S. Buss, S., S. Cook, A. Gupta, and V. Ramachandran, An Optimal Parallel Algorithm for Formula Evaluation, SIAM J. Comput., 21(4) (1992), pp. 755--780.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... for Boolean Formul Evaluation nd for Tree Contraction 115 [7] N. A. LYNCH, Log space recognition and translation of parenthesis languages J. Assoc. Cornput. Mach. 24 (1977) pp. 583 590. 8] G. L. MILLER AND J. H. REIF, Parallel tree contraction and its appli cation, in Proceedings of the 26th IEEE Symposium on Foundations of Computer Science, IEEE ....

....for Boolean Formul Evaluation nd for Tree Contraction 115 [7] N. A. LYNCH, Log space recognition and translation of parenthesis languages J. Assoc. Cornput. Mach. 24 (1977) pp. 583 590. 8] G. L. MILLER AND J. H. REIF, Parallel tree contraction and its appli cation, in Proceedings of the 26th IEEE Symposium on Foundations of Computer Science, IEEE Computer Society, 1985, pp. 478 489. Department of Mathematics University of California, San Diego La Jolla, California ....

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S. R. Buss, S. A. COOK, A. GUPTA, AND V. RAMACHANDRAN, An optimal parallel algorithm for formula evaluation. To appear in SIAM Journal on Computation.

....the hierarchy of complexity classes. Among their results we list in Table 1 those which are most significant to our work. This is one more way of characterizing complexity classes in algebraic terms, which comes after the problem of evaluating formulas and circuits over the Boolean semiring (see [11, 7, 8]) and the computational models of programs over monoids (see [3, 4] and leaf languages (see [6] among others. Using tensor calculus in this context is especially appealing, if only because of the many applications matrix algebra finds in various areas, such as the specification of parallel ....

S.R. Buss, S. Cook, A. Gupta, and V. Ramachandran. An optimal parallel algorithm for formula evaluation. SIAM Journal on Computing, 21(4):755--780, 1992.

....algorithm. Such an algorithm takes as input a description of a formula, with all of its inputs speci ed, and produces as output the value of the formula. Parallel algorithms for this problem have been proposed by Gupta [6] Miller and Reif [9] Buss [2] Buss, Cook, Gupta, and Ramachandran [3]; and Kosaraju and Delcher [8] These yield NC algorithms for the problem that also produce, for any given formula of size S, a circuit of depth O(log S) When these circuits are expressed as formulas, the sizes are S ) for various 2. In the case of divisionfree formulas, the exponents ....

S. Buss, S. Cook, A. Gupta, and V. Ramachandran, An optimal parallel algorithm for formula evaluation, SIAM J. Comput., 21(4) (1992), pp. 755-780.

....the value ae of the root of T equals #PATH(a; b) Below we sketch how ae can be computed in logarithmic space. Let p 1 p 2 : be the standard enumeration of primes. The prime number theorem implies Q p i n m p i = e (n m) 1 o(1) #PATH(a; b) 1) Using the log space algorithm of [7] one can transform T into a binary tree T 0 of depth O(log z) representing an arithmetic expression with the same value ae as T . We evaluate T 0 mod p i using the algorithm in [4] For p i n m this algorithm works in space O(log z log(n m) O(log n) By inequality (1) taking all p i ....

S. Buss, S. Cook, A. Gupta, V. Ramachandran, An Optimal Parallel Algorithm for Formula Evaluation, SIAM J. Comput. 21, 1992, 755-780.

....for NC 1 respectively, under DLOGTIME reductions. Our groupoid in the case of NC 1 has order 10, again in contrast with Barrington s NC 1 complete word problem over the smallest non solvable group [Ba86] this traces the somewhat obscure difference between the NC 1 complete Formula Value [Bu87, BuCoGuRa89] and Width 5 Graph Accessibility [Ba86, BaImSt88] problems to the structural difference between groupoids and monoids. Our second extension, from monoid M to monoid family fM n g, yields much more unusual complexity classes. In the present paper we restrict our attention to the case of ....

S.R. Buss, S. Cook, A. Gupta and V. Ramachandran, An optimal parallel algorithm for formula evaluation, preprint, 1989, to appear in SIAM Journal on Computing.

....the value ae of the root of T equals #PATH(a; b) Below we sketch how ae can be computed in logarithmic space. Let p 1 p 2 : be the standard enumeration of primes. The prime number theorem implies Y p i n m p i = e (n m) 1 o(1) #PATH(a; b) 2) Using the log space algorithm of [7] one can tranform T into a binary tree T 0 of depth O(log z) representing an arithmetic expression with the same value ae as T . We evaluate T 0 mod p i using the algorithm in [4] For p i n m this algorithm works in space O(log z log(n m) O(log n) By inequality (2) taking all p i ....

S. Buss, S. Cook, A. Gupta, V. Ramachandran, An Optimal Parallel Algorithm for Formula Evaluation, SIAM J. Comput. 21, 1992, 755-780.

....the general problem of evaluating a f ; 2g formula over the natural numbers is #NC 1 complete. Beyond a struggle with DLOGTIME uniformity, this question again seems to bring to the fore the question of how efficiently an algebraic formula over a semiring can be balanced (see for instance [14]) In Section 5 of this paper, we have refined leaf language characterizations of complexity classes. In particular, we now have delicate characterizations of PSPACE in terms of restricted context free languages and (projections over) automata. What can be made of these How much further, if at ....

S. Buss, S. Cook, A. Gupta and V. Ramachandran, An optimal parallel algorithm for formula evaluation, SIAM Journal on Computing, 21:4:755-780, 1992.

....of S (given explicitly by enumeration or implicitly by a condition) decide whether T evaluates to an element in the accepting set. Many special cases of this problem have been studied. For example, intricate NC 1 upper bounds are known when S is the Boolean [13] or a more general semiring [14], the case of groups was crucial Partially supported by Deutsche Forschungsgemeinschaft grant Me 1077 14 1. Part of the work was done while the author was at Universitat Trier, Germany. y Supported by the Qu ebec FCAR and by the NSERC of Canada. to elucidating the power of bounded width ....

S.R. Buss, S. Cook, A. Gupta, and V. Ramachandran. An optimal parallel algorithm for formula evaluation. SIAM Journal on Computing, 21(4):755-- 780, 1992.

....occurring before shorter ones. Because we picture finite state machines as moving left to right, we reverse the sentence so that it is in prefix notation with shorter operands first, but still use the name PLOF. The problem remains complete for NC 1 under the restriction to PLOF sentences [Bus87, BCGR92]. Theorem 4.3 The BSVP restricted to PLOF sentences is n to n=2 contractible by an SSM that needs only odd even state information. There exist regular languages that are complete for NC 1 [Bar89] but the finite automata accepting them maintain the whole symmetric (or alternating) permutation ....

S. Buss, S. Cook, A. Gupta, and V. Ramachandran. An optimal parallel algorithm for formula evaluation. SIAM J. Comput., 21:755--780, 1992.

....can be recognized in linear time by a deterministic Turing machine. The Turing machine that recognizes the direct connection language of fCng will be referred to as the uniformity machine for fCng. The above notion of uniformity is the one that is generally used for small complexity classes (see [6, 12, 23]) However, we are going to use a slightly less restrictive notion of uniformity for our results. Our notion of uniformity can be informally referred to as Polylogtime uniformity. The reason that we use this notion is that we are dealing with circuits of possibly superpolynomial size and the ....

S. Buss, S. Cook, A. Gupta, and V. Ramachandran, An optimal parallel algorithm for formula evaluation, SIAM J. Comput., 21 (1992), pp. 755--780.

....the general problem of evaluating a f ; Thetag formula over the natural numbers is #NC 1 complete. Beyond a struggle with DLOGTIME uniformity, this question again seems to bring to the fore the question of how efficiently an algebraic formula over a semiring can be balanced (see for instance [14]) In Section 5 of this paper, we have refined leaf language characterizations of complexity classes. In particular, we now have delicate characterizations of PSPACE in terms of restricted context free languages and (projections over) automata. What can be made of these How much further, if at ....

S. Buss, S. Cook, A. Gupta and V. Ramachandran, An optimal parallel algorithm for formula evaluation, SIAM Journal on Computing, 21:4:755-780, 1992.

....is a language, then the oracle gate computes the characteristic function of the language. Our results do not depend very much on the particular uniformity condition used. Logspace uniformity is sufficient, although the theorems also hold if more restrictive uniformity conditions are used; see [12, 10] for discussions of uniformity issues involved in low level circuit complexity. It will turn out to be useful to consider certain refinements of AC 0 reducibility. Following [4] see also [6] define AC 0 i (f) to be the class of languages accepted by AC 0 circuits with oracle gates for f , ....

S. Buss, S. Cook, A. Gupta, and V. Ramachandran, An optimal parallel algorithm for formula evaluation, SIAM Journal on Computing 21 (1992) 755--780.

....iterating S recognizes D 1 in O(log n) passes and linear overall work. The original proof by Buss [Bus87] that the Boolean formula value problem belongs to NC 1 reduced the problem to that of evaluating a Boolean sentence OE written in postfix form with longer operands first, called PLOF in [Bus87, BCGR92]. Call the latter problem BSVP with PLOF promise. The PLOF form was later shown unnecessary by Buss [Bus91] via the tree contraction method of [ADKP89] while Kosaraju and Delcher [KD90, KD92] obtained circuits of depth O(log n) and size O(n log O(1) n) for BSVP without the promise. For CFAs ....

S. Buss, S. Cook, A. Gupta, and V. Ramachandran. An optimal parallel algorithm for formula evaluation. SIAM J. Comp., 21:755--780, 1992.

....is always provided the length of the input (in binary) on the work tape as part of its initial configuration on a particular input. This convention has been introduced to simplify the proof. It is worthwhile to note that the input length can be computed deterministically in logarithmic time (see [BCGR92]) but this requires multiple accesses to the input along a given computation path. We consider ATMs that access their input only at the leaves. That is, the only configurations that depend on the input are halting configurations. These are of two types: those that accept if and only if bit i of ....

S. Buss, S. Cook, A. Gupta, and V. Ramachandran. An optimal parallel algorithm for formula evaluation. SIAM Journal on Computing, 1992. To appear.

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S. R. Buss, S. A. Cook, A. Gupta, and V. Ramachandran, An optimal parallel algorithm for formula evaluation, SIAM J. Comput., 21 (1992), pp. 755--780.

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Samuel R. Buss, Steven A. Cook, Arvind Gupta, and Vijaya Ramachandran. An optimal parallel algorithm for formula evaluation. SIAM Journal on Computing, 21:755--780, 1992.

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S. R. Buss, S. A. Cook, A. Gupta, and V. Ramachandran, An optimal parallel algorithm for formula evaluation, SIAM J. Comput., 21 (1992), pp. 755--780.

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S. R. Buss, S. A. Cook, A. Gupta, and V. Ramachandran, An optimal parallel algorithm for formula evaluation, SIAM J. Comput., 21 (1992), pp. 755--780.

....question of separating Frege and extended Frege systems and the question of separating N( from P. Nanlely, the lines in a polynomial size Frege proof consist of polynomial size propositional formulas and it is known that polynomial size formulas can express precisely properties in (nonuniform) N( [30,3,7,6]. Likewise, because of the ability to use abbreviations for long formulas, the lines in a polynomial size extended Frege proof are essentially polynomial size circuits and thus can express properties that are in nonuniform P [23] Thus, one can intuitively view polynomial size Frege proofs as ....

S. R. Buss, S. A. COOK, A. GUPTA, AND V. RAMACItANDRAN, An optimal parallel algorithm for' formula evaluation, SIAM J. Cornput., 21 (1992), pp. 755 780.

....We will assume that the reader is familiar with these capabilities of Alogtime, and also is familiar with both the circuit characterization and the game characterization of Alogtime. For more information on these aspects of alternating logtime, the reader can consult the introductory portions of [3, 5, 4]. There are logspace algorithms for converting pointer representations of trees into string representations of tree, and vice versa, so for logspace and for more powerful complexity classes, the use of string representations is equivalent to the use of pointer representations. 4 Definition When ....

.... ut 6 Conclusions Our main results show that tree isomorphism, tree comparison and tree canonization are in alternating logarithmic time; improving on the logarithmic space algorithms of [8] There are number of other problems known to be in Alogtime, including the Boolean Formula Value Problem [3, 5, 4], and the word problem for S 5 [2] These problems are also known to be complete for Alogtime under deterministic log time reductions. It is still open whether the tree isomorphism, comparison and canonization problems are also complete for Alogtime. A problem with a similar name to the tree ....

S. R. Buss, S. A. Cook, A. Gupta, and V. Ramachandran, An optimal parallel algorithm for formula evaluation, SIAM J. Comput., 21 (1992), pp. 755--780.

....of separating Frege and extended Frege systems and the question of separating NC 1 from P . Namely, the lines in a polynomial size Frege proof consist of polynomial size propositional formulas and it is known that polynomial size formulas can express precisely properties in (nonuniform) NC 1 [24,3,7,6]. Likewise, because of the ability to use abbreviations for long formulas, the lines in a polynomial size extended Frege proof are essentially polynomial size circuits and thus can express properties that are in nonuniform P [18] Thus, one can intuitively view polynomial size Frege proofs as ....

S. R. Buss, S. A. Cook, A. Gupta, and V. Ramachandran, An optimal parallel algorithm for formula evaluation, SIAM J. Comput., 21 (1992), pp. 755--780.

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S. Buss, S., S. Cook, A. Gupta, and V. Ramachandran, An Optimal Parallel Algorithm for Formula Evaluation, SIAM J. Comput., 21(4) (1992), pp. 755--780.

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S. Buss, S. Cook, A. Gupta, , and V. Ramachandran. An optimal parallel algorithm for formula evaluation. SIAM Journal on Computing, 21:755--780, 1992.

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S. Buss, S. Cook, A. Gupta, , and V. Ramachandran. An optimal parallel algorithm for formula evaluation. SIAM Journal on Computing, 21:755--780, 1992. 82 BIBLIOGRAPHY 83

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S. R. Buss and S. A. Cook and A. Gupta and V. Ramachandran, An optimal parallel algorithm for formula evaluation, SIAM Journal on Computing, 21:4, pp. 755-780, 1992.

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