The Schwartz-Zippel Lemma states that if a low-degree multivariate polynomial with coefficients in a field is not zero everywhere in the field, then it has few roots on every finite subcube of the field. This fundamental fact about multivariate polynomials has found many applications in algorithms, complexity theory, coding theory, ... more >>>

Strong algebraic proof systems such as IPS (Ideal Proof System; Grochow-Pitassi JACM 2018) offer a general model for
deriving polynomials in an ideal and refuting unsatisfiable propositional formulas, subsuming most standard propositional proof systems. A major approach for lower bounding the size of IPS refutations is the Functional Lower Bound ... more >>>

We develop the theory of cryptographic nondeterministic-secure pseudorandomness beyond the point reached by Rudich's original work (Rudich 1997), and apply it to draw new consequences in average-case complexity and proof complexity. Specifically, we show the following:

?*Demi-bit stretch*: Super-bits and demi-bits are variants of cryptographic pseudorandom generators which are ... more >>>

We prove super-polynomial lower bounds on the size of propositional proof systems operating with constant-depth algebraic circuits over fields of zero characteristic. Specifically, we show that the subset-sum variant $\sum_{i,j,k,l\in[n]} z_{ijkl}x_ix_jx_kx_l-\beta = 0$, for Boolean variables, does not have polynomial-size IPS refutations where the refutations are multilinear and written as ... more >>>

We propose a diagonalization-based approach to several important questions in proof complexity. We illustrate this approach in the context of the algebraic proof system IPS and in the context of propositional proof systems more generally.

We give an explicit sequence of CNF formulas $\{\phi_n\}$ such that VNP$\neq$VP iff there are ... more >>>

We introduce the `binary value principle' which is a simple subset-sum instance expressing that a natural number written in binary cannot be negative, relating it to central problems in proof and algebraic complexity. We prove conditional superpolynomial lower bounds on the Ideal Proof System (IPS) refutation size of this instance, ... more >>>

Aiming to provide weak as possible axiomatic assumptions in which one can develop basic linear algebra, we give a uniform and integral version of the short propositional proofs for the determinant identities demonstrated over $GF(2)$ in Hrubes-Tzameret [SICOMP'15]. Specifically, we show that the multiplicativity of the determinant function and the ... more >>>

Resolution over linear equations (introduced in [RT08]) emerged recently as an important object of study. This refutation system, denoted Res(lin$_R$), operates with disjunction of linear equations over a ring $R$. On the one hand, the system captures a natural ``minimal'' extension of resolution in which efficient counting can be achieved; ... more >>>

We survey recent progress in the proof complexity of strong proof systems and its connection to algebraic circuit complexity, showing how the synergy between the two gives rise to new approaches to fundamental open questions, solutions to old problems, and new directions of research. In particular, we focus on tight ... more >>>

We give upper and lower bounds on the power of subsystems of the Ideal Proof System (IPS), the algebraic proof system recently proposed by Grochow and Pitassi, where the circuits comprising the proof come from various restricted algebraic circuit classes. This mimics an established research direction in the ... more >>>

Does every Boolean tautology have a short propositional-calculus proof? Here, a propositional-calculus (i.e., Frege) proof is any proof starting from a set of axioms and deriving new Boolean formulas using a fixed set of sound derivation rules. Establishing any super-polynomial size lower bound on Frege proofs (in terms of the ... more >>>

Motivated by the fundamental lower bounds questions in proof complexity, we investigate the complexity of generating identities of matrix rings, and related problems. Specifically, for a field $\mathbb{F}$ let $A$ be a non-commutative (associative) $\mathbb{F}$-algebra (e.g., the algebra Mat$_d(\mathbb{F})\;$ of $d\times d$ matrices over $\mathbb{F}$). We say that a non-commutative ... more >>>

We formalize a combinatorial principle, called the 3XOR principle, due to Feige, Kim and Ofek (2006), as a family of unsatisfiable propositional formulas for which refutations of small size in any propositional proof system that possesses the feasible interpolation property imply an efficient *deterministic* refutation algorithm for random 3SAT with ... more >>>

Can complexity classes be characterized in terms of efficient reducibility to the (undecidable) set of Kolmogorov-random strings? Although this might seem improbable, a series of papers has recently provided evidence that this may be the case. In particular, it is known that there is a class of problems $C$ defined ... more >>>

We study arithmetic proof systems $\mathbb{P}_c(\mathbb{F})$ and $\mathbb{P}_f(\mathbb{F})$ operating with arithmetic circuits and arithmetic formulas, respectively, that prove polynomial identities over a field $\mathbb{F}$. We establish a series of structural theorems about these proof systems, the main one stating that $\mathbb{P}_c(\mathbb{F})$ proofs can be balanced: if a polynomial identity of ... more >>>

Separating different propositional proof systems---that is, demonstrating that one proof system cannot efficiently simulate another proof system---is one of the main goals of proof complexity. Nevertheless, all known separation results between non-abstract proof systems are for specific families of hard tautologies: for what we know, in the average case all ... more >>>

We study possible formulations of algebraic propositional proof systems operating with noncommutative formulas. We observe that a simple formulation gives rise to systems at least as strong as Frege--yielding a semantic way to define a Cook-Reckhow (i.e., polynomially verifiable) algebraic analogue of Frege proofs, different from that given in Buss ... more >>>

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. ... more >>>

We introduce an algebraic proof system that manipulates 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 the following:

1. Algebraic proofs manipulating depth 2 multilinear arithmetic formulas polynomially simulate Resolution, Polynomial ... more >>>