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The P versus NP problem
 Clay Mathematical Institute; The Millennium Prize Problem
, 2000
"... The P versus NP problem is to determine whether every language accepted by some nondeterministic algorithm in polynomial time is also accepted by some (deterministic) algorithm in polynomial time. To define the problem precisely it is necessary to give a formal model of a computer. The standard comp ..."
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The P versus NP problem is to determine whether every language accepted by some nondeterministic algorithm in polynomial time is also accepted by some (deterministic) algorithm in polynomial time. To define the problem precisely it is necessary to give a formal model of a computer. The standard computer model in computability theory is the Turing machine, introduced by Alan Turing in 1936 [37]. Although the model was introduced before physical computers were built, it nevertheless continues to be accepted as the proper computer model for the purpose of defining the notion of computable function. Informally the class P is the class of decision problems solvable by some algorithm within a number of steps bounded by some fixed polynomial in the length of the input. Turing was not concerned with the efficiency of his machines, rather his concern was whether they can simulate arbitrary algorithms given sufficient time. It turns out, however, Turing machines can generally simulate more efficient computer models (for example, machines equipped with many tapes or an unbounded random access memory) by at most squaring or cubing the computation time. Thus P is a
Classification of Search Problems and Their Definability in Bounded Arithmetic
, 2001
"... Classication of Search Problems and Their Denability in Bounded Arithmetic Tsuyoshi Morioka Master of Science Graduate Department of Computer Science University of Toronto 2001 We present a new framework for the study of search problems and their denability in bounded arithmetic. We identify t ..."
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Classication of Search Problems and Their Denability in Bounded Arithmetic Tsuyoshi Morioka Master of Science Graduate Department of Computer Science University of Toronto 2001 We present a new framework for the study of search problems and their denability in bounded arithmetic. We identify two notions of complexity of search problems: veri cation complexity and computational complexity. Notions of exact solvability and exact reducibility are developed, and exact b i denability of search problems in bounded arithmetic is introduced. We specify a new machine model called the oblivious witnessoracle Turing machines.
Generic Separations and Leaf Languages
 Mathematical Logic Quaterly
, 2001
"... In the early nineties of the previous century, leaf languages were introduced as a means for the uniform characterization of many complexity classes, mainly in the range between P (polynomial time) and PSPACE (polynomial space). It was shown that the separability of two complexity classes can be ..."
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In the early nineties of the previous century, leaf languages were introduced as a means for the uniform characterization of many complexity classes, mainly in the range between P (polynomial time) and PSPACE (polynomial space). It was shown that the separability of two complexity classes can be reduced to a combinatorial property of the corresponding dening leaf languages. In the present paper, it is shown that every separation obtained in this way holds for every generic oracle in the sense of Blum and Impagliazzo. We obtain several consequences of this result, regarding, e.g., simultaneous separations and universal oracles, resourcebounded genericity, and type2 complexity. Keywords: computational and structural complexity, leaf language, oracle separation, generic oracle, type2 complexity theory. 1
The Relative Complexity of Local Search Heuristics And The Iteration Principle
, 2003
"... Johnson, Papadimitriou and Yannakakis introduce the class PLS consisting of optimization problems for which ecient localsearch heuristics exist. We formulate a type2 problem ITER that characterizes PLS in style of Beame et al., and prove a criterion for type2 problems to be nonreducible to ITER. As ..."
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Johnson, Papadimitriou and Yannakakis introduce the class PLS consisting of optimization problems for which ecient localsearch heuristics exist. We formulate a type2 problem ITER that characterizes PLS in style of Beame et al., and prove a criterion for type2 problems to be nonreducible to ITER. As a corollary, we obtain the rst relative separation of PLS from Papadimitriou's classes PPA, PPAD, PPADS, and PPP. Based on the criterion, we derive a special case of Riis's independence criterion for the Bounded Arithmetic 2 (L). We also prove that PLS is closed under Turing reducibility.