The prospects for mathematical logic in the twenty-first century

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.Comment: Association for Symbolic Logi

Computability and analysis: the legacy of Alan Turing

We discuss the legacy of Alan Turing and his impact on computability and analysis.Comment: 49 page

Semi-galois Categories II: An arithmetic analogue of Christol's theorem

In connection with our previous work on semi-galois categories, this paper proves an arithmetic analogue of Christol's theorem concerning an automata-theoretic characterization of when a formal power series over finite field is algebraic over the polynomial ring. There are by now several variants of Christol's theorem, all of which are concerned with rings of positive characteristic. This paper provides an arithmetic (or F_1) variant of Christol's theorem in the sense that it replaces the polynomial ring with the ring of integers of a number field and the ring of formal power series with the ring of Witt vectors. We also study some related problems.Comment: 20 pages. Made corrections in the last parts of the proofs of Proposition 2 and Lemma 3.2; fixed some typos and wordin