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Register a new userAn Effective Property of $omega$-Rational Functions
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Added by
Olivier Finkel
Publication date
2018
fields
Informatics Engineering
and research's language is
English
Authors
Olivier Finkel
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We prove that $omega$-regular languages accepted by Buchi or Muller automata satisfy an effective automata-theoretic version of the Baire property. Then we use this result to obtain a new effective property of rational functions over infinite words which are realized by finite state Buchi transducers: for each such function $F: Sigma^omega rightarrow Gamma^omega$, one can construct a deterministic Buchi automaton $mathcal{A}$ accepting a dense ${bf Pi}^0_2$-subset of $Sigma^omega$ such that the restriction of $F$ to $L(mathcal{A})$ is continuous.
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We begin to study classical dimension theory from the computable analysis (TTE) point of view. For computable metric spaces, several effectivisations of zero-dimensionality are shown to be equivalent. The part of this characterisation that concerns covering dimension extends to higher dimensions and to closed shrinkings of finite open covers. To deal with zero-dimensional subspaces uniformly, four operations (relative to the space and a class of subspaces) are defined; these correspond to definitions of inductive and covering dimensions and a countable basis condition. Finally, an effective retract characterisation of zero-dimensionality is proven under an effective compactness condition. In one direction this uses a version of the construction of bilocated sets.
A wide range of intuitionistic type theories may be presented as equational theories within a logical framework. This method was formulated by Per Martin-L{o}f in the mid-1980s and further developed by Uemura, who used it to prove an initiality result for a class of models. Herein is presented a logical framework for type theories that includes an extensional equality type so that a type theory may be given by a signature of constants. The framework is illustrated by a number of examples of type-theoretic concepts, including identity and equality types, and a hierarchy of universes.
Fix 2<n<omega. Let L_n denote first order logic restricted to the first n variables. CA_n denotes the class of cylindric algebras of dimension n and for m>n, Nr_nCA_m(subseteq CA_n) denotes the class of n-neat reducts of CA_ms. The existence of certain finite relation algebras and finite CA_ns lacking relativized complete representations is shown to imply that the omitting types theorem (OTT) fails for L_n with respect to clique guarded semantics (which is an equivalent formalism of its packed fragments), and for the multi-dimensional modal logic S5^n. Several such relation and cylindric algebras are explicitly exhibited using rainbow constructions and Monk-like algebras. Certain CA_n constructed to show non-atom canonicity of the variety SNr_nCA_{n+3} are used to show that Vaughts theorem (VT) for L_{omega, omega}, looked upon as a special case of OTT for L_{omega, omega}, fails almost everywhere (a notion to be defined below) when restricted to L_n. That VT fails everywhere for L_n, which is stronger than failing almost everywhere as the name suggests, is reduced to the existence, for each n<m<omega, of a finite relation algebra R_m having a so-called m-1 strong blur, but R_m has no m-dimensional relational basis. VT for other modal fragments and expansions of L_n, like its guarded fragments, n-products of uni-modal logics like K^n, and first order definable expansions, is approached. It is shown that any multi-modal canonical logic L, such that $K^nsubseteq Lsubseteq S5^n$, L cannot be axiomatized by canonical equations. In particular, L is not Sahlqvist. Elementary generation and di-completeness for L_n and its clique guarded fragments are proved. Positive omitting types theorems are proved for L_n with respect to standard semantics.
We show that the results we had obtained on diagonals of nine and ten parameters families of rational functions using creative telescoping, yielding modular forms expressed as pullbacked $ _2F_1$ hypergeometric functions, can be obtained, much more efficiently, calculating the $ j$-invariant of an elliptic curve canonically associated with the denominator of the rational functions. In the case where creative telescoping yields pullbacked $ _2F_1$ hypergeometric functions, we generalize this result to other families of rational functions in three, and even more than three, variables. We also generalise this result to rational functions in more than three variables when the denominator can be associated to an algebraic variety corresponding to products of elliptic curves, foliation in elliptic curves. We also extend these results to rational functions in three variables when the denominator is associated with a {em genus-two curve such that its Jacobian is a split Jacobian} corresponding to the product of two elliptic curves. We sketch the situation where the denominator of the rational function is associated with algebraic varieties that are not of the general type, having an infinite set of birational automorphisms. We finally provide some examples of rational functions in more than three variables, where the telescopers have pullbacked $ _2F_1$ hypergeometric solutions, the denominator corresponding to an algebraic variety being not simply foliated in elliptic curves, but having a selected elliptic curve in the variety explaining the pullbacked $ _2F_1$ hypergeometric solution.
The concept of topological gyrogroups is a generalization of a topological group. In this work, ones prove that a topological gyrogroup G is metrizable iff G has an {omega}{omega}-base and G is Frechet-Urysohn. Moreover, in topological gyrogroups, every (countably, sequentially) compact subset being strictly (strongly) Frechet-Urysohn and having an {omega}{omega}-base are all weakly three-space properties with H a closed L-subgyrogroup
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