Results 11  20
of
36
Nondeterministic NC¹ computation
"... We define the counting classes #NC 1 , GapNC 1 , PNC 1 and C=NC 1 . We prove that boolean circuits, algebraic circuits, programs over nondeterministic finite automata, and programs over constant integer matrices yield equivalent definitions of the latter three classes. We investigate closure ..."
Abstract

Cited by 15 (4 self)
 Add to MetaCart
We define the counting classes #NC 1 , GapNC 1 , PNC 1 and C=NC 1 . We prove that boolean circuits, algebraic circuits, programs over nondeterministic finite automata, and programs over constant integer matrices yield equivalent definitions of the latter three classes. We investigate closure properties. We observe that #NC 1 ` #L and that C=NC 1 ` L. Then we exploit our finite automaton model and extend the padding techniques used to investigate leaf languages. Finally, we draw some consequences from the resulting body of leaf language characterizations of complexity classes, including the unconditional separation of ACC 0 from MODPH as well as that of TC 0 from the counting hierarchy. Moreover we obtain that dlogtimeuniformity and logspaceuniformity for AC 0 coincide if and only if the polynomial time hierarchy equals PSPACE .
On Monotone Planar Circuits
, 1999
"... 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 nonuniform AC 0 . This provides a striking contrast to the nonplanar case, where exactly NC 1 i ..."
Abstract

Cited by 13 (2 self)
 Add to MetaCart
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 nonuniform AC 0 . This provides a striking contrast to the nonplanar case, where exactly NC 1 is computed. We show that the circuit value problem for monotone planar circuits, with inputs on the outer face only, can be solved in LOGDCFL ` SC, improving a LOGCFL upper bound due to Dymond and Cook. We show that for monotone planar circuits, with inputs on the outer face only, excessive depth compared to width is useless; any function computed by a monotone planar circuit of width w with inputs on the outer face can be computed by a monotone planar circuit of width O(w) and depth w O(1) . Finally, we show that monotone planar readonce circuits, with inputs on the outer face only, can be efficiently learned using membership queries. 1 Introduction In this paper, we prove a number of ...
The Complexity of Computing over Quasigroups
, 1994
"... In [7] the notions of recognition by semigroups and by programs over semigroups were extended to groupoids. This led to a new characterization of the contextfree languages and the class SAC¹. In this paper, we investigate the classes of languages obtained when the groupoids are restricted to be ..."
Abstract

Cited by 11 (6 self)
 Add to MetaCart
In [7] the notions of recognition by semigroups and by programs over semigroups were extended to groupoids. This led to a new characterization of the contextfree languages and the class SAC¹. In this paper, we investigate the classes of languages obtained when the groupoids are restricted to be quasigroups (i.e. the multiplication table forms a latin square). We prove that languages recognized by quasigroups are regular and that programs over quasigroups characterize NC¹. We introduce the notions of linear recognition by groupoids and by programs over groupoids, and characterize the linear contextfree languages and NL. Here again, when quasigroups are used, only regular languages and languages in NC¹ can be obtained. We also consider the problem of evaluating a wellparenthesized expression over a finite loop (a quasigroup with an identity). This problem is in NC¹ for any finite loop, and we give algebraic conditions for its completeness. In particular, we prove that it is sufficient that the loop be nonsolvable, extending a wellknown theorem of Barrington ([3]).
Some Problems Involving RazborovSmolensky Polynomials
, 1991
"... 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 polyno ..."
Abstract

Cited by 11 (2 self)
 Add to MetaCart
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 groups. Here we pose a number of conjectures on the behavior of such polynomials over rings and groups, and present some partial results toward proving them. 1. Introduction 1.1. Polynomials and Circuit Complexity The representation of Boolean functions as polynomials over the finite field Z 2 = f0; 1g dates back to early work in switching theory [?]. A formal language L can be identified with the family of functions f i : Z i 2 ! Z 2 , where f i (x 1 ; : : : ; x i ) = 1 iff x 1 : : : x i 2 L. Each of these functions can be written as a polynomial in the variables x 1 ; : : : ; x n . We can consider algebraic formulas or circuits with...
Uniform Characterizations of Complexity Classes
 Complexity Theory Column 23, ACMSIGACT News
, 1999
"... In the past few years, generalized operators (a. k. a. leaf languages) in the context of polynomial time machines, and gates computing arbitrary groupoidal functions in the context of Boolean circuits have raised some interest. We survey results from both areas, point out connections between them, ..."
Abstract

Cited by 7 (3 self)
 Add to MetaCart
In the past few years, generalized operators (a. k. a. leaf languages) in the context of polynomial time machines, and gates computing arbitrary groupoidal functions in the context of Boolean circuits have raised some interest. We survey results from both areas, point out connections between them, and present relations to a generalized quantifier concept known from finite model theory. 1 Introduction There is an "amusing and instructive way of looking at [...] diverse complexity classes" [Pap94a, p. 504] that are of current focal interest in computational complexity theory. This way makes instrumental use of characterizations of classes in terms of conditions on computations trees of nondeterministic polynomialtime Turing machines. As an example, let us look at the class NP. By definition, a language A 2 NP is given by a nondeterministic polynomialtime machine (NPTM) M such that for all inputs x, we have that x belongs to A if and only if in the computation tree that M produces wh...
Succinct Inputs, Lindström Quantifiers, and a General Complexity Theoretic Operator Concept
 IN READERS OF THE NINTH EUROPEAN SUMMER SCHOOL IN LOGIC, LANGUAGE AND INFORMATION, CHAPTER CL7. CNRS AIXENPROVENCE AND THE EUROPEAN ASSOCIATION FOR LOGIC, LANGUAGE AND INFORMATION
, 1996
"... We address the question of the power of several logics with Lindstrom quantifiers over finite ordered structures. We will see that in the firstorder case this nicely fits into the framework of Barrington, Immerman, and Straubing's examination of constant depth circuit classes. In the secondorder c ..."
Abstract

Cited by 7 (2 self)
 Add to MetaCart
We address the question of the power of several logics with Lindstrom quantifiers over finite ordered structures. We will see that in the firstorder case this nicely fits into the framework of Barrington, Immerman, and Straubing's examination of constant depth circuit classes. In the secondorder case we get a strong relationship to succinct encodings of languages via circuits. Some of these logics can be characterized as closures of succinct encodings under appropriate reducibilities, others by certain hierarchies of circuit classes. We will see that in a special case secondorder Lindstrom quantifiers can equivalently be expressed in firstorder logic, while in the general case this equivalence seems unlikely.
On Serializable Languages
 University of Rochester, Department of Computer Science
, 1994
"... Cai and Furst introduced the notion of bottleneck Turing machines and showed that the languages recognized by width5 bottleneck Turing machines are exactly those in PSPACE. Computational power of bottleneck Turing machines with width fewer than 5 is investigated. It is shown that width2 bottleneck ..."
Abstract

Cited by 7 (1 self)
 Add to MetaCart
Cai and Furst introduced the notion of bottleneck Turing machines and showed that the languages recognized by width5 bottleneck Turing machines are exactly those in PSPACE. Computational power of bottleneck Turing machines with width fewer than 5 is investigated. It is shown that width2 bottleneck Turing machines capture polynomialtime manyone closure of nearly neartestable sets. For languages recognized by bottleneck Turing machines with intermediate width 3 and 4, some lower and upperbounds are shown. 1 Introduction Branching program is one of the most interesting topics in complexity theory. For k 2, a widthk branching program for nbit inputs is a sequence of instructions f(p i ; f i ; g i )g m i=1 such that for each i; 1 i m, 1 p i n and f i ; g i 2 F k , where F k is the monoid consisting of all mappings of [k] = f1; \Delta \Delta \Delta ; kg to itself. Given an input x 2 \Sigma = f0; 1g of length n, for each i, let h i = f i if the p i th bit of x is a 1 ...
Circuits on Cylinders
, 2002
"... We consider the computational power of constant width polynomial size cylindrical circuits and nondeterministic branching programs. We show that every function computed by a #2 circuit can also be computed by a constant width polynomial size cylindrical nondeterministic branching program (o ..."
Abstract

Cited by 7 (1 self)
 Add to MetaCart
We consider the computational power of constant width polynomial size cylindrical circuits and nondeterministic branching programs. We show that every function computed by a #2 circuit can also be computed by a constant width polynomial size cylindrical nondeterministic branching program (or cylindrical circuit) and that every function computed by a constant width polynomial size cylindrical circuit belongs to ACC .
Arithmetic Complexity, Kleene Closure, and Formal Power Series
, 1999
"... The aim of this paper is to use formal power series techniques to study the structure of small arithmetic complexity classes such as GapNC¹ and GapL. More precisely, we apply the Kleene closure of languages and the formal power series operations of inversion and root extraction to these complexity c ..."
Abstract

Cited by 6 (2 self)
 Add to MetaCart
The aim of this paper is to use formal power series techniques to study the structure of small arithmetic complexity classes such as GapNC¹ and GapL. More precisely, we apply the Kleene closure of languages and the formal power series operations of inversion and root extraction to these complexity classes. We define a counting version of Kleene closure and show that it is intimately related to inversion and root extraction within GapNC¹ and GapL. We prove that Kleene closure, inversion, and root extraction are all hard operations in the following sense: There is a language in AC 0 for which inversion and root extraction are GapLcomplete, and there is a finite set for which inversion and root extraction are GapNC¹complete, with respect to appropriate reducibilities. The latter result raises the question of classifying finite languages so that their inverses fall within interesting subclasses of GapNC¹, such as GapAC^0. We initiate work in this direction by classifyi...
A new parallel vector model, with exact characterizations of NC k
 in Proc. 11th Annual Symposium on Theoretical Aspects of Computer Science
, 1994
"... This paper develops a new and natural parallel vector model, and shows that for all k ≥ 1, the languages recognizable in O(log k n) time and polynomial work in the model are exactly those in NC k. Some improvements to other simulations in parallel models and reversal complexity are given. 1 ..."
Abstract

Cited by 5 (4 self)
 Add to MetaCart
This paper develops a new and natural parallel vector model, and shows that for all k ≥ 1, the languages recognizable in O(log k n) time and polynomial work in the model are exactly those in NC k. Some improvements to other simulations in parallel models and reversal complexity are given. 1