4 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
Characterizing Omega-Regularity Through Finite-Memory Determinacy of Games on Infinite Graphs
We consider zero-sum games on infinite graphs, with objectives specified as sets of infinite words over some alphabet of colors. A well-studied class of objectives is the one of ?-regular objectives, due to its relation to many natural problems in theoretical computer science. We focus on the strategy complexity question: given an objective, how much memory does each player require to play as well as possible? A classical result is that finite-memory strategies suffice for both players when the objective is ?-regular. We show a reciprocal of that statement: when both players can play optimally with a chromatic finite-memory structure (i.e., whose updates can only observe colors) in all infinite game graphs, then the objective must be ?-regular. This provides a game-theoretic characterization of ?-regular objectives, and this characterization can help in obtaining memory bounds. Moreover, a by-product of our characterization is a new one-to-two-player lift: to show that chromatic finite-memory structures suffice to play optimally in two-player games on infinite graphs, it suffices to show it in the simpler case of one-player games on infinite graphs. We illustrate our results with the family of discounted-sum objectives, for which ?-regularity depends on the value of some parameters
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
Homogeneity and Homogenizability: Hard Problems for the Logic SNP
We show that the question whether a given SNP sentence defines a
homogenizable class of finite structures is undecidable, even if the sentence
comes from the connected Datalog fragment and uses at most binary relation
symbols. As a byproduct of our proof, we also get the undecidability of some
other properties for Datalog programs, e.g., whether they can be rewritten in
MMSNP, whether they solve some finite-domain CSP, or whether they define the
age of a reduct of a homogeneous Ramsey structure in a finite relational
signature. We subsequently show that the closely related problem of testing the
amalgamation property for finitely bounded classes is EXPSPACE-hard or
PSPACE-hard, depending on whether the input is specified by a universal
sentence or a set of forbidden substructures.Comment: 34 pages, 3 figure