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33
Search problems in the decision tree model
 SIAM JOURNAL ON DISCRETE MATHEMATICS
, 1995
"... We study the relative power of determinism, randomness and nondeterminism for search problems in the Boolean decision tree model. We show that the gaps between the nondeterministic, the randomized and the deterministic complexities can be arbitrary large for search problems. We also mention an inter ..."
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We study the relative power of determinism, randomness and nondeterminism for search problems in the Boolean decision tree model. We show that the gaps between the nondeterministic, the randomized and the deterministic complexities can be arbitrary large for search problems. We also mention an interesting connection of this model to the complexity of resolution proofs.
A General Method to Construct Oracles Realizing Given Relationships between Complexity Classes
, 1994
"... We present a method to prove oracle theorems of the following type. ..."
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Cited by 9 (1 self)
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We present a method to prove oracle theorems of the following type.
Search versus Decision for Election Manipulation Problems
, 2013
"... Most theoretical definitions about the complexity of manipulating elections focus on the decision problem of recognizing which instances can be successfully manipulated, rather than the search problem of finding the successful manipulative actions. Since the latter is a far more natural goal for man ..."
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Most theoretical definitions about the complexity of manipulating elections focus on the decision problem of recognizing which instances can be successfully manipulated, rather than the search problem of finding the successful manipulative actions. Since the latter is a far more natural goal for manipulators, that definitional focus may be misguided if these two complexities can differ. Our main result is that they probably do differ: If integer factoring is hard, then for election manipulation, election bribery, and some types of election control, there are election systems for which recognizing which instances can be successfully manipulated is in polynomial time but producing the successful manipulations cannot be done in polynomial time.
A Tight Relationship between Generic Oracles and Type2 Complexity Theory
, 1997
"... We show that any two complexity classes satisfying some general conditions are distinct relative to a generic oracle iff the corresponding type2 classes are distinct. ..."
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Cited by 7 (1 self)
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We show that any two complexity classes satisfying some general conditions are distinct relative to a generic oracle iff the corresponding type2 classes are distinct.
ArthurMerlin Games in Boolean Decision Trees
, 1997
"... It is well known that probabilistic boolean decision trees cannot be much more powerful than deterministic ones (N. Nisan, SIAM Journal on Computing, 20(6):9991007, 1991). Motivated by a question if randomization can significantly speed up a nondeterministic computation via a boolean decision tree ..."
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It is well known that probabilistic boolean decision trees cannot be much more powerful than deterministic ones (N. Nisan, SIAM Journal on Computing, 20(6):9991007, 1991). Motivated by a question if randomization can significantly speed up a nondeterministic computation via a boolean decision tree, we address structural properties of ArthurMerlin games in this model and prove some lower bounds. We consider two cases of interest, the first when the length of communication between the players is bounded and the second if it is not. While in the first case we can carry over the relations between the corresponding Turing complexity classes, in the second case we observe in contrast with Turing complexity that a one round MerlinArthur protocol is as powerful as a general interactive proof system and, in particular, can simulate a oneround ArthurMerlin protocol. Moreover, we show that sometimes a MerlinArthur protocol can be more efficient than an ArthurMerlin protocol, and than a Me...
Nondeterministic functions and the existence of optimal proof systems
 THEOR. COMPUT. SCI
, 2009
"... Abstract. We provide new characterizations of two previously studied questions on nondeterministic function classes: Q1: Do nondeterministic functions admit efficient deterministic refinements? Q2: Do nondeterministic function classes contain complete functions? We show that Q1 for the class NPMVt i ..."
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Abstract. We provide new characterizations of two previously studied questions on nondeterministic function classes: Q1: Do nondeterministic functions admit efficient deterministic refinements? Q2: Do nondeterministic function classes contain complete functions? We show that Q1 for the class NPMVt is equivalent to the question whether the standard proof system for SAT is poptimal, and to the assumption that every optimal proof system is poptimal. Assuming only the existence of a poptimal proof system for SAT, we show that every set with an optimal proof system has a poptimal proof system. Under the latter assumption, we also obtain a positive answer to Q2 for the class NPMVt. An alternative view on nondeterministic functions is provided by disjoint sets and tuples. We pursue this approach for disjoint NPpairs and its generalizations to tuples of sets from NP and coNP with disjointness conditions of varying strength. In this way, we obtain new characterizations of Q2 for the class NPSV. Question Q1 for NPSV is equivalent to the question whether every disjoint NPpair is easy to separate. In addition, we characterize this problem by the question whether every propositional proof system has the effective interpolation property. Again, these interpolation properties are intimately connected to disjoint NPpairs, and we show how different interpolation properties can be modeled by NPpairs associated with the underlying proof system.
On The BPP Hierarchy Problem
, 1997
"... In this paper we give arguments both for and against the existence of an oracle A, relative to which BPP equals probabilistic linear time. First, we prove a structure theorem for probabilistic oracle machines, which says that either we can fix the output of the machine by setting the answer to only ..."
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In this paper we give arguments both for and against the existence of an oracle A, relative to which BPP equals probabilistic linear time. First, we prove a structure theorem for probabilistic oracle machines, which says that either we can fix the output of the machine by setting the answer to only polynomially many oracle strings, or else we can set part of the oracle such that the machine becomes improper. This theorem could help complete the construction of the oracle A, which was proposed by Fortnow and Sipser in [2]. Second, we show that there are previously unknown problems with this construction. Thus the question whether probabilistic polynomial time has a hierarchy relative to all oracles remains completely open.
Inverting Onto Functions and Polynomial Hierarchy
"... Abstract. The class TFNP, defined by Megiddo and Papadimitriou, consists of multivalued functions with values that are polynomially verifiable and guaranteed to exist. Do we have evidence that such functions are hard, for example, if TFNP is computable in polynomialtime does this imply the polynomi ..."
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Abstract. The class TFNP, defined by Megiddo and Papadimitriou, consists of multivalued functions with values that are polynomially verifiable and guaranteed to exist. Do we have evidence that such functions are hard, for example, if TFNP is computable in polynomialtime does this imply the polynomialtime hierarchy collapses? By computing a multivalued function in deterministic polynomialtime we mean on every input producing one of the possible values of that function on that input. We give a relativized negative answer to this question by exhibiting an oracle under which TFNP functions are easy to compute but the polynomialtime hierarchy is infinite. We also show that relative to this same oracle, P ̸ = UP and TFNP NP functions are not computable in polynomialtime with an NP oracle. 1
Feasibly Continuous TypeTwo Functionals
, 1997
"... A wellknown theorem of typetwo recursion theory states that a functional is continuous if and only if it is computable relative to some oracle. We show that a feasible analogue of this theorem holds, using techniques originally developed in the study of Boolean decision tree complexity. 1 Introduc ..."
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A wellknown theorem of typetwo recursion theory states that a functional is continuous if and only if it is computable relative to some oracle. We show that a feasible analogue of this theorem holds, using techniques originally developed in the study of Boolean decision tree complexity. 1 Introduction Typetwo computability theory deals with the computability of functionals, which take functions and numbers as input, and produce numbers as output. A surprising and pleasing aspect of typetwo computability is its close connections with topology on Baire space. Notions of relative typetwo computability (that is, computability with respect to some oracle,) can be characterized using purely topological notions. In particular, a typetwo functional is computable relative to an oracle if and only if it is continuous. While the theory of typetwo computability has been widely successful, relatively little work has been done on the development of a complexity theory for typetwo functionals...