Lower Bounds for (Non-Monotone) Comparator Circuits

Comparator circuits are a natural circuit model for studying the concept of bounded fan-out computations, which intuitively corresponds to whether or not a computational model can make "copies" of intermediate computational steps. Comparator circuits are believed to be weaker than general Boolean circuits, but they can simulate Branching Programs and Boolean formulas. In this paper we prove the first superlinear lower bounds in the general (non-monotone) version of this model for an explicitly defined function. More precisely, we prove that the n-bit Element Distinctness function requires ?((n/ log n)^(3/2)) size comparator circuits

Colourful TFNP and Propositional Proofs

Recent work has shown that many of the standard TFNP classes - such as PLS, PPADS, PPAD, SOPL, and EOPL - have corresponding proof systems in propositional proof complexity, in the sense that a total search problem is in the class if and only if the totality of the problem can be efficiently proved by the corresponding proof system. We build on this line of work by studying coloured variants of these TFNP classes: C-PLS, C-PPADS, C-PPAD, C-SOPL, and C-EOPL. While C-PLS has been studied in the literature before, the coloured variants of the other classes are introduced here for the first time. We give a family of results showing that these coloured TFNP classes are natural objects of study, and that the correspondence between TFNP and natural propositional proof systems is not an exceptional phenomenon isolated to weak TFNP classes. Namely, we show that: - Each of the classes C-PLS, C-PPADS, and C-SOPL have corresponding proof systems characterizing them. Specifically, the proof systems for these classes are obtained by adding depth to the formulas in the corresponding proof system for the uncoloured class. For instance, while it was previously known that PLS is characterized by bounded-width Resolution (i.e. depth 0.5 Frege), we prove that C-PLS is characterized by depth-1.5 Frege (Res(polylog(n)). - The classes C-PPAD and C-EOPL coincide exactly with the uncoloured classes PPADS and SOPL, respectively. Thus, both of these classes also have corresponding proof systems: unary Sherali-Adams and Reversible Resolution, respectively. - Finally, we prove a coloured intersection theorem for the coloured sink classes, showing C-PLS ? C-PPADS = C-SOPL, generalizing the intersection theorem PLS ? PPADS = SOPL. However, while it is known in the uncoloured world that PLS ? PPAD = EOPL = CLS, we prove that this equality fails in the coloured world in the black-box setting. More precisely, we show that there is an oracle O such that C-PLS^O ? C-PPAD^O ? C-EOPL^O. To prove our results, we introduce an abstract multivalued proof system - the Blockwise Calculus - which may be of independent interest

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

Extremely Deep Proofs

We further the study of supercritical tradeoffs in proof and circuit complexity, which is a type of tradeoff between complexity parameters where restricting one complexity parameter forces another to exceed its worst-case upper bound. In particular, we prove a new family of supercritical tradeoffs between depth and size for Resolution, Res(k), and Cutting Planes proofs. For each of these proof systems we construct, for each c ? n^{1-?}, a formula with n^{O(c)} clauses and n variables that has a proof of size n^{O(c)} but in which any proof of size no more than roughly exponential in n^{1-?}/c must necessarily have depth ? n^c. By setting c = o(n^{1-?}) we therefore obtain exponential lower bounds on proof depth; this far exceeds the trivial worst-case upper bound of n. In doing so we give a simplified proof of a supercritical depth/width tradeoff for tree-like Resolution from [Alexander A. Razborov, 2016]. Finally, we outline several conjectures that would imply similar supercritical tradeoffs between size and depth in circuit complexity via lifting theorems

Intersection Classes in TFNP and Proof Complexity

A recent breakthrough in the theory of total NP search problems (TFNP) by Fearnley, Goldberg, Hollender, and Savani has shown that CLS = PLS ∩ PPAD, or, in other words, the class of problems reducible to gradient descent are exactly those problems in the intersection of the complexity classes PLS and PPAD. Since this result, two more intersection theorems have been discovered in this theory: EOPL = PLS ∩ PPAD and SOPL = PLS ∩ PPADS. It is natural to wonder if this exhausts the list of intersection classes in TFNP, or, if other intersections exist. In this work, we completely classify all intersection classes involved among the classical TFNP classes PLS, PPAD, and PPA, giving new complete problems for the newly-introduced intersections. Following the close links between the theory of TFNP and propositional proof complexity, we develop new proof systems - each of which is a generalization of the classical Resolution proof system - that characterize all of the classes, in the sense that a query total search problem is in the intersection class if and only if a tautology associated with the search problem has a short proof in the proof system. We complement these new characterizations with black-box separations between all of the newly introduced classes and prior classes, thus giving strong evidence that no further collapse occurs. Finally, we characterize arbitrary intersections and joins of the PPA_q classes for q ≥ 2 in terms of the Nullstellensatz proof systems

Average Case Lower Bounds for Monotone Switching Networks

An approximate computation of a function f : {0, 1} n → {0, 1} by a computaional model M is a computation in which M computes f correctly on the majority of the inputs (rather than on all inputs). Lower bounds for approximate computations are also known as average case hardness results. We obtain the first average case monotone depth lower bounds for a function in monotone P, tolerating errors that are asymptotically the best possible for monotone circuits. Specifically, we prove average case exponential lower bounds on the size of monotone switching networks for the GEN function. As a corollary, we establish that for every i, there are functions computed with no error in monotone NC i+1 , but that cannot be computed without large error by monotone circuits in NC i