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48
Algebrization: A new barrier in complexity theory
 MIT Theory of Computing Colloquium
, 2007
"... Any proof of P � = NP will have to overcome two barriers: relativization and natural proofs. Yet over the last decade, we have seen circuit lower bounds (for example, that PP does not have linearsize circuits) that overcome both barriers simultaneously. So the question arises of whether there is a ..."
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Cited by 51 (3 self)
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Any proof of P � = NP will have to overcome two barriers: relativization and natural proofs. Yet over the last decade, we have seen circuit lower bounds (for example, that PP does not have linearsize circuits) that overcome both barriers simultaneously. So the question arises of whether there is a third barrier to progress on the central questions in complexity theory. In this paper we present such a barrier, which we call algebraic relativization or algebrization. The idea is that, when we relativize some complexity class inclusion, we should give the simulating machine access not only to an oracle A, but also to a lowdegree extension of A over a finite field or ring. We systematically go through basic results and open problems in complexity theory to delineate the power of the new algebrization barrier. First, we show that all known nonrelativizing results based on arithmetization—both inclusions such as IP = PSPACE and MIP = NEXP, and separations such as MAEXP � ⊂ P/poly —do indeed algebrize. Second, we show that almost all of the major open problems—including P versus NP, P versus RP, and NEXP versus P/poly—will require nonalgebrizing techniques. In some cases algebrization seems to explain exactly why progress stopped where it did: for example, why we have superlinear circuit lower bounds for PromiseMA but not for NP. Our second set of results follows from lower bounds in a new model of algebraic query complexity, which we introduce in this paper and which is interesting in its own right. Some of our lower bounds use direct combinatorial and algebraic arguments, while others stem from a surprising connection between our model and communication complexity. Using this connection, we are also able to give an MAprotocol for the Inner Product function with O ( √ n log n) communication (essentially matching a lower bound of Klauck), as well as a communication complexity conjecture whose truth would imply NL � = NP. 1
TimeSpace Tradeoffs for Satisfiability
 Journal of Computer and System Sciences
, 1997
"... We give the first nontrivial modelindependent timespace tradeoffs for satisfiability. Namely, we show that SAT cannot be solved simultaneously in n 1+o(1) time and n 1\Gammaffl space for any ffl ? 0 on general randomaccess nondeterministic Turing machines. In particular, SAT cannot be solved ..."
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Cited by 37 (1 self)
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We give the first nontrivial modelindependent timespace tradeoffs for satisfiability. Namely, we show that SAT cannot be solved simultaneously in n 1+o(1) time and n 1\Gammaffl space for any ffl ? 0 on general randomaccess nondeterministic Turing machines. In particular, SAT cannot be solved deterministically by a Turing machine using quasilinear time and p n space. We also give lower bounds for logspace uniform NC 1 circuits and branching programs. Our proof uses two basic ideas. First we show that if SAT can be solved nondeterministically with a small amount of time then we can collapse a nonconstant number of levels of the polynomialtime hierarchy. We combine this work with a result of Nepomnjascii that shows that a nondeterministic computation of super linear time and sublinear space can be simulated in alternating linear time. A simple diagonalization yields our main result. We discuss how these bounds lead to a new approach to separating the complexity classes NL a...
Reductions in Circuit Complexity: An Isomorphism Theorem and a Gap Theorem
 Journal of Computer and System Sciences
"... We show that all sets that arecomplete for NP under nonuniform AC are isomorphic under nonuniform AC computable isomorphisms. Furthermore, these sets remain NPcomplete even under nonuniform NC reductions. ..."
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Cited by 31 (12 self)
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We show that all sets that arecomplete for NP under nonuniform AC are isomorphic under nonuniform AC computable isomorphisms. Furthermore, these sets remain NPcomplete even under nonuniform NC reductions.
On Limited versus Polynomial Nondeterminism
, 1997
"... In this paper, we show that efficient algorithms for some problems that require limited nondeterminism imply the subexponential simulation of nondeterministic computation by deterministic computation. In particular, if cliques of size O(log n) can be found in polynomial time, then nondeterministic t ..."
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Cited by 28 (0 self)
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In this paper, we show that efficient algorithms for some problems that require limited nondeterminism imply the subexponential simulation of nondeterministic computation by deterministic computation. In particular, if cliques of size O(log n) can be found in polynomial time, then nondeterministic time f(n) is contained in deterministic time 2 O( p f(n)polylogf(n)) .
On the Complexity of SAT
, 1999
"... We show that nondeterministic time NT IME(n) is not contained in deterministic time n # 2# and polylogarithmic space, for any # > 0. This implies that (infinitely often) satisfiability cannot be solved in time O(n # 2# ) and polylogarithmic space. A similar result is presented for uniform cir ..."
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Cited by 25 (1 self)
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We show that nondeterministic time NT IME(n) is not contained in deterministic time n # 2# and polylogarithmic space, for any # > 0. This implies that (infinitely often) satisfiability cannot be solved in time O(n # 2# ) and polylogarithmic space. A similar result is presented for uniform circuits.
Nearly linear time
 In Logic at Botik ’89, Symposium on Logical Foundations of Computer Science
, 1989
"... The notion of lineartime computability is very sensitive to machine model. In this connection, we introduce a class NLT of functions computable in nearly linear time n(log n) O(1) on random access computers. NLT is very robust and does not depend on the particular choice of random access computers. ..."
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Cited by 18 (1 self)
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The notion of lineartime computability is very sensitive to machine model. In this connection, we introduce a class NLT of functions computable in nearly linear time n(log n) O(1) on random access computers. NLT is very robust and does not depend on the particular choice of random access computers. Kolmogorov machines, Schonhage machines, random access Turing machines, etc. also compute exactly NLT functions in nearly linear time. It is not known whether usual multitape Turing machines are able to compute all NLT functions in nearly linear time. We do not believe they are and do not consider them necessarily appropriate for this relatively low complexity level. It turns out, however, that nondeterministic Turing machines accept exactly the languages in the nondeterministic version of NLT. We give also a machineindependent denition of NLT and a natural problem complete for NLT.
Transparent Proofs and Limits to Approximation
, 1994
"... We survey a major collective accomplishment of the theoretical computer science community on efficiently verifiable proofs. Informally, a formal proof is transparent (or holographic) if it can be verified with large confidence by a small number of spotchecks. Recent work by a large group of researc ..."
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Cited by 16 (0 self)
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We survey a major collective accomplishment of the theoretical computer science community on efficiently verifiable proofs. Informally, a formal proof is transparent (or holographic) if it can be verified with large confidence by a small number of spotchecks. Recent work by a large group of researchers has shown that this seemingly paradoxical concept can be formalized and is feasible in a remarkably strong sense; every formal proof in ZF, say, can be rewritten in transparent format (proving the same theorem in a different proof system) without increasing the length of the proof by too much. This result in turn has surprising implications for the intractability of approximate solutions of a wide range of discrete optimization problems, extending the pessimistic predictions of the PNP theory to approximate solvability. We discuss the main results on transparent proofs and their implications to discrete optimization. We give an account of several links between the two subjects as well ...
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 sat ..."
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Cited by 15 (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.
Inductive TimeSpace Lower Bounds for SAT and Related Problems
 Computational Complexity
, 2005
"... Abstract. We improve upon indirect diagonalization arguments for lower bounds on explicit problems within the polynomial hierarchy. Our contributions are summarized as follows. 1. We present a technique that uniformly improves upon most known nonlinear time lower bounds for nondeterminism andalterna ..."
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Cited by 15 (4 self)
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Abstract. We improve upon indirect diagonalization arguments for lower bounds on explicit problems within the polynomial hierarchy. Our contributions are summarized as follows. 1. We present a technique that uniformly improves upon most known nonlinear time lower bounds for nondeterminism andalternating computation, on both subpolynomial (n o(1) ) space RAMs and sequential onetape machines with random access to the input. We obtain improved lower bounds for Boolean satisfiability (SAT), as well as all NPcomplete problems that have efficient reductions from SAT, and ΣkSAT, for constant k ≥ 2. For example, SAT cannot be solved by random access machines using n √ 3 time and subpolynomial space. 2. We show how indirect diagonalization leads to timespace lower bounds for computation with bounded nondeterminism. For both the random access and multitape Turing machine models, we prove that for all k ≥ 1, there is a constant ck> 1 such that linear time with n 1/k nondeterministic bits is not contained in deterministic n ck time with subpolynomial space. This is used to prove that satisfiability of Boolean circuits with n inputs and n k size cannot be solved by deterministic multitape Turing machines running in n k·ck time and subpolynomial space.
Algorithms and Resource Requirements for Fundamental Problems
, 2007
"... no. DGE0234630. The views and conclusions contained in this document are those of the author and should not be interpreted as representing the official policies, either expressed or implied, of any sponsoring institution, the U.S. government or any other entity. ..."
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Cited by 12 (6 self)
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no. DGE0234630. The views and conclusions contained in this document are those of the author and should not be interpreted as representing the official policies, either expressed or implied, of any sponsoring institution, the U.S. government or any other entity.