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Graph Nonisomorphism Has Subexponential Size Proofs Unless The PolynomialTime Hierarchy Collapses
 SIAM Journal on Computing
, 1998
"... We establish hardness versus randomness tradeoffs for a broad class of randomized procedures. In particular, we create efficient nondeterministic simulations of bounded round ArthurMerlin games using a language in exponential time that cannot be decided by polynomial size oracle circuits with acce ..."
Abstract

Cited by 110 (4 self)
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We establish hardness versus randomness tradeoffs for a broad class of randomized procedures. In particular, we create efficient nondeterministic simulations of bounded round ArthurMerlin games using a language in exponential time that cannot be decided by polynomial size oracle circuits with access to satisfiability. We show that every language with a bounded round ArthurMerlin game has subexponential size membership proofs for infinitely many input lengths unless exponential time coincides with the third level of the polynomialtime hierarchy (and hence the polynomialtime hierarchy collapses). This provides the first strong evidence that graph nonisomorphism has subexponential size proofs. We set up a general framework for derandomization which encompasses more than the traditional model of randomized computation. For a randomized procedure to fit within this framework, we only require that for any fixed input the complexity of checking whether the procedure succeeds on a given ...
SuperPolynomial versus HalfExponential Circuit Size in the Exponential Hierarchy
, 1999
"... . Circuit size lower bounds were previously established for functions in p 2 , ZPP NP , exp 2 , ZPEXP NP and MA exp . We ask the general question: Given a time bound t(n). What is the best circuit size lower bound that can be currently shown for the classes MATIME[t(n)], ZPTIME NP [t(n ..."
Abstract

Cited by 18 (4 self)
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. Circuit size lower bounds were previously established for functions in p 2 , ZPP NP , exp 2 , ZPEXP NP and MA exp . We ask the general question: Given a time bound t(n). What is the best circuit size lower bound that can be currently shown for the classes MATIME[t(n)], ZPTIME NP [t(n)]; . . .? For the classes MA exp , ZPEXP NP and exp 2 , the answer turns out to be \halfexponential". Informally, a function f is said to be halfexponential when f f is exponential. Such functions were constructed by Szekeres. 1 Introduction One main issue of complexity theory is how powerful nonuniform (e.g. circuit based) computation is, compared to uniform (machine based) computation. In particular, a 64K dollar question is whether exponential time has polynomial size circuits. This being a challenging open question, a series of papers have looked at circuit size of functions further up the exponential hierarchy. In the early eighties, Kannan [11] established that there is ...