Descriptive complexity of circuit-based counting classes

We study Boolean circuits as a representation of Boolean functions and consider different equivalence, audit, and enumeration problems. For a number of restricted sets of gate types (bases) we obtain efficient algorithms, while for all other gate types we show these problems are at least NP-hard.

Information Processing Letters, 1990

We use Karchmer and Wigderson's recent characterization of circuit depth in terms of communication complexity to design shallow Boolean circuits for the counting functions. We show that the MOD, counting function on n arguments can be computed by Boolean networks which contain negations and binary OR-and AND-gates in depth c logrn, where c A 2.881. This is an improvement over the obvious depth upper bound of 3 logan. We can also design circuits for the MOD, and MOD,, functions having depth 3.475 logan and 4.930 logan, respectively.

Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, 2014

We study algorithms for the satisfiability problem for quantified Boolean formulas (QBFs), and consequences of faster algorithms for circuit complexity. • We show that satisfiability of quantified 3-CNFs with m clauses, n variables, and two quantifier blocks (one existential block and one universal) can be solved deterministically in time 2 n−Ω(√ n) • poly(m). For the case of multiple quantifier blocks (alternations), we show that satisfiability of quantified CNFs of size poly(n) on n variables with q quantifier blocks can be solved in 2 n−n 1/(q+1) • poly(n) time by a zero-error randomized algorithm. These are the first provable improvements over brute force search in the general case, even for quantified polynomial-sized CNFs with two quantifier blocks. A second zero-error randomized algorithm solves QBF on circuits of size s in 2 n−Ω(q) • poly(s) time when the number of quantifier blocks is q. • We complement these algorithms by showing that improvements on them would imply new circuit complexity lower bounds. For example, if satisfiability of quantified CNF formulas with n variables, poly(n) size and at most q quantifier blocks can be solved in time 2 n−n ωq (1/q) , then the complexity class NEXP does not have O(log n) depth circuits of polynomial size. Furthermore, solving satisfiability of quantified CNF formulas with n variables, poly(n) size and O(log n) quantifier blocks in time 2 n−ω(log(n)) time would imply the same circuit complexity lower bound. The proofs of these results proceed by establishing strong relationships between the time complexity of QBF satisfiability over CNF formulas and the time complexity of QBF satisfiability over arbitrary Boolean formulas.

2006

2010

The class NC 1 of problems solvable by bounded fan-in circuit families of logarithmic depth is known to be contained in logarithmic space L, but not much about the converse is known. In this paper we examine the structure of classes in between NC 1 and L based on counting functions or, equivalently, based on arithmetic circuits. The classes PNC 1 and C = NC 1 , defined by a test for positivity and a test for zero, respectively, of arithmetic circuit families of logarithmic depth, sit in this complexity interval. We study the landscape of Boolean hierarchies, constant-depth oracle hierarchies, and logarithmic-depth oracle hierarchies over PNC 1 and C = NC 1. We provide complete problems, obtain the upper bound L for all these hierarchies, and prove partial hierarchy collapses. In particular, the constant-depth oracle hierarchy over PNC 1 collapses to its first level PNC 1 , and the constant-depth oracle hierarchy over C = NC 1 collapses to its second level.

2002

Rice's Theorem states that all nontrivial language properties of recursively enumerable sets are undecidable. Borchert and Stephan (BSOO) started the search for complexity-theoretic analogs of Rice's Theorem, and proved that every nontrivial counting property of boolean circuits is UP-hard. Hemaspaandra and Rothe (HROO] improved the UP-hardness lower bound to UPocwhardness. The present paper raises the lower bound for nontrivial counting properties from UP O(l)-hardness to FewPhardness, i.e., from constant-ambiguity nondeterminism to polynomialambiguity nondeterminism. Furthermore, we prove that this lower bound is rather tight with respect to relativizable techniques, i.e., no relativizable technique can raise this lower bound to FewP-:s;f-tt-hardness. We also prove a Rice-style theorem for NP, namely that every nontrivial language property of NP sets is NP-hard.

2007

The classes of the W-hierarchy are the most important classes of intractable problems in parameterized complexity. These classes were originally defined via the weighted satisfiability problem for Boolean circuits. The analysis of the parameterized majority vertex cover problem and other parameterized problems led us to study circuits that contain connectives such as majority, not-all-equal, and unique, instead of (or in addition to) the Boolean connectives. For example, a gate labelled by the majority connective outputs TRUE if more than half of the inputs are TRUE. For any finite set C of connectives we construct the corresponding W(C)-hierarchy. We derive some general conditions which guarantee that the W-hierarchy and the W(C)-hierarchy coincide levelwise. Surprisingly, if C contains only the majority connective (i.e., no boolean connectives), then the first levels coincide. We use this to show that the majority vertex cover problem is W[1]-complete.

2008

In this thesis, we present some results in computational complexity. We consider two approaches for showing that #P has polynomial-size circuits. These approaches use ideas from the interactive proof for #3-SAT. We show that these approaches fail. We discuss whether there are instance checkers for languages complete for the class of approximate counting problems. We provide evidence that such instance checkers do not exist. We discuss the extent to which proofs of hierarchy theorems are constructive. We examine the problems that arise when trying to make the proof of Fortnow and Santhanam’s nonuniform BPP hierarchy theorem more constructive. ii Acknowledgements First, I would like to thank my supervisor, Charles Rackoff. Working with Charlie has been an intellectually stimulating and enjoyable experience. I greatly appreciate the many hours that Charlie spent explaining new concepts to me and suggesting ideas for this thesis. I would like to thank my second reader, Stephen Cook, for...

We use Karchmer and Wigderson's recent characterization of circuit depth in terms of communication complexity to design shallow Boolean circuits for the counting functions.

Computational Complexity, 1995

Vie consider planar circuits, formulas and multilective planar circuits. It is shown that planar circuits and formulas are incomparable. An ~(n log n) lower bound is given for the multilective planar circuit complexity of a decision problem and an 12(n 3/2) lower bound is given for the multilective planar circuit complexity of a multiple output function.

2007

By introducing a parallel extension rule that is aware of independence of the introduced extension variables, a calculus for quantified propositional logic is obtained where heights of derivations correspond to heights of appropriate circuits. Adding an uninterpreted predicate on bit-strings (analog to an oracle in relativised complexity classes) this statement can be made precise in the sense that the height of the most shallow proof that a circuit can be evaluated is, up to an additive constant, the height of that circuit. The main tool for showing lower bounds on proof heights is a variant of an iteration principle studied by Takeuti. This reformulation might be of independent interest, as it allows for polynomial size formulae in the relativised language that require proofs of exponential height.

1997

1986

A function of boolean arguments is symmetric if its value depends solely on the number of l's among its arguments. In the first part of this paper we partially characterize those symmetric functions that can be computed by constant-depth polynomial-size sequences of boolean circuits, and discuss the complete characterization. (We treat both uniform and non-uniform sequences of circuits.) Our results imply that these circuits can compute functions that are not definable in first-order logic. In the second part of the paper we generalize from circuits computing symmetric functions to circuits recognizing first-order structures. By imposing fairly natural restrictions we develop a circuit model with precisely the power of first-order logic: a class of structures is first-order definable if and only if it can be recognized by a constant-depth polynomial-time sequence of such circuits. © 1986 Academic Press, Inc.

We examine the complexity of the formula value problem for Boolean formulas, which is the following decision problem: Given a Boolean formula without variables, does it evaluate to true? We show that the complexity of this problem is determined by certain closure properties of the connectives allowed to build the formula, and achieve a complete classification: The formula value problem is either in LOGTIME, complete for one of the classes NLOGTIME, coNLOGTIME or NC 1 , or equivalent to counting modulo 2 under very strict reductions.

Information Processing Letters, 2008

Propositional circumscription, asking for the minimal models of a Boolean formula, is an important problem in artificial intelligence, in data mining, in coding theory, and in the model checking based procedures in automated reasoning. We consider the counting problems of propositional circumscription for several subclasses with respect to the structure of the formula. We prove that the counting problem of propositional circumscription for dual Horn, bijunctive, and affine formulas is #P-complete for a particular case of Turing reduction, whereas for Horn and 2affine formulas it is in FP. As a corollary, we obtain also the #P-completeness result for the counting problem of hypergraph transversal. * É quipe de Logique Mathématique -CNRS UMR 7056, Université Denis Diderot -Paris 7, UFR de mathmatiques case 7012, site Chevaleret 75205 Paris Cedex 13 -France.