D. Mix Barrington, N. Immerman, and H. Straubing, "On uniformity in NC , J. Comput. System Sci. 41 (1990) 274--306.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....0 , following [Clo93] ALV 0 is similar to PV, except that the function symbols range over NC 1 functions instead of polynomial time functions. The NC 1 function symbols are introduced in a manner based on Barrington s characterization of NC 1 in terms of bounded width branching programs [Bar89, BIS90], and adapted by Clote [Clo93] to show that the NC 1 functions are the least class including certain initial functions and closed under composition, concatenation recursion on notation, and K bounded recursion on notation (for constant K) In more detail, the initial functions of ALV 0 ....

D. M. Barrington, N. Immerman, and H. Straubing, On uniformity in NC 1 , J. Comp. Systems Sci. 41 (1990), no. 3, 274--306.

....analogues, including the polynomial hierarchy. The quantifiers of SBH have the same length as those of the log time hierarchy, which is equivalent to logtime uniform AC 0 . The most elegant characterization of uniform AC 0 is in terms of first order definability, as discovered by Immerman [3, 34, 35]. The SBH can be characterized similarly, by generalizing Immerman s definition of first order definability to treat an arbitrary class of relations as atomic. We show that when QL is used as the atomic class, the result is precisely SBH (QL) Definition 13 Let C be a class of relations on N. A ....

D. Mix Barrington, N. Immerman, and H. Straubing, "On uniformity in NC 1 , J. Comput. System Sci. 41 (1990) 274--306.

....circuit model lies in the robustness of this approach for the low level classes AC 0 and AC 1 . Indeed, the class AC 0 , as defined above, has been shown to be equal to to a number of quite differently defined classes, such as first order definable functions [15] the logtime hierarchy lh [2], and a function algebra A 0 independently developed [7] and has arguably the best claim to be the most natural definition of constant parallel time. To characterize parallel classes, we need the additional initial function bit(x; i) giving the coefficient of 2 i in the binary representation of ....

....[21] Stockmeyer and Vishkin related the concurrent read, concurrent write crcw pram to unbounded fan in boolean circuit families. In [15] Immerman introduced the concurrent random access machine cram, a slight modification of the crcw pram which admits a unit cost shift instruction. There and in [2], Immerman extended the Stockmeyer Vishkin result to prove that languages computable in uniform cram time O(log k (n) with processor bound n O(1) are exactly those in first order uniform unbounded fan in boolean circuits of depth O(log k (n) and size n O(1) for all k 0. In this ....

D. Mix Barrington, N. Immerman, and H. Straubing. On uniformity in NC 1 . J. Comp. Syst. Sci., 41(3):274--306, 1990.

....analogues, including the polynomial hierarchy. The quantifiers of SBH have the same length as those of the log time hierarchy, which is equivalent to logtime uniform AC 0 . The most elegant characterization of uniform AC 0 is in terms of first order definability, as discovered by Immerman [3, 36, 37]. The SBH can be characterized similarly, by generalizing Immerman s definition of first order definability to treat an arbitrary class of relations as atomic. We show that when QL is used as the atomic class, the result is precisely SBH (QL) Definition 12 Let C be a class of relations on N. A ....

D. Mix Barrington, N. Immerman, and H. Straubing, "On uniformity in NC 1 , J. Comput. System Sci. 41 (1990) 274--306.

....(v) H consists of numerical predicates which are constructible or recognizable within some complexity bound, such as polynomialtime or logarithmic space. Such uniformity conditions were considered in a series of papers by Barrington, Compton, Gurevich, Immerman, Lewis, Straubing, Th erien, e.g. [GL84, BCST92, BIS90, BI94, Str94]. vi) H consists of DEF (L 1 ) i.e. all the classes definable in a logic L 1 (without auxiliary predicates) ORD is FOL definable in this sense (ORD 2 DEF (FOL) but SUCC is not (as we can not express the absence of cycles) In section 4 we shall discuss the impact of this choice for various ....

....iff K has polynomial circuit complexity. Our uniformity condition concerns in the complexity or definability of the advice sequence, not of the circuits. A priori, this is a weaker assumption, as it does not affect the circuit size. For uniformity conditions imposed on the circuit families, cf. [BIS90, BI94]. However, the proof of [BS90, theorem 2.1] shows that for polynomial size circuits the two uniformity conditions are related, as the circuits can be computed from the Turing machine. We shall elaborate on the exact relationship between the two kinds of uniformity in subsequent paper. Theorem 4 ....

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D.A.M. Barrington, N. Immerman, and H. Straubing. On uniformity in nc 1 . JCSS, 41:274--306, 1990.

.... if this can be done for dyadic numerical predicates the result will be quite significant: The natural uniform versions of the classes ACC(q) and CC(q) are defined using the single numerical predicate BIT (x; y) which is interpreted to mean the x th bit of the binary representation of y is on [BIS]. Thus an extension of Theorem 2.1 to the dyadic case would resolve most of the open questions about the relations among AC 0 ; CC and ACC in the uniform setting. My suspicion is that the monadic case is very special, and that any extension beyond this will encounter all of the difficulties ....

D. Barrington, N. Immerman and H. Straubing, On uniformity in NC 1 , J. Comp. Syst. Sci., 41 (1990), 274-306.

....sets. We prove these results by reducing questions about the Turing machine complexity of locally self reducible sets to questions about the circuit complexity of multiplication in finite monoids, and then applying results of Barrington, Immerman, Straubing, and Th erien on circuits and monoids [3, 5, 4]. This algebraic approach to circuit complexity is proving to be a valuable tool in the study of Turing machine based complexity classes; see, for example, 7, 11] 2. A Connection to Algebra Let A be k locally self reducible, so there is a polynomialtime algorithm A that takes x 1 and A (x ....

.... of length jxj: ffl The i th element of ff(x) depends only on the f jxj (i) th bit of x: ffl If x 2 L; then Q ff(x) the product in S 5 of the elements of ff(x) is equal to b: If x = 2 L; then Q ff(x) c: Barrington in his original paper, and later Barrington, Immerman and Straubing [5] studied the complexity of this circuit simulation for uniform NC 1 : For our purposes, we only need the following facts, easily derivable from Barrington s original construction: Suppose we have a fan in 2 circuit whose underlying graph is the full binary tree of depth d: Thus there are 2 d ....

D. Barrington, N. Immerman, and H. Straubing. On Uniformity in NC 1 . JCSS, 41:274--306, 1990.

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D. Mix Barrington, N. Immerman, and H. Straubing, "On uniformity in NC , J. Comput. System Sci. 41 (1990) 274--306.

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D.A.M. Barrington, N. Immerman, and H. Straubing. On uniformity in nc 1 . JCSS, 41:274--306, 1990.

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