This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NP-completeness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NP-Completeness,’ ’ W. H. Freeman & Co., New York, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, cross-references will be given to that book and the list of problems (NP-complete and harder) presented there. Readers who have results they would like mentioned (NP-hardness, PSPACE-hardness, polynomial-time-solvability, etc.) or open problems they would like publicized, should

this paper, we consider circuits of bounded depth in the basis f; \Phig.

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

this article, I have attempted to organize and describe this literature, including an occasional opinion about the most fruitful directions, but no technical details. In the first half of this century, work on the power of formal systems led to the formalization of the notion of algorithm and the realization that certain problems are algorithmically unsolvable. At around this time, forerunners of the programmable computing machine were beginning to appear. As mathematicians contemplated the practical capabilities and limitations of such devices, computational complexity theory emerged from the theory of algorithmic unsolvability. Early on, a particular type of computational task became evident, where one is seeking an object which lies

Inexpressibility results in Finite Model Theory are often proved by showing that Duplicator, one of the two players of an Ehrenfeucht game, has a winning strategy on certain structures. In this article a new method is introduced that allows, under certain conditions, the extension of a winning strategy of Duplicator on some small parts of two finite structures to a global winning strategy. As applications of this technique it is shown that (*) Graph Connectivity is not expressible in existential monadic second-order logic (MonNP), even in the presence of a built-in linear order, (*) Graph Connectivity is not expressible in MonNP even in the presence of arbitrary built-in relations of degree n^o(1), and (*) the presence of a built-in linear order gives MonNP more expressive power than the presence of a built-in successor relation.

Russell Impagliazzo , Ramamohan Paturi , Michael E. Saks y Abstract The following size--depth trade--off for threshold circuits is obtained: any threshold circuit of depth d that computes the parity function on n variables must have at least n 1+c` \Gammad edges, where constants c ? 0 and ` 3 are constants independent of n and d. Previously known constructions show that up to the choice of c and ` this bound is best possible. In particular, the lower bound implies an affirmative answer to the conjecture of Paturi and Saks that a bounded depth threshold circuit that computes parity requires a super--linear number of edges. This is the first super-- linear lower bound for an explicit function that holds for any fixed depth, and the first that applies to threshold circuits with unrestricted weights. The trade-off is obtained as a consequence of a general restriction theorem for threshold circuits with a small number of edges: For any threshold circuit with n inputs, de...

In this paper we prove a perhaps unexpected relationship between the complexity class of the boolean functions that have linear size circuits, and n-party private protocols.

Abstract not found

this paper. Transparency 2 indicates the kind of provable security that I would like to see eventually reached in cryptography. By way of contrast, transparency 3 indicates the kind of provable security that most people talk about today. From a complexity viewpoint, transparency 2 deals with a non-uniform complexity measure while transparency 3 deals with a uniform complexity measure #i.e., the same algorithm must compute all instances of the function#. In the former case, it makes sense to talk about a function #i.e., one and only one instance of a #function"# being di#cult, whereas in the latter case one must always talk about the di#culty of an in#nite sequence of functions. My lecture was aimed at the former kind of di#culty.

In this work, we are interested in non-trivial upper bounds on the spectral norm of binary matrices M from . It is known that the distributed Boolean function represented by M is hard to compute in various restricted models of computation if the spectral norm is bounded from above by N , where # > 0 denotes a fixed constant.