E. W. Allender. P-uniform circuit complexity. Journal of the Association for Computing Machinery, 36(4):912--928, October 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....accepts LDC in time bounded by O(log n) BIS90] Similarly we speak of polylogtime uniformity, when the run time is bounded by some O( log n) k ) k 2 IN) Bar92] and of P uniformity, if the run time is bounded by some polynomial. This latter uniformity condition was considered by Allender in [All89] Some connections between these different conditions are known, e.g. for polynomial size and depth at least the square of the logarithm, logspace and logtime 2 uniformity coincide [Ruz81] But in general, different conditions may yield different classes. Uniformity conditions directly reflect ....

.... 0 = PLH, qACC 0 = MOD PLH, and qTC 0 = PLCH) The question which now arises is of course: Can logspace (or polylogspace) uniform circuit classes also be characterized on sublinear time Turing machines The following proposition can be proved with methods similar to those used by Allender in [All89] which in turn rely on simulations given in [Ruz81] Proposition 2. All logspace uniform circuit classes (polylogspace uniform circuit classes, resp. mentioned above can be characterized by logspace bounded machines (polylogspace bounded machines, resp. of the same type as in the ....

E. Allender. P-uniform circuit complexity. Journal of the Association for Computing Machinery, 36:912--928, 1989.

....D 80290 Munchen, Germany x Supported by DFG SFB 0342 KLARA over a one letter alphabet, recognizable in polynomial time. A similar statement holds for L uniform logdepth circuits and moreover for A uniform logdepth circuits where A is a class fulfilling some weak closure properties. Allender [1] studied the same question concerning P uniform polylogarithmic depth circuits (polynomial size and depth log O(1) Our results are similar to Allender s, but the proofs are more subtle. This is because we build on DLOGTIME uniformity 1 and because our results are applicable to more general ....

....class of languages accepted by DLOGTIME uniform circuits of depth (log n) O(1) and size O(n O(1) With the same translation technique as above we can prove: Corollary 10 If NC 1 = P uniform NC 1 then NC = P uniform NC and PSPACE = DTIME(2 n O(1) Proof By Corollary 6. 7 of Allender [1] we have NC = P uniform NC if and only if PSPACE = DTIME(2 n O(1) Therefore it suffices to show that NC 1 = P uniform NC 1 implies NC = P uniform NC. Obviously ALINTIME ATISP(n O(1) n) ASPACE(n) By Corollary 3.5 of Chandra et al. 6] for s(n) log n holds ASPACE(s(n) S ....

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E. W. Allender. P-uniform circuit complexity. Journal of the Association for Computing Machinery, 36(4):912--928, October 1989.

....With the same translation technique we can prove: Corollary 10 If NC 1 = P uniform NC 1 then NC = P uniform NC and PSPA CE = DT IME(2 n O(1) Proof It suffices to show that NC 1 = P uniform NC 1 implies NC = P uniform NC. The other equality follows from Corollary 6. 7 of Allender [2]. By Theorem 9, NC 1 = P uniform NC 1 iff ALINT IME = DT IME(2 O(n) This implies AT ISP(n O(1) n) DT IME(2 O(n) ASPA CE(n) and using Corollary 6.6 of Allender [2] the claim follows. 2 5 Other Applications 5.1 Logspace Printability and Logspace Uniformity Tally languages ....

....NC 1 = P uniform NC 1 implies NC = P uniform NC. The other equality follows from Corollary 6.7 of Allender [2] By Theorem 9, NC 1 = P uniform NC 1 iff ALINT IME = DT IME(2 O(n) This implies AT ISP(n O(1) n) DT IME(2 O(n) ASPA CE(n) and using Corollary 6. 6 of Allender [2] the claim follows. 2 5 Other Applications 5.1 Logspace Printability and Logspace Uniformity Tally languages have often been studied in connection with other types of sparse sets. For L uniform circuits the results in Section 4 can be generalized to the class Lprint of printable sets (see [9] ....

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E. W. Allender. P-uniform circuit complexity. Journal of the Association for Computing Machinery, 36(4):912--928, October 1989.

....618 1 1 z Fakultat fur Informatik, Technische Universitat Munchen, Arcisstr. 21, D 80290 Munchen, Germany x Supported by DFG SFB 0342 KLARA for L uniform logdepth circuits and moreover for A uniform logdepth circuits where A is a class fulfilling some weak closure properties. Allender [1] studied the same question concerning P uniform polylogarithmic depth circuits (polynomial size and depth log O(1) Our results are similar to Allender s, but the proofs are more subtle. This is because we build on DLOGTIME uniformity 1 and because our results are applicable to more general ....

....the class of languages accepted by DLOGTIME uniform circuits of depth (log n) O(1) and size O(n O(1) With the same translation technique as above we can prove: Corollary 10 If NC 1 = P uniform NC 1 then NC = P uniform NC and PSPACE = DTIME(2 n O(1) Proof By Corollary 6. 7 of [1] we have NC = P uniform NC if and only if PSPACE = DTIME(2 n O(1) Therefore it suffices to show that NC 1 = P uniform NC 1 implies NC = P uniform NC. Obviously ALINTIME ATISP(n O(1) n) ASPACE(n) By Corollary 3.5 of [6] for s(n) log n holds ASPACE(s(n) S c 0 DTIME(2 ....

[Article contains additional citation context not shown here]

E. W. Allender. P-uniform circuit complexity. Journal of the Association for Computing Machinery, 36(4):912--928, October 1989.

....shown to be in NC; in particular, it is known that nondeterministic logspace (NLOG) is in NC 2 [Sud75, Ruz80] Pippenger [Pip79] and Ruzzo [Ruz81] have given alternate characterizations of NC. Another related class that has been considered is the P uniform version of NC called PUNC. Allender [All89b] studied this class and provided alternate characterizations in terms of alternating Turing machines and other parallel machines. He provides evidence that NC does not adequately model the notion of feasible parallelism and argues that PUNC is a better candidate. Logspace uniformity is a fairly ....

E. Allender. P-uniform circuit complexity. J. Assoc. Comput. Mach., 36:912--928, 1989.

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E. Allender. P-uniform circuit complexity. J. ACM, 36:912--928, 1989.

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E. Allender. P-uniform circuit complexity. J. ACM 36:912--928, 1989.

....there is an efficient algorithm for constructing C n , given n, then the family is said to be uniform, where different notions of efficient give rise to different notions of uniformity. We will consider P uniform, Logspace uniform, and Dlogtime uniform circuit families. For P uniform circuits [BCH86, All89], the mapping n 7 C n is computable in polynomial time, for Logspace uniform circuits [Ruz81] the mapping is computable in Logspace. Dlogtimeuniformity requires a somewhat more careful definition; we refer the reader to [BIS90] Although Dlogtime uniformity is widely regarded as being the ....

E. Allender. P-uniform circuit complexity. J. ACM, 36:912--928, 1989.

....realizable pair via a computation that consumes no input, and evaluates to ; otherwise. The rest of the construction is similar to the construction sketched above for 1 LOGCFL. Related observations concerning AuxPDAs that have limits on the number of times they move their input heads are made in [Al89, ABP92]; providing a full proof is routine, using ideas presented there. Corollary 7.4 A language L is accepted by an unambiguous 1 LOGCFL machine iff there is a polynomial size, polynomial degree circuit generating L unambiguously. A language L is be accepted by an unambiguous 1 AuxPDA machine iff ....

E. Allender, P-uniform circuit complexity, J. ACM 36 (1989) 912--928.

....is to remove the P uniformity condition. The following proposition indicates that this is unlikely. Proposition 5.1. There is a 1 NAuxPDA M such that NC is equal to Puniform NC if and only if the maximal word function for M is computable by logspace uniform NC circuits. Proof. It was shown in [3] that there is a tally set T 2 P such that T is in NC if and only if P uniform NC is equal to NC; and it was also observed there that every tally set in P is accepted by a 1 NAuxPDA. Let M be a 1 NAuxPDA accepting T . Clearly, deciding if 0 n 2 T reduces to the problem of computing the maximal ....

E. Allender, P--Uniform Circuit Complexity. J. Assoc. Comput. Mach. 36 (1989), 912--928. Maximal word functions 21

.... complexity go hand in hand with significant progress being made in understanding the relationships that exist among various subclasses of PSPACE, such as the polynomial hierarchy, PP, and the counting hierarchy, starting with the seminal result of Toda (1991) These connections are surveyed in Allender and Wagner (1990), and similar connections are explored in Kannan et al. 1991) Amid all of this recent work showing the surprising power of Quasi TC 0 circuits, it has been suggested that even apparently larger complexity classes such as NC 1 might be contained in Quasi TC 0 . Note that, by standard ....

Allender, E. (1989a), P-uniform circuit complexity, J. ACM 36, 912--928.

....C n has at most s(n) gates; it has depth d(n) if the length of the longest path from input to output in C n is at most d(n) A family fC n g is uniform if the function n 7 C n is easy to compute in some sense. In this paper, we will consider only Dlogtime uniformity [BIS90] and P uniformity [Al89] (in addition to non uniform circuit families) A function f is said to be in AC 0 if there is a circuit family fC n g of size n O(1) and depth O(1) consisting of unbounded fan in AND and OR and NOT gates such that for each input x of length n, the output of C n on input x is f(x) Note that, ....

E. Allender, P-uniform circuit complexity, J. ACM 36 (1989) 912--928.

....C n has at most s(n) gates; it has depth d(n) if the length of the longest path from input to output in C n is at most d(n) A family fC n g is uniform if the function n 7 C n is easy to compute in some sense. In this paper, we will consider only Dlogtime uniformity [BIS90] and P uniformity [Al89] (in addition to non uniform circuit families) A function f is said to be in AC 0 if there is a circuit family fC n g of size n O(1) and depth O(1) consisting of unbounded fan in AND and OR and NOT gates such that for each input x 3 of length n, the output of C n on input x is f(x) ....

E. Allender, P-uniform circuit complexity, J. ACM 36 (1989) 912--928.

.... complexity go hand in hand with significant progress being made in understanding the relationships that exist among various subclasses of PSPACE, such as the polynomial hierarchy, PP, and the counting hierarchy, starting with the seminal result of Toda (1991) These connections are surveyed in Allender and Wagner (1990), and similar connections are explored in Kannan et al. 1991) Amid all of this recent work showing the surprising power of Quasi TC 0 circuits, it has been suggested that even apparently larger complexity classes such as NC 1 might be contained in Quasi TC 0 . Note that, by standard ....

Allender, E. (1989a), P-uniform circuit complexity, J. ACM 36, 912--928.

....there is an efficient algorithm for constructing Cn , given n, then the family is said to be uniform, where different notions of efficient give rise to different notions of uniformity. We will consider P uniform, logspace uniform, and Dlogtime uniform circuit families. For P uniform circuits [BCH86, All89], the mapping n 7 Cn is computable in polynomial time, for logspace uniform circuits [Ruz81] the mapping is computable in logspace. Dlogtime uniformity requires a somewhat more careful definition; we refer the reader to [BIS90] Although Dlogtime uniformity is widely regarded as being the ....

E. Allender. P-uniform circuit complexity. J. ACM, 36:912--928, 1989.

....circuit Cn has at most s(n) gates; it has depth d(n) if the length of the longest path from input to output in Cn is at most d(n) A family fCng is uniform if the function n 7 Cn is easy to compute in some sense. In this paper, we will consider only Dlogtime uniformity [BIS90] and P uniformity [Al89] (in addition to non uniform circuit families) A function f is said to be in AC 0 if there is a circuit family fCng of size n O(1) and depth O(1) consisting of unbounded fan in AND and OR and NOT gates such that for each input x of length n, the output of Cn on input x is f(x) Note that, ....

E. Allender, P-uniform circuit complexity, J. ACM 36 (1989) 912--928.

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E. W. Allender. P-uniform circuit complexity. Journal of the Association for Computing Machinery, 36(4):912--928, October 1989.

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E. W. Allender. P-uniform circuit complexity. Journal of the Association for Computing Machinery, 36(4):912--928, October 1989.

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Eric W. Allender. P-uniform circuit complexity. Journal of the ACM, 36(4):912--928, October 1989.

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Eric W. Allender. P-uniform circuit complexity. Journal of the ACM, 36(4):912--928, October 1989.

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Eric Allender (1989), P-Uniform Circuit Complexity, JACM, 36, 912 - 928.

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