E. Allender, D. A. M. Barrington, and W. Hesse. Uniform circuits for division: consequences and problems. In Proc. 16th Conf. on Computational Complexity, pages 150-159, Los Alamitos, CA, USA, 2001. IEEE Computer Society Press.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....problem is rst order de nable and hence decidable in logarithmic space. For the conditional collapse results in Theorem 6 we build on Balc azar s work [8] on logspace self reducible context free languages, on Venkateswaran s characterization [62] of , and on Allender, Barrington, and Hesse s [3] recent algorithm for converting Chinese remainder representations into binary representations in logarithmic space. For the result on GAP and logspace O(1) mc sets, we adapt the proof techniques used by Agrawal, Arvind, Beigel, Kummer, Ogihara, and Stephan [2, 13, 46] to logarithmic space. This ....

....an integer, s and t are vertices of G, is a binary integer such that 0 k 1, and the number of paths in G from s to t is congruent to modulo k. Since the number k is given in unary and G is topologically sorted, B is logspace self reducible. Recent work of Allender, Barrington, and Hesse [3] shows that converting the Chinese remainder representation of a number into its binary representation can be done in logspace uniform NC , and thus in logspace. Suppose we wish to compute the number X of paths in an n vertex topologically sorted graph G from vertex 1 to vertex n. Since X ....

E. Allender, D. A. M. Barrington, and W. Hesse. Uniform circuits for division: consequences and problems. In Proc. 16th Conf. on Computational Complexity, pages 150-159, Los Alamitos, CA, USA, 2001. IEEE Computer Society Press.

....NC 1 many one reducibility. This is true with the exception of Theorem 5.5 and Corollaries 5.6 and 5.7. In this cases the proofs make use of the recent NC 1 algorithm for reconstructing a number from its Chinese reminder representation [12] which is only known to be logarithmic space uniform [1]. Because of this, the (uniform) hardness results in the mentioned cases only hold for logarithmic space reducibilities. 2.2. Graph Isomorphism An automorphism in an undirected graph G = V; E) is a permutation of the nodes, that preserves adjacency. That is, for every u; v 2 V; u; v) 2 E , ....

E. Allender and D. Mix Barrington, Uniform circuits for division: consequences and problems, personal communication, June 2000.

....can be solved by fully uniform families of threshold circuits of constant depth and polynomial size. Equivalently, all three are reducible to integer multiplication by fully uniform circuits of constant depth and polynomial size. There are two parts to our proof. First (in work first reported in [7]) we show that the non uniformity necessary for the construction 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 ....

.....DLOGTIME uniform is equal to Constable s class K [23] The 1 theorems of C 2 do not have Craig interpolants of polynomial circuit size, unless the Diffie Hellman key exchange protocol is insecure. The complexity classes #AC and GapAC were introduced in [1] and have been studied in [7, 5, 56]. The main motivation for introducing and studying these classes comes from the fact that they give rise to several characterizations of TC . However, there was a problem with these characterizations some of them were not known to hold in the uniform setting. For instance, four different ....

E. Allender, D. A. Mix Barrington, and W. Hesse. Uniform circuits for division: Consequences and problems. Proceedings of the 16th Annual IEEE Conference on Computational Complexity (CCC-2001), 150-159. IEEE Computer Society, 2001.

....x i h i m i , each of which is between 0 and m i . The computation of this rank function is central to the argument of [21] that Division is in L uniform TC . It is computable in logspace [25, 43] and in fact the algorithms can be adapted to put it in FOM POW. For more detail on this see [6]. Here we present a self contained argument, without computing rank directly, that conversion from CRR to binary is in FOM POW. First we note again that we can carry out the other conversion, from binary to CRR. Lemma 4.1. If X,m 1 , m k are each given in binary and X M , we can ....

E. Allender and D. A. M. Barrington. Uniform circuits for division: Consequences and problems. Electronic Colloquium on Computational Complexity 7:065 (2000). Preliminary version of this paper.

....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 ....

Eric Allender, David A. Mix Barrington, and William Hesse. Uniform circuits for division: Consequences and problems. In Proceedings of the 16th Annual IEEE Conference on Computational Complexity, pages 150--159, 2001.

....division and iterated multiplication. Allender and Barrington reinterpreted those results in the framework of descriptive complexity, and showed that the only di#culty in expressing iterated multiplication and division in FO(M) was the di#culty of raising numbers to a power modulo a small prime [2]. The current paper completes this e#ort by showing that this power predicate lies in FO. As division is complete for FO(M) via FO Turing reductions, it is unlikely that the complexity of division can be further reduced. 4 Division reduces to POW The key problem examined by this paper is POW, ....

....consider the descriptive complexity of this problem as if it had input size n. We ask whether this predicate can be represented by FO or FO(M) formulas over the universe 0, n. Allender, Barrington, and the author showed that DIVISION and IMULT are in FO(M) if and only if POW is in FO(M) [2]. They did this by showing that DIVISION and IMULT are FO Turing reducible to POW. A version of this proof, with additional simplifications, is in the full version of this paper. The predicate POW is used to convert inputs from binary to CRR, and to find discrete logarithms in the multiplicative ....

[Article contains additional citation context not shown here]

E. Allender, D. A. Mix Barrington, and W. Hesse. Uniform circuits for division: Consequences and problems. To appear in Proceedings of the 16th Annual IEEE Conference on Computational Complexity (CCC-

....x i h i m i , each of which is between 0 and m i . The computation of this rank function is central to the argument of [11] that DIVISION is in L uniform TC 0 . It is computable in logspace [13, 21] and in fact the algorithms can be adapted to put it in FOM POW. For more detail on this see [2]. Here we present a self contained argument, without computing rank directly, that conversion from CRR to binary and DIVISION are in FOM POW. First we note again that we can carry out the other conversion, from binary to CRR. Lemma 3.1 If X,m 1 , m k are each given in binary and X M ....

E. Allender and D. A. M. Barrington. Uniform circuits for division: Consequences and problems. Electronic Colloquium on Computational Complexity 7:065 (2000). Preliminary version of this paper.

....the possible non integrality of n # , n # ,etc. of the description of g and h, i.e. in time linear in n. Step 2 Conversion of the arithmetic circuit into a Boolean threshold circuit Now we can use the recent result that iterated multiplication is in logspace uniform TC 0 ( 9] see also [3, 17]) along with the fact that iterated addition is in Dlogtime uniform TC 0 ( 6] to convert the arithmetic circuit to a constant depth Boolean threshold circuit. Let us assume that the output of the arithmetic circuit has n s bits. We replace each gate with the appropriate constant depth ....

E. Allender, D. Mix Barrington, and W. Hesse. Uniform circuits for division: Consequences and problems. To appear in Proc. IEEE Conference on Computational Complexity, 2001.

....i modulo m i . It is easy to verify that X is congruent modulo M to P k i=1 x i h i C i . In fact X is equal, as an integer, to ( P k i=1 x i h i C i ) rM for some particular number r, called the rank of X with respect to M (denoted rank M (X) Here I am following the convention introduced in [6] of using capital letters (such as X;M , etc. to refer to numbers with bit length polynomially related to n (call these long numbers ) and lower case letters (such as r; x i , etc. to refer to numbers with bit length O(log n) call these short numbers ) In particular, note that r is a short ....

....ITERATED MULTIPLICATION lie in uniform NC 1 . In this survey, I ll sketch for now only a proof that these problems lie in Luniform TC 0 . As observed in the previous section, it is sufficient to show that one can convert from CRR to binary in L uniform TC 0 . Following the development in [6], I ll actually state and sketch a slightly stronger result. Let POW(a; i; b; p) be defined to be true if and only if a i b (mod p) where a, b, i and p each have O(log n) bits, and p is prime. We ll show that converting from CRR to binary is in FOM POW. It is easy to see that POW is ....

[Article contains additional citation context not shown here]

E. Allender, D. Mix Barrington, and W. Hesse. Uniform circuits for division: Consequences and problems. To appear in Proc. IEEE Conference on Computational Complexity, 2001.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC