 by means of the standard polynomial-ring axioms applied to any subformula. We introduce fragments of symbolic proofs named analytic symbolic proofs, enjoying a natural property: a symbolic proof is analytic if one cannot introduce arbitrary new formulas anywhere in the proof (that is, formulas computing the zero polynomial which do not originate, in a precise manner, from the initial arithmetic formula). We establish exponential lower bounds on the number of steps in analytic symbolic proofs operating with depth-3 arithmetic formulas, under a certain regularity condition on the structure of proofs (roughly, mimicking a tree-like structure). The hard instances are explicit and rely on small formulas for the symmetric polynomials. ALTERNATIVE MODELS OF REFUTATION – PROMISE PROPOSITIONAL PROOFS: We study the problem of certifying unsatisfiability of CNF formulas under the promise that any satisfiable formula has many satisfying assignments, where “many” stands for an explicitly specified function Λ in the number of variables n. To this end, we develop propositional proof systems under different measures of promises (that is, different Λ) as extensions of resolution. This is done by augmenting resolution with axioms that, roughly, can eliminate sets of truth assignments defined by Boolean circuits. We then investigate the complexity of such systems, obtaining an exponential separation in the average-case between resolution under different size promises: (i) Resolution has polynomial-size refutations for all unsatisfiable 3CNF formulas when the promise is ε ·2n, for any constant 0 < ε < 1; (ii) There are no sub-exponential size resolution refutations for random 3CNF formulas, when the promise is 2 (and the number of clauses isO(n3/2− ), for 0 < < 1 2 ), for any constant 0 < δ < 1.