scholarly journals Algebraic methods in the theory of lower bounds for Boolean circuit complexity

1987 ◽  
Author(s):  
R. Smolensky
Keyword(s):  
Lower Bounds ◽  
Circuit Complexity ◽  
Algebraic Methods ◽  
Boolean Circuit

scholarly journals Computational and Proof Complexity of Partial String Avoidability

2021 ◽  
Vol 13 (1) ◽  
pp. 1-25
Author(s):  
Dmitry Itsykson ◽  
Alexander Okhotin ◽  
Vsevolod Oparin
Keyword(s):  
Lower Bounds ◽  
Circuit Complexity ◽  
Conjunctive Normal Form ◽  
Proof Complexity ◽  
Proof Systems ◽  
Finite Set ◽  
Propositional Proof Systems ◽  
Classical Proof ◽  
Exponential Size ◽  
Infinite String

The partial string avoidability problem is stated as follows: given a finite set of strings with possible “holes” (wildcard symbols), determine whether there exists a two-sided infinite string containing no substrings from this set, assuming that a hole matches every symbol. The problem is known to be NP-hard and in PSPACE, and this article establishes its PSPACE-completeness. Next, string avoidability over the binary alphabet is interpreted as a version of conjunctive normal form satisfiability problem, where each clause has infinitely many shifted variants. Non-satisfiability of these formulas can be proved using variants of classical propositional proof systems, augmented with derivation rules for shifting proof lines (such as clauses, inequalities, polynomials, etc.). First, it is proved that there is a particular formula that has a short refutation in Resolution with a shift rule but requires classical proofs of exponential size. At the same time, it is shown that exponential lower bounds for classical proof systems can be translated for their shifted versions. Finally, it is shown that superpolynomial lower bounds on the size of shifted proofs would separate NP from PSPACE; a connection to lower bounds on circuit complexity is also established.

New upper bounds on the Boolean circuit complexity of symmetric functions

2010 ◽  
Vol 110 (7) ◽  
pp. 264-267 ◽  
Author(s):  
E. Demenkov ◽  
A. Kojevnikov ◽  
A. Kulikov ◽  
G. Yaroslavtsev
Keyword(s):  
Symmetric Functions ◽  
Circuit Complexity ◽  
Upper Bounds ◽  
Boolean Circuit

scholarly journals Information Flows, Graphs and their Guessing Numbers

10.37236/962 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Søren Riis
Keyword(s):  
Network Coding ◽  
Circuit Complexity ◽  
Information Flows ◽  
Boolean Circuit ◽  
Graph Theoretic ◽  
Common Information ◽  
Shift Problem ◽  
Cyclic Shifts ◽  
The Common ◽  
Multiuser Information Theory

Valiant's shift problem asks whether all $n$ cyclic shifts on $n$ bits can be realized if $n^{(1+\epsilon)}$ input output pairs ($\epsilon < 1$) are directly connected and there are additionally $m$ common bits available that can be arbitrary functions of all the inputs. If it could be shown that this is not realizable with $m = O({n\over \log \log n})$ common bits then a significant breakthrough in Boolean circuit complexity would follow. In this paper it is shown that in certain cases all cyclic shifts are realizable with $m=(n- n^{\epsilon})/2$ common bits. Previously, no solution with $m < n - o(n)$ was known, and Valiant had conjectured that $m < n/2$ was not achievable. The construction therefore establishes a novel way of realizing communication in the manner of Network Coding for the shift problem, but leaves the viability of the common information approach to proving lower bounds in Circuit Complexity open. The construction uses the graph-theoretic notion of guessing number. As a by-product the paper also establish an interesting link between Circuit Complexity and Network Coding, a new direction of research in multiuser information theory.

scholarly journals Complexity measures in QFT and constrained geometric actions

2021 ◽  
Vol 2021 (9) ◽  
Author(s):  
Pablo Bueno ◽  
Javier M. Magán ◽  
C. S. Shahbazi
Keyword(s):  
Hilbert Space ◽  
Lower Bounds ◽  
Quantum Dynamics ◽  
Circuit Complexity ◽  
Coadjoint Orbit ◽  
Cost Functions ◽  
Complexity Measures ◽  
Minimal Geodesic ◽  
Quantum Action ◽  
Explicit Realization

Abstract We study the conditions under which, given a generic quantum system, complexity metrics provide actual lower bounds to the circuit complexity associated to a set of quantum gates. Inhomogeneous cost functions — many examples of which have been recently proposed in the literature — are ruled out by our analysis. Such measures are shown to be unrelated to circuit complexity in general and to produce severe violations of Lloyd’s bound in simple situations. Among the metrics which do provide lower bounds, the idea is to select those which produce the tightest possible ones. This establishes a hierarchy of cost functions and considerably reduces the list of candidate complexity measures. In particular, the criterion suggests a canonical way of dealing with penalties, consisting in assigning infinite costs to directions not belonging to the gate set. We discuss how this can be implemented through the use of Lagrange multipliers. We argue that one of the surviving cost functions defines a particularly canonical notion in the sense that: i) it straightforwardly follows from the standard Hermitian metric in Hilbert space; ii) its associated complexity functional is closely related to Kirillov’s coadjoint orbit action, providing an explicit realization of the “complexity equals action” idea; iii) it arises from a Hamilton-Jacobi analysis of the “quantum action” describing quantum dynamics in the phase space canonically associated to every Hilbert space. Finally, we explain how these structures provide a natural framework for characterizing chaos in classical and quantum systems on an equal footing, find the minimal geodesic connecting two nearby trajectories, and describe how complexity measures are sensitive to Lyapunov exponents.

scholarly journals Sunflower Theorems in Monotone Circuit Complexity

2021 ◽  
Author(s):  
Bruno Pasqualotto Cavalar ◽  
Yoshiharu Kohayakawa
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Circuit Complexity ◽  
Restricted Range ◽  
Significant Progress ◽  
Randomness Extractors ◽  
Monotone Circuits ◽  
Monotone Boolean Function ◽  
Sunflower Lemma ◽  
Clique Problem

Alexander Razborov (1985) developed the approximation method to obtain lower bounds on the size of monotone circuits deciding if a graph contains a clique. Given a "small" circuit, this technique consists in finding a monotone Boolean function which approximates the circuit in a distribution of interest, but makes computation errors in that same distribution. To prove that such a function is indeed a good approximation, Razborov used the sunflower lemma of Erd\H{o}s and Rado (1960). This technique was improved by Alon and Boppana (1987) to show lower bounds for a larger class of monotone computational problems. In that same work, the authors also improved the result of Razborov for the clique problem, using a relaxed variant of sunflowers. More recently, Rossman (2010) developed another variant of sunflowers, now called "robust sunflowers", to obtain lower bounds for the clique problem in random graphs. In the following years, the concept of robust sunflowers found applications in many areas of computational complexity, such as DNF sparsification, randomness extractors and lifting theorems. Even more recent was the breakthrough result of Alweiss, Lovett, Wu and Zhang (2020), which improved Rossman's bound on the size of hypergraphs without robust sunflowers. This result was employed to obtain a significant progress on the sunflower conjecture. In this work, we will show how the recent progress in sunflower theorems can be applied to improve monotone circuit lower bounds. In particular, we will show the best monotone circuit lower bound obtained up to now, breaking a 20-year old record of Harnik and Raz (2000). We will also improve the lower bound of Alon and Boppana for the clique function in a slightly more restricted range of clique sizes. Our exposition is self-contained. These results were obtained in a collaboration with Benjamin Rossman and Mrinal Kumar.

Guest Column

2021 ◽  
Vol 52 (4) ◽  
pp. 56-73
Author(s):  
Ben Volk
Keyword(s):  
Recent Work ◽  
Complexity Theory ◽  
Lower Bounds ◽  
Open Problem ◽  
Computational Models ◽  
Boolean Circuit ◽  
Geometric Complexity ◽  
Algebraic Version ◽  
Circuit Lower Bounds ◽  
Geometric Complexity Theory

Algebraic Natural Proofs is a recent framework which formalizes the type of reasoning used for proving most lower bounds on algebraic computational models. This concept is similar to and inspired by the famous natural proofs notion of Razborov and Rudich [RR97] for boolean circuit lower bounds, but, unlike in the boolean case, it is an open problem whether this constitutes a barrier for proving super-polynomial lower bounds for strong models of algebraic computation. From an algebraic-geometric viewpoint, it is also related to basic questions in Geometric Complexity Theory (GCT), and from a meta-complexity theoretic viewpoint, it can be seen as an algebraic version of the MCSP problem. We survey the recent work around this concept which provides some evidence both for and against the existence of an algebraic natural proofs barrier, with an emphasis on the di erent viewpoints and the connections to other areas.

Bounds on an exponential sum arising in Boolean circuit complexity

2005 ◽  
Vol 341 (5) ◽  
pp. 279-282 ◽  
Author(s):  
Frederic Green ◽  
Amitabha Roy ◽  
Howard Straubing
Keyword(s):  
Circuit Complexity ◽  
Exponential Sum ◽  
Boolean Circuit
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