Nonrelativizing separations

Buhrman, Harry; Fortnow, Lance; Thierauf, Thomas

doi:10.1109/ccc.1998.694585

Cited by 51 publications

(65 citation statements)

References 14 publications

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“…In particular, we do not know of any Karp-Lipton style collapse 5 with that setting of s. We may attempt to replace SAT with a seemingly harder language, say some language in a class C, where C is contained in the class for which we want to show a lower bound. Unfortunately this issue cannot be fixed this way; for otherwise we would not have the current gap between the known super-polynomial size lower bounds (which hold with respect to classes that are contained in the second level, namely MA-EXP (Buhrman, Fortnow, and Thierauf (1998))), and the known exponential-size lower bounds (which only hold with respect to classes that are in the third level of the hierarchy, namely E Σ P 2 (Kannan (1982))). Indeed, it was argued by Miltersen, Vinodchandran, and Watanabe (1999) that Karp-Lipton style collapses that are needed for Kannan's strategy hold with respect to size functions up to half-exponential (a function s is half-exponential if s(s(n)) ∈ 2 Θ(n) ) but do not seem to carry over to larger size bounds, to 2…”

Section: +ω(1)mentioning

confidence: 99%

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Derandomizing Arthur-Merlin Games and Approximate Counting Implies Exponential-Size Lower Bounds

Gutfreund

Kawachi

2011

comput. complex.

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Abstract. We show that if Arthur-Merlin protocols can be derandomized, then there is a language computable in deterministic exponentialtime with access to an NP oracle, that requires circuits of exponential size. More formally, if every promise problem in prAM, the class of promise problems that have Arthur-Merlin protocols, can be computed by a deterministic polynomial-time algorithm with access to an NP oracle then there is a language in E NP that requires circuits of size Ω(2 n /n). The lower bound in the conclusion of our theorem suffices to construct pseudorandom generators with exponential stretch. We also show that the same conclusion holds if the following two related problems can be computed in polynomial time with access to an NP-oracle: (i) approximately counting the number of accepted inputs of a circuit, up to multiplicative factors; and (ii) recognizing an approximate lower bound on the number of accepted inputs of a circuit, up to multiplicative factors.

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Section: +ω(1)mentioning

confidence: 99%

“…The first result of this flavor was given by Buhrman, Fortnow, and Thierauf (1998) who showed that if the class MA is equal to NP then NEXP ⊆ P/poly.…”

mentioning

confidence: 99%

Derandomizing Arthur-Merlin Games and Approximate Counting Implies Exponential-Size Lower Bounds

Gutfreund

Kawachi

2011

comput. complex.

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“…The class of functions #EXP is defined analogously to #P, except with T a non-deterministic exponential-time machine. We will deal with a decision version of #EXP, PEXP, the set of problems solvable by nondeterministic Turing machine in exponential time, where the acceptance condition is that more than a half of computation paths accept [BFT98].…”

Section: Proof Notice That Ifmentioning

confidence: 99%

Two Variable vs. Linear Temporal Logic in Model Checking and Games

Benedikt¹,

Lenhardt²,

Worrell³

2013

Logical Methods in Computer Science

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Abstract. Model checking linear-time properties expressed in first-order logic has nonelementary complexity, and thus various restricted logical languages are employed. In this paper we consider two such restricted specification logics, linear temporal logic (LTL) and two-variable first-order logic (FO 2 ). LTL is more expressive but FO 2 can be more succinct, and hence it is not clear which should be easier to verify. We take a comprehensive look at the issue, giving a comparison of verification problems for FO 2 , LTL, and various sublogics thereof across a wide range of models. In particular, we look at unary temporal logic (UTL), a subset of LTL that is expressively equivalent to FO 2 ; we also consider the stutterfree fragment of FO 2 , obtained by omitting the successor relation, and the expressively equivalent fragment of UTL, obtained by omitting the next and previous connectives.We give three logic-to-automata translations which can be used to give upper bounds for FO 2 and UTL and various sub-logics. We apply these to get new bounds for both nondeterministic systems (hierarchical and recursive state machines, games) and for probabilistic systems (Markov chains, recursive Markov chains, and Markov decision processes). We couple these with matching lower-bound arguments.Next, we look at combining FO 2 verification techniques with those for LTL. We present here a language that subsumes both FO 2 and LTL, and inherits the model checking properties of both languages. Our results give both a unified approach to understanding the behaviour of FO 2 and LTL, along with a nearly comprehensive picture of the complexity of verification for these logics and their sublogics.

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“…However, even non-uniform classes such as P/poly can be separated from large enough classes by means of diagonalization. Diagonalization combined with "arithmetization 1 " yields the best-known result along these lines: the result of Buhrman, Fortnow and Thierauf [14] that MA EXP is not contained in P/poly (and hence is also not contained in (non-uniform) TC 0 ). Is there hope that these techniques might lead to better lower bounds for TC 0 ?…”

Section: Tried and True Techniquesmentioning

confidence: 95%

“…Diagonalization is the canonical example of a "relativizing" proof technique, and even when when combined with arithmetization techniques the known separation results "algebrize" (using the terminology introduced by Aaronson and Wigderson [1]). Aaronson and Wigderson show that algebrizing proof techniques are not strong enough to prove that NEXP is not in P/poly (so that the lower bound of [14] mentioned above is close to the best that can be obtained using these techniques).…”

Section: Tried and True Techniquesmentioning

confidence: 99%

Cracks in the Defenses: Scouting Out Approaches on Circuit Lower Bounds

Allender

Computer Science – Theory and Applications

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Abstract. Razborov and Rudich identified an imposing barrier that stands in the way of progress toward the goal of proving superpolynomial lower bounds on circuit size. Their work on "natural proofs" applies to a large class of arguments that have been used in complexity theory, and shows that no such argument can prove that a problem requires circuits of superpolynomial size, even for some very restricted classes of circuits (under reasonable cryptographic assumptions). This barrier is so daunting, that some researchers have decided to focus their attentions elsewhere. Yet the goal of proving circuit lower bounds is of such importance, that some in the community have proposed concrete strategies for surmounting the obstacle. This lecture will discuss some of these strategies, and will dwell at length on a recent approach proposed by Michal Koucký and the author.

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Nonrelativizing separations

Cited by 51 publications

References 14 publications

Derandomizing Arthur-Merlin Games and Approximate Counting Implies Exponential-Size Lower Bounds

Derandomizing Arthur-Merlin Games and Approximate Counting Implies Exponential-Size Lower Bounds

Two Variable vs. Linear Temporal Logic in Model Checking and Games

Cracks in the Defenses: Scouting Out Approaches on Circuit Lower Bounds

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