Effective Genericity and Differentiability

We prove that a real x is 1-generic if and only if every differentiable computable function has continuous derivative at x. This provides a counterpart to recent results connecting effective notions of randomness with differentiability. We also consider multiply differentiable computable functions and polynomial time computable functions.Comment: Revision: added sections 6-8; minor correction

The descriptive theory of represented spaces

This is a survey on the ongoing development of a descriptive theory of represented spaces, which is intended as an extension of both classical and effective descriptive set theory to deal with both sets and functions between represented spaces. Most material is from work-in-progress, and thus there may be a stronger focus on projects involving the author than an objective survey would merit.Comment: survey of work-in-progres

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

On Projective Ordinals

We study in this paper the projective ordinals δ^1_n, where δ^1_n = sup{ξ: ξ is the length of ɑ Δ^1_n prewellordering of the continuum}. These ordinals were introduced by Moschovakis in [8] to serve as a measure of the "definable length" of the continuum. We prove first in §2 that projective determinacy implies δ^1_n 0 (the same result for odd n is due to Moschovakis). Next, in the context of full determinacy, we partly generalize (in §3) the classical fact that δ^1_1 ℵ_l and the result of Martin that δ^1_3 = ℵ_(ω + 1) by proving that δ^1_(n2+1) = λ^+_(2n+1), where λ_(2n+1) is a cardinal of cofinality ω. Finally we discuss in §4 the connection between the projective ordinals and Solovay's uniform indiscernibles. We prove among other things that ∀α (α^# exists) implies that every δ^1_n with n ≥ 3 is a fixed point of the increasing enumeration of the uniform indiscernibles

Bisimilarity is not Borel

We prove that the relation of bisimilarity between countable labelled transition systems is $\Sigma_1^1$-complete (hence not Borel), by reducing the set of non-wellorders over the natural numbers continuously to it. This has an impact on the theory of probabilistic and nondeterministic processes over uncountable spaces, since logical characterizations of bisimilarity (as, for instance, those based on the unique structure theorem for analytic spaces) require a countable logic whose formulas have measurable semantics. Our reduction shows that such a logic does not exist in the case of image-infinite processes.Comment: 20 pages, 1 figure; proof of Sigma_1^1 completeness added with extended comments. I acknowledge careful reading by the referees. Major changes in Introduction, Conclusion, and motivation for NLMP. Proof for Lemma 22 added, simpler proofs for Lemma 17 and Theorem 30. Added references. Part of this work was presented at Dagstuhl Seminar 12411 on Coalgebraic Logic