D. Barrington, N. Immerman, H. Straubing, On uniformity within NC Journal of Computer and System Sciences 41 (1990) 274-306. Home/Search Document Not in Database Summary Related Articles Check |
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Completeness results for Graph Isomorphism - Jenner, Köbler, al. (2002) (1 citation) (Correct)
....set of automorphisms in G, that is the set of bijections from V 1 onto itself satisfying GA is the problem of deciding whether a given graph has a non trivial (different from the identity) automorphism. 2. 2 Reducibilities We state our results using DLOGTIME uniform many one AC reducibility [20]. For languages A; B , we say that A is many one AC reducible to B if there is a function f computed by a DLOGTIME uniform family of circuits having the property that, for all x 2 , x 2 A i f(x) 2 B. We write in this case A m B. 2.3 Graphs and representations For simplicity ....
....is NC complete. We rst consider colored trees for which the hardness proof is simpler. We denote by CTI the isomorphism problem for colored trees. Lemma 3.1 In the string representation, CTI is NC hard under reducibility. Proof. Barrington, Immerman and Straubing show in [20] that an NC circuit can be simulated by a balanced DLOGTIME uniform family of Boolean expressions made up of alternating layers of ANDs and ORs. Because of this fact, it suces to reduce the evaluation problem for these expressions to CTI. The core of the reduction is the simple construction ....
D. Barrington, N. Immerman, H. Straubing, On uniformity within NC Journal of Computer and System Sciences 41 (1990) 274-306.
Searching constant width mazes captures the AC0 hierarchy - Barrington, Lu, Miltersen, .. (1998) (4 citations) Self-citation (Barrington) (Correct)
....k is the class of languages recognizable by circuits with k alternating levels of unbounded fan in AND and OR gates, with the output an OR gate and a zeroth level of input gates and their negations. Non uniform Pi k is defined analogously, but with the output gate being an AND gate. Following [1], we define a uniform version of the hiearchy as follows: Uniform Pi k ( Sigma k ) is the class of languages accepted by alternating Turing machines running in logarithmic time and making exactly k alternations, the first being universal (existential) An appropriate class of reductions to use ....
...., where the C i s are Pi k Gamma1 formulae. Construct the width k Gamma 1 graphs C(G i ) corresponding to the C i s and let G(C) be the width k 1 graph of Figure 5. The correctness of the construction is easily checked. 2 5 Uniformity considerations As in Barrington, Immerman, and Straubing [1], we define a log time Turing machine to have a read only input tape of length n, a constant number of read write work tapes of total length O(log n) and a read write input address tape of length log n. On a given time step the machine has access to the bit of the input tape denoted by the ....
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D. A. M. Barrington, N. Immerman and H. Straubing. On uniformity within NC 1 Journal of Computer and System Sciences, 41(3):274--306.
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