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34
Communication complexity lower bounds by polynomials
 In Proc. of the 16th Conf. on Computational Complexity (CCC
, 2001
"... The quantum version of communication complexity allows Alice and Bob to communicate qubits and/or to make use of prior entanglement (shared EPRpairs). Some lower bound techniques are available for qubit communication [17, 11, 2], but except for the inner product function [11], no bounds are known f ..."
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Cited by 60 (12 self)
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The quantum version of communication complexity allows Alice and Bob to communicate qubits and/or to make use of prior entanglement (shared EPRpairs). Some lower bound techniques are available for qubit communication [17, 11, 2], but except for the inner product function [11], no bounds are known for the model with unlimited prior entanglement. We show that the “log rank ” lower bound extends to the strongest model (qubit communication + prior entanglement). By relating the rank of the communication matrix to properties of polynomials, we are able to derive some strong bounds for exact protocols. In particular, we prove both the “logrank conjecture ” and the polynomial equivalence of quantum and classical communication complexity for various classes of functions. We also derive some weaker bounds for boundederror protocols. 1
Quantum Communication and Complexity
 Theoretical Computer Science
, 2000
"... In the setting of communication complexity, two distributed parties want to compute a function depending on both their inputs, using as little communication as possible. The required communication can sometimes be significantly lowered if we allow the parties the use of quantum communication. We sur ..."
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Cited by 32 (13 self)
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In the setting of communication complexity, two distributed parties want to compute a function depending on both their inputs, using as little communication as possible. The required communication can sometimes be significantly lowered if we allow the parties the use of quantum communication. We survey the main results of the young area of quantum communication complexity: its relation to teleportation and dense coding, the main examples of fast quantum communication protocols, lower bounds, and some applications. 1 Introduction The area of communication complexity deals with the following type of problem. There are two separated parties, called Alice and Bob. Alice receives some input x 2 X, Bob receives some y 2 Y , and together they want to compute some function f(x; y). As the value f(x; y) will generally depend on both x and y, neither Alice nor Bob will have sufficient information to do the computation by themselves, so they will have to communicate in order to achieve their go...
On quantum and probabilistic communication: Las Vegas and oneway protocols
 in Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, 2000
, 2000
"... We investigate the power of quantum communication protocols compared to classical probabilistic protocols. In our first result we describe a total Boolean function that has a quantum Las Vegas protocol communicating at most O(N^{10/11+ epsilon}) qubits for all epsilon > 0, while any classical probab ..."
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Cited by 30 (5 self)
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We investigate the power of quantum communication protocols compared to classical probabilistic protocols. In our first result we describe a total Boolean function that has a quantum Las Vegas protocol communicating at most O(N^{10/11+ epsilon}) qubits for all epsilon > 0, while any classical probabilistic protocol (with bounded error) needs Omega(N/log N) bits. Then we investigate quantum oneway communication complexity. First we show that the VCdimension lower bound on oneway probabilistic communication of [26] holds for quantum protocols, too. Then we prove that for oneway protocols computing total functions quantum Las Vegas communication is asymptotically as efficient as exact quantum communication, which is exactly as efficient as deterministic communication. We describe applications of the lower bounds for oneway communication complexity to quantum finite automata and quantum formulae.
Towards Regular Languages Over Infinite Alphabets
, 2001
"... Motivated by formal models recently proposed in the context of XML, we study automata and logics on strings over infinite alphabets. These are conservative extensions of classical automata and logics defining the regular languages on finite alphabets. Specically, we consider register and pebble auto ..."
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Cited by 24 (4 self)
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Motivated by formal models recently proposed in the context of XML, we study automata and logics on strings over infinite alphabets. These are conservative extensions of classical automata and logics defining the regular languages on finite alphabets. Specically, we consider register and pebble automata, and extensions of firstorder logic and monadic secondorder logic. For each type of automaton we consider oneway and twoway variants, as well as deterministic, nondeterministic, and alternating control. We investigate the expressiveness and complexity of the automata, their connection to the logics, as well as standard decision problems. Some of our results answer open questions of Kaminski and Francez on register automata.
Quantum communication complexity
 In Proc. Intl. Colloquium on Automata, Languages, and Programming (ICALP
, 2000
"... This paper surveys the field of quantum communication complexity. Some interesting recent results are collected concerning relations to classical communication, lower bound methods, oneway communication, and applications of quantum communication complexity. 1 ..."
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Cited by 18 (2 self)
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This paper surveys the field of quantum communication complexity. Some interesting recent results are collected concerning relations to classical communication, lower bound methods, oneway communication, and applications of quantum communication complexity. 1
A Lower Bound for Randomized ReadkTimes Branching Programs
 Electr. Coll. on Comp. Compl
, 1997
"... In this paper, we are concerned with randomized OBDDs and randomized readktimes branching programs. We present an example of a Boolean function which has polynomial size randomized OBDDs with small, onesided error, but only nondeterministic readonce branching programs of exponential size. Further ..."
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Cited by 15 (8 self)
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In this paper, we are concerned with randomized OBDDs and randomized readktimes branching programs. We present an example of a Boolean function which has polynomial size randomized OBDDs with small, onesided error, but only nondeterministic readonce branching programs of exponential size. Furthermore, we discuss a lower bound technique for randomized OBDDs with twosided error and prove an exponential lower bound of this type. Our main result is an exponential lower bound for randomized readktimes branching programs with twosided error. 1 Introduction Branching programs are a theoretically and practically interesting data structure for the representation of Boolean functions. In complexity theory, among other problems, lower bounds for the size of branching programs for explicitly defined functions and the relations of the various branching program models are investigated. A branching program (BP) on the variable set fx 1 ; : : : ; x n g is a directed acyclic graph with one sour...
On the Size of Randomized OBDDs and ReadOnce Branching Programs for kStable Functions
 In Proc. of the 16th Ann. Symp. on Theoretical Aspects of Computer Science (STACS), LNCS 1563
, 1999
"... In this paper, a simple technique which unifies the known approaches for proving lower bound results on the size of deterministic, nondeterministic, and randomized OBDDs and kOBDDs is described. ..."
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Cited by 12 (9 self)
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In this paper, a simple technique which unifies the known approaches for proving lower bound results on the size of deterministic, nondeterministic, and randomized OBDDs and kOBDDs is described.
Complexity Theoretical Results for Randomized Branching Programs
, 1998
"... This work is settled in the area of complexity theory for restricted variants of branching programs. Today, branching programs can be considered one of the standard nonuniform models of computation. One reason for their popularity is that they allow to describe computations in an intuitively straigh ..."
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Cited by 9 (8 self)
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This work is settled in the area of complexity theory for restricted variants of branching programs. Today, branching programs can be considered one of the standard nonuniform models of computation. One reason for their popularity is that they allow to describe computations in an intuitively straightforward way and promise to be easier to analyze than the traditional models. In complexity theory, we are mainly interested in upper and lower bounds on the size of branching programs. Although proving superpolynomial lower bounds on the size of general branching programs still remains a challenging open problem, there has been considerable success in the study of lower bound techniques for various restricted variants, most notably perhaps readonce branching programs and OBDDs (ordered binary decision diagrams). Surprisingly, OBDDs have also turned out to be extremely useful in practical applications as a data structure for Boolean functions. So far, research has concentrated on determinis...
Communication Complexity Method for Measuring Nondeterminism in Finite Automata
, 2000
"... While deterministic finite automata seem to be well understood, surprisingly many important problems concerning nondeterministic finite automata (nfa's) remain open. One such problem ..."
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Cited by 7 (1 self)
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While deterministic finite automata seem to be well understood, surprisingly many important problems concerning nondeterministic finite automata (nfa's) remain open. One such problem
Finding lower bounds for nondeterministic state complexity is hard (extended abstract)
 PROCEEDINGS OF THE 10TH INTERNATIONAL CONFERENCE ON DEVELOPMENTS IN LANGUAGE THEORY
, 2006
"... Abstract. We investigate the following lower bound methods for regular languages: The fooling set technique, the extended fooling set technique, and the biclique edge cover technique. It is shown that the maximal attainable lower bound for each of the above mentioned techniques can be algorithmicall ..."
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Cited by 6 (4 self)
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Abstract. We investigate the following lower bound methods for regular languages: The fooling set technique, the extended fooling set technique, and the biclique edge cover technique. It is shown that the maximal attainable lower bound for each of the above mentioned techniques can be algorithmically deduced from a canonical finite graph, the so called dependency graph of a regular language. This graph is very helpful when comparing the techniques with each other and with nondeterministic state complexity. In most cases it is shown that for any two techniques the gap between the best bounds can be arbitrarily large. The only exception is the biclique edge cover technique which is always as good as the logarithm of the deterministic or nondeterministic state complexity. Moreover, we show that deciding whether a certain lower bound w.r.t. one of the investigated techniques can be achieved is in most cases computationally hard, i.e., PSPACEcomplete and hence is as hard as minimizing nondeterministic finite automata. 1