On language equations with concatenation and various sets of Boolean operations

Abstract

Systems of equations of the form X i = ϕ i (X 1 , ... , X n ), for 1 ⩽ i ⩽ n , in which the unknowns X i are formal languages, and the right-hand sides ϕ i may contain concatenation and union, are known for representing context-free grammars. If, instead of union only, another set of Boolean operations is used, the expressive power of such equations may change: for example, using both union and intersection leads to conjunctive grammars [A. Okhotin, J. Automata, Languages and Combinatorics 6 (2001) 519–535], whereas using all Boolean operations allows all recursive sets to be expressed by unique solutions [A. Okhotin, Decision problems for language equations with Boolean operations, Automata, Languages and Programming, ICALP 2003, Eindhoven, The Netherlands, 239–251]. This paper investigates the expressive power of such equations with any possible set of Boolean operations. It is determined that different sets of Boolean operations give rise to exactly seven families of formal languages: the recursive languages, the conjunctive languages, the context-free languages, a certain family incomparable with the context-free languages, as well as three subregular families.