Non-uniform automata over groups

Barrington, David A. Mix; Thérien, Denis

doi:10.1007/3-540-18088-5_13

Cited by 29 publications

(32 citation statements)

References 21 publications

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“…This seems to be a natural conjecture dual to the fact that MOD q cannot be efficiently computed by AND and OR. The conjecture is indeed true when q = p k is a prime power [13]: any constant depth circuit consisting of only MOD q gates, q = p k , is equivalent to a constant degree polynomial over Z p and therefore cannot compute the AND function at all regardless of its size. On the other hand, when q is not a prime power exponentially large CC 0 circuits can compute any Boolean function [13].…”

Section: Introductionmentioning

confidence: 99%

“…The conjecture is indeed true when q = p k is a prime power [13]: any constant depth circuit consisting of only MOD q gates, q = p k , is equivalent to a constant degree polynomial over Z p and therefore cannot compute the AND function at all regardless of its size. On the other hand, when q is not a prime power exponentially large CC 0 circuits can compute any Boolean function [13]. Improving on results of Barrington [10] and Smolensky [40], Thérien [44] shows that CC 0 circuits computing AND require at least Ω(n) gates at the bottom level.…”

Section: Introductionmentioning

confidence: 99%

“…In fact, Barrington, Straubing and Thérien [13] conjectured that even the Boolean AND function cannot be computed by polynomial size bounded depth circuits consisting entirely of MOD q gates -CC 0 circuits. This seems to be a natural conjecture dual to the fact that MOD q cannot be efficiently computed by AND and OR.…”

Section: Introductionmentioning

confidence: 99%

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confidence: 99%

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A New Characterization of ACC^0 and Probabilistic CC^0

Hansen¹,

Koucky²

2009

2009 24th Annual IEEE Conference on Computational Complexity

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Abstract-Barrington, Straubing and Thérien (1990) conjectured that the Boolean AND function can not be computed by polynomial size constant depth circuits built from modular counting gates, i.e., by CC 0 circuits. In this work we show that the AND function can be computed by uniform probabilistic CC 0 circuits that use only O(log n) random bits. This may be viewed as evidence contrary to the conjecture.As a consequence of our construction we get that all of ACC 0 can be computed by probabilistic CC 0 circuits that use only O(log n) random bits. Thus, if one were able to derandomize such circuits, we would obtain a collapse of circuit classes giving ACC 0 = CC 0 . We present a derandomization of probabilistic CC 0 circuits using AND and OR gates to obtain ACC 0 = AND • OR • CC 0 = OR • AND • CC 0 . AND and OR gates of sublinear fan-in suffice.Both these results hold for uniform as well as non-uniform circuit classes. For non-uniform circuits we obtain the stronger conclusion that ACC 0 = rand − ACC 0 = rand − CC 0 = rand(log n)−CC 0 , i.e., probabilistic ACC 0 circuits can be simulated by probabilistic CC 0 circuits using only O(log n) random bits.As an application of our results we obtain a characterization of ACC 0 by constant width planar nondeterministic branching programs, improving a previous characterization for the quasipolynomial size setting.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…

…”

mentioning

confidence: 99%

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A New Characterization of ACC^0 and Probabilistic CC^0

Hansen¹,

Koucky²

2009

2009 24th Annual IEEE Conference on Computational Complexity

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“…There has been substantial work on trying to prove lower bounds for circuits with MOD 6 gates, and in general, circuits with MOD pq gates where p and q are distinct primes [BST90,Gro01,CGPT06]. Circuits with such gates appear to be possibly far more powerful than circuits with merely AND, OR, NOT, and MOD p gates (for some fixed prime p).…”

Section: Other Related Workmentioning

confidence: 99%

Time-Space Tradeoffs for Counting NP Solutions Modulo Integers

Williams

2007

Twenty-Second Annual IEEE Conference on Computational Complexity (CCC'07)

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Abstract.We prove the first time-space tradeoffs for counting the number of solutions to an NP problem modulo small integers, and also improve upon known time-space tradeoffs for Sat. Let m > 0 be an integer, and define MOD m -Sat 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 < 2 cos(π/7) ≈ 1.801, there is a d > 0 such that MOD p -Sat 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 MOD 6 -Sat, as well as MOD m -Sat 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.

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Exponential Size Lower Bounds for Some Depth 3 Circuits

Yan

Parberry

1994

Information and Computation

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Non-uniform automata over groups

Cited by 29 publications

References 21 publications

A New Characterization of ACC^0 and Probabilistic CC^0

A New Characterization of ACC^0 and Probabilistic CC^0

Time-Space Tradeoffs for Counting NP Solutions Modulo Integers

Exponential Size Lower Bounds for Some Depth 3 Circuits

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