Independent Languages and Witnesses of Dependence (Non-satisfaction)

We consider the independent language property defined by the language equation φ(X)=\emptyset, where the expression φ involves the language variable X, constant languages, and standard regular operations including transductions, but not complementation. This property consists of all languages L satisfying φ(L)=\emptyset. Depending on the choice of the operations in φ, the equation defines a broad class of codes, including combinations of standard variable-length codes and error-detecting codes. We show that any φ-independence is a Jürgensen independence, we define what a witness of non-satisfaction of φ(X)=\emptyset is, for a language L, and we show how to compute a witness of non-satisfaction when L is regular. We also discuss the complexity of the problem, showing that the decision version of the problem is PSPACE-complete.