Proceedings of the nineteenth annual ACM symposium on Theory of computing最新文献

{"title":"The Boolean formula value problem is in ALOGTIME","authors":"S. Buss","doi":"10.1145/28395.28409","DOIUrl":"https://doi.org/10.1145/28395.28409","url":null,"abstract":"The Boolean formula value problem is to determine the truth value of a variable-free Boolean formula, or equivalently, to recognize the true Boolean sentences. N. Lynch [ll] gave log space algorithms for the Boolean formula value problem and for the more general problem of recognizing a parenthesis context-free grammar. This paper shows that these problems have alternating log time algorithms. This answers the question of Cook [5] of whether the Boolean formula value problem is log space complete it is not, unless log space and alternating log time are identical. Our results are optimal since, for an appropriately defined notion of fog time reductions, the Boolean formula value problem is complete for alternating log time under deterministic log time reductions; consequently, it is al30 complete for alternating log time under AC0 reductions. It follows that the Boolean formula value problem is not in the log time hierarchy. There are two reasons why the Boolean formula value problem is interesting. First, a Boolean (or propositional) formula is a very fundamental concept of logic. The computational complexity of evaluating a Boolean formula is therefore of interest. Indeed, the results below will give a precise characterisation of the computational complexity of determining the truth value of a Boolean formula. Second, the existence of an alternating log time algorithm for the Boolean formula problem implies the existence of log depth, polynomial size circuits for this problem and hence there are (at least theoretically) good parallel algorithms for determining the value of a Boolean sentence. As mentioned above, N. Lynch [ll] first studied the complexity of the Boolean formula problem. It follows from Lynch’s work that the Boolean formula value problem is in NC’, since Borodin [l] showed that LOCSPACE C NCZ. Another early significant result on this problem was due to Spira [17] who showed that for every formula of size n, there is an equivalent formula of size O(n2) and depth O(log n). An improved construction, which also applied to the evaluation of rational expressions, was obtained by Brent [z]. Spira’s result WAS significant in part because because it implied that there might be a family of polynomial size, log depth circuits for recognizing true Boolean formulas. In other words, that the Boolean formula value problem might be in (non-uniform) NC’. However, it was not known if the transformations of formulas defined by Brent and Spira could","PeriodicalId":161795,"journal":{"name":"Proceedings of the nineteenth annual ACM symposium on Theory of computing","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1987-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125614214","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A model for hierarchical memory","authors":"A. Aggarwal, B. Alpern, A. K. Chandra, M. Snir","doi":"10.1145/28395.28428","DOIUrl":"https://doi.org/10.1145/28395.28428","url":null,"abstract":"In this paper we introduce the Hierarchical Memory Model (HMM) of computation. It is intended to model computers with multiple levels in the memory hierarchy. Access to memory location x is assumed to take time ⌈ log x ⌉. Tight lower and upper bounds are given in this model for the time complexity of searching, sorting, matrix multiplication and FFT. Efficient algorithms in this model utilize locality of reference by bringing data into fast memory and using them several times before returning them to slower memory. It is shown that the circuit simulation problem has inherently poor locality of reference. The results are extended to HMM's where memory access time is given by an arbitrary (nondecreasing) function. Tight upper and lower bounds are obtained for HMM's with polynomial memory access time; the algorithms for searching, FFT and matrix multiplication are shown to be optimal for arbitrary memory access time. On-line memory management algorithms for the HMM model are also considered. An algorithm that uses LRU policy at the successive “levels” of the memory hierarchy is shown to be optimal.","PeriodicalId":161795,"journal":{"name":"Proceedings of the nineteenth annual ACM symposium on Theory of computing","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1987-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129184408","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Matrix multiplication via arithmetic progressions","authors":"D. Coppersmith, S. Winograd","doi":"10.1145/28395.28396","DOIUrl":"https://doi.org/10.1145/28395.28396","url":null,"abstract":"We present a new method for accelerating matrix multiplication asymptotically. This work builds on recent ideas of Volker Strassen, by using a basic trilinear form which is not a matrix product. We make novel use of the Salem-Spencer Theorem, which gives a fairly dense set of integers with no three-term arithmetic progression. Our resulting matrix exponent is 2.376.","PeriodicalId":161795,"journal":{"name":"Proceedings of the nineteenth annual ACM symposium on Theory of computing","volume":"37 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1987-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127752816","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Finite monoids and the fine structure of NC1","authors":"D. M. Barrington, D. Thérien","doi":"10.1145/28395.28407","DOIUrl":"https://doi.org/10.1145/28395.28407","url":null,"abstract":"Recently a new connection was discovered between the parallel complexity class NC1 and the theory of finite automata, in the work of Barrington [Ba86] on bounded width branching programs. There (non-uniform) NC1 was characterized as those languages recognized by a certain non-uniform version of a DFA. Here we extend this characterization to show that the internal structures of NC1 and the class of automata are closely related. In particular, using Thérien's classification of finite monoids [Th81], we give new characterizations of the classes AC0, depth-k AC0, and ACC, the last being the AC0 closure of the mod q functions for all constant q. We settle some of the open questions in [Ba86], give a new proof that the dot-depth hierarchy of algebraic automata theory is infinite [BK78], and offer a new framework for understanding the internal structure of NC1.","PeriodicalId":161795,"journal":{"name":"Proceedings of the nineteenth annual ACM symposium on Theory of computing","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1987-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114736946","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"How to play ANY mental game","authors":"Oded Goldreich, S. Micali, A. Wigderson","doi":"10.1145/28395.28420","DOIUrl":"https://doi.org/10.1145/28395.28420","url":null,"abstract":"We present a polynomial-time algorithm that, given as a input the description of a game with incomplete information and any number of players, produces a protocol for playing the game that leaks no partial information, provided the majority of the players is honest. Our algorithm automatically solves all the multi-party protocol problems addressed in complexity-based cryptography during the last 10 years. It actually is a completeness theorem for the class of distributed protocols with honest majority. Such completeness theorem is optimal in the sense that, if the majority of the players is not honest, some protocol problems have no efficient solution [C].","PeriodicalId":161795,"journal":{"name":"Proceedings of the nineteenth annual ACM symposium on Theory of computing","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1987-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125090228","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The strong exponential hierarchy collapses","authors":"Lane A. Hemachandra","doi":"10.1145/28395.28408","DOIUrl":"https://doi.org/10.1145/28395.28408","url":null,"abstract":"The polynomial hierarchy, composed of the levels P, NP, PNP, NPNP, etc., plays a central role in classifying the complexity of feasible computations. It is not known whether the polynomial hierarchy collapses. We resolve the question of collapse for an exponential-time analogue of the polynomial-time hierarchy. Composed of the levels E (i.e., ⋓c DTIME[2cn]), NE, PNE, NPNE, etc., the strong exponential hierarchy collapses to its &Dgr;2 level. E ≠ PNE = NPNE ⋓ NPNPNE ⋓ ··· Our proof stresses the use of partial census information and the exploitation of nondeterminism. Extending our techniques, we also derive new quantitative relativization results. We show that if the (weak) exponential hierarchy's &Dgr;j+1 and &Sgr;j+1 levels, respectively E&Sgr;pj and NE&Sgr;pj, do separate, this is due to the large number of queries NE makes to its &Sgr;pj database.1 Our technique provide a successful method of proving the collapse of certain complexity classes.","PeriodicalId":161795,"journal":{"name":"Proceedings of the nineteenth annual ACM symposium on Theory of computing","volume":"37 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1987-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129498395","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Imperfect random sources and discrete controlled processes","authors":"David Lichtenstein, N. Linial, M. Saks","doi":"10.1145/28395.28414","DOIUrl":"https://doi.org/10.1145/28395.28414","url":null,"abstract":"We consider a simple model for a class of discrete control processes, motivated in part by recent work about the behavior of imperfect random sources in computer algorithms. The process produces a string of characters from {0, 1} of length n and is a “success” or “failure” depending on whether the string produced belongs to a prespecified set L. In an uninfluenced process each character is chosen by a fair coin toss, and hence the probability of success is |L|/2n. We are interested in the effect on the probability of success in the presence of a player (controller) who can intervene in the process by specifying the value of certain characters in the string. We answer the following questions in both worst and average case: (1) how much can the player increase the probability of success given a fixed number of interventions? (2) in terms of |L| what is the expected number of interventions needed to guarantee success? In particular our results imply that if |L|/2n = 1/w(n) where w(n) tends to infinity with n (so the probability of success with no interventions is o(1)) then with &Ogr;(√nlogw(n)) interventions the probability of success is 1-o(1). Our main results and the proof techniques are related to a well-known theorem of Kruskal, Katona, and Harper in extremal set theory.","PeriodicalId":161795,"journal":{"name":"Proceedings of the nineteenth annual ACM symposium on Theory of computing","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1987-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129744435","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Recognizing primes in random polynomial time","authors":"L. Adleman, Ming-Deh A. Huang","doi":"10.1145/28395.28445","DOIUrl":"https://doi.org/10.1145/28395.28445","url":null,"abstract":"This paper is the first in a sequence of papers which will prove the existence of a random polynomial time algorithm for the set of primes. The techniques used are from arithmetic algebraic geometry and to a lesser extent algebraic and analytic number theory. The result complements the well known result of Strassen and Soloway that there exists a random polynomial time algorithm for the set of composites.","PeriodicalId":161795,"journal":{"name":"Proceedings of the nineteenth annual ACM symposium on Theory of computing","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1987-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130303132","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Deterministic simulation in LOGSPACE","authors":"M. Ajtai, J. Komlos, E. Szemerédi","doi":"10.1145/28395.28410","DOIUrl":"https://doi.org/10.1145/28395.28410","url":null,"abstract":"In this paper we show that a wide class of probabilistic algorithms can be simulated by deterministic algorithms. Namely if there is a test in LOGSPACE so that a random sequence of length (log n)2 / log log n passes the test with probability at least 1/n then a deterministic sequence can be constructed in LOGSPACE which also passes the test. It is important that the machine performing the test gets each bit of the sequence only once. The theorem remains valid if both the test and the machine constructing the satisfying sequence have access to the same oracle of polynomial size. The sequence that we construct does not really depend on the test, in the sense that a polynomial family of sequences is constructed so that at least one of them passes any test. This family is the same even if the test is allowed to use an oracle of polynomial size, and it can be constructed in LOGSPACE (without using an oracle).","PeriodicalId":161795,"journal":{"name":"Proceedings of the nineteenth annual ACM symposium on Theory of computing","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1987-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131779604","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Simple algebras are difficult","authors":"L. Rónyai","doi":"10.1145/28395.28438","DOIUrl":"https://doi.org/10.1145/28395.28438","url":null,"abstract":"Let F be a finite field or an algebraic number field. In previous work we have shown how to find the basic building blocks (the radical and the simple components) of a finite dimensional algebra over F in polynomial time (deterministically in characteristic zero and Las Vegas in the finite case). Here we address the more general problem of finding zero divisors in A. This problem is equivalent to finding a nontrivial common invariant subspace of a set of linear operators and includes, as a subcase, the problem of factoring polynomials over the field in question. In [FR] the problem of zero divisors has been reduced, in polynomial time (Las Vegas in the finite case), to the case of simple algebras. We show that, while zero divisors can be found in Las Vegas polynomial time if F is finite, the problem over the rationals might be substantially more difficult. We link the problem to hard number theoretic problems such as quadratic residuosity modulo a composite number. We show that assuming the Generalized Riemann Hypothesis, there exists a randomized polynomial time reduction from quadratic residuosity to determining whether or not a given 4-dimensional algebra over Q has zero divisors. It will follow that finding a pair of zero divisors is at least as hard as factoring squarefree integers. As for the finite case, we give a polynomial time Las Vegas method to construct explicit isomorphisms of matrix algebras. Applications include an algorithm to solve the problem of finding common invariant subspaces for a set of linear operators. Another application answers a question of W. M. Kantor on permutation groups. Finally, as another application of the GRH, we mention a partial result on deterministic factoring over finite fields.","PeriodicalId":161795,"journal":{"name":"Proceedings of the nineteenth annual ACM symposium on Theory of computing","volume":"81 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1987-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114473800","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}