Narrow Proofs May Be Maximally Long

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. Moreover, our lower bound generalizes to polynomial calculus resolution (PCR) and Sherali-Adams, implying that the corresponding size upper bounds in terms of degree and rank are tight as well. Our results do not extend all the way to Lasserre, however, where the formulas we study have proofs of constant rank and size polynomial in both n and w

Proof Complexity and Beyond

Proof complexity is a multi-disciplinary intellectual endeavor that addresses questions of the general form “how difficult is it to prove certain mathematical facts?” The current workshop focused on recent advances in our understanding of logic-based proof systems and on connections to algorithms, geometry and combinatorics research, such as the analysis of approximation algorithms, or the size of linear or semidefinite programming formulations of combinatorial optimization problems, to name just two important examples

Narrow proofs may be maximally long

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. Moreover, our lower bound generalizes to polynomial calculus resolution and Sherali-Adams, implying that the corresponding size upper bounds in terms of degree and rank are tight as well. The lower bound does not extend all the way to Lasserre, however, since we show that there the formulas we study have proofs of constant rank and size polynomial in both n and w.Peer ReviewedPostprint (author's final draft

Approximability and proof complexity

This work is concerned with the proof-complexity of certifying that optimization problems do \emph{not} have good solutions. Specifically we consider bounded-degree "Sum of Squares" (SOS) proofs, a powerful algebraic proof system introduced in 1999 by Grigoriev and Vorobjov. Work of Shor, Lasserre, and Parrilo shows that this proof system is automatizable using semidefinite programming (SDP), meaning that any $n$-variable degree-$d$ proof can be found in time $n^{O(d)}$. Furthermore, the SDP is dual to the well-known Lasserre SDP hierarchy, meaning that the "$d/2$-round Lasserre value" of an optimization problem is equal to the best bound provable using a degree-$d$ SOS proof. These ideas were exploited in a recent paper by Barak et al.\ (STOC 2012) which shows that the known "hard instances" for the Unique-Games problem are in fact solved close to optimally by a constant level of the Lasserre SDP hierarchy. We continue the study of the power of SOS proofs in the context of difficult optimization problems. In particular, we show that the Balanced-Separator integrality gap instances proposed by Devanur et al.\ can have their optimal value certified by a degree-4 SOS proof. The key ingredient is an SOS proof of the KKL Theorem. We also investigate the extent to which the Khot--Vishnoi Max-Cut integrality gap instances can have their optimum value certified by an SOS proof. We show they can be certified to within a factor .952 ($> .878$) using a constant-degree proof. These investigations also raise an interesting mathematical question: is there a constant-degree SOS proof of the Central Limit Theorem?Comment: 34 page

Size bounds for algebraic and semialgebraic proof systems

This thesis concerns the proof complexity of algebraic and semialgebraic proof systems Polynomial Calculus, Sums-of-Squares and Sherali-Adams. The most studied complexity measure for these systems is the degree of the proofs. This thesis concentrates on other possible complexity measures of interest to proof complexity, monomial-size and bit-complexity. We aim to showcase that there is a reasonably well-behaved theory for these measures also. Firstly we tie the complexity measures of degree and monomial size together by proving a size-degree trade-off for Sums-of-Squares and Sherali-Adams. We show that if there is a refutation with at most s many monomials, then there is a refutation whose degree is of order square root of n log s plus k, where k is the maximum degree of the constraints and n is the number of variables. For Polynomial Calculus similar trade-off was obtained earlier by Impagliazzo, Pudlák and Sgall. Secondly we prove a feasible interpolation property for all three systems. We show that for each system there is a polynomial time algorithm that given two sets P(x,z) and Q(y,z) of polynomial constraints in disjoint sequences x,y and z of variables, a refutation of the union of P(x,z) and Q(y,z), and an assignment a to the z-variables, finds either a refutation of P(x,a) or a refutation of Q(y,a). Finally we consider the relation between monomial-size and bit-complexity in Polynomial Calculus and Sums-of-Squares. We show that there is an unsatisfiable set of polynomial constraints that has both Polynomial Calculus and Sums-of-Squares refutations of polynomial monomial-size, but for which any Polynomial Calculus or Sums-of-Squares refutation requires exponential bit-complexity. Besides the emphasis on complexity measures other than degree, another unifying theme in all the three results is the use of semantic characterizations of resource-bounded proofs and refutations. All results make heavy use of the completeness properties of such characterizations. All in all, the work on these semantic characterizations presents itself as the fourth central contribution of this thesis.Aquesta tesi tracta de la complexitat de les proves en els sistemes de prova algebraics i semialgebraics Càlcul Polinomial (Polynomial Calculus), Sumes de Quadrats (Sums of Squares), i Sherali-Adams. La mesura de complexitat més estudiada per a aquests sistemes és el grau dels polinomis. Aquesta tesi se centra en altres possibles mesures de complexitat d'interès per a la complexitat de proves: el nombre de monomis i la longitud de representació en nombre de bits. Pretenem demostrar que aquestes mesures admeten una teoria comparable i complementària a la teoria del grau com a mesura de complexitat. En primer lloc, establim una relació entre les mesures de grau i de nombre de monomis demostrant una propietat d'intercanvi (trade-off) entre les dues mesures per als sistemes Sumes de Quadrats i Sherali-Adams. Demostrem que si hi ha una refutació amb com a màxim s monomis, aleshores hi ha una refutació el grau de la qual és d'ordre de l'arrel quadrada de n.log(s) més k, on k és el grau màxim de les restriccions i n és el nombre de variables. Per al Càlcul Polinomial, una propietat d'intercanvi similar va ser obtinguda per Impagliazzo, Pudlák i Sgall. En segon lloc, demostrem que els tres sistemes admeten la propietat d'interpolació eficient. Mostrem que, per a cadascun dels sistemes, hi ha un algorisme de temps polinomial que, donat dos conjunts P(x,z) i Q(y,z) de restriccions polinomials en successions disjuntes de variables x, y i z, donada una refutació de la unió de les restriccions de P(x,z) i Q(y,z), i donada una assignació per a les variables z, troba una refutació de P(x,a) o una refutació de Q(y,a). Finalment considerem la relació entre el nombre de monomis i la longitud de representació en bits per al Càlcul Polinomial i per a Sumes de Quadrats. Mostrem que hi ha un conjunt insatisfactible de restriccions polinomials que admet refutacions tant en Càlcul Polinomial com en Sumes de Quadrats amb un nombre polinòmic de monomis, però per a les quals qualsevol refutació en Càlcul Polinomial o en Sumes de Quadrats requereix complexitat en nombre de bits exponencial. A més de l'èmfasi en les mesures de complexitat diferents del grau, un altre tema unificador en els tres resultats és l'ús de certes caracteritzacions semàntiques de proves i refutacions limitades en recursos. Tots els resultats fan un ús clau de la propietat de completesa d'aquestes caracteritzacions. Amb tot, el treball sobre aquestes caracteritzacions semàntiques es presenta com la quarta aportació central d'aquesta tesi.Postprint (published version

Stabbing Planes

We introduce and develop a new semi-algebraic proof system, called Stabbing Planes that is in the style of DPLL-based modern SAT solvers. As with DPLL, there is only one rule: the current polytope can be subdivided by branching on an inequality and its "integer negation." That is, we can (nondeterministically choose) a hyperplane a x >= b with integer coefficients, which partitions the polytope into three pieces: the points in the polytope satisfying a x >= b, the points satisfying a x <= b-1, and the middle slab b-1 < a x < b. Since the middle slab contains no integer points it can be safely discarded, and the algorithm proceeds recursively on the other two branches. Each path terminates when the current polytope is empty, which is polynomial-time checkable. Among our results, we show somewhat surprisingly that Stabbing Planes can efficiently simulate Cutting Planes, and moreover, is strictly stronger than Cutting Planes under a reasonable conjecture. We prove linear lower bounds on the rank of Stabbing Planes refutations, by adapting a lifting argument in communication complexity

Automating Resolution is NP-Hard

We show that the problem of finding a Resolution refutation that is at most polynomially longer than a shortest one is NP-hard. In the parlance of proof complexity, Resolution is not automatizable unless P = NP. Indeed, we show it is NP-hard to distinguish between formulas that have Resolution refutations of polynomial length and those that do not have subexponential length refutations. This also implies that Resolution is not automatizable in subexponential time or quasi-polynomial time unless NP is included in SUBEXP or QP, respectively

A Finite-Model-Theoretic View on Propositional Proof Complexity

We establish new, and surprisingly tight, connections between propositional proof complexity and finite model theory. Specifically, we show that the power of several propositional proof systems, such as Horn resolution, bounded-width resolution, and the polynomial calculus of bounded degree, can be characterised in a precise sense by variants of fixed-point logics that are of fundamental importance in descriptive complexity theory. Our main results are that Horn resolution has the same expressive power as least fixed-point logic, that bounded-width resolution captures existential least fixed-point logic, and that the polynomial calculus with bounded degree over the rationals solves precisely the problems definable in fixed-point logic with counting. By exploring these connections further, we establish finite-model-theoretic tools for proving lower bounds for the polynomial calculus over the rationals and over finite fields