Michael Alekhnovich, Eli Ben-Sasson, Alexander Razborov, Avi Wigderson

We study space complexity in the framework of

propositional proofs. We consider a natural model analogous to

Turing machines with a read-only input tape, and such

popular propositional proof systems as Resolution, Polynomial

Calculus and Frege systems. We propose two different space measures,

corresponding to the maximal number of bits, ...
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Michael Alekhnovich, Eli Ben-Sasson, Alexander Razborov, Avi Wigderson

We call a pseudorandom generator $G_n:\{0,1\}^n\to \{0,1\}^m$ {\em

hard} for a propositional proof system $P$ if $P$ can not efficiently

prove the (properly encoded) statement $G_n(x_1,\ldots,x_n)\neq b$ for

{\em any} string $b\in\{0,1\}^m$. We consider a variety of

``combinatorial'' pseudorandom generators inspired by the

Nisan-Wigderson generator on the one hand, and ...
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Dima Grigoriev, Edward Hirsch

We introduce two algebraic propositional proof systems F-NS

and F-PC. The main difference of our systems from (customary)

Nullstellensatz and Polynomial Calculus is that the polynomials

are represented as arbitrary formulas (rather than sums of

monomials). Short proofs of Tseitin's tautologies in the

constant-depth version of F-NS provide ...
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Ran Raz, Iddo Tzameret

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 >>>

Nicola Galesi, Massimo Lauria

We introduce an algebraic proof system Pcrk, which combines together {\em Polynomial Calculus} (Pc) and {\em $k$-DNF Resolution} (Resk).

This is a natural generalization to Resk of the well-known {\em Polynomial Calculus with Resolution} (Pcr) system which combines together Pc and Resolution.

We study the complexity of proofs in such ... more >>>

Massimo Lauria

We study the space required by Polynomial Calculus refutations of random $k$-CNFs. We are interested in how many monomials one needs to keep in memory to carry on a refutation. More precisely we show that for $k \geq 4$ a refutation of a random $k$-CNF of $\Delta n$ clauses and ... more >>>

Iddo Tzameret

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 >>>

Yuval Filmus, Massimo Lauria, Jakob Nordström, Noga Ron-Zewi, Neil Thapen

During the last decade, an active line of research in proof complexity has been to study space complexity and time-space trade-offs for proofs. Besides being a natural complexity measure of intrinsic interest, space is also an important issue in SAT solving, and so research has mostly focused on weak systems ... more >>>

Yuval Filmus, Massimo Lauria, Mladen Mikša, Jakob Nordström, Marc Vinyals

In 2003, Atserias and Dalmau resolved a major open question about the resolution proof system by establishing that the space complexity of CNF formulas is always an upper bound on the width needed to refute them. Their proof is beautiful but somewhat mysterious in that it relies heavily on tools ... more >>>

Albert Atserias, Massimo Lauria, Jakob Nordström

We prove that there are 3-CNF formulas over n variables that can be refuted in resolution in width w but require resolution proofs of size n^Omega(w). This shows that the simple counting argument that any formula refutable in width w must have a proof in size n^O(w) is essentially tight. ... more >>>

Ilario Bonacina, Nicola Galesi, Tony Huynh, Paul Wollan

We investigate the space complexity of refuting $3$-CNFs in Resolution and algebraic systems. No lower bound for refuting any family of $3$-CNFs was previously known for the total space in resolution or for the monomial space in algebraic systems. We prove that every Polynomial Calculus with Resolution refutation of a ... more >>>

Mladen Mikša, Jakob Nordström

We study the problem of obtaining lower bounds for polynomial calculus (PC) and polynomial calculus resolution (PCR) on proof degree, and hence by [Impagliazzo et al. '99] also on proof size. [Alekhnovich and Razborov '03] established that if the clause-variable incidence graph of a CNF formula F is a good ... more >>>

Christoph Berkholz

We relate different approaches for proving the unsatisfiability of a system of real polynomial equations over Boolean variables. On the one hand, there are the static proof systems Sherali-Adams and sum-of-squares (a.k.a. Lasserre), which are based on linear and semi-definite programming relaxations. On the other hand, we consider polynomial calculus, ... more >>>

Nicola Galesi, Leszek Kolodziejczyk, Neil Thapen

We show that if a $k$-CNF requires width $w$ to refute in resolution, then it requires space $\sqrt w$ to refute in polynomial calculus, where the space of a polynomial calculus refutation is the number of monomials that must be kept in memory when working through the proof. This is ... more >>>

Dmitry Sokolov

One of the major open problems in proof complexity is to prove lower bounds on $AC_0[p]$-Frege proof

systems. As a step toward this goal Impagliazzo, Mouli and Pitassi in a recent paper suggested to prove

lower bounds on the size for Polynomial Calculus over the $\{\pm 1\}$ basis. In this ...
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Erfan Khaniki

The refutation system ${Res}_R({PC}_d)$ is a natural extension of resolution refutation system such that it operates with disjunctions of degree $d$ polynomials over ring $R$ with boolean variables. For $d=1$, this system is called ${Res}_R({lin})$. Based on properties of $R$, ${Res}_R({lin})$ systems can be too strong to prove lower ... more >>>

Per Austrin, Kilian Risse

We study the complexity of proving that a sparse random regular graph on an odd number of vertices does not have a perfect matching, and related problems involving each vertex being matched some pre-specified number of times. We show that this requires proofs of degree $\Omega(n/\log n)$ in the Polynomial ... more >>>

tatsuie tsukiji

This paper aims to derandomize the following problems in the smoothed analysis of Spielman and Teng. Learn Disjunctive Normal Form (DNF), invert Fourier Transforms (FT), and verify small circuits' unsatisfiability. Learning algorithms must predict a future observation from the only $m$ i.i.d. samples of a fixed but unknown joint-distribution $P(G(x),y)$ ... more >>>

Ilario Bonacina, Nicola Galesi, Massimo Lauria

Vanishing sums of roots of unity can be seen as a natural generalization of knapsack from Boolean variables to variables taking values over the roots of unity. We show that these sums are hard to prove for polynomial calculus and for sum-of-squares, both in terms of degree and size.

more >>>Sam Buss, Noah Fleming, Russell Impagliazzo

Connections between proof complexity and circuit complexity have become major tools for obtaining lower bounds in both areas. These connections -- which take the form of interpolation theorems and query-to-communication lifting theorems -- translate efficient proofs into small circuits, and vice versa, allowing tools from one area to be applied ... more >>>

Yogesh Dahiya, Meena Mahajan, Sasank Mouli

In this paper, we obtain new size lower bounds for proofs in the

Polynomial Calculus (PC) proof system, in two different settings.

1. When the Boolean variables are encoded using $\pm 1$ (as opposed

to $0,1$): We establish a lifting theorem using an asymmetric gadget

$G$, showing ...
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Olaf Beyersdorff, Kaspar Kasche, Luc Nicolas Spachmann

We initiate an in-depth proof-complexity analysis of polynomial calculus (Q-PC) for Quantified Boolean Formulas (QBF). In the course of this we establish a tight proof-size characterisation of Q-PC in terms of a suitable circuit model (polynomial decision lists). Using this correspondence we show a size-degree relation for Q-PC, similar in ... more >>>

Sasank Mouli

For every $n >0$, we show the existence of a CNF tautology over $O(n^2)$ variables of width $O(\log n)$ such that it has a Polynomial Calculus Resolution refutation over $\{0,1\}$ variables of size $O(n^3polylog(n))$ but any Polynomial Calculus refutation over $\{+1,-1\}$ variables requires size $2^{\Omega(n)}$. This shows that Polynomial Calculus ... more >>>