Nonsofic

Nonsofic is a term used in mathematics to describe objects that are not sofic. Sofic objects arise in two closely related areas: symbolic dynamics and group theory.

In symbolic dynamics, a subshift is called sofic if its set of allowed finite blocks (its language) can be recognized by a finite-state automaton; equivalently, the subshift is the image, under a sliding-block code, of a shift of finite type. A nonsofic subshift is therefore a subshift whose language cannot be captured by any finite automaton. It is known that there exist nonsofic subshifts; explicit constructions use combinatorial or substitution methods.

In group theory, a group is called sofic if it can be approximated, in a precise ultralimit sense, by finite symmetric groups. Sofic groups include all amenable groups and all residually finite groups, and the class is closed under many standard constructions. A nonsofic group would be a group that is not sofic; as of current knowledge, no explicit nonsofic group is known, and whether every group is sofic remains an open problem in mathematics.

Nonsofic objects highlight the boundary between finite-state descriptions and more intricate, non-regular behavior. The study of nonsofic examples informs questions about decidability, entropy, and structural complexity in dynamical systems and group actions. The term underscores ongoing efforts to delineate what kinds of dynamical and algebraic structures can be captured by finite models versus those that require more expansive descriptions.