Decidability of unrestricted regular string constraints with concatenation-only relational constraints

Determine whether the satisfiability problem for regular string constraints without the acyclicity restriction is decidable when relational constraints use only concatenation instead of shuffle; specifically, decide the satisfiability of constraints where each variable has a regular (NFA-recognizable) membership requirement and each relational constraint asserts that a variable’s value is a scattered subword of a concatenation of other variables’ values, in the setting without rational transductions.

Background

The paper studies satisfiability of string constraints under subword-ordering with shuffle and rational transductions, establishing decidability for an acyclic fragment and tight complexity bounds. When replacing shuffle by concatenation, the landscape changes: although shuffle can simulate concatenation with polynomial blow-up (preserving acyclicity), the status of several problems under concatenation-only remains unsettled in the prior setting of [9] without transductions.

In particular, for the unrestricted (i.e., potentially cyclic) case with only regular membership and no transductions, the authors note that it is unknown whether satisfiability remains decidable when the relational operator is restricted to concatenation instead of shuffle. They contrast this with their own setting with transductions, where they show undecidability under concatenation, leaving the no-transducer case open.