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.

References

The problem of deciding whether two-sided finite-type shifts of sequences are conjugate remains open (see, for example, ).

— One-sided Hom shifts  (2509.24754 - Béal et al., 29 Sep 2025) in Section 1 (Introduction)

Can we decide SSE equivalence over $\mathbb{Z}_+$ ?

— Combinatorial Search for Strong Shift Equivalence  (2609.03567 - Jeandel, 3 Sep 2026) in Problem environment, Section 2, “Main open problem of symbolic dynamics”; also Introduction

At the same time, the decidability problem for strong shift equivalence, and hence for conjugacy of shifts of finite type, remains one of the central open problems in symbolic dynamics (see for recent perspectives).

— Graded classification of Leavitt path algebras in terms of strong shift equivalence  (2609.11834 - Bilich et al., 10 Sep 2026) in Section 1, Introduction