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Section: New Results

Computable SFTs

Previous works by the two participants have shown that there is a striking similarity between subshifts of finite type (tilings, coloring of the plane that do not contain a given set of patterns) and finitely presented groups (finitely generated groups with a finite number of equations).

This analogy can be described intuitively as follows: colors in subshifts corresponds to the generators of the groups, forbidden patterns correspond to the equations. Finite type is the same as finite presentation, and minimal subshifts correspond to simple groups.

The article [29] develops this analogy to computable objects: It is well known by the Higman-Thompson theorem that a finitely generated group is computable iff it is a subgroup of a simple group which is itself a subgroup of a finitely presented group. In this article, we give an equivalent for subshifts : a subshift is computable iff it is the restriction of a minimal subshift which is itself the restriction of a subshift of finie type.