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43
Nondeterministic NC¹ computation
"... We define the counting classes #NC¹, GapNC¹, PNC¹ and C=NC¹. 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 ..."
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Cited by 18 (6 self)
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We define the counting classes #NC¹, GapNC¹, PNC¹ and C=NC¹. 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 #NC¹ ` #L and that C=NC¹ ` 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 ACC⁰ from MODPH as well as that of TC⁰ from the counting hierarchy. Moreover we obtain that dlogtimeuniformity and logspaceuniformity for AC⁰ coincide if and only if the polynomial time hierarchy equals PSPACE .
Dynamic Word Problems
, 1993
"... Let M be a fixed finite monoid. We consider the problem of implementing a data type containing a vector x = (x1 ; x2 ; : : : ; xn) 2 M n , initially (1; 1; : : : ; 1), with two kinds of operations, for each i 2 f1; : : : ; ng and a 2 M , an operation change i;a which changes x i to a and a singl ..."
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Cited by 18 (6 self)
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Let M be a fixed finite monoid. We consider the problem of implementing a data type containing a vector x = (x1 ; x2 ; : : : ; xn) 2 M n , initially (1; 1; : : : ; 1), with two kinds of operations, for each i 2 f1; : : : ; ng and a 2 M , an operation change i;a which changes x i to a and a single operation product returning Q n i=1 x i . This is the dynamic word problem for M . If we in addition for each j 2 f1; : : : ; ng have an operation prefix j returning Q j i=1 x i , we get the dynamic prefix problem for M . We analyze the complexity of these problems in the cell probe or decision assignment tree model for two natural cell sizes, 1 bit and log n bits. We obtain a partial classification of the complexity based on algebraic properties of M .
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 ..."
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Cited by 13 (7 self)
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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]).
On TC⁰, AC⁰, and Arithmetic Circuits
 JOURNAL OF COMPUTER AND SYSTEM SCIENCES
, 2000
"... Continuing a line of investigation that has studied the function classes #P [Val79b], #SAC¹ [Val79a, Vin91, AJMV], #L [AJ93b, Vin91, AO94], and #NC¹ [CMTV96], we study the class of functions #AC⁰. One way to define #AC⁰ is as the class of functions computed by constantdepth polynomialsize arith ..."
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Cited by 13 (3 self)
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Continuing a line of investigation that has studied the function classes #P [Val79b], #SAC¹ [Val79a, Vin91, AJMV], #L [AJ93b, Vin91, AO94], and #NC¹ [CMTV96], we study the class of functions #AC⁰. One way to define #AC⁰ is as the class of functions computed by constantdepth polynomialsize arithmetic circuits of unbounded fanin addition and multiplication gates. In contrast to the preceding
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 ..."
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Cited by 11 (2 self)
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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...
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 ..."
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Cited by 9 (3 self)
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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 .
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 ..."
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Cited by 7 (1 self)
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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 ...
Arithmetic Complexity, Kleene Closure, and Formal Power Series
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, REPORT NO. 8 (1999)
, 1999
"... The aim of this paper is to use formal power series techniques to study the structure of small arithmetic complexity classes such as GapNC 1 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 ..."
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Cited by 7 (3 self)
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The aim of this paper is to use formal power series techniques to study the structure of small arithmetic complexity classes such as GapNC 1 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 GapNC 1 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 GapNC 1complete, with respect to appropriate reducibilities. The latter result raises the question of classifying finite languages so that their inverses fall within interesting subclasses of GapNC 1, such as GapAC 0. We initiate work in this direction by classifying the complexity of the Kleene closure of finite languages. We formulate the problem in terms of finite monoids and relate its complexity to the internal structure of the monoid.
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, ..."
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Cited by 7 (3 self)
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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 secondor ..."
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Cited by 7 (2 self)
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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.