Results 1  10
of
17
Lower Bounds for Deterministic and Nondeterministic Branching Programs
 in Proceedings of the FCT'91, Lecture Notes in Computer Science
, 1991
"... We survey lower bounds established for the complexity of computing explicitly given Boolean functions by switchingandrectifier networks, branching programs and switching networks. We first consider the unrestricted case and then proceed to various restricted models. Among these are monotone networ ..."
Abstract

Cited by 57 (4 self)
 Add to MetaCart
We survey lower bounds established for the complexity of computing explicitly given Boolean functions by switchingandrectifier networks, branching programs and switching networks. We first consider the unrestricted case and then proceed to various restricted models. Among these are monotone networks, boundedwidth devices , oblivious devices and readk times only devices. 1 Introduction The main goal of the Boolean complexity theory is to prove lower bounds on the complexity of computing "explicitly given" Boolean functions in interesting computational models. By "explicitly given" researchers usually mean "belonging to the class NP ". This is a very plausible interpretation since on the one hand this class contains the overwhelming majority of interesting Boolean functions and on the other hand it is small enough to prevent us from the necessity to take into account counting arguments. To illustrate the second point, let me remind the reader that already the class \Delta p 2 ,...
Circuit Complexity before the Dawn of the New Millennium
, 1997
"... The 1980's saw rapid and exciting development of techniques for proving lower bounds in circuit complexity. This pace has slowed recently, and there has even been work indicating that quite different proof techniques must be employed to advance beyond the current frontier of circuit lower bounds. Al ..."
Abstract

Cited by 30 (3 self)
 Add to MetaCart
The 1980's saw rapid and exciting development of techniques for proving lower bounds in circuit complexity. This pace has slowed recently, and there has even been work indicating that quite different proof techniques must be employed to advance beyond the current frontier of circuit lower bounds. Although this has engendered pessimism in some quarters, there have in fact been many positive developments in the past few years showing that significant progress is possible on many fronts. This paper is a (necessarily incomplete) survey of the state of circuit complexity as we await the dawn of the new millennium.
Nonuniform ACC circuit lower bounds
, 2010
"... The class ACC consists of circuit families with constant depth over unbounded fanin AND, OR, NOT, and MODm gates, where m> 1 is an arbitrary constant. We prove: • NTIME[2 n] does not have nonuniform ACC circuits of polynomial size. The size lower bound can be slightly strengthened to quasipolynom ..."
Abstract

Cited by 16 (4 self)
 Add to MetaCart
The class ACC consists of circuit families with constant depth over unbounded fanin AND, OR, NOT, and MODm gates, where m> 1 is an arbitrary constant. We prove: • NTIME[2 n] does not have nonuniform ACC circuits of polynomial size. The size lower bound can be slightly strengthened to quasipolynomials and other less natural functions. • ENP, the class of languages recognized in 2O(n) time with an NP oracle, doesn’t have nonuniform ACC circuits of 2no(1) size. The lower bound gives an exponential sizedepth tradeoff: for every d there is a δ> 0 such that ENP doesn’t have depthd ACC circuits of size 2nδ. Previously, it was not known whether EXP NP had depth3 polynomial size circuits made out of only MOD6 gates. The highlevel strategy is to design faster algorithms for the circuit satisfiability problem over ACC circuits, then prove that such algorithms entail the above lower bounds. The algorithm combines known properties of ACC with fast rectangular matrix multiplication and dynamic programming, while the second step requires a subtle strengthening of the author’s prior work [STOC’10]. Supported by the Josef Raviv Memorial Fellowship.
Machine Models and Linear Time Complexity
 SIGACT News
, 1993
"... wer bounds. Machine models. Suppose that for every machine M 1 in model M 1 running in time t = t(n) there is a machine M 2 in M 2 which computes the same partial function in time g = g(t; n). If g = O(t)+O(n) we say that model M 2 simulates M 1 linearly. If g = O(t) the simulation has constantf ..."
Abstract

Cited by 5 (3 self)
 Add to MetaCart
wer bounds. Machine models. Suppose that for every machine M 1 in model M 1 running in time t = t(n) there is a machine M 2 in M 2 which computes the same partial function in time g = g(t; n). If g = O(t)+O(n) we say that model M 2 simulates M 1 linearly. If g = O(t) the simulation has constantfactor overhead ; if g = O(t log t) it has a factorofO(log t) overhead , and so on. The simulation is online if each step of M 1 i
Definability of Languages by Generalized FirstOrder Formulas over (N
 In 23rd Symp. on Theoretical Aspects of Comp. Sci. (STACS’06
, 2006
"... Abstract. We consider an extension of firstorder logic by modular quantifiers of a fixed modulus q. Drawing on collapse results from finite model theory and techniques of finite semigroup theory, we show that if the only available numerical predicate is addition, then sentences in this logic cannot ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
Abstract. We consider an extension of firstorder logic by modular quantifiers of a fixed modulus q. Drawing on collapse results from finite model theory and techniques of finite semigroup theory, we show that if the only available numerical predicate is addition, then sentences in this logic cannot define the set of bit strings in which the number of 1’s is divisible by a prime p that does not divide q. More generally, we completely characterize the regular languages definable in this logic. The corresponding statement, with addition replaced by arbitrary numerical predicates, is equivalent to the conjectured separation of the circuit complexity class ACC from NC 1. Thus our theorem can be viewed as proving a highly uniform version of the conjecture. 1
A note on a theorem of Barrington, Straubing and Thérien
, 1996
"... We show that the result of Barrington, Straubing and Thérien [5] provides, as a direct corollary, an exponential lower bound for the size of depthtwo MOD 6 circuits computing the AND function. This problem was solved, in a more general way, by Krause and Waack [8]. We point out that all known lower ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
We show that the result of Barrington, Straubing and Thérien [5] provides, as a direct corollary, an exponential lower bound for the size of depthtwo MOD 6 circuits computing the AND function. This problem was solved, in a more general way, by Krause and Waack [8]. We point out that all known lower bounds rely on the special form of the MOD 6 gate occurring at the bottom of the circuits, so that in fact, proving a lower bound for "general" MOD 6 circuits of depth two is still an open question.
Languages Defined With Modular Counting Quantifiers
 Information and Computation
, 2001
"... . We prove that a regular language defined by a boolean combination of generalized \Sigma 1sentences built using modular counting quantifiers can be defined by a boolean combination of \Sigma 1sentences in which only regular numerical predicates appear. The same statement, with "\Sigma 1 " replace ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
. We prove that a regular language defined by a boolean combination of generalized \Sigma 1sentences built using modular counting quantifiers can be defined by a boolean combination of \Sigma 1sentences in which only regular numerical predicates appear. The same statement, with "\Sigma 1 " replaced by "firstorder" is equivalent to the conjecture that the nonuniform circuit complexity class ACC is strictly contained in NC 1 : The argument introduces some new techniques, based on a combination of semigroup theory and Ramsey theory, which may shed some light on the general case. A preliminary version of this paper appeared in the Proceedings of the 1998 STACS conference. 1 Background 1.1 Lower bounds questions for smalldepth circuit families This paper was motivated by some open problems about the computational power of families of boolean circuits. As it turns out, we will not mention circuits at all after this introductory section. Nonetheless, our main result represents a pos...
Circuit Complexity and the Expressive Power of Generalized FirstOrder Formulas
"... . The circuit complexity classes AC 0 ; ACC; and CC (also called pureACC) can be characterized as the classes of languages definable in certain extensions of firstorder logic. All of the known and conjectured inclusions among these classes have been shown to be equivalent to a single conjecture ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
. The circuit complexity classes AC 0 ; ACC; and CC (also called pureACC) can be characterized as the classes of languages definable in certain extensions of firstorder logic. All of the known and conjectured inclusions among these classes have been shown to be equivalent to a single conjecture concerning the form of the formulas required to define the regular languages they contain. (The conjecture states, roughly, that when a formula defines a regular language, predicates representing numerical relations on the positions in a string can be replaced by predicates computed by finite state automata.) Here this conjecture is established in a special case: It is shown that the conjecture holds for the subclasses of AC 0 ; ACC; and CC defined by restricting all the numerical predicates occurring in the logical formulas to be either unary relations, or the order relation ! : 1 Introduction Certain formulas of predicate logic can be interpreted in a natural way in strings over a fini...
An algebraic point of view on the cranebeach conjecture, 2006. Document in preparation
"... Abstract. A letter e ∈ Σ is said to be neutral for a language L if it can be inserted and deleted at will in a word without affecting membership in L. The CraneBeach Conjecture, which was recently disproved, stated that any language containing a neutral letter and definable in FO is in fact FO[<] d ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Abstract. A letter e ∈ Σ is said to be neutral for a language L if it can be inserted and deleted at will in a word without affecting membership in L. The CraneBeach Conjecture, which was recently disproved, stated that any language containing a neutral letter and definable in FO is in fact FO[<] definable and is thus a regular, starfree language. More generally, we say that a logic or a computational model has the Crane Beach property if the only languages with neutral letter that it can define/compute are regular. We develop an algebraic point of view on the Crane Beach properties using the program over monoid formalism which has proved of importance in circuit complexity. Using recent communication complexity results we establish a number of Crane Beach results for programs over specific classes of monoids. These can be viewed as Crane Beach theorems for classes of boundedwidth branching programs. We also apply this to a standard extension of FO using modularcounting quantifiers and show that the boolean closure of this logic’s Σ1 fragment has the CBP. 1