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.

In the first part of the paper we discuss some conceptual problems related to the notion of proof. In the second part we survey five major open problems in Provability Logic as well as possible directions for future research in this area. 1

this paper we deal only with the case where fluents do not to contain predicates from the event calculus, so that we need not use the program for the truth predicate

Recursion in grammar and performance In the last 50 years of cognitive science, linguistic theory has proposed more and more articulated structures, while computer science has shown that simpler, flatter structures are more easily processed. If we are interested in adequate models of human linguistic abilities, models that explain the very rapid and accurate human recognition and production of ordinary fluent speech, it seems we need to come to some appropriate understandingoftherelationshipbetweentheseapparently opposing pressures for more and less structure. Here we show how the apparent conflict disappears when it is considered more carefully. Even when we regard the linguists ’ project as a psychological one, there is no pressure for linguists to abandon their rather deep structures in order to account for our easy production and recognition of fluent speech. The deeper, more recursive structures reflect insights into similarities among linguistic constituents and operations, but a processor can compute exactly these structures without the extra effort that deeper analyses might seem to require. To show how this works, we