The Complexity of the Descriptiveness of Boolean Circuits over Different Sets of Gates (original) (raw)

The Complexity of Boolean Functions in Different Characteristics

computational complexity, 2010

Every Boolean function on n variables can be expressed as a unique multivariate polynomial modulo p for every prime p. In this work, we study how the degree of a function in one characteristic affects its complexity in other characteristics. We establish the following general principle: functions with low degree modulo p must have high complexity in every other characteristic q. More precisely, we show the following results about Boolean functions f : {0, 1} n → {0, 1} which depend on all n variables, and distinct primes p, q:

Beating Exhaustive Search for Quantified Boolean Formulas and Connections to Circuit Complexity

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.

An upper bound for the circuit complexity of existentially quantified Boolean formulas

Theoretical Computer Science, 2010

The expressive power of existentially quantified Boolean formulas ∃CNF with free variables is investigated. We introduce a hierarchy of subclasses ∃MU * (k) of ∃CNF formulas based on the maximum deficiency k of minimal unsatisfiable subformulas of the bound part of the formulas. We will establish an upper bound of the size of minimally equivalent circuits. It will be shown, that there are constants a and b, such that for every formula in ∃MU * (k) of length m of the bound part and length l of the free part of the formula there is an equivalent circuit of size less than l + a • m b(log 2 (m)+k) 2 .

New upper bounds on the Boolean circuit complexity of symmetric functions

Information Processing Letters, 2010

In this note, we present improved upper bounds on the circuit complexity of symmetric Boolean functions. In particular, we describe circuits of size 4.5n + o(n) for any symmetric function of n variables, as well as circuits of size 3n for MOD n 3 function.

Data structures, minimization and complexity of boolean functions

1996

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The Complexity of the Boolean Formula Value Problem

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.