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The multiparty communication complexity of ExactT: Improved bounds and new problems
 In Proc. of 31st MFCS
, 2006
"... Abstract Let x1,..., xk be nbit numbers and T 2 N. Assume that P1,..., Pk are players such that Pi knows all of the numbers except xi. The players want to determine if Pkj=1 xj = T bybroadcasting as few bits as possible. Chandra, Furst, and Lipton obtained an upper bound of O(pn) bits for the k = 3 ..."
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Cited by 6 (3 self)
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Abstract Let x1,..., xk be nbit numbers and T 2 N. Assume that P1,..., Pk are players such that Pi knows all of the numbers except xi. The players want to determine if Pkj=1 xj = T bybroadcasting as few bits as possible. Chandra, Furst, and Lipton obtained an upper bound of O(pn) bits for the k = 3 case, and a lower bound of!(1) for k> = 3 when T = \Theta (2n). We obtain(1) for general k> = 3 an upper bound of k + O(n1/(k1)), (2) for k = 3, T = \Theta (2n), a lowerbound of \Omega (log log n), (3) a generalization of the protocol to abelian groups, (4) lower boundson the multiparty communication complexity of some regular languages, (5) lower bounds on branching programs, and (6) empirical results for the k = 3 case. 1 Introduction Multiparty communication complexity was first defined by Chandra, Furst, and Lipton [8] and used to obtain lower bounds on branching programs. Since then it has been used to get additional lower bounds and tradeoffs for branching programs [1, 5], lower bounds on problems in data structures [5], timespace tradeoffs for restricted Turing machines [1], and unconditional pseudorandom generators for logspace [1]. Def 1.1 Let f: {{0, 1}n}k! {0, 1}. Assume, for 1 < = i < = k, Pi has all of the inputs except xi. Let d(f) be the total number of bits broadcast in the optimal deterministic protocol for f. This is called the multiparty communication complexity of f. The scenario is called the forehead model.
Relating Branching Program Size and Formula Size over the Full Binary Basis
 In Proc. of 16th STACS, LNCS 1563
, 1998
"... Circuit size, branching program size, and formula size of Boolean functions, denoted by C(f), BP(f), and L(f), are the most important complexity measures for Boolean functions. Often also the formula size L (f) over the restricted basis f; ; :g is considered. It is wellknown that C(f) 3 BP(f), ..."
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Cited by 5 (2 self)
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Circuit size, branching program size, and formula size of Boolean functions, denoted by C(f), BP(f), and L(f), are the most important complexity measures for Boolean functions. Often also the formula size L (f) over the restricted basis f; ; :g is considered. It is wellknown that C(f) 3 BP(f), BP(f) L (f), L (f) L(f) 2 , and C(f) L(f) \Gamma 1. These estimates are optimal. But the inequality BP(f) L(f) 2 can be improved to BP(f) 1:360 L(f) fi , where fi = log 4 (3 + p 5) ! 1:195. 1
Letting Alice and Bob choose which problem to solve: Implications to the study of cellular automata ✩
"... In previous works we found necessary conditions for a cellular automaton (CA) in order to be intrinsically universal (a CA is said to be intrinsically universal if it can simulate any other). The idea was to introduce different canonical communication problems, all of them parameterized by a CA. The ..."
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In previous works we found necessary conditions for a cellular automaton (CA) in order to be intrinsically universal (a CA is said to be intrinsically universal if it can simulate any other). The idea was to introduce different canonical communication problems, all of them parameterized by a CA. The necessary condition was the following: if Ψ is an intrinsically universal CA then the communication complexity of all the canonical problems, when parameterized by Ψ, must be maximal. In this paper, instead of introducing a new canonical problem, we study the setting where they can all be used simultaneously. Roughly speaking, when Alice and Bob –the two parties of the communication complexity model – receive their inputs they may choose online which canonical problem to solve. We give results showing that such freedom makes this new problem, that we call Ovrl, a very strong filter for ruling out CAs from being intrinsically universal. More precisely, there are some CAs having high complexity in all the canonical problems but have much lower complexity in Ovrl. Key words: automata. communication complexity, intrinsic universality, cellular 1.
Birkhiiuser Verlag, Basel TWO TAPES VERSUS ONE FOR OFFL INE TURING MACHINES
"... Abst rac t. We prove the first superlinear lower bound for a concrete, polynomial time recognizable d cision problem on a Taring machine with one work tape and a twoway input tape (also called offline 1tape Turing machine). In particular, for offline Turing machines we show that two tapes are bet ..."
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Abst rac t. We prove the first superlinear lower bound for a concrete, polynomial time recognizable d cision problem on a Taring machine with one work tape and a twoway input tape (also called offline 1tape Turing machine). In particular, for offline Turing machines we show that two tapes are better than one and that three pushdown stores are better than two (both in the deterministic and in the nondeterministic case). Key words, offline 1tape Turing machines; two tapes; lower bounds; time; nondeterminism. Subject classifications. 68Q05, 68Q25. 1. In t roduct ion A 1tape offline Turing machine (see Hennie 1965, p.166) is a Turing machine (TM) with one work tape and an additional twoway input tape, i.e., an input tape with end markers on which the associated readonly input head can move without restriction in both directions. These TM's are used as the standard model for the analysis of the space complexity of TMcomputations. In addition, they are of interest as an intermediate model between the relatively slow 1tape TM without input tape and the relatively powerful 2tape TM. No nontrivial ower bounds are known for the recognition of polynomial time computable languages on 2tape Turing machines. On the other hand, lower bound arguments for concrete languages on restricted TM's have progressed from 1tape TM's without input tape (Hennie 1965, Rabin 1963) to
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"... Let x1,..., xk be nbit numbers and T ∈ N. Assume that P1,..., Pk are players such that Pi knows all of the numbers except xi. The players want to determine if � k j=1 xj = T by broadcasting as few bits as possible. Chandra, Furst, and Lipton obtained an upper bound of O ( √ n) bits for the k = 3 c ..."
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Let x1,..., xk be nbit numbers and T ∈ N. Assume that P1,..., Pk are players such that Pi knows all of the numbers except xi. The players want to determine if � k j=1 xj = T by broadcasting as few bits as possible. Chandra, Furst, and Lipton obtained an upper bound of O ( √ n) bits for the k = 3 case, and a lower bound of ω(1) for k ≥ 3 when T = Θ(2 n). We obtain (1) for general k ≥ 3 an upper bound of k + O(n 1/(k−1)), (2) for k = 3, T = Θ(2 n), a lower bound of Ω(log log n), (3) a generalization of the protocol to abelian groups, (4) lower bounds on the multiparty communication complexity of some regular languages, (5) lower bounds on branching programs, and (6) empirical results for the k = 3 case. 1
Abstract On the Power of Automata Based Proof Systems
"... One way to address the NP = co − NP question is to consider the length of proofs of tautologies in various proof systems. In this work we consider proof systems defined by appropriate classes of automata. In general, starting from a given class of automata we can define a corresponding proof system ..."
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One way to address the NP = co − NP question is to consider the length of proofs of tautologies in various proof systems. In this work we consider proof systems defined by appropriate classes of automata. In general, starting from a given class of automata we can define a corresponding proof system in a natural way. An interesting new proof system that we consider is based on the class of push down automata. We present an exponential lower bound for oblivious readonce branching programs which implies that the new proof system based on push down automata is, in a certain sense, more powerful than oblivious regular resolution. 1
On the Power of Automata Based Proof Systems
, 1999
"... One way to address the NP = co  NP question is to consider the length of proofs of tautologies in various proof systems. In this work we consider proof systems defined by appropriate classes of automata. In general, starting from a given class of automata we can define a corresponding proof syste ..."
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One way to address the NP = co  NP question is to consider the length of proofs of tautologies in various proof systems. In this work we consider proof systems defined by appropriate classes of automata. In general, starting from a given class of automata we can define a corresponding proof system in a natural way. An interesting new proof system that we consider is based on the class of push down automata. We present an exponential lower bound for oblivious readonce branching programs which implies that the new proof system based on push down automata is, in a certain sense, more powerful than oblivious regular resolution. 1 Introduction One of the famous open questions of complexity theory is: does NP equal coNP ? Put another way do tautologies always have "short" proofs? If proof is taken in its most general form, i.e. does some nondeterministic polynomial time Turing Machine correctly accept exactly the class of tautologies, then the question seems to be completely beyond ou...
Acknowledgements
, 1998
"... iii This thesis reports an unusual and unexpected journey through intellectual territory entirely new to me. Who is prepared for such journeys? I was not. Thus my debt is great to those who have helped me along the way, without whose help I would have been completely lost and uninspired. There is no ..."
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iii This thesis reports an unusual and unexpected journey through intellectual territory entirely new to me. Who is prepared for such journeys? I was not. Thus my debt is great to those who have helped me along the way, without whose help I would have been completely lost and uninspired. There is no way to give sufficient thanks to my advisor, John Hopfield. His encouragement for me to get my feet wet, and his advice cutting to the bone of each issue, have been invaluable. John’s policy has been, in his own words, to give his students “enough rope to hang themselves with. ” But he knows full well that a lot of rope is needed to weave macrame. Perhaps the only possible repayment is in kind: to maintain a high standard of integrity, and when my turn comes, to provide a nurturing environment for other young minds. Len Adleman and Ned Seeman have each been mentors during my thesis work. In many ways, my research can be seen as the direct offspring of their work, combining the notion of using DNA for computation with the ability to design DNA structures with artificial topology. Both Len and Ned have provided encouragement, support, and valuable feedback throughout this project. Iwould like additionally to thank Ned Seeman and Xiaoping Yang for teaching me how to
Ramsey Theory Applications
"... There are many interesting applications of Ramsey theory, these include results in number theory, algebra, geometry, topology, set theory, logic, ergodic theory, information theory and theoretical computer science. Relations of Ramseytype theorems to various fields in mathematics are well documente ..."
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There are many interesting applications of Ramsey theory, these include results in number theory, algebra, geometry, topology, set theory, logic, ergodic theory, information theory and theoretical computer science. Relations of Ramseytype theorems to various fields in mathematics are well documented in published books and monographs. The main objective of this survey is to list applications mostly in theoretical computer science of the last two decades not contained in these. 1