Nondeterministic circuits, space complexity and quasigroups

Wolf, Marty J.

doi:10.1016/0304-3975(92)00014-i

Cited by 16 publications

(14 citation statements)

References 7 publications

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“…A stronger result showing that GroupIso can be solved in space O(log 2 n) was given in [LSZ76]. The same result for the case of quasigroups was obtained later by Wolf in [Wol94]. In spite of these facts, no deterministic polynomial time algorithm for these problems is known although they seem far from being NP-complete 1 .…”

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confidence: 80%

“…Observe that Tarjan's algorithm can in fact be converted into a polynomial time nondeterministic procedure for QGroupIso that uses only log 2 n nondeterministic bits, by guessing the mapping from the generator set in G 1 to G 2 instead of testing all possible 1-1 mappings, and then extend this partial map to the whole quasigroup. This observation is mentioned explicitly in [PY96,Wol94]. Papadimitriou and Yannakakis [PY96] show that the quasigroup isomorphism problem is in β 2 P, a restricted version of NP, where on input of length n, a polynomial time bounded Turing machine has access to O(log 2 n) non-deterministic bits (more detail is given in the preliminaries section).…”

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confidence: 99%

“…Papadimitriou and Yannakakis [PY96] show that the quasigroup isomorphism problem is in β 2 P, a restricted version of NP, where on input of length n, a polynomial time bounded Turing machine has access to O(log 2 n) non-deterministic bits (more detail is given in the preliminaries section). In [AT06] some evidence is given indicating that QGroupIso is probably not complete for β 2 P. Wolf [Wol94] improved the nondeterministic complexity of the problem by showing that QGroupIso ∈ β 2 NC 2 the class of problems computed by NC 2 circuits having additionally O(log 2 n) non-deterministic bits on inputs of size n. As in the β 2 P upper bound, the circuit can guess the generators of both quasigroups as well as a bijection between both generator sets. Wolf shows that checking whether this partial bijection can be extended to an isomorphism, can be done by an NC 2 circuit.…”

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confidence: 99%

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Graph Isomorphism is Not AC0-Reducible to Group Isomorphism

Chattopadhyay

Torán

Wagner

2013

ACM Trans. Comput. Theory

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We give a new upper bound for the Group and Quasigroup Isomorphism problems when the input structures are given explicitly by multiplication tables. We show that these problems can be computed by polynomial size nondeterministic circuits of unbounded fan-in with O(log log n) depth and O(log 2 n) nondeterministic bits, where n is the number of group elements. The input of the Group Isomorphism problem GroupIso consists of two groups G 1 and G 2 of order n given by multiplication tables (n × n matrices of integers between 1 and n) and it is asked whether the groups are isomorphic, that is, whether there is a bijection ϕ between the elements of both groups satisfying for every pair of elements i, j, ϕ(ij) = ϕ(i)ϕ(j) (for convenience, we represent in both groups the group operation by concatenation). A quasigroup is an algebraic structure (Ω, ·) where the set Ω is closed under a binary operation · that has the following property: for each pair of elements a, b, there exists unique elements c L and c R such that c L · a = b and a · c R = b. In contrast to groups, a quasigroup is not necessarily associative and does not need to have an identity. The Quasigroup Isomorphism problem QGroupIso is defined as GroupIso but the input structures are multiplication tables of quasigroups, also called Latin squares. GroupIso is trivially reducible to QGroupIso but a reduction in the other direction is not known. The complexity of both problems has been studied for more than three decades. Groups and quasigroups of order n have generator sets of size bounded by log n. Because of this fact an isomorphism algorithm for GroupIso or QGroupIso running in time n log n+O(1) can be obtained by computing a generator set of size log n in G 1 , mapping this set in every possible way to a set of elements in G 2 and 1 supported by a NSERC postdoctoral fellowship and research grants of Prof. Toniann Pitassi.© Arkadev Chattopadhyay and Jacobo Torán and Fabian Wagner; licensed under Creative Commons License NC-ND

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Graph Isomorphism is Not AC0-Reducible to Group Isomorphism

Chattopadhyay

Torán

Wagner

2013

ACM Trans. Comput. Theory

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“…We abusively denote NNC i (poly) when f (n) is a polynomial function. If f (n) = log n then the amount of non-deterministic variables can be described by a polynomial number of NC gates [Wol94]:…”

Section: -Circuit Classes -mentioning

confidence: 99%

“…We study a non-deterministic extension of the parallel Curry-Howard isomorphism in a uniform setting always by giving a uniform depth-preserving simulation of each classes. On one hand there are several characterizations of non-determinism in circuits [Ven92,Wol94]. We use NNC (poly) a class equivalent to NP , which is defined in the same way as NC but using at most a polynomial amount of non-deterministic variables.…”

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Uniform Circuits, & Boolean Proof Nets

Mogbil

Rahli

2007

Logical Foundations of Computer Science

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Abstract. The relationship between Boolean proof nets of multiplicative linear logic (AP N) and Boolean circuits has been studied [Ter04] in a non-uniform setting. We refine this results by taking care of uniformity: the relationship can be expressed in term of the (Turing) polynomial hierarchy. We give a proofs-as-programs correspondence between proof nets and deterministic as well as non-deterministic Boolean circuits with a uniform depth-preserving simulation of each other. The Boolean proof nets class m BN (poly) is built on multiplicative and additive linear logic with a polynomial amount of additive connectives as the nondeterministic circuit class NNC (poly) is with non-deterministic variables. We obtain uniform-AP N = NC and m BN (poly) = NNC (poly) = NP .

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On Limited Nondeterminism and the Complexity of the V-C Dimension

Papadimitriou

Yannakakis

1996

Journal of Computer and System Sciences

157

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We characterize precisely the complexity of several natural computational problems in NP, which have been proposed but not categorized satisfactorily in the literature: Computing the Vapnik Chervonenkis dimension of a 0 1 matrix; finding the minimum dominating set of a tournament; satisfying a Boolean expression by perturbing the default truth assignment; and several others. These problems can be solved in n O(log n) time, and thus, they are probably not NP-complete. We define two new complexity classes between P and NP, very much in the spirit of MAXNP and MAXSNP. We show that computing the V C dimension is complete for the more general class, while the other two problems are complete for the weaker class. ]

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Nondeterministic circuits, space complexity and quasigroups

Cited by 16 publications

References 7 publications

Graph Isomorphism is Not AC0-Reducible to Group Isomorphism

Graph Isomorphism is Not AC0-Reducible to Group Isomorphism

Uniform Circuits, & Boolean Proof Nets

On Limited Nondeterminism and the Complexity of the V-C Dimension

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