Frege systems for quantified Boolean logic

We define and investigate Frege systems for quantified Boolean formulas (QBF). For these new proof systems, we develop a lower bound technique that directly lifts circuit lower bounds for a circuit class C to the QBF Frege system operating with lines from C. Such a direct transfer from circuit to proof complexity lower bounds has often been postulated for propositional systems but had not been formally established in such generality for any proof systems prior to this work. This leads to strong lower bounds for restricted versions of QBF Frege, in particular an exponential lower bound for QBF Frege systems operating with AC0[p] circuits. In contrast, any non-trivial lower bound for propositional AC0[p]-Frege constitutes a major open problem. Improving these lower bounds to unrestricted QBF Frege tightly corresponds to the major problems in circuit complexity and propositional proof complexity. In particular, proving a lower bound for QBF Frege systems operating with arbitrary P/poly circuits is equivalent to either showing a lower bound for P/poly or for propositional extended Frege (which operates with P/poly circuits). We also compare our new QBF Frege systems to standard sequent calculi for QBF and establish a correspondence to intuitionistic bounded arithmetic.This research was supported by grant nos. 48138 and 60842 from the John Templeton Foundation, EPSRC grant EP/L024233/1, and a Doctoral Prize Fellowship from EPSRC (third author). The second author was funded by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013)/ERC grant agreement no. 279611 and under the European Union’s Horizon 2020 Research and Innovation Programme/ERC grant agreement no. 648276 AUTAR. The fourth author was supported by the Austrian Science Fund (FWF) under project number P28699 and by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2014)/ERC Grant Agreement no. 61507. Part of this work was done when Beyersdorff and Pich were at the University of Leeds and Bonacina at Sapienza University Rome.Peer ReviewedPostprint (published version

Reordering Rule Makes OBDD Proof Systems Stronger

Atserias, Kolaitis, and Vardi showed that the proof system of Ordered Binary Decision Diagrams with conjunction and weakening, OBDD(^, weakening), simulates CP^* (Cutting Planes with unary coefficients). We show that OBDD(^, weakening) can give exponentially shorter proofs than dag-like cutting planes. This is proved by showing that the Clique-Coloring tautologies have polynomial size proofs in the OBDD(^, weakening) system. The reordering rule allows changing the variable order for OBDDs. We show that OBDD(^, weakening, reordering) is strictly stronger than OBDD(^, weakening). This is proved using the Clique-Coloring tautologies, and by transforming tautologies using coded permutations and orification. We also give CNF formulas which have polynomial size OBDD(^) proofs but require superpolynomial (actually, quasipolynomial size) resolution proofs, and thus we partially resolve an open question proposed by Groote and Zantema. Applying dag-like and tree-like lifting techniques to the mentioned results, we completely analyze which of the systems among CP^*, OBDD(^), OBDD(^, reordering), OBDD(^, weakening) and OBDD(^, weakening, reordering) polynomially simulate each other. For dag-like proof systems, some of our separations are quasipolynomial and some are exponential; for tree-like systems, all of our separations are exponential

Rank Lower Bounds in Propositional Proof Systems Based on Integer Linear Programming Methods

The work of this thesis is in the area of proof complexity, an area which looks to uncover the limitations of proof systems. In this thesis we investigate the rank complexity of tautologies for several of the most important proof systems based on integer linear programming methods. The three main contributions of this thesis are as follows: Firstly we develop the first rank lower bounds for the proof system based on the Sherali-Adams operator and show that both the Pigeonhole and Least Number Principles require linear rank in this system. We also demonstrate a link between the complexity measures of Sherali-Adams rank and Resolution width. Secondly we present a novel method for deriving rank lower bounds in the well-studied Cutting Planes proof system. We use this technique to show that the Cutting Plane rank of the Pigeonhole Principle is logarithmic. Finally we separate the complexity measures of Resolution width and Sherali-Adams rank from the complexity measures of Lovasz and Schrijver rank and Cutting Planes rank

Reasons for Hardness in QBF Proof Complexity

Quantified Boolean Formulas (QBF) extend the canonical NP-complete satisfiability problem by including Boolean quantifiers. Determining the truth of a QBF is PSPACE-complete; this is expected to be a harder problem than satisfiability, and hence QBF solving has much wider applications in practice. QBF proof complexity forms the theoretical basis for understanding QBF solving, as well as providing insights into more general complexity theory, but is less well understood than propositional proof complexity. We begin this thesis by looking at the reasons underlying QBF hardness, and in particular when the hardness is propositional in nature, rather than arising due to the quantifiers. We introduce relaxing QU-Res, a previous model for identifying such propositional hardness, and construct an example where relaxing QU-Res is unsuccessful in this regard. We then provide a new model for identifying such hardness which we prove captures this concept. Now equipped with a means of identifying ‘genuine’ QBF hardness, we prove a new lower bound technique for tree-like QBF proof systems. Lower bounds using this technique allows us to show a new separation between tree-like and dag-like systems. We give a characterisation of lower bounds for a large class of tree-like proof systems, in which such lower bounds play a prominent role. Further to the tree-like bound, we provide a new lower bound technique for QBF proof systems in general. This technique has some similarities to the above technique for tree-like systems, but requires some refinement to provide bounds for dag-like systems. We give applications of this new technique by proving lower bounds across several systems. The first such lower bounds are for a very simple family of QBFs. We then provide a construction to combine false QBFs to give formulas for which we can show lower bounds in this way, allowing the generation of the first random QBF proof complexity lower bounds

Monotone Proofs of the Pigeon Hole Principle

Lecture Notes in Computer Science. Geneva, Switzerland, July 9-15

Exponential Separations Using Guarded Extension Variables

We study the complexity of proof systems augmenting resolution with inference rules that allow, given a formula ? in conjunctive normal form, deriving clauses that are not necessarily logically implied by ? but whose addition to ? preserves satisfiability. When the derived clauses are allowed to introduce variables not occurring in ?, the systems we consider become equivalent to extended resolution. We are concerned with the versions of these systems without new variables. They are called BC?, RAT?, SBC?, and GER?, denoting respectively blocked clauses, resolution asymmetric tautologies, set-blocked clauses, and generalized extended resolution. Each of these systems formalizes some restricted version of the ability to make assumptions that hold "without loss of generality," which is commonly used informally to simplify or shorten proofs. Except for SBC?, these systems are known to be exponentially weaker than extended resolution. They are, however, all equivalent to it under a relaxed notion of simulation that allows the translation of the formula along with the proof when moving between proof systems. By taking advantage of this fact, we construct formulas that separate RAT? from GER? and vice versa. With the same strategy, we also separate SBC? from RAT?. Additionally, we give polynomial-size SBC? proofs of the pigeonhole principle, which separates SBC? from GER? by a previously known lower bound. These results also separate the three systems from BC? since they all simulate it. We thus give an almost complete picture of their relative strengths

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: 1. exponential lower bounds and separations for the QBF proof system ACo[p]-Frege+∀red for all primes p; 2. an exponential separation of ACo[p]-Frege+∀red from TCo/d-Frege+∀red; 3. 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

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