Ilario Bonacina

Proofs of Space: When Space Is of the Essence

Proofs of computational effort were devised to control denial of service attacks. Dwork and Naor (CRYPTO ’92), for example, proposed to use such proofs to discourage spam. The idea is to couple each email message with a proof of work that demonstrates the sender performed some computational task. A proof of work can be either CPU-bound or memory-bound. In a CPU-bound proof, the prover must compute a CPU-intensive function that is easy to check by the verifier. A memory-bound proof, instead, forces the prover to access the main memory several times, effectively replacing CPU cycles with memory accesses.

Lower Bounds: From Circuits to QBF Proof Systems

A general and long-standing belief in the proof complexity community asserts that there is a close connection between progress in lower bounds for Boolean circuits and progress in proof size lower bounds for strong propositional proof systems. Although there are famous examples where a transfer from ideas and techniques from circuit complexity to proof complexity has been effective, a formal connection between the two areas has never been established so far. Here we provide such a formal relation between lower bounds for circuit classes and lower bounds for Frege systems for quantified Boolean formulas (QBF). Starting from a propositional proof system P we exhibit a general method how to obtain a QBF proof system P+∀red{P}, which is inspired by the transition from resolution to Q-resolution. For us the most important case is a new and natural hierarchy of QBF Frege systems C-Frege+∀red that parallels the well-studied propositional hierarchy of C-Frege systems, where lines in proofs are restricted to belong to a circuit class C. Building on earlier work for resolution [Beyersdorff, Chew and Janota, 2015a] we establish a lower bound technique via strategy extraction that transfers arbitrary lower bounds for the circuit class C to lower bounds in C-Frege+∀red. By using the full spectrum of state-of-the-art circuit lower bounds, our new lower bound method leads to very strong lower bounds for QBF \FREGE systems: exponential lower bounds and separations for the QBF proof system ACo[p]-Frege+∀red for all primes p; an exponential separation of ACo[p]-Frege+∀red from TCo/d-Frege+∀red; an exponential separation of the hierarchy of constant-depth systems ACo/d-Frege+∀red by formulas of depth independent of d. In the propositional case, all these results correspond to major open problems.

Pseudo-partitions, transversality and locality: a combinatorial characterization for the space measure in algebraic proof systems

We devise a new combinatorial framework for proving space lower bounds in algebraic proof systems like Polynomial Calculus (Pc) and Polynomial Calculus with Resolution (Pcr). Our method can be thought as a Spoiler-Duplicator game, which is capturing boolean reasoning on polynomials instead that clauses as in the case of Resolution. Hence, for the first time, we move the problem of studying the space complexity for algebraic proof systems in the range of 2-players games, as is the case for Resolution. A very simple case of our method allows us to obtain all the currently known space lower bounds for Pc/Pcr (CTn, PHPmn, BIT-PHPmn, XOR-PHPmn). The way our method applies to all these examples explains how and why all the known examples of space lower bounds for Pc/Pcr are an application of the method originally given by [Alekhnovich et al 2002] that holds for set of contradictory polynomials having high degree. Our approach unifies in a clear way under a common combinatorial framework and language the proofs of the space lower bounds known so far for Pc/Pcr. More importantly, using our approach in its full potentiality, we answer to the open problem [Alekhnovich et al 2002, Filmus et al 2012] of proving space lower bounds in Polynomial Calculus and Polynomials Calculus with Resolution for the polynomial encoding of randomly chosen k-CNF formulas. Our result holds for k>= 4. Then, as proved for Resolution in [BenSasson and Galesi 2003], also in Pc and in Pcr refuting a random k-CNF over n variables requires high space measure of the order of Omega(n). Our method also applies to the Graph-PHPmn, which is a PHPmn defined over a constant (left) degree bipartite expander graph. We develop a common language for the two examples.

A Framework for Space Complexity in Algebraic Proof Systems

Algebraic proof systems, such as Polynomial Calculus (PC) and Polynomial Calculus with Resolution (PCR), refute contradictions using polynomials. Space complexity for such systems measures the number of distinct monomials to be kept in memory while verifying a proof. We introduce a new combinatorial framework for proving space lower bounds in algebraic proof systems. As an immediate application, we obtain the space lower bounds previously provided for PC/PCR [Alekhnovich et al. 2002; Filmus et al. 2012]. More importantly, using our approach in its full potential, we prove Ω(n) space lower bounds in PC/PCR for random k-CNFs (k≥ 4) in n variables, thus solving an open problem posed in Alekhnovich et al. [2002] and Filmus et al. [2012]. Our method also applies to the Graph Pigeonhole Principle, which is a variant of the Pigeonhole Principle defined over a constant (left) degree expander graph.

Space proof complexity for random 3-CNFs☆

We investigate the space complexity of refuting

Total Space in Resolution

3

Strong ETH and Resolution via Games and the Multiplicity of Strategies

-CNFs in Resolution and algebraic systems. We prove that every Polynomial Calculus with Resolution refutation of a random

Improving resolution width lower bounds for k-CNFs with applications to the Strong Exponential Time Hypothesis

3

Lower bounds: from circuits to QBF proof systems.

-CNF

Space proof complexity for random

\phi