1 Objectives

1 Objectives The objective is to investigate continuous-time models of computation and the complexity thereof. Two complementary approaches will be followed. On one hand, at a more abstract level, the focus is on the theory of real recursive functions. The main objectives are: 1) clarify the algebraic properties of the relations between analog and standard notions of complexity; 2) find analog characterizations of low time complexity classes; 3) address open problems in computational complexity in a continuous setting; 4) investigate further the connections of classes of real recursive functions with continuous dynamical systems. On the other hand, the focus is on certain continuous dynamical systems. In particular, the dynamics of continuous time neural networks and the sensitivity to noise of analog circuits will be investigated. It is expected that the research will clarify how suitable models of computation may be proposed under these paradigms and which is the right notion of complexity for such systems. 2 The project and the long term objectives of the principal investigator (PI) Previous work of the PI focused mainly on the theory of real recursive functions, on the characterization, in terms of computacional complexity, of algebras of real recursive functions and on their relations to analog circuits and continuous dynamical systems. It is expected that this line of reseach will contribute further to understanding the relationship between standard and analog computation and to clarify the notion of computational complexity for dynamical systems. Besides extending the previous work of the PI on foundations, namely on complexity theory and computable analysis, it is important to explore the same issues for specific continuous time models of computation, which model actual information processing systems. The research team includes both undergraduates working on recursion over the reals and analog circuits, and graduate members with expertise on category theory and on continuous time neural networks , who will contribute to pursue the goals of the investigation.