Theoretical computer science treats any computational subject for which a good model can be created. Research on formal models of computation was initiated in the 1930s and 1940s by Turing, Post, Kleene, Church, and others. In the 1950s and 1960s programming languages, language translators, and operating systems were under development and therefore became both the subject and basis for a great deal of theoretical work. The power of computers of this period was limited by slow processors and small amounts of memory, and thus theories (models, algorithms, and analysis) were developed to explore the efficient use of computers as well as the inherent complexity of problems. The former subject is known today as algorithms and data structures, the latter computational complexity. The focus of theoretical computer scientists in the 1960s on languages is reflected in the first textbook on the subject, Formal Languages and Their Relation to Automata by John Hopcroft and Jeffrey Ullman. This influential book led to the creation of many languagecentered theoretical computer science courses; many introductory theory courses today continue to reflect the content of this book and the interests of theoreticians of the 1960s and early 1970s. Although

We survey lower bounds established for the complexity of computing explicitly given Boolean functions by switching-and-rectifier networks, branching programs and switching networks. We first consider the unrestricted case and then proceed to various restricted models. Among these are monotone networks, bounded-width devices , oblivious devices and read-k times only devices. 1 Introduction The main goal of the Boolean complexity theory is to prove lower bounds on the complexity of computing "explicitly given" Boolean functions in interesting computational models. By "explicitly given" researchers usually mean "belonging to the class NP ". This is a very plausible interpretation since on the one hand this class contains the overwhelming majority of interesting Boolean functions and on the other hand it is small enough to prevent us from the necessity to take into account counting arguments. To illustrate the second point, let me remind the reader that already the class \Delta p 2 ,...

Abstract: We give an exponential lower bound for the size of any linear-time Boolean branching program computing an explicitly given function. More precisely, we prove that for all positive integers k and for all sufficiently small ε> 0, if n is sufficiently large then there is no Boolean (or 2-way) branching program of size less than 2 εn which, for all inputs X ⊆ {0,1,...,n − 1}, computes in time kn the parity of the number of elements of the set of all pairs 〈x,y 〉 with the property x ∈ X, y ∈ X, x < y, x + y ∈ X. For the proof of this fact we show that if A = (ai, j) n i=0, j=0 is a random n by n matrix over the field with 2 elements with the condition that “A is constant on each minor diagonal,” then with high probability the rank of each δn by δn submatrix of A is at least cδ|logδ | −2n, where c> 0 is an absolute constant and n is sufficiently large with respect to δ.

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We obtain the first non-trivial time-space tradeoff lower bound for functions f : {0, 1}^n → {0, 1} on general branching programs by exhibiting a Boolean function f that requires exponential size to be computed by any branching program of length (1 + ε)n, for some constant ε > 0. We also give the first separation result between the syntactic and semantic read-k models [BRS93] for k > 1 by showing that polynomial-size semantic read-twice branching programs can compute functions that require exponential size on any syntactic read-k branching program. We also show...

We prove the first time-space lower bound tradeoffs for randomized computation of decision problems.

We prove the first time-space lower bound tradeoffs for randomized computation of decision problems. The bounds hold even in the case that the computation is allowed to have arbitrary probability of error on a small fraction of inputs. Our techniques are an extension of those used by Ajtai [Ajt99a, Ajt99b] in his time-space tradeoffs for deterministic RAM algorithms computing element distinctness and for Boolean branching programs computing a natural quadratic form. Ajtai's bounds were of the following form...

An optimal R(n2) lower bound is shown for the time-space product of any R-way branching program that determines those values which occur exactly once in a list of n integers in the range [l, R] where R 1 n. This Q(n2) tradeoff also applies to the sorting problem and thus improves the previous time-space tradeoffs for sorting. Because the R-way branching program is a such a powerful model these time-space product tradeoffs also apply to all models of sequential computation that have a fair measure of space such as off-line multi-tape Turing machines and off-line log-cost RAMS. 1

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We consider the classical problem of clock synchronization in distributed systems. Previously, this problem was solved optimally and efficiently only in the case when all individual clocks are non-drifting, i.e., only for systems where all clocks advance at the rate of real time. In this paper, we present a new algorithm for systems with drifting clocks, which is the first optimal algorithm to solve the problem efficiently: clock drift bounds and message latency bounds may be arbitrary; the computational complexity depends on the communication pattern of the system in a way which is bounded by a polynomial in the network size for most systems. More specifically, the complexity is polynomial in the maximal number of messages known to be sent but not received, the relative system speed, and time-stamp s...