18 research outputs found

    List of R.V. Book's publications

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    Unconditional Lower Bounds against Advice

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    We show several unconditional lower bounds for exponential time classes against polynomial time classes with advice, including: 1. For any constant c, NEXP ̸ ⊆ P NP[nc

    Computational complexity theory and the philosophy of mathematics

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    Computational complexity theory is a subfield of computer science originating in computability theory and the study of algorithms for solving practical mathematical problems. Amongst its aims is classifying problems by their degree of difficulty — i.e., how hard they are to solve computationally. This paper highlights the significance of complexity theory relative to questions traditionally asked by philosophers of mathematics while also attempting to isolate some new ones — e.g., about the notion of feasibility in mathematics, the P≠NP problem and why it has proven hard to resolve, and the role of non-classical modes of computation and proof

    The complexity of parameters for probabilistic and quantum computation

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    In this dissertation we study some effects of allowing computational models that use parameters whose own computational complexity has a strong effect on the computational complexity of the languages computable from the model. We show that in the probabilistic and quantum models there are parameter sets that allow one to obtain noncomputable outcomes;In Chapter 3 we define BP[beta]P the BPP class based on a coin with bias [beta]. We then show that if [beta] is BPP-computable then it is the case that BP[beta]P = BPP. We also show that each language L in P/CLog is in BP[beta]P for some [beta]. Hence there are some [beta] from which we can compute noncomputable languages. We also examine the robustness of the class BPP with respect to small variations from fairness in the coin;In Chapter 4 we consider measures that are based on polynomial-time computable sequences of biased coins in which the biases are bounded away from both zero and one (strongly positive P-sequences). We show that such a sequence [vector][beta] generates a measure [mu][vector][beta] equivalent to the uniform measure in the sense that if C is a class of languages closed under positive, polynomial-time, truth-table reductions with queries of linear length then C has [mu][vector][beta]-measure zero if and only if it has measure zero relative to the uniform measure [mu]. The classes P, NP, BPP, P/Poly, PH, and PSPACE are among those to which this result applies. Thus the measures of these much-studied classes are robust with respect to changes of this type in the underlying probability measure;In Chapter 5 we introduce the quantum computation model and the quantum complexity class BQP. We claim that the computational complexity of the amplitudes is a critical factor in determining the languages computable using the quantum model. Using results from chapter 3 we show that the quantum model can also compute noncomputable languages from some amplitude sets. Finally, we determine a restriction on the amplitude set to limit the model to the range of languages implicit in others\u27 typical meaning of the class BQP

    Two Applications of Inductive Counting for Complementation Problems

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