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 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.

This paper introduces communicating branching programs, and develops a general technique for demonstrating communication-space tradeoffs for pairs of communicating branching programs. This technique is then used to prove communication-space tradeoffs for any pair of communicating branching programs that hashes according to a universal family of hash functions. Other tradeoffs follow from this result. As an example, any pair of communicating Boolean branching programs that computes matrix-vector products over GF(2) requires communication-space product Ω(n 2), provided the space used is o(n / log n). These are the first examples of communication-space tradeoffs on a completely general model of communicating processes.

Some Topics in Parallel Computation and Branching Programs by Rakesh Kumar Sinha Chairperson of the Supervisory Committee: Professor Paul Beame Department of Computer Science and Engineering There are two parts of this thesis: the first part gives two constructions of branching programs; the second part contains three results on models of parallel machines. The branching program model has turned out to be very useful for understanding the computational behavior of problems. In addition, several restrictions of branching programs, for example ordered binary decision diagrams, have proven to be successful data structures in several VLSI design and verification applications. We construct a branching program of o(n log 3 n) nodes for computing any threshold function on n variables and a branching program of o(n log 4 n) nodes for determining the sum of n variables modulo a fixed divisor. These are improvements over constructions of size 2(n 3=2 ) due to Lupanov [Lup65]. The second p...

It is well established that the problem of detecting a triangle in a graph can be reduced to Boolean matrix multiplication (BMM). Many have asked if there is a reduction in the other direction: can a fast triangle detection algorithm be used to solve BMM faster? The general intuition has been that such a reduction is impossible: for example, triangle detection returns one bit, while a BMM algorithm returns n 2 bits. Similar reasoning goes for other matrix products and their corresponding triangle problems. We show this intuition is false, and present a new generic strategy for efficiently computing matrix products over algebraic structures used in optimization. We say an algorithm on n × n matrices (or n-node graphs) is truly subcubic if it runs in O(n 3−δ · poly(log M)) time for some δ> 0, where M is the absolute value of the largest entry (or the largest edge weight). We prove an equivalence between the existence of truly subcubic algorithms for any (min, ⊙) matrix product, the corresponding matrix product verification problem, and a corresponding triangle detection problem. Our work simplifies and unifies prior work, and has some new consequences: • The following problems either all have truly subcubic algorithms, or none of them do: