2 research outputs found
Computable Stone spaces
We investigate computable metrizability of Polish spaces up to homeomorphism.
In this paper we focus on Stone spaces. We use Stone duality to construct the
first known example of a computable topological Polish space not homeomorphic
to any computably metrized space. In fact, in our proof we construct a
right-c.e. metrized Stone space which is not homeomorphic to any computably
metrized space. Then we introduce a new notion of effective categoricity for
effectively compact spaces and prove that effectively categorical Stone spaces
are exactly the duals of computably categorical Boolean algebras. Finally, we
prove that, for a Stone space , the Banach space has a
computable presentation if, and only if, is homeomorphic to a computably
metrized space. This gives an unexpected positive partial answer to a question
recently posed by McNicholl.Comment: 16 page
Computable classifications of continuous, transducer, and regular functions
We develop a systematic algorithmic framework that unites global and local
classification problems for functional separable spaces and apply it to attack
classification problems concerning the Banach space C[0,1] of real-valued
continuous functions on the unit interval. We prove that the classification
problem for continuous (binary) regular functions among almost everywhere
linear, pointwise linear-time Lipshitz functions is -complete. We
show that a function is (binary)
transducer if and only if it is continuous regular; interestingly, this
peculiar and nontrivial fact was overlooked by experts in automata theory. As
one of many consequences, our -completeness result covers the class
of transducer functions as well. Finally, we show that the Banach space
of real-valued continuous functions admits an arithmetical
classification among separable Banach spaces. Our proofs combine methods of
abstract computability theory, automata theory, and functional analysis.Comment: Revised argument in Section 5; results unchange