Regularity Preserving but not Reflecting Encodings
J\"org Endrullis, Clemens Grabmayer, Dimitri Hendriks

TL;DR
The paper constructs bijective encodings that preserve regularity of languages in a way that the inverse does not, challenging assumptions about model equivalences and recognizability in computation and language theory.
Contribution
It proves the existence of bijective encodings that preserve regularity in one direction but not the other, with significant implications for computational models and recognizability.
Findings
Existence of bijective encodings preserving regularity in images
Finite-state automata can simulate Turing machines under certain encodings
Universal recognizability of recursive sets with a specific number representation
Abstract
Encodings, that is, injective functions from words to words, have been studied extensively in several settings. In computability theory the notion of encoding is crucial for defining computability on arbitrary domains, as well as for comparing the power of models of computation. In language theory much attention has been devoted to regularity preserving functions. A natural question arising in these contexts is: Is there a bijective encoding such that its image function preserves regularity of languages, but its pre-image function does not? Our main result answers this question in the affirmative: For every countable class C of languages there exists a bijective encoding f such that for every language L in C its image f[L] is regular. Our construction of such encodings has several noteworthy consequences. Firstly, anomalies arise when models of computation are compared with respect…
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Taxonomy
Topicssemigroups and automata theory · Computability, Logic, AI Algorithms · Advanced Algebra and Logic
