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

Decidability of cutpoint isolation for letter-monotonic probabilistic finite automata

We show the surprising result that the cutpoint isolation problem is decidable for probabilistic finite automata where input words are taken from a letter-bounded context-free language. A context-free language $L$ is letter-bounded when $L \subseteq a_1^*a_2^* \cdots a_k^*$ for some finite $k > 0$ where each letter is distinct. A cutpoint is isolated when it cannot be approached arbitrarily closely. The decidability of this problem is in marked contrast to the situation for the (strict) emptiness problem for PFA which is undecidable under the even more severe restrictions of PFA with polynomial ambiguity, commutative matrices and input over a letter-bounded language as well as to the injectivity problem which is undecidable for PFA over letter-bounded languages. We provide a constructive nondeterministic algorithm to solve the cutpoint isolation problem, which holds even when the PFA is exponentially ambiguous. We also show that the problem is at least NP-hard and use our decision procedure to solve several related problems

Cyclic isogenies and nonstandard arithmetic

AbstractFor a prime N we denote by X0(N)(K) the set of K-rational points on the modul curve of elliptic curves with isogenies of degree N. We formulate arithmetical axioms for number fields K that imply finiteness properties of X0(N)(K). To prove the results we use the nonstandard version of the Siegel-Mahler theorem (A. Robinson and P. Roquette, J. Number Theory 7 (1975), 121–176) and the nonstandard interpretation of a sum formula derived from the local heights on elliptic curves