Roots and Powers in Regular Languages: Recognizing Nonregular Properties by Finite Automata

It is well known that the set of powers of any given order, for example squares, in a regular language need not be regular. Nevertheless, finite automata can identify them via their roots. More precisely, we recall that, given a regular language L, the set of square roots of L is regular. The same holds true for the nth roots for any n and for the set of all nontrivial roots; we give a concrete construction for all of them. Using the above result, we obtain decision algorithms for many natural problems on powers. For example, it is decidable, given two regular languages, whether they contain the same number of squares at each length. Finally, we give an exponential lower bound on the size of automata identifying powers in regular languages. Moreover, we highlight interesting behavior differences between taking fractional powers of regular languages and taking prefixes of a fractional length. Indeed, fractional roots in a regular language can typically not be identified by finite automata.

Roots and Powers in Regular Languages: Recognizing Nonregular Properties by Finite Automata

It is well known that the set of powers of any given order, for example squares, in a regular language need not be regular. Nevertheless, finite automata can identify them via their roots. More precisely, we recall that, given a regular language L, the set of square roots of L is regular. The same holds true for the nth roots for any n and for the set of all nontrivial roots; we give a concrete construction for all of them. Using the above result, we obtain decision algorithms for many natural problems on powers. For example, it is decidable, given two regular languages, whether they contain the same number of squares at each length. Finally, we give an exponential lower bound on the size of automata identifying powers in regular languages. Moreover, we highlight interesting behavior differences between taking fractional powers of regular languages and taking prefixes of a fractional length. Indeed, fractional roots in a regular language can typically not be identified by finite automata.

A Dual Cavity Fabry–Perot Device for High Precision Doppler Measurements in Astronomy

We propose a dual cavity Fabry–Perot interferometer as a wavelength calibrator and a stability tracking device for astronomical spectrograph. The FPI consists of two adjoining cavities engraved on a low expansion monoblock spacer. A low-finesse astro-cavity is intended for generating a uniform grid of reference lines to calibrate the spectrograph and a high-finesse lock-cavity is meant for tracking the stability of the reference lines using optical frequency standards. The differential length changes in two cavities due to temperature and vibration perturbations are quantitatively analyzed using finite element method. An optimized mounting geometry with fractional length changes [Formula: see text] is suggested. We also identify conditions necessary to suppress relative length variations between two cavities well below [Formula: see text], thus facilitating accurate dimension tracking and generation of stable reference spectra for Doppler measurement at [Formula: see text] level.