A. Borodin, S. A. Cook, P. W. Dymond, W. L. Ruzzo, and M. Tompa. "Two Applications of Inductive Counting for Complementation Problems". SIAM Journal of Computing, 18:559--578, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... [S] V1] Properties and characterizations of LOGCFL are studied in [C] SV] Semi unbounded fan in circuits of larger depths correspond to extensions of context free languages [C, I, Ru, V1] An interesting property of classes defined by semi unbounded fan in circuits, proved by Borodin et al. [BCDRT], is that they are closed under complementation for all depths that are Omega Gammae 2 n) We consider the arithmetic analogs of the above complexity classes, defined by semi unbounded fan in arithmetic circuits. These classes have been studied for example in [AJ] The class PhiSAC , the ....

A. Borodin, S. A. Cook, P. W. Dymond, W. L. Ruzzo, M. Tompa, "Two applications of inductive counting for complementation problems," SIAM J. Comput., Vol. 18, No. 3, (1989), pp. 559-578.

....under intersection what contradicts Theorem 17. 2 Corollary 19 Let k 2 be a constant, then L (k SN) and L (SN) are not closed under set di erence. The non closure under complement yields another proof of the properness of the inclusion L (SN) LOGCFL by means of di erent closure properties. In [1] the closure of LOGCFL under complement has been shown. ....

Borodin, A., Cook, S. A., Dymond, P. W., Ruzzo, W. L., and Tompa, M. Two applications of inductive counting for complementation problems. SIAM J. Comput. 18 (1989), 559-578.

....RL, which depending on the author of the paper, has one of two distinctly di#erent meanings. This section proposes a reasonably comprehensive and consistent set of notation that does not di#er too much from common usage; the conventions are inspired by and adapted from the taxonomy proposed in [8] for randomized log space complexity classes, with some modifications that seem appropriate for extending these definitions to general space bounded classes. At the end of this section we include a short discussion relating the notation here to notation elsewhere. If M is a PTM, the language L ....

....of P rSPACE(s) and P r HSPACE(s) For NSPACE(s) and hence for RSPACE(s) closure under complementation is, of course, from [18, 48] Finally for SSPACE(s) the result follows from a very clever direct reduction [35] from the problem co USTCON to USTCON . This implies an earlier result of [8], which was proved using inductive counting arguments in the spirit of [18, 48] that SSPACE(s) RHSPACE(s) i.e. ZPHSPACE(s) Conspicuous by its omission from the list of classes in Theorem 3.2 is the class RHSPACE(s) Indeed, the following is a very tantalizing open question, which ....

[Article contains additional citation context not shown here]

A. Borodin, S. Cook, P. Dymond, W. Ruzzo, and M. Tompa. Two applications of inductive counting for complementation problems. SIAM Journal of Computing, 18:559--578, 1989.

.... that SAC 1 is the class of languages accepted by polynomial size circuits of depth O(log n) having unbounded fan in OR gates and bounded fan in AND gates; it is equal to LogCFL: the class of problems logspace reducible to a context free language [27] Since LogCFL is closed under complement [4], it can also be defined in terms of bounded fan in OR gates and unbounded fan in AND gates. Recall also that NC 1 # L # NL # SAC 1 . 3 Main Result Our main result extends the time space tradeoff of [14] where instead of giving lower bounds for solving SATwith nondeterministic ....

....accept input x, otherwise we reject it. # Remark: Although SAC 1 # NTiSp(n O(1) log 2 n) 23] it is not known how to evaluate a nearly linear size SAC 1 circuit in nearly linear nondeterministic time, using less than linear space. Similarly, although SAC 1 is closed under complement [4], known constructions involve squaring the circuit size. A simple corollary to the previous lemma is the following statement that is similar in flavor to (but much easier than) the version of the Karp Lipton collapse that we used in the proof of Theorem 8. Note that the uniformity condition is ....

A. Borodin, S. A. Cook, P. W. Dymond, W. L. Ruzzo, and M. Tompa. Two applications of inductive counting for complementation problems. SIAM Journal on Computing, 18:559--578, 1989.

....to it. The machine may access any bit on this tape several times. In this paper we focus our attention on this last issue: the access to randomness. For discussion of some of the other issues, and of the various 2 complexity classes defined by the different choices, we refer the reader e.g. to [BCD 89, Gil77, BCP83, KV85, RST84, Hon80]. In this note we only consider machines which halt always, i.e. machines with running time which is at most singly exponential in the space bound. Randomized space S machines which are not required to do so may run for exp(exp(S) time and are known to be quite strong, even if no error is ....

....(for zero error, one sided error, and bounded two sided error resp. and the classes obtained by multiple access by ZP SPACE(S) R SPACE(S) and BP SPACE(S) Most of our discussion will be focused on Logspace machines and the classes obtained by them: ZPL, RL, BPL, ZP L, R L, BP L (Note: in [BCD 89] the first three classes are denoted by ZPLP , RLP , and BPLP resp. Everything which we say, however, applies to any other nice (e.g. space constructible) space bound. The read once classes are the ones which are usually defined as the randomized analogs of Logspace, while multiple access to ....

[Article contains additional citation context not shown here]

A. Borodin, S.A. Cook, P.W. Dymond, W.L. Ruzzo, and M. Tompa. Two applications of inductive counting for complementation problems. SIAM J. Comput., 18(3):559--578, 1989.

....time limit. This time limit is polynomially related to n, and is easily computable in LOGSPACE from the description of G. The random algorithm has one sided error with small probability, it fails to determine that S and T are connected. A Las Vegas algorithm for reachability is designed in [4], but its expected running time is much larger than the running time of [2] s algorithm. It follows that connectivity can also be decided with one sided error in RLOGSPACE, and the time required is at most a factor of n higher than that required for reachability. But can we do better In this ....

A. Borodin, S. Cook, P. Dymond, W. Ruzzo, and M. Tompa. "Two applications of inductive counting for complementation problems". SIAM J. Comput., 18(3):559-578, 1989.

....a walk at S. If T is reached within 3mn steps, declare that S and T are connected. Otherwise not connected. The error probability can be made arbitrarily small by independent repetition of the algorithm. A logspace Las Vegas randomized algorithm with no error was subsequently designed in [BCDRT]. It is apparent that the random walk algorithm is wasteful in terms of time, as each edge is visited on average n times. Can the time complexity of the algorithm be improved, at the price of only a modest increase in the space complexity Aleliunas et al. AKLLR] asked whether there is a ....

A. Borodin, S. Cook, P. Dymond, W. Ruzzo, and M. Tompa. "Two applications of inductive counting for complementation problems". SIAM J. Comput., 18(3):559-578, 1989. 22

....characterization of the class LOGCFL of problems logspace reducible to context free languages. Let SAC i denote the class of boolean functions computable by semiunbounded fan in circuit families of polynomial size and depth O(log i n) so that NC i SAC i AC i . Since Borodin et al. [2] have shown that SAC i is closed under complementation for every i, this is equal to the class Supported by the DFG Emmy Noether Program under grant No. Jo 291 2 1 1 of functions computable by polynomial size, depth O(log i n) circuit families of fan in 2 and unbounded fan in. The ....

A. Borodin, S. A. Cook, P. W. Dymond, W. L. Ruzzo, and M. Tompa. Two applications of inductive counting for complementation problems. SIAM Journal of Computing, 18:559-578, 1989.

....can be simulated by logarithmically space bounded probabilistic Turing machines and vice versa. The classes ZPL, RL, BPL, and PL are defined as the classes of languages decidable by probabilistic uniform Turing machines with logarithmic space bound using the respective error model (see, e.g. [25] or [72] Ch. 2) Here we consider the nonuniform counterparts of these classes. Theorem 2.27: ZPP GBP = ZPL= Poly; RP GBP = RL= Poly; BPP GBP = BPL= Poly; PP GBP = PL= Poly : Proof: We adapt the proof of the Theorem of Pudlak and Zak (see Chapter 1) to the randomized setting. We only consider ....

A. Borodin, S. A. Cook, P. W. Dymond, W. L. Ruzzo, and M. Tompa. Two applications of inductive counting for complementation problems. SIAM J. Comp., 18(3):559--578, 1989.

....a natural extension of the study of monotone complexity classes, the study of non cancellative classes provides insight about the role of cancellation in structural issues, such as closure properties of complexity classes. For instance, while NL and SAC 1 are known to be closed under complement [12, 23, 4], we show that their non cancellative analogues are not. As mentioned earlier, non cancellative circuits can compute all Boolean functions, given enough size. It is therefore meaningful to ask how much size is necessary before cancellation becomes useless. To answer this, we show that the ....

....1 . But USTCONN is known to be in mNL. Since mNL SAC 1 , USTCONN 2 SAC 1 . 2 Corollary 5.1 NL is not closed under complement. Proof: Since NL SAC 1 , USTCONN 62 NL. But USTCONN is in mNL and therefore in NL. 2 The classes NL and SAC 1 are both known to be closed under complement [12, 23, 4]. The above result therefore implies that cancellations are critical to attain closure under complementation of these classes. For monotone classes that are known to be closed under complement, we do not know if the closure result also holds for the analogous non cancellative class. For example, ....

A. Borodin, S. Cook, P. Dymond, W. Ruzzo, M. Tompa, Two applications of inductive counting for complementation problems, SIAM J. Comput., 18 (1989), 559-578.

....of O(log n) space and 2 O(d) time [24] 93] There are other characterizations of these language classes in terms of alternating Turing machines and first order formulae [44] 81] 93] These equivalences hold for both the uniform and the nonuniform versions of the models. For a survey see [21]. An interesting property of classes defined by semi unbounded fan in circuits, proved by Borodin et al. 21] is that they are closed under complementation for all depths that are Omega Gammae 1 n) We consider the arithmetic analogs of the above complexity classes, defined by semi unbounded ....

....in terms of alternating Turing machines and first order formulae [44] 81] 93] These equivalences hold for both the uniform and the nonuniform versions of the models. For a survey see [21] An interesting property of classes defined by semi unbounded fan in circuits, proved by Borodin et al. [21], is that they are closed under complementation for all depths that are Omega Gammae 1 n) We consider the arithmetic analogs of the above complexity classes, defined by semi unbounded fan in arithmetic circuits. These classes have been studied for example in [6] PhiSAC k denotes the class ....

A. Borodin, S. A. Cook, P. W. Dymond, W. L. Ruzzo, M. Tompa, "Two applications of inductive counting for complementation problems," SIAM J. Comput., Vol. 18, No. 3, 1989, pp. 559-578. 66

....that combines both features: runs in polynomial time and poly logarithmic space. For st connectivity in undirected graphs two more types of algorithms are known. In [AKL 79] a randomized Logspace (and polynomial time) algorithm is given. A zero error version of this type of algorithm is given in [BCD 89]. As for deterministic algorithms, Barnes and Ruzzo [BR91] recently presented an algorithm for undirected st connectivity that runs in polynomial time and n ffl space (for any fixed ffl 0) Dept. of CS, Hebrew University, Jerusalem. Supported by BSF 89 00126 and by a Wolfson research award. ....

....problem: Input: An n by n transition probability matrix M , an integer t, and a rational ffl. t and ffl are given in unary. Output: A matrix A such that kA Gamma M t k ffl. 1 For an overview of the subtleties involved in the definition of randomized space bounded algorithms see e.g. [BCD 89, Nis93]. We follow the notation of [Nis93] In the notation of [BCD 89] we are talking about BPLP . 2 The algorithm runs in time poly(N) and space O(log 2 N) where N = n 2 t ffl Gamma1 is the input size. The simulation of randomized algorithms is based upon a careful usage of the ....

[Article contains additional citation context not shown here]

A. Borodin, S.A. Cook, P.W. Dymond, W.L. Ruzzo, and M. Tompa. Two applications of inductive counting for complementation problems. SIAM J. Comput., 18(3):559--578, 1989.

.... having a one way read only input tape, a pushdown tape and a log space bounded twoway read write work tape [31] It is known that the class of languages accepted by two way polynomial time NAuxPDA corresponds to the class of decision problems logspace reducible to context free recognition (see [12] for further references and results) while the class of languages accepted by polynomial time 1 NAuxPDA corresponds exactly to the class of decision problems that are reducible to c.f. recognition via one way logspace reductions [34, 14] Given a 1 NAuxPDA M , we define by d M (x) the number of ....

A. Borodin, S. A. Cook, P. W. Dymond, W. L. Ruzzo, and M. Tompa. Two applications of inductive counting for complementation problems. SIAM Journal on Computing, 18(6):559--578, 1989.

....small space simultaneously. See Tompa [32] and Edmonds and Poon [22] for lower bounds, and Barnes et al. 5] for an upper bound. In contrast, undirected graphs can be traversed in polynomial time and logarithmic space probabilistically by using a random walk (Aleliunas et al. 2] Borodin et al. [15]) this implies similar resource bounds on (nonuniform) deterministic algorithms (Aleliunas et al. 2] More recent work presents uniform deterministic polynomial time algorithms for the undirected case using sublinear space (Barnes and Ruzzo [8] and even O(log n) space (Nisan [28] as well ....

A. Borodin, S. A. Cook, P. W. Dymond, W. L. Ruzzo, and M. Tompa. Two applications of inductive counting for complementation problems. SIAM Journal on Computing, 18(3):559-- 578, June 1989.

No context found.

A. Borodin, S. A. Cook, P. W. Dymond, W. L. Ruzzo, and M. Tompa. "Two Applications of Inductive Counting for Complementation Problems". SIAM Journal of Computing, 18:559--578, 1989.

No context found.

Borodin, A., Cook, S. A., Dymond, P. W., Ruzzo, W. L., and Tompa, M. (1989). Two applications of inductive counting for complementation problems. SIAM Journal on Computing, 18(3):559--578.

No context found.

A. Borodin, S. A. Cook, P. W. Dymond, W. L. Ruzzo, and M. Tompa. Two applications of inductive counting for complementation problems. SIAM Journal on Computing, 18(3):559-- 578, 1989.

No context found.

A. Borodin, S. A. Cook, P. W. Dymond, W. L. Ruzzo, and M. Tompa. Two applications of inductive counting for complementation problems. SIAM Journal on Computing, 18(3):559--578, 1989.

No context found.

A. Borodin, S.A. Cook, P.W. Dymond, W.L. Ruzzo, and M. Tompa. Two applications of inductive counting for complementation problems. SIAM Journal on Computing, 18(3):559--578, 1989.

No context found.

A. Borodin, S. A. Cook, P. W. Dymond, W. L. Ruzzo, and M. Tompa. "Two Applications of Inductive Counting for Complementation Problems". SIAM Journal of Computing, 18:559--578, 1989.

No context found.

A. Borodin, S. A. Cook, P. W. Dymond, W. L. Ruzzo, and M. Tompa. Two applications of inductive counting for complementaion problems. SIAM J. Computing, 13:559-578, 1989.

No context found.

A. Borodin, S.A. Cook, P.W. Dymond, W.L. Ruzzo, and M. Tompa. Two applications of inductive counting for complementation problems. SIAM Journal on Computing, 18(3):559--578, 1989.

No context found.

A. Borodin, S. A. Cook, P. W. Dymond, W. L. Ruzzo, and M. Tompa. Two applications of inductive counting for complementation problems. SIAM Journal on Computing, 18(3):559--578, June 1989. See also 18(6): 1283, December 1989.

No context found.

A. Borodin, S.A. Cook, P.W. Dymond, W.L. Ruzzo, and M. Tompa. Two applications of inductive counting for complementation problems. SIAM Journal on Computing, 18(3):559--578, 1989.

No context found.

A. Borodin, S.A. Cook, P.W. Dymond, W.L. Ruzzo, and M. Tompa. Two applications of inductive counting for complementation problems. SIAM Journal on Computing, 18(3):559--578, 1989. 8

First 50 documents

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