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CODING IN GRAPHS AND LINEAR ORDERINGS

Part of: Computability and recursion theory Model theory

Published online by Cambridge University Press:  18 June 2020

JULIA F. KNIGHT ,
ALEXANDRA A. SOSKOVA  and
STEFAN V. VATEV
JULIA F. KNIGHT
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF NOTRE DAME NOTRE DAME, IN, USAE-mail: knight.1@nd.edu
ALEXANDRA A. SOSKOVA
Affiliation:
DEPARTMENT OF MATHEMATICAL LOGIC SOFIA UNIVERSITYSOFIA, BULGARIAE-mail: asoskova@fmi.uni-sofia.bgE-mail: stefanv@fmi.uni-sofia.bg
STEFAN V. VATEV
Affiliation:
DEPARTMENT OF MATHEMATICAL LOGIC SOFIA UNIVERSITYSOFIA, BULGARIAE-mail: asoskova@fmi.uni-sofia.bgE-mail: stefanv@fmi.uni-sofia.bg
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Abstract

There is a Turing computable embedding $\Phi $ of directed graphs $\mathcal {A}$ in undirected graphs (see [15]). Moreover, there is a fixed tuple of formulas that give a uniform effective interpretation; i.e., for all directed graphs $\mathcal {A}$ , these formulas interpret $\mathcal {A}$ in $\Phi (\mathcal {A})$ . It follows that $\mathcal {A}$ is Medvedev reducible to $\Phi (\mathcal {A})$ uniformly; i.e., $\mathcal {A}\leq _s\Phi (\mathcal {A})$ with a fixed Turing operator that serves for all $\mathcal {A}$ . We observe that there is a graph G that is not Medvedev reducible to any linear ordering. Hence, G is not effectively interpreted in any linear ordering. Similarly, there is a graph that is not interpreted in any linear ordering using computable $\Sigma _2$ formulas. Any graph can be interpreted in a linear ordering using computable $\Sigma _3$ formulas. Friedman and Stanley [4] gave a Turing computable embedding L of directed graphs in linear orderings. We show that there is no fixed tuple of $L_{\omega _1\omega }$ -formulas that, for all G, interpret the input graph G in the output linear ordering $L(G)$ . Harrison-Trainor and Montalbán [7] have also shown this, by a quite different proof.

Keywords

effective interpretabilityturing computable embedding

MSC classification

Primary: 03C57: Effective and recursion-theoretic model theory
Secondary: 03D45: Theory of numerations, effectively presented structures
Type
Articles
Information
The Journal of Symbolic Logic , Volume 85 , Issue 2 , June 2020 , pp. 673 - 690
Copyright
© The Association for Symbolic Logic 2020

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