Decidability of conjugacy for two-sided shifts of finite type

Determine whether there exists an algorithm that, given two two-sided shifts of finite type (subshifts of A^Z specified by finite sets of forbidden blocks over finite alphabets), decides whether they are topologically conjugate via a shift-commuting homeomorphism.

Background

The paper studies conjugacy questions for specific subclasses of shift spaces and tree-shifts, proving decidability results for one-sided shifts of finite type and for tree-shifts, including Hom and directed Hom tree-shifts. These results build on Williams's theory for one-sided shifts and analogous constructions for tree automata.

In contrast to the one-sided and tree settings, the classical problem of deciding whether two given two-sided shifts of finite type are conjugate has remained unresolved in symbolic dynamics. The authors explicitly note that this general conjugacy decision problem for two-sided SFTs is still open.