Results 1  10
of
14
An oracle builder’s toolkit
, 2002
"... We show how to use various notions of genericity as tools in oracle creation. In particular, 1. we give an abstract definition of genericity that encompasses a large collection of different generic notions; 2. we consider a new complexity class AWPP, which contains BQP (quantum polynomial time), and ..."
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Cited by 47 (10 self)
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We show how to use various notions of genericity as tools in oracle creation. In particular, 1. we give an abstract definition of genericity that encompasses a large collection of different generic notions; 2. we consider a new complexity class AWPP, which contains BQP (quantum polynomial time), and infer several strong collapses relative to SPgenerics; 3. we show that under additional assumptions these collapses also occur relative to Cohen generics; 4. we show that relative to SPgenerics, ULIN ∩ coULIN ̸ ⊆ DTIME(n k) for any k, where ULIN is unambiguous linear time, despite the fact that UP ∪ (NP ∩ coNP) ⊆ P relative to these generics; 5. we show that there is an oracle relative to which NP/1∩coNP/1 ̸ ⊆ (NP∩coNP)/poly; and 6. we use a specialized notion of genericity to create an oracle relative to which NP BPP ̸ ⊇ MA.
Two queries
 In CCC
, 1999
"... We consider the question whether two queries to SAT are as powerful as one query. We show that if P NP�℄� P NP�℄then Locally either NP�coNP or NP has polynomialsize circuits. ..."
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Cited by 32 (7 self)
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We consider the question whether two queries to SAT are as powerful as one query. We show that if P NP�℄� P NP�℄then Locally either NP�coNP or NP has polynomialsize circuits.
TimeSpace Tradeoffs for Counting NP Solutions Modulo Integers
 In Proceedings of the 22nd IEEE Conference on Computational Complexity
, 2007
"... We prove the first timespace tradeoffs for counting the number of solutions to an NP problem modulo small integers, and also improve upon known timespace tradeoffs for Sat. Let m> 0 be an integer, and define MODmSat to be the problem of determining if a given Boolean formula has exactly km satisf ..."
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Cited by 11 (5 self)
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We prove the first timespace tradeoffs for counting the number of solutions to an NP problem modulo small integers, and also improve upon known timespace tradeoffs for Sat. Let m> 0 be an integer, and define MODmSat to be the problem of determining if a given Boolean formula has exactly km satisfying assignments, for some integer k. We show for all primes p except for possibly one of them, and for all c < 2cos(π/7) ≈ 1.801, there is a d> 0 such that MODpSat is not solvable in n c time and n d space by general algorithms. That is, there is at most one prime p that does not satisfy the tradeoff. We prove that the same limitation holds for Sat and MOD6Sat, as well as MODmSat for any composite m that is not a prime power. Our main tool is a general method for rapidly simulating deterministic computations with restricted space, by counting the number of solutions to NP predicates modulo integers. The simulation converts an ordinary algorithm into a “canonical ” one that consumes roughly the same amount of time and space, yet canonical algorithms have nice properties suitable for counting.
Randomness is Hard
 SIAM Journal on Computing
, 2000
"... We study the set of incompressible strings for various resource bounded versions of Kolmogorov complexity. The resource bounded versions of Kolmogorov complexity we study are: polynomial time CD complexity dened by Sipser, the nondeterministic variant due to Buhrman and Fortnow, and the polynomi ..."
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Cited by 6 (3 self)
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We study the set of incompressible strings for various resource bounded versions of Kolmogorov complexity. The resource bounded versions of Kolmogorov complexity we study are: polynomial time CD complexity dened by Sipser, the nondeterministic variant due to Buhrman and Fortnow, and the polynomial space bounded Kolmogorov complexity, CS introduced by Hartmanis. For all of these measures we dene the set of random strings R CD t , R CND t , and R CS s as the set of strings x such that CD t (x), CND t (x), and CS s (x) is greater than or equal to the length of x, for s and t polynomials. We show the following: MA NP R CD t , where MA is the class of MerlinArthur games dened by Babai. AM NP R CND t , where AM is the class of ArthurMerlin games. PSPACE NP cR CS s . In the last item cR CS s is the set of pairs <x; y> so that x is random given y. These results show that the set of random strings for various resource bounds is hard for...
The Complexity of Symmetric Boolean Parity Holant Problems (Extended Abstract)
"... Abstract. For certain subclasses of NP, ⊕P or #P characterized by local constraints, it is known that if there exist any problems that are not polynomial time computable within that subclass, then those problems are NP, ⊕P or #Pcomplete. Such dichotomy results have been proved for characterizatio ..."
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Cited by 6 (3 self)
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Abstract. For certain subclasses of NP, ⊕P or #P characterized by local constraints, it is known that if there exist any problems that are not polynomial time computable within that subclass, then those problems are NP, ⊕P or #Pcomplete. Such dichotomy results have been proved for characterizations such as Constraint Satisfaction Problems, and directed and undirected Graph Homomorphism Problems, often with additional restrictions. Here we give a dichotomy result for the more expressive framework of Holant Problems. These additionally allow for the expression of matching problems, which have had pivotal roles in complexity theory. As our main result we prove the dichotomy theorem that, for the class ⊕P, every set of boolean symmetric Holant signatures of any arities that is not polynomial time computable is ⊕Pcomplete. The result exploits some special properties of the class ⊕P and characterizes four distinct tractable subclasses within ⊕P. It leaves open the corresponding questions for NP, #P and #kP for k ̸ = 2. 1
Two Oracles that Force a Big Crunch
, 1999
"... The central theme of this paper is the construction of an oracle A such that NEXP A = P NP A . The construction of this oracle answers a long standing open question rst posed by Heller, and unsuccessfully attacked many times since. For the rst construction of the oracle, we present a new ty ..."
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Cited by 5 (1 self)
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The central theme of this paper is the construction of an oracle A such that NEXP A = P NP A . The construction of this oracle answers a long standing open question rst posed by Heller, and unsuccessfully attacked many times since. For the rst construction of the oracle, we present a new type of injury argument that we call \resource bounded injury." In the special case of the construction of this oracle, a tree method can be used to transform unbounded search into exponentially bounded, hence recursive, search. This transformation of the construction can be interleaved with another construction so that relative to the new combined oracle also P = UP = NP\coNP. This leads to the curious situation where LOW(NP) = P, but LOW(P NP ) = NEXP, and the complete p m degree for P NP collapses to a single pisomorphism type. 1 Introduction In 1978, Seiferas, Fischer and Meyer [SFM78] showed a very strong separation theorem for nondeterministic time: For time constru...
Degree bounds on polynomials and relativization theory
 In Proceedings of the 3rd IFIP International Conference on Theoretical Computer Science
, 2003
"... Abstract We demonstrate the applicability of the polynomial degree bound technique to notions such as the nonexistence of Turinghard sets in some relativized world, (non)uniform gapdefinability, and relativized separations. This way, we settle certain open questions of Hemaspaandra, Ramachandran & ..."
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Cited by 3 (2 self)
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Abstract We demonstrate the applicability of the polynomial degree bound technique to notions such as the nonexistence of Turinghard sets in some relativized world, (non)uniform gapdefinability, and relativized separations. This way, we settle certain open questions of Hemaspaandra, Ramachandran & Zimand [HRZ95] and Fenner, Fortnow & Kurtz [FFK94], extend results of Hemaspaandra, Jain & Vereshchagin [HJV93] and construct oracles achieving desired results.
Approximation of Boolean Functions by Combinatorial Rectangles
 Electr. Coll. on Comp. Compl
, 2000
"... This paper deals with the number of monochromatic combinatorial rectangles required to approximate a Boolean function on a constant fraction of all inputs, where each rectangle may be defined with respect to its own partition of the input variables. The main result of the paper is that the number of ..."
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Cited by 2 (2 self)
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This paper deals with the number of monochromatic combinatorial rectangles required to approximate a Boolean function on a constant fraction of all inputs, where each rectangle may be defined with respect to its own partition of the input variables. The main result of the paper is that the number of rectangles required for the approximation of Boolean functions in this model is very sensitive to the allowed error: There is an explicitly defined sequence of functions f n : {0, 1} n # {0, 1} such that f n has rectangle approximations with a constant number of rectangles and onesided error 1/3+o(1) or twosided error 1/4+2 #(n) , but, on the other hand, f n requires exponentially many rectangles if the error bounds are decreased by an arbitrarily small constant. Rectangle partitions and rectangle approximations with the same partition of the input variables for all rectangles have been thoroughly investigated in communication complexity theory. The complexity measures where each r...
FixedPolynomial Size Circuit Bounds
"... Abstract—In 1982, Kannan showed that Σ P 2 does not have n ksized circuits for any k. Do smaller classes also admit such circuit lower bounds? Despite several improvements of Kannan’s result, we still cannot prove that P NP does not have linear size circuits. Work of Aaronson and Wigderson provides ..."
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Cited by 2 (0 self)
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Abstract—In 1982, Kannan showed that Σ P 2 does not have n ksized circuits for any k. Do smaller classes also admit such circuit lower bounds? Despite several improvements of Kannan’s result, we still cannot prove that P NP does not have linear size circuits. Work of Aaronson and Wigderson provides strong evidence – the “algebrization ” barrier – that current techniques have inherent limitations in this respect. We explore questions about fixedpolynomial size circuit lower bounds around and beyond the algebrization barrier. We find several connections, including The following are equivalent: – NP is in SIZE(n k) (has O(n k)size circuit families) for some k – For each c, P NP[nc] k is in SIZE(n) for some k – ONP/1 is in SIZE(n k) for some k, where ONP is the class of languages accepted obliviously by NP machines, with witnesses for “yes ” instances depending only on the input length. For a large number of natural classes C and all k � 1, C is in SIZE(n k) if and only if C/1 ∩P/poly is in SIZE(n k). If there is a d such that MATIME(n) ⊆ NTIME(n d), then P NP does not have O(n k) size circuits for any k> 0. One cannot show n 2size circuit lower bounds for ⊕P without new nonrelativizing techniques. In particular, the proof that PP ̸ ⊆ SIZE(n k) for all k relies on the (relativizing) result that P PP ⊆ MA = ⇒ PP ̸ ⊆ SIZE(n k), and we give an oracle relative to which P ⊕P ⊆ MA and ⊕P ⊆ SIZE(n 2) both hold. I.
Circuit Lower Bounds Collapse Relativized Complexity Classes
, 1998
"... Since the publication of Furst, Saxe, and Sipser's seminal paper connecting AC 0 with the polynomial hierarchy [FSS84], it has been well known that circuit lower bounds allow you to construct oracles that separate complexity classes. We will show that similar circuit lower bounds allow you to cons ..."
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Since the publication of Furst, Saxe, and Sipser's seminal paper connecting AC 0 with the polynomial hierarchy [FSS84], it has been well known that circuit lower bounds allow you to construct oracles that separate complexity classes. We will show that similar circuit lower bounds allow you to construct oracles that collapse complexity classes. For example, based on Hastad's parity lower bound, we construct an oracle such that P = PH ae \PhiP = EXP. Dept. of Electrical Engineering and Computer Science, University of Illinois at Chicago, Rm. 1120, SEO Building, M/C 154, 851 S. Morgan St., Chicago, IL 606077053. beigel@eecs.uic.edu. www.eecs.uic.edu/¸beigel. Part of this work was performed while this author was at the Department of Computer Science, UMIACS, and the HumanComputer Interaction Laboratory, University of Maryland, College Park, on sabbatical from the Yale University Department of Computer Science. Supported in part by the National Science Foundation under grants CCR895...