Nowadays computer science is surpassing mathematics as the primary field of logic applications, but logic is not tuned properly to the new role. In particular, classical logic is preoccupied mostly with infinite static structures whereas many objects of interest in computer science are dynamic objects with bounded resources. This chapter consists of two independent parts. The first part is devoted to finite model theory; it is mostly a survey of logics tailored for computational complexity. The second part is devoted to dynamic structures with bounded resources. In particular, we use dynamic structures with bounded resources to model Pascal.

Machines and Recursive Definitions 2.1 Abstract Machines The best-known model of mechanical computation is (still) the first, introduced by Turing [18], and after half a century of study, few doubt the truth of the fundamental Church-Turing Thesis : A function f : N # N on the natural numbers (or, more generally, on strings from a finite alphabet) is computable in principle exactly when it can be computed by a Turing Machine. The Church-Turing Thesis grounds proofs of undecidability and it is essential for the most important applications of logic. On the other hand, it cannot be argued seriously that Turing machines model faithfully all algorithms on the natural numbers. If, for example, we code the input n in binary (rather than unary) notation, then the time needed for the computation of f(n) can sometimes be considerably shortened; and if we let the machine use two tapes rather than one, then (in some cases) we may gain a quadratic speedup of the computation, see [8]. This mea...

. Finding a natural meeting ground between the highly developed complexity theory of computer science ---with its historical roots in logic and the discrete mathematics of the integers--- and the traditional domain of real computation, the more eclectic less foundational field of numerical analysis ---with its rich history and longstanding traditions in the continuous mathematics of analysis--- presents a compelling challenge. Here we illustrate the issues and pose our perspective toward resolution. This article is essentially the introduction of a book with the same title (to be published by Springer) to appear shortly. Webster: A public declaration of intentions, motives, or views. k Partially supported by NSF grants. y International Computer Science Institute, 1947 Center St., Berkeley, CA 94704, U.S.A., lblum@icsi.berkeley.edu. Partially supported by the Letts-Villard Chair at Mills College. z Universitat Pompeu Fabra, Balmes 132, Barcelona 08008, SPAIN, cucker@upf.es. P...

The four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.

My purpose in this lecture is to explain how the representation of algorithms by recursive programs can be used in complexity theory, especially in the derivation of lower bounds for worst-case time complexity, which apply to all—or, at least, a very large class of—algorithms. It may be argued that recursive programs are not a new computational paradigm, since their manifestation as Herbrand-Gödel-Kleene systems was present at the very beginning of the modern theory of computability, in 1934. But they have been dissed as tools for complexity analysis, and part of my mission here is to rehabilitate them. I will draw my examples primarily from van den Dries ’ [1] and the joint work in [3, 2], incidentally providing some publicity for the fine results in those papers. Some of these results are stated in Section 3; before that, I will set the stage in Sections 1 and 2, and in the last Section 4 of this abstract I will outline very briefly some conclusions about recursion and complexity which I believe that they support. 1 Partial Algebras

Yiannis Moschovakis argues that some algorithms, and in particular the mergesort algorithm, cannot be adequately described in terms of machines acting on states. We show how to describe the mergesort algorithm, on its natural level of abstraction, in terms of distributed abstract state machines.