The monadic theory of toric words

Valérie Berthé, Toghrul Karimov, Joris Nieuwveld, Joël Ouaknine, Mihir Vahanwala, James Worrell

Abstract: For which unary predicates P1,…,Pm<math><msub is="true"><mrow is="true"><mi is="true">P</mi></mrow><mrow is="true"><mn is="true">1</mn></mrow></msub><mo is="true">,</mo><mo is="true">…</mo><mo is="true">,</mo><msub is="true"><mrow is="true"><mi is="true">P</mi></mrow><mrow is="true"><mi is="true">m</mi></mrow></msub></math> is the MSO theory of the structure 〈N;<,P1,…,Pm〉<math><mo stretchy="false" is="true">〈</mo><mi mathvariant="double-struck" is="true">N</mi><mo is="true">;</mo><mo linebreak="badbreak" linebreakstyle="after" is="true">&lt;</mo><mo is="true">,</mo><msub is="true"><mrow is="true"><mi is="true">P</mi></mrow><mrow is="true"><mn is="true">1</mn></mrow></msub><mo is="true">,</mo><mo is="true">…</mo><mo is="true">,</mo><msub is="true"><mrow is="true"><mi is="true">P</mi></mrow><mrow is="true"><mi is="true">m</mi></mrow></msub><mo stretchy="false" is="true">〉</mo></math> decidable? We survey the state of the art, leading us to investigate combinatorial properties of almost-periodic, morphic, and toric words. In doing so, we show that if each Pi<math><msub is="true"><mrow is="true"><mi is="true">P</mi></mrow><mrow is="true"><mi is="true">i</mi></mrow></msub></math> can be generated by a toric dynamical system of a certain kind, then the attendant MSO theory is decidable. We give various applications of toric words, including the recent result of [1] that the MSO theory of 〈N;<,{2n:n∈N},{3n:n∈N}〉<math><mo stretchy="false" is="true">〈</mo><mi mathvariant="double-struck" is="true">N</mi><mo is="true">;</mo><mo linebreak="badbreak" linebreakstyle="after" is="true">&lt;</mo><mo is="true">,</mo><mo stretchy="false" is="true">{</mo><msup is="true"><mrow is="true"><mn is="true">2</mn></mrow><mrow is="true"><mi is="true">n</mi></mrow></msup><mo is="true">:</mo><mi is="true">n</mi><mo is="true">∈</mo><mi mathvariant="double-struck" is="true">N</mi><mo stretchy="false" is="true">}</mo><mo is="true">,</mo><mo stretchy="false" is="true">{</mo><msup is="true"><mrow is="true"><mn is="true">3</mn></mrow><mrow is="true"><mi is="true">n</mi></mrow></msup><mo is="true">:</mo><mi is="true">n</mi><mo is="true">∈</mo><mi mathvariant="double-struck" is="true">N</mi><mo stretchy="false" is="true">}</mo><mo stretchy="false" is="true">〉</mo></math> is decidable.