David A. Mix Barrington

• Ph.D., M.I.T. 1986
• University of Massachusetts Amherst

In this corrigendum, we retract part of our Corollary 6.6, which was presented as an immediate and obvious consequence of our main theorem, which showed that division lies in Dlogtime-uniform TC0TC0.

We study the complexity of restricted versions ofs-t-connectivity, which is the standard complete problem forNL. In particular, we focus on different classes ofplanar graphs, of which grid graphs are an important special case. Our main results are: • Reachability in graphs of genus one is logspace-equivalent to reachability in grid graphs (and in p...

Continuing the study of the relationship between TC Continuing the study of the relationship between TC 0, AC 0, AC 0 and arithmetic circuits, started by Agrawal et al. [1], we answer a few questions left open in this paper. Our main result 0 and arithmetic circuits, started by Agrawal et al. [1], we answer a few questions left open in this paper....

In this paper we study the power of constant-depth circuits containing negation gates, unobunded fan-in AND and OR gates, and a small number of MAJORITY gates. It is easy to show that a depth 2 circuit of size O(n) (where n is the number of inputs) containing O(n) MAJORITY gates can determine whether the sum of the input bits is divisible by k, for...

We prove that constant depth circuits, with one layer of MOD m gates at the inputs, followed by a fixed number of layers of MOD p gates, where p is prime, require exponential size to compute the MOD q function, if q is a prime that divides neither p nor q.

We define the notion of an oracle branching program in order to investigate space-bounded computation. Within this new framework we examine the P-complete problem GEN which consists of determining membership in a subalgebra of a general (not necessarily associative) binary algebra (input as a multiplication table). Our work begins with the statemen...

A new model, non-uniform deterministic finite automata (NUDFA's) over general finite monoids, has recently been developed as a strong link between the theory of finite automata and low-level parallel complexity. Achievements of this model include the proof that width 5 branching programs recognize exactly the languages in non-uniform NC 1 [Ba86], N...

We study the complexity of restricted versions ofst- connectivity, which is the standard complete problem for NL. Grid graphs are a useful tool in this regard, since reachability on grid graphs is logspace-equivalent to reachability in general planar digraphs, and reachability on certain classes of grid graphs gives natural examples of problems tha...

A language L over an alphabet A is said to have a neutral letter if there is a letter e∈A such that inserting or deleting e's from any word in A* does not change its membership or non-membership in L.The presence of a neutral letter affects the definability of a language in first-order logic. It was conjectured that it renders all numerical predica...

We study the complexity of reachability problems on various classes of grid graphs. Reachability on certain classes of grid graphs gives natural examples of problems that are hard for NC¹ under AC⁰ reductions but are not known to be hard far L; they thus give insight into the structure of L. In addition to explicating the stru...

It has been known since the mid-1980s (SIAM J. Comput. 15 (1986) 994; SIAM J. Comput. 21 (1992) 896) that integer division can be performed by poly-time uniform constant-depth circuits of Majority gates; equivalently, the division problem lies in P-uniform TC0. Recently, this was improved to L-uniform TC0 (RAIRO Theoret. Inform. Appl. 35 (2001) 259...

this paper. 2.1. Circuit Classes We begin by formally defining the three circuit complexity classes that will concern us here. These are given by combinatorial restrictions on the circuits of the family. We will then define the uniformity restrictions we will use. Finally, we will give the equivalent formulations of uniform circuit complexity class...

The Cayley group membership problem (CGM) is to input a groupoid (binary algebra) G given as a multiplication table, a subset X of G, and an element t of G and to determine whether t can be expressed as a product of elements of X. For general groupoids CGM is P-complete, and for associative algebras (semigroups) it is NL-complete. Here we investiga...

Integer division has been known to lie in P-uniform TC⁰ since the mid-1980s, and recently this was improved to L-uniform TC⁰. At the time that the results in this paper were proved and submitted for conference presentation, it was unknown whether division lay in DLOGTIME-uniform TC⁰ (also known as FOM). We obtain ti...

Integer division has been known to lie in P-uniform TC 0 since the mid-1980's, and recently this was improved to L- uniform TC 0 . At the time that the results in this paper were proved and submitted for conference presentation, it was unknown whether division lay in DLOGTIME-uniform TC 0 (also known as FOM). We obtain tight bounds on the uniformit...

A language L over an alphabet A is said to have a neutral letter if there is a letter e∈A such that inserting or deleting e's from any word in A* does not change its membership (or non-membership) in L. The presence of a neutral letter affects the definability of a language in first-order logic. It was conjectured that it renders all numerical pred...

. Constant-depth arithmetic circuits have been defined and studied in [AAD97,ABL98]; these circuits yield the function classes #AC 0 and GapAC 0 . These function classes in turn provide new characterizations of the computational power of threshold circuits, and provide a link between the circuit classes AC 0 (where many lower bounds are known) and...

We study the computational complexity of solving equations and of determining the satisfiability of programs over a fixed finite monoid. We partially answer an open problem of [4] by exhibiting quasi-polynomial time algorithms for a subclass of solvable non-nilpotent groups and relate this question to a natural circuit complexity conjecture. In the...

We study the computational complexity of solving equations and of determining the satisfiability of programs over a fixed finite monoid. We partially answer an open problem of [4] by exhibiting quasi-polynomial time algorithms for a sub-class of solvable non-nilpotent groups and relate this question to a natural circuit complexity conjecture.

We prove that constant depth circuits, with one layer of M O D m gates at the inputs, followed by a fixed number of layers of M O D p gates, where p is prime, require exponential size to compute the M O D q function, if q is a prime that divides neither p nor m.

We prove that the set of properties describable by a uniform sequence of firstorder sentences using at most k + 1 distinct variables is exactly equal to the set of properties checkable by a Turing machine in DSPACE [n k ] (where n is the size of the universe). This set is also equal to the set of properties describable using an iterative definition...

In this paper we show several results about monotone planar circuits. We show that monotone planar circuits of bounded width, with access to negated input variables, compute exactly the functions in non-uniform AC 0 . This provides a striking contrast to the non-planar case, where exactly NC 1 is computed. We show that the circuit value problem for...

In this paper we show several results about monotone planar circuits. We show that monotone planar circuits of bounded width, with access to negated input variables, compute exactly the functions in non-uniform AC⁰. This provides a striking contrast to the non-planar case, where exactly NC¹ is computed. We show that the circui...

Constant-depth arithmetic circuits have been defined and studied in [AAD97,ABL98]; these circuits yield the function classes #AC0 and GapAC0. These function classes in turn provide new characterizations of the computational power of threshold circuits, and provide a link between the circuit classes AC0 (where many lower bounds are known) and TC0 (w...

We show that searching a width k maze is complete for Π k, i.e., for the k'th level of the AC 0 hierarchy. Equivalently, st-connectivity for width k grid graphs is complete for Π k. As an application, we show that there is a data structure solving dynamic st-connectivity for con stant width grid graphs with time bound O(log log n) per operation on...

We examine the computational power of modular counting, where the modulus m is not a prime power, in the setting of polynomials in Boolean variables over Z m . In particular, we say that a polynomial P weakly represents a Boolean function f (both have n variables) if for any inputs x and y in {0,1}n, we have whenever . Barrington et al. (1994) in...

We use algebraic techniques to obtain superlinear lower bounds on the size of bounded-width branching programs to solve a number of problems. In particular, we show that any bounded-width branching program computing a nonconstant threshold function has length Ω(n log log n), improving on the previous lower bounds known to apply to all such threshol...

The SB-PRAM is a shared memory parallel machine under construction in Saarbrtucken. With the help of simulations we have evaluated the performance of transaction systems on thus machine. We use the well known DEBIT/CREDIT benchmark as workload. ...

We prove that constant depth circuits, with one layer of MODm gates at the inputs, followed by a fixed number of layers of MODp gates, where p is prime, require exponential size to compute the MODq function, if q is a prime that divides neither p nor m.

We examine the computational power of modular counting, where the modulus m is not a prime power, in the setting of polynomials in boolean variables over Z_m. In particular, we say that a polynomial P weakly represents a boolean function f (both have n variables) if for any inputs x and y in {0, 1}ⁿ we have P(x)≠P(y) whenever f...

We describe three orthogonal complexity measures: parallel time, amount of hardware, and degree of non-uniformity, which together parametrize most complexity classes. We show that the descriptive complexity framework neatly captures these measures using the parameters: quantifier depth, number of variable bits, and type of numeric predicates respec...

We introduce a natural set of arithmetic expressions and define the complexity class AE to consist of all those arithmetic functions (over the fieldsF 2n) that are described by these expressions. We show that AE coincides with the class of functions that are computable with constant depth and polynomial-size unbounded fan-in arithmetic circuits sat...

Define the MOD m -degree of a boolean functionF to be the smallest degree of any polynomialP, over the ring of integers modulom, such that for all 0–1 assignments [(x)\vec]\vec x , F([(x)\vec]) = 0F(\vec x) = 0 iff P([(x)\vec]) = 0P(\vec x) = 0 . We obtain the unexpected result that the MOD m -degree of the OR ofN variables is O(\tau Ö{N})O(\sqrt[\...

Several recent results in circuit complexity theory have used a representation of Boolean functions by polynomials over finite fields. Our current inability to extend these results to superficially similar situations may be related to properties of these polynomials which do not extend to polynomials over general finite rings or finite abelian grou...

Two important measures of the computational complexity of a regular language are the type of finite automaton needed to recognize it and the type of logical expression needed to describe it. Important connections between these measures were studied by Buchi and McNaughton as early as 1960. In this survey we describe the logical formalism used, outl...

We review the existing results showing languages to be outside of complexity classes defined by tight constraints on boolean circuits, using the new characterizations of these classes in terms of automata theory [BT88] and formal logic [BIS88]. We outline the methods and results in the case of three types of classes defined by circuits of constant...

We examine the size complexity of the symmetric boolean functions in two circuit models containing threshold gates: the d-perceptron model [BRS, ABFR] (a single threshold function of constant-depth AND/OR circuits) and the parity-threshold model studied by Bruck [Br] (a single threshold function of exclusive-ORs). These models are intermediate betw...

Fagin et al. characterized those symmetric Boolean functions which can be computed by small AND/OR circuits of constant depth and unbounded fan-in. Here we provide a similar characterization for d- perceptrons --- AND/OR circuits of constant depth and unbounded fan-in with a single MAJORITY gate at the output. We show that a symmetric function has...

R. Fagin, M. M. Klawe, N. J. Pippenger and L. Stockmeyer [Theor. Comput. Sci. 36, 239-250 (1985; Zbl 0574.94024)] characterized those symmetric Boolean functions which can be computed by small AND/OR circuits of constant depth and unbounded fan-in. Here we provide a similar characterization for d-perceptrons – AND/OR circuits of constant depth and...

By considering the size of the logical network needed to perform a given computational task, the intrinsic difficulty of that task can be examined. Boolean function complexity, the combinatorial study of such networks, is a subject that started back in the 1950s and has today become one of the most challenging and vigorous areas of theoretical comp...

We give several characterizations, in terms of formal logic, semigroup theory, and operations on languages, of the regular languages in the circuit complexity class AC0, thus answering a question of Chandra, Fortune, and Lipton. As a by-product, we are able to determine effectively whether a given regular language is in AC0 and to solve in part an...

We study the power of constant-depth circuits containing negation gates, unbounded fan-in AND and OR gates, and a small number of MAJORITY gates. It is easy to show that a depth 2 circuit of sizeO(n) (wheren is the number of inputs) containingO(n) MAJORITY gates can determine whether the sum of the input bits is divisible byk, for any fixedk>1, whe...

Circuit complexity theory has tried to understand which problems can be solved by `small' circuits of constant depth. Normally `small' has meant `polynomial in the input size', but a number of recent results have dealt with circuits of size 2 to the log n ⁰⁽¹⁾ power, or quasipolynomial size. The author summarizes the reasons for thinking...

The authors use algebraic techniques to obtain superlinear lower bounds on the size of bounded-width branching programs to solve a number of problems. In particular, they show that any bounded-width branching program computing a nonconstant threshold function has length Ω( n log log n ), improving on the previous lower bounds known to apply to all...

ACC0 (also called ACC) is the class of languages recognized by circuit families with polynomial size, constant depth, and unbounded fan-in, where gates may calculate the AND, OR, or MOD c function for constant c. Robust uniformity definitions for ACC0 and related classes were given by Barrington, Immerman and Straubing [3]. Here we show that unifor...

We introduce a natural set of arithmetic expressions and define the complexity class AE to consist of all those arithmetic functions (over the fields F_(2)n) that are described by these expressions. We show that AE coincides with the class of functions that are computable with constant depth and polynomial size unbounded fan-in arithmetic circuits...

A new model, non-uniform deterministic finite automata (NUDFA's) over general finite monoids, has recently been developed as a strong link between the theory of finite automata and low-level parallel complexity. Achievements of this model include the proof that width 5 branching programs recognize exactly the languages in non-uniform NC1, NUDFA cha...

In order to study circuit complexity classes within NC1 in a uniform setting, we need a uniformity condition which is more restrictive than those in common use. Two such conditions, stricter than NC1 uniformity, have appeared in recent research: Immerman's families of circuits defined by first-order formulas and a uniformity corresponding to Buss'...

In order to study circuit complexity classes within NC¹ in a uniform setting, we need a uniformity condition which is more restrictive than those in common use. Two such conditions, stricter than NC¹ uniformity, have appeared in recent research: Immerman's families of circuits defined by first-order formulas and a uniformity corresponding to Buss'...

A new model, non-uniform deterministic finite automata (NUDFA's) over general finite monoids, has recently been developed as a strong link between the theory of finite automata and low-level parallel complexity. Achievements of this model include the proof that width 5 branching programs recognize exactly the languages in non-uniform NC¹, NUDFA cha...

We consider the relative complexity of a number of languages known to be in uniform NC1, using the descriptive framework of Barrington, Immerman, and Straubing (1988). In particular, we sharpen several results of Ibarra, Jiang, and Ravikumar (1988). We show that the one-sided Dyck languages, structured CFL's, and bracketed CFL's are recognizable by...

We show that any language recognized by an NC1 circuit (fan-in 2, depth O(log n)) can be recognized by a width-5 polynomial-size branching program. As any bounded-width polynomial-size branching program can be simulated by an NC1 circuit, we have that the class of languages recognized by such programs is exactly nonuniform NC1. Further, following R...

We show that any language recognized by an NC¹ circuit (fan-in 2, depth O(log n)) can be recognized by a width-5 polynomial-size branching program. As any bounded-width polynomial-size branching program can be simulated by an NC¹ circuit, we have that the class of languages recognized by such programs is exactly nonuniform NC¹. Further, following R...

We define the notion of an oracle branching program in order to investigate space-bounded computation. Within this new framework we examine the P-complete problem GEN which consists of determining membership in a subalgebra of a general (not necessarily associative) binary algebra (input as a multiplication table). Our work begins with the statemen...

Recently a new connection was discovered between the parallel complexity class NC ¹ and the theory of finite automata in the work of Barrington on bounded width branching programs. There (nonuniform) NC ¹ was characterized as those languages recognized by a certain nonuniform version of a DFA. Here we extend this characterization to show that the i...

The study of circuit complexity classes within NC¹ in a uniform setting requires a uniformity condition that is more restrictive than those in common use. Two such conditions, stricter than NC¹ uniformity, have appeared in recent research. It is shown that the two notions are equivalent, leading to a natural notion of uniform...

We show that any language recognized by an NC ’ circuit (fan-in 2, depth O(log n)) can be recognized by a width-5 polynomial-size branching program. As any bounded-width polynomial-size branching program can be simulated by an NC ’ circuit, we have that the class of languages recognized by such programs is exactly nonuniform NC’. Further, following

The essential idea in the fast parallel computation of division and related problems is that of Chinese remainder representation (CRR) -- storing a number in the form of its residues modulo many small primes. Integer division provides one of the few natural examples of problems for which all currently-known constructions of e#cient circuits rely on...