Results 1  10
of
31
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 ..."
Abstract

Cited by 120 (13 self)
 Add to MetaCart
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...
Relationships Among PL, L, and the Determinant
, 1996
"... Recent results byToda, Vinay, Damm, and Valianthave shown that the complexity of the determinantischaracterized by the complexity of counting the number of accepting computations of a nondeterministic logspacebounded machine. #This class of functions is known as #L.# By using that characterizati ..."
Abstract

Cited by 32 (8 self)
 Add to MetaCart
Recent results byToda, Vinay, Damm, and Valianthave shown that the complexity of the determinantischaracterized by the complexity of counting the number of accepting computations of a nondeterministic logspacebounded machine. #This class of functions is known as #L.# By using that characterization and by establishing a few elementary closure properties, we giveavery simple proof of a theorem of Jung, showing that probabilistic logspacebounded #PL# machines lose none of their computational power if they are restricted to run in polynomial time.
On the Power of NumberTheoretic Operations with Respect to Counting
 IN PROCEEDINGS 10TH STRUCTURE IN COMPLEXITY THEORY
, 1995
"... We investigate function classes h#Pi f which are defined as the closure of #P under the operation f and a set of known closure properties of #P, e.g. summation over an exponential range. First, we examine operations f under which #P is closed (i.e., h#Pi f = #P) in every relativization. We obtain t ..."
Abstract

Cited by 32 (9 self)
 Add to MetaCart
We investigate function classes h#Pi f which are defined as the closure of #P under the operation f and a set of known closure properties of #P, e.g. summation over an exponential range. First, we examine operations f under which #P is closed (i.e., h#Pi f = #P) in every relativization. We obtain the following complete characterization of these operations: #P is closed under f in every relativization if and only if f is a finite sum of binomial coefficients over constants. Second, we characterize operations f with respect to their power in the counting context in the unrelativized case. For closure properties f of #P, we have h#Pi f = #P. The other end of the range is marked by operations f for which h#Pi f corresponds to the counting hierarchy. We call these operations counting hard and give general criteria for hardness. For many operations f we show that h#Pi f corresponds to some subclass C of the counting hierarchy. This will then imply that #P is closed under f if and only if ...
Upward Separation for FewP and Related Classes
, 1994
"... This paper studies the range of application of the upward separation technique that has been introduced by Hartmanis to relate certain structural properties of polynomialtime complexity classes to their exponentialtime analogs and was first applied to NP [Har83]. Later work revealed the limitation ..."
Abstract

Cited by 14 (3 self)
 Add to MetaCart
This paper studies the range of application of the upward separation technique that has been introduced by Hartmanis to relate certain structural properties of polynomialtime complexity classes to their exponentialtime analogs and was first applied to NP [Har83]. Later work revealed the limitations of the technique and identified classes defying upward separation. In particular, it is known that coNP as well as certain promise classes such as BPP, R, and ZPP do not possess upward separation in all relativized worlds [HIS85; HJ93], and it had been suspected that this was also the case for other promise classes such as UP and FewP [All91]. In this paper, we refute this conjecture by proving that, in particular, FewP does display upward separation, thus providing the first upward separation result for a promise class. In fact, this follows from a more general result the proof of which heavily draws on Buhrman, Longpr'e, and Spaan's recently discovered tally encoding of sparse sets. As ...
Closure Properties and Witness Reduction
 IN PROCEEDINGS OF THE 6TH ANNUAL IEEE STRUCTURE IN COMPLEXITY THEORY CONFERENCE
, 1995
"... Witness reduction has played a crucial role in several recent results in complexity theory. These include Toda's result that PH ` BP \Delta \PhiP, the "collapsing" of PH into \PhiP with a high probability; Toda and Ogiwara's results which "collapses" PH into various counting classes with a high prob ..."
Abstract

Cited by 10 (1 self)
 Add to MetaCart
Witness reduction has played a crucial role in several recent results in complexity theory. These include Toda's result that PH ` BP \Delta \PhiP, the "collapsing" of PH into \PhiP with a high probability; Toda and Ogiwara's results which "collapses" PH into various counting classes with a high probability; and hard functions for various function classes studied by Ogiwara and Hemachandra. Ogiwara and Hemachandra's results establish a connection between functions being hard for #P and functions interacting with the class to effect witness reduction. In fact, we believe that the ability to achieve some form of witness reduction is what makes a function hard for a class of functions. To support our thesis we define new function classes and obtain results analogous to those of Ogiwara and Hemachandra. We also introduce the notion of randomly hard functions and obtain similar results.
Bounded Depth Arithmetic Circuits: Counting and Closure
, 1999
"... Constantdepth arithmetic circuits have been defined and studied in [AAD97, ABL98]; these circuits yield the function classes #AC . These function classes in turn provide new characterizations of the computational power of threshold circuits, and provide a link between the circuit classes AC ..."
Abstract

Cited by 9 (3 self)
 Add to MetaCart
Constantdepth arithmetic circuits have been defined and studied in [AAD97, ABL98]; these circuits yield the function classes #AC . These function classes in turn provide new characterizations of the computational power of threshold circuits, and provide a link between the circuit classes AC (where many lower bounds are known) and TC (where essentially no lower bounds are known). In this paper, we resolve several questions regarding the closure properties of #AC .
On Cluster Machines and Function Classes
, 1997
"... We consider a special kind of nondeterministic Turing machines. Cluster machines are distinguished by a neighbourhood relationship between accepting paths. Based on a formalization using equivalence relations some subtle properties of these machines are proven. Moreover, by abstraction we gain the ..."
Abstract

Cited by 9 (1 self)
 Add to MetaCart
We consider a special kind of nondeterministic Turing machines. Cluster machines are distinguished by a neighbourhood relationship between accepting paths. Based on a formalization using equivalence relations some subtle properties of these machines are proven. Moreover, by abstraction we gain the machineindependend concept of cluster sets which is the starting point to establish cluster operators. Cluster operators map complexity classes of sets into complexity classes of functions where for the domain classes only cluster sets are allowed. For the counting operator c#\Delta and the optimization operators cmax\Delta and cmin\Delta the structural relationships between images resulting from these operators on the polynomialtime hierarchy are investigated. Furthermore, we compare cluster operators with the corresponding common operators #\Delta, max\Delta and min\Delta [Tod90b, HW97].
Reducing the Number of Solutions of NP Functions
, 2000
"... We study whether one can prune solutions from NP functions. Though it is known that, unless surprising complexity class collapses occur, one cannot reduce the number of accepting paths of NP machines [OH93], we nonetheless show that it often is possible to reduce the number of solutions of NP functi ..."
Abstract

Cited by 7 (4 self)
 Add to MetaCart
We study whether one can prune solutions from NP functions. Though it is known that, unless surprising complexity class collapses occur, one cannot reduce the number of accepting paths of NP machines [OH93], we nonetheless show that it often is possible to reduce the number of solutions of NP functions. For finite cardinality types, we give a sufficient condition for such solution reduction. We also give absolute and conditional necessary conditions for solution reduction, and in particular we show that in many cases solution reduction is impossible unless the polynomial hierarchy collapses.
Promise Problems and Access to Unambiguous Computation
, 1992
"... This paper studies the power of three types of access to unambiguous computation: nonadaptive access, faulttolerant access, and guarded access. (1) Though for NP it is known that nonadaptive access has exponentially terse adaptive simulations, we show that UP has no such relativizable simulations: ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
This paper studies the power of three types of access to unambiguous computation: nonadaptive access, faulttolerant access, and guarded access. (1) Though for NP it is known that nonadaptive access has exponentially terse adaptive simulations, we show that UP has no such relativizable simulations: there are worlds in which (k + 1)truthtable access to UP is not subsumed by kTuring access to UP. (2) Though faulttolerant access (i.e., "lhelping" access) to NP is known to be no more powerful than NP itself, we give both structural and relativized evidence that fault tolerant access to UP suffices to recognize even sets beyond UP. Furthermore, we completely characterize, in terms of locally positive reductions, the sets that faulttolerantly reduce to UP. (3) In guarded access, Grollmann and Selman's natural notion of access to unambiguous computation, a deterministic polynomialtime Turing machine asks questions to a nondeterministic polynomialtime Turing machine in such a way that the nondeterministic machine never accepts ambiguously. In contrast to guarded access, the standard notion of access to unambiguous computation is that of access to a set that is uniformly unambiguouseven for queries that it never will be asked by its questioner, it must be unambiguous. We show that these notions, though the same for nonadaptive reductions, differ for Turing and strong nondeterministic reductions.
Generalized Theorems on the Relationships among Reducibility Notions to Certain Complexity Classes
 Mathematical Systems Theory
, 1994
"... In this paper, we give several generalized theorems concerning reducibility notions to certain complexity classes. We study classes that are either (I) closed under NP manyone reductions and polynomial time conjunctive reductions or (II) closed under coNP manyone reductions and polynomial time dis ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
In this paper, we give several generalized theorems concerning reducibility notions to certain complexity classes. We study classes that are either (I) closed under NP manyone reductions and polynomial time conjunctive reductions or (II) closed under coNP manyone reductions and polynomial time disjunctive reductions. We prove that for such a class K, reducibility notions of sets to K under polynomial time constantround truthtable reducibility, polynomial time logTuring reducibility, logspace constantround truthtable reducibility, logspace logTuring reducibility and logspace Turing reducibility are all equivalent and (2) every set that is polynomial time positive Turing reducible to a set in K is already in K. From these results, we derive some observations on the reducibility notions to C=P and NP.