W. Hesse. Division is in uniform TC . In 28th International Colloquium on Automata, Languages and Programming, volume 2076 of Lecture Notes in Computer Science, pages 104--114. Springer-Verlag, 2001.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....Out of the familiar arithmetical operations, the complexity of division was the most dicult to classify. In fact it was not until recently that division found its complexity home , a long process nished by Bill Hesse in showing that the predicate DIVISION(X;Y; i) is complete for DLOGTIME TC [Hes01]. 6.2 Preview In the next chapter, we will be examining the complexity of addition and multiplication of short numbers. These problems will be represented respectively by the predicates PLUS(i; j; k) and TIMES(i; j; k) where i; j; k are variables thought of as representing positions in a ....

W. Hesse. Division is in uniform TC0. ICALP

....Theory is whether Fermat s Little Theorem is provable in I Delta 0 . Berarducci and Intrigila [2] point out that one important difficulty is that the modular exponentiation relation x y j z (mod n) is not known to be Delta 0 definable. The situation has changed, however. Very recently, Hesse [11] proved that the modular exponentiation relation on numbers of O(log(n) bits is first order definable. A well known translational argument shows then that x y j z (mod n) is Delta 0 definable. The proof of this result, however, seems to rely on Fermat s Little Theorem, and therefore it is not ....

W. Hesse. Division is in uniform TC 0 . To appear in ICALP'01, 2001.

....to a tight lower bound on the complexity of a problem. In the case of division, defining the right complexity class takes a bit of explanation, as does defining the notion of completeness . I ll provide the necessary definitions later. For now, let s state the result: Breakthrough number 2: [18] Division is complete for DLOGTIME uniform TC 0 . This latest breakthrough will be presented at ICALP 2001 by Bill Hesse, a student at the University of Massachusetts. He will be receiving the best paper award for Track A at ICALP 2001 (combined with the best student paper award) All of these ....

....known for a while that multiplication reduces to division, and thus division was known to be hard for TC 0 . In fact, division had been known to be in P uniform TC 0 ever since it was observed in [27, 28] that the algorithm of [9] can be implemented in P uniform TC 0 . The breakthrough of [18] is that division is in DLOGTIME uniform TC 0 . 3 Background on Division All of the recent work on division builds on the work of Beame, Cook, and Hoover [9] Beame, Cook, and Hoover make use of the fact that, for small enough u, 1= 1 u) P i=0 u i . Thus to divide x by y, we first let j ....

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W. Hesse. Division is in Uniform TC 0 . In ICALP 2001: Twenty-Eighth International Colloquium on Automata, Languages and Programming (July 2001), to appear.

....of [21] is quite limited: In Immerman s descriptive complexity setting [39] we need only first order formulas with Majority quantifiers and a single extra numerical predicate. This predicate expresses powering of integers modulo a prime of O(log n) bits. Next (in work first reported in [38]) we show that powering modulo any small prime is in DLOGTIME uniform AC . Equivalently, it can be expressed in first order logic on ordered structures, with addition and multiplication. We also consider the implications of the new division algorithm for the study of smallspace complexity ....

W. Hesse. Division is in Uniform TC . In ICALP 2001: Twenty-Eighth International Colloquium on Automata, Languages and Programming Lecture Notes in Computer Science 2076, Springer-Verlag, 2001, pp. 104--114.

....that the updates computed by circuits in our algorithm can be computed by first order query languages with aggregate operators, and to help show that the steps in our algorithm can be computed by a TC circuit. We will use a recent result that polynomial evaluation is in DLOGTIME uniform TC [8]. 5 A Dynamic Algorithm for REACH Our dynamic algorithm for REACH keeps track of the number of paths p i,j (k)of length k from vertex i to vertex j, for every 0 k, i, j n. The number p i,j (k) is always less than n , so it is representable as an n log n bit binary number. On adding or ....

....only state that there exist polynomialtime uniform TC circuits to evaluate them. A series of results by Chiu, Davida, Litow, Allender, Barrington, and Hesse shows that the circuits for division, powering, and iterated multiplication used to evaluate polynomials can be made DLOGTIME uniform [1,8,3,4]. In particular, finding the product of n (polynomially many) numbers, each with n bits, can be done by a DLOGTIME uniform TC circuit. Evaluating the above polynomial requires us to raise numbers of O(n ) bits 11 to powers up to n 2, multiply the results by other numbers, add n 1 ....

William Hesse. Division is in uniform TC . In Automata, Languages and Programming: 28th International Colloquium; Proceeedings (ICALP 2001.

.... POW obtained by augmenting FOM with a predicate for powering modulo small primes. We also show that, under a well known number theoretic conjecture (that there are many smooth primes) POW (and hence division) lies in FOM. Building on this work, Hesse has shown recently that division is in FOM [17]. The essential idea in the fast parallel computation of division and related problems is that of Chinese remainder representation (CRR) storing a number in the form of its residues modulo many small primes. The fact that CRR operations can be carried out in log space has interesting ....

....we show that the special case of powering modulo a smooth prime is in fully uniform TC 0 . It follows that under a widely believed but unproven hypothesis about the density of smooth primes, division and related problems actually are in fully uniform TC 0 . Subsequent to this work, Hesse [17] has shown unconditionally that powering modulo any small prime is in fullyuniform TC 0 (in fact it is in fully uniform AC 0 ) Thus integer division and the related problems considered here are actually all in fully uniform TC 0 . This work will not be described here but will appear ....

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W. Hesse. Division is in Uniform TC 0 . In ICALP

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W. Hesse. Division is in uniform TC . In 28th International Colloquium on Automata, Languages and Programming, volume 2076 of Lecture Notes in Computer Science, pages 104--114. Springer-Verlag, 2001.

No context found.

W. Hesse. Division is in uniform TC . In 28th International Colloquium on Automata, Languages and Programming, volume 2076 of Lecture Notes in Computer Science, pages 104--114. Springer-Verlag, 2001.

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