Complement-free forbidden-pattern characterization of regular subshifts

Identify whether there exists, for every level of the quantifier alternation hierarchy (Δn, Σn, Πn), a characterization of regular subshifts by sets of forbidden patterns that are accepted by plane-walking automata without taking complements of languages.

Background

In two dimensions, recognizable picture languages are not closed under complement, and the paper’s acceptance notions for regular subshifts via plane-walking automata rely on complements of languages of finite patterns.

The authors explicitly ask whether one can avoid complementation and still define, at each alternation level, regular subshifts directly by forbidden patterns recognized by appropriate plane-walking automata.