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

Consider the following communication problem. Alice gets a word x 2 f0; 1g n and Bob gets a word y 2 f0; 1g n . Alice and Bob are told that x 6= y. Their goal is to find an index 1 i n such that x i 6= y i (the index i should be known to both of them). This problem is one of the most basic communication problems. It arises naturally from the correspondence between circuit depth and communication complexity discovered by Karchmer and Wigderson. We present three protocols using which Alice and Bob can solve the problem by exchanging at most n + 2 bits. One of this protocols is due to Rudich and Tardos. These protocols improve the previous upper bound of n + log n, obtained by Karchmer. We also show that any protocol for solving the problem must exchange, in the worst case, at least n+ 1 bits. This improves a simple lower bound of n \Gamma 1 obtained by Karchmer. Our protocols, therefore, are at most one bit away from optimality. The three n + 2 bit protocols use two completely d...

In contrast to machine models like Turing machines or random access machines, circuits are a static computational model. The internal information flow of a computation is fixed in advance, independent of the actual input. Therefore, the size and the depth are natural and simple measures for circuits and provide a worst case measure. We consider a model where an internal gate can be evaluated when the result is determined only by parts of its input. So we obtain a dynamic definition of delay. In [JRS94] we have defined an average case measure for the time complexity of circuits. Using this notion tight upper and lower bounds could be obtained for the average case complexity of several basic Boolean functions. Here, we will examine the asymptotic average case complexity of the set of all n-ary Boolean functions. In contrast to worst case analyses a simple counting argument does not work. We prove that almost all Boolean function require at least n \Gamma log n \Gamma log log n expected t...

Abstract. We consider the relationship between size and depth for layered Boolean circuits, synchronous circuits and planar circuits as well as classes of circuits with small separators. In particular, we show that every layered Boolean circuit of size s can be simulated by a layered Boolean circuit of depth O ( √ s log s). For planar circuits and synchronous circuits of size s, we obtain simulations of depth O ( √ s). The best known result so far was by Paterson and Valiant [16], and Dymond and Tompa [6], which holds for general Boolean circuits and states that D(f) = O(C(f) / log C(f)), where C(f) and D(f) are the minimum size and depth, respectively, of Boolean circuits computing f. The proof of our main result uses an adaptive strategy based on the two-person pebble game introduced by Dymond and Tompa [6]. Improving any of our results by polylog factors would immediately improve the bounds for general circuits. Key words: Boolean circuits, circuit size, circuit depth, pebble games 1