Results 1  10
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44
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 ..."
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Cited by 35 (8 self)
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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 ..."
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Cited by 32 (9 self)
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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 ..."
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Cited by 13 (2 self)
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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 cou ..."
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Cited by 11 (1 self)
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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.
ErrorBounded Probabilistic Computations Between MA and AM
 IN PROCEEDINGS 28TH MATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCE
, 2002
"... We introduce the probabilistic class SBP which is defined in a BPPlike manner. This class emerges from BPP by keeping the promise of a probability gap but decreasing the probability limit from 1/2 to exponentially small values. We show ..."
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Cited by 11 (1 self)
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We introduce the probabilistic class SBP which is defined in a BPPlike manner. This class emerges from BPP by keeping the promise of a probability gap but decreasing the probability limit from 1/2 to exponentially small values. We show
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 ..."
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Cited by 10 (3 self)
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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 ..."
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Cited by 9 (1 self)
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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].
Looking for an analogue of Rice's Theorem in circuit complexity theory
 Mathematical Logic Quarterly
, 1989
"... Abstract. Rice’s Theorem says that every nontrivial semantic property of programs is undecidable. In this spirit we show the following: Every nontrivial absolute (gap, relative) counting property of circuits is UPhard with respect to polynomialtime Turing reductions. For generators [31] we show a ..."
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Cited by 7 (1 self)
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Abstract. Rice’s Theorem says that every nontrivial semantic property of programs is undecidable. In this spirit we show the following: Every nontrivial absolute (gap, relative) counting property of circuits is UPhard with respect to polynomialtime Turing reductions. For generators [31] we show a perfect analogue of Rice’s Theorem. Mathematics Subject Classification: 03D15, 68Q15. Keywords: Rice’s Theorem, Counting problems, Promise classes, UPhard, NPhard, generators.
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 ..."
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Cited by 7 (4 self)
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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.