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This is a survey of results in descriptive set theory for domains and similar spaces, with the emphasis on the ω-algebraic domains. We try to demonstrate that the subject is interesting in its own right and is closely related to some areas of theoretical computer science. Since the subject is still in its beginning, we discuss in detail several open… (More)

The structure of the Wadge degrees on zero-dimensional spaces is very simple (almost well-ordered), but for many other natural non-zero-dimensional spaces (including the space of reals) this structure is much more complicated. We consider weaker notions of reducibility, including the so-called ∆ 0 α-reductions, and try to find for various natural… (More)

We prove that the homomorphic quasiorder of finite k-labeled forests has a hereditary undecidable first-order theory for k ≥ 3, in contrast to the known decidability result for k = 2. We establish also hereditary undecidability (again for every k ≥ 3) of first-order theories of two other relevant structures: the homomorphic quasiorder of finite k-labeled… (More)

We propose a new, logical, approach to the decidability problem for the Straubing and Brzozowski hierarchies based on the preservation theorems from model theory, on a theorem of Higman, and on the Rabin tree theorem. In this way, we get purely logical, short proofs for some known facts on decidability, which might be of methodological interest. Our… (More)