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41
GapDefinable Counting Classes
, 1991
"... The function class #P lacks an important closure property: it is not closed under subtraction. To remedy this problem, we introduce the function class GapP as a natural alternative to #P. GapP is the closure of #P under subtraction, and has all the other useful closure properties of #P as well. We s ..."
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Cited by 122 (13 self)
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The function class #P lacks an important closure property: it is not closed under subtraction. To remedy this problem, we introduce the function class GapP as a natural alternative to #P. GapP is the closure of #P under subtraction, and has all the other useful closure properties of #P as well. We show that most previously studied counting classes, including PP, C=P, and Mod k P, are "gapdefinable," i.e., definable using the values of GapP functions alone. We show that there is a smallest gapdefinable class, SPP, which is still large enough to contain Few. We also show that SPP consists of exactly those languages low for GapP, and thus SPP languages are low for any gapdefinable class. These results unify and improve earlier disparate results of Cai & Hemachandra [7] and Kobler, Schoning, Toda, & Tor'an [15]. We show further that any countable collection of languages is contained in a unique minimum gapdefinable class, which implies that the gapdefinable classes form a lattice un...
A Taxonomy of Complexity Classes of Functions
 Journal of Computer and System Sciences
, 1992
"... This paper comprises a systematic comparison of several complexity classes of functions that are computed nondeterministically in polynomial time or with an oracle in NP. There are three components to this work. ffl A taxonomy is presented that demonstrates all known inclusion relations of these cla ..."
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Cited by 88 (12 self)
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This paper comprises a systematic comparison of several complexity classes of functions that are computed nondeterministically in polynomial time or with an oracle in NP. There are three components to this work. ffl A taxonomy is presented that demonstrates all known inclusion relations of these classes. For (nearly) each inclusion that is not shown to hold, evidence is presented to indicate that the inclusion is false. As an example, consider FewPF, the class of multivalued functions that are nondeterministically computable in polynomial time such that for each x, there is a polynomial bound on the number of distinct output values of f(x). We show that FewPF ` PF NP tt . However, we show PF NP tt ` FewPF if and only if NP = coNP, and thus PF NP tt ` FewPF is likely to be false. ffl Whereas it is known that P NP (O(log n)) = P NP tt ` P NP [Hem87, Wagb, BH88], we show that PF NP (O(log n)) = PF NP tt implies P = FewP and R = NP. Also, we show that PF NP tt = PF ...
Some Connections between Bounded Query Classes and NonUniform Complexity
 In Proceedings of the 5th Structure in Complexity Theory Conference
, 1990
"... This paper is dedicated to the memory of Ronald V. Book, 19371997. ..."
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Cited by 71 (23 self)
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This paper is dedicated to the memory of Ronald V. Book, 19371997.
Counting Classes: Thresholds, Parity, Mods, and Fewness
, 1996
"... Counting classes consist of languages defined in terms of the number of accepting computations of nondeterministic polynomialtime Turing machines. Well known examples of counting classes are NP, coNP, \PhiP, and PP. Every counting class is a subset of P #P[1] , the class of languages computable ..."
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Cited by 61 (13 self)
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Counting classes consist of languages defined in terms of the number of accepting computations of nondeterministic polynomialtime Turing machines. Well known examples of counting classes are NP, coNP, \PhiP, and PP. Every counting class is a subset of P #P[1] , the class of languages computable in polynomial time using a single call to an oracle capable of determining the number of accepting paths of an NP machine. Using closure properties of #P, we systematically develop a complexity theory for counting classes defined in terms of thresholds and moduli. An unexpected result is that MOD k iP = MOD k P for prime k. Finally, we improve a result of Cai and Hemachandra by showing that recognizing languages in the class Few is as easy as distinguishing uniquely satisfiable formulas from unsatisfiable formulas (or detecting unique solutions, as in [28]). 1. Introduction Valiant [27] defined the class #P of functions whose values equal the number of accepting paths of polynomialtime bo...
Representing Boolean Functions As Polynomials Modulo Composite Numbers
 Computational Complexity
, 1994
"... . Define the MODm degree of a boolean function F to be the smallest degree of any polynomial P , over the ring of integers modulo m, such that for all 01 assignments ~x, F (~x) = 0 iff P (~x) = 0. We obtain the unexpected result that the MODm degree of the OR of N variables is O( r p N ), wher ..."
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Cited by 56 (6 self)
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. Define the MODm degree of a boolean function F to be the smallest degree of any polynomial P , over the ring of integers modulo m, such that for all 01 assignments ~x, F (~x) = 0 iff P (~x) = 0. We obtain the unexpected result that the MODm degree of the OR of N variables is O( r p N ), where r is the number of distinct prime factors of m. This is optimal in the case of representation by symmetric polynomials. The MOD n function is 0 if the number of input ones is a multiple of n and is one otherwise. We show that the MODm degree of both the MOD n and :MOD n functions is N\Omega\Gamma1/ exactly when there is a prime dividing n but not m. The MODm degree of the MODm function is 1; we show that the MODm degree of :MODm is N\Omega\Gamma30 if m is not a power of a prime, O(1) otherwise. A corollary is that there exists an oracle relative to which the MODmP classes (such as \PhiP) have this structure: MODmP is closed under complementation and union iff m is a prime power, and...
A Complexity Theory for Feasible Closure Properties
, 1991
"... The study of the complexity of sets encompasses two complementary aims: (1) establishing  usually via explicit construction of algorithms  that sets are feasible, and (2) studying the relative complexity of sets that plausibly might be feasible but are not currently known to be feasible (such as ..."
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Cited by 47 (3 self)
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The study of the complexity of sets encompasses two complementary aims: (1) establishing  usually via explicit construction of algorithms  that sets are feasible, and (2) studying the relative complexity of sets that plausibly might be feasible but are not currently known to be feasible (such as the NPcomplete sets and the PSPACEcomplete sets). For the study of the complexity of closure properties, a recent urry of results [21, 33, 49, 6, 7, 16] has established an analog of (1); these papers explicitly demonstrate many closure properties possessed by PP and C=P (and the proofs implicitly give closure properties of the function class #P). The present paper presents and develops, for function classes such as #P, SpanP, OptP, and MidP, an analog of (2): a general theory of the complexity of closure properties. In particular, we show that subtraction is hard for the closure properties of each of these classes: each is closed under subtraction if and only if it is closed under every polynom...
Structure and Importance of LogspaceMODClasses
, 1992
"... . We refine the techniques of Beigel, Gill, Hertrampf [4] who investigated polynomial time counting classes, in order to make them applicable to the case of logarithmic space. We define the complexity classes MOD k L and demonstrate their significance by proving that all standard problems of linear ..."
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Cited by 40 (1 self)
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. We refine the techniques of Beigel, Gill, Hertrampf [4] who investigated polynomial time counting classes, in order to make them applicable to the case of logarithmic space. We define the complexity classes MOD k L and demonstrate their significance by proving that all standard problems of linear algebra over the finite rings Z/kZ are complete for these classes. We then define new complexity classes LogFew and LogFewNL and identify them as adequate logspace versions of Few and FewP. We show that LogFewNL is contained in MODZ k L and that LogFew is contained in MOD k L for all k. Also an upper bound for L #L in terms of computation of integer determinants is given from which we conclude that all logspace counting classes are contained in NC 2 . 1 Introduction Valiant [21] defined the class #P of functions f such that there is a nondeterministic polynomial time Turing machine which, on input x, has exactly f(x) accepting computation paths. Many complexity classes in the area betw...
Relativizable And Nonrelativizable Theorems In The Polynomial Theory Of Algorithms
 In Russian
, 1993
"... . Starting with the paper of Baker, Gill and Solovay [BGS 75] in complexity theory, many results have been proved which separate certain relativized complexity classes or show that they have no complete language. All results of this kind were, in fact, based on lower bounds for boolean decision tree ..."
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Cited by 36 (0 self)
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. Starting with the paper of Baker, Gill and Solovay [BGS 75] in complexity theory, many results have been proved which separate certain relativized complexity classes or show that they have no complete language. All results of this kind were, in fact, based on lower bounds for boolean decision trees of a certain type or for machines with polylogarithmic restrictions on time. The following question arises: Are these methods of proving "relativized" results universal? In the first part of the present paper we propose a general framework in which assertions of universality of this kind may be formulated and proved as convenient criteria. Using these criteria we obtain, as easy consequences of the known results on boolean decision trees, some new "relativized" results and new proofs of some known results. In the second part of the present paper we apply these general criteria to many particular cases. For example, for many of the complexity classes studied in the literature all relativiza...
PolynomialTime Membership Comparable Sets
, 1994
"... This paper studies a notion called polynomialtime membership comparable sets. For a function g, a set A is polynomialtime gmembership comparable if there is a polynomialtime computable function f such that for any x 1 ; \Delta \Delta \Delta ; xm with m g(maxfjx 1 j; \Delta \Delta \Delta ; jx m j ..."
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Cited by 31 (4 self)
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This paper studies a notion called polynomialtime membership comparable sets. For a function g, a set A is polynomialtime gmembership comparable if there is a polynomialtime computable function f such that for any x 1 ; \Delta \Delta \Delta ; xm with m g(maxfjx 1 j; \Delta \Delta \Delta ; jx m jg), outputs b 2 f0; 1g m such that (A(x 1 ); \Delta \Delta \Delta ; A(xm )) 6= b. The following is a list of major results proven in the paper. 1. Polynomialtime membership comparable sets construct a proper hierarchy according to the bound on the number of arguments. 2. Polynomialtime membership comparable sets have polynomialsize circuits. 3. For any function f and for any constant c ? 0, if a set is p f(n)tt reducible to a Pselective set, then the set is polynomialtime (1 + c) log f(n)membership comparable. 4. For any C chosen from fPSPACE;UP;FewP;NP;C=P;PP;MOD 2 P; MOD 3 P; \Delta \Delta \Deltag, if C ` Pmc(c log n) for some c ! 1, then C = P. As a corollary of the last tw...
Unambiguity and Fewness for Logarithmic Space
, 1991
"... We consider various types of unambiguity for logarithmic space bounded Turing machines and polynomial time bounded log space auxiliary pushdown automata. In particular, we introduce the notions of (general), reach, and strong unambiguity. We demonstrate that closure under complement of unambiguo ..."
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Cited by 23 (6 self)
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We consider various types of unambiguity for logarithmic space bounded Turing machines and polynomial time bounded log space auxiliary pushdown automata. In particular, we introduce the notions of (general), reach, and strong unambiguity. We demonstrate that closure under complement of unambiguous classes implies equivalence of unambiguity and "unambiguous fewness". This, as we will show, applies in the cases of reach and strong unambiguity for logspace. Among the many relations we exhibit, we show that the unambiguous linear contextfree languages, which are not known to be contained in LOGSPACE, nevertheless are contained in strongly unambiguous logspace, and, consequently, in LOGDCFL. In fact, this turns out to be the case for all logspace languages with reach unambiguous fewness. In addition, we show that general unambiguity and fewness of logspace classes can be simulated by reach unambiguity and fewness of auxiliary pushdown automata. 1 Introduction Although the pow...