The strong independence of substitution and homomorphic replication

— The opérations of homomorphic repHcation and substitution are strongly independent forfull semiAFLs in the sense that if ££ is afull semiAFL closed under neither homomorphic repHcation nor substitution, then the closure of ^ under homomorphic repHcation («&,(&)) is incomparable with the closure of <£ under substitution {Jta(<£)). The leastfull AFL containing afull semiAFL ££ and closed under homomorphic repHcation (J%(.j£?)) is closed under substitution if and only if either S£ s #r(REGL) (REGL is the famüy of regular languages) or <£ =Jlr{3?) = #r{&)=Jia{S?). 1. The opération of homomorphic replication (first introduced in [13]) has been used in se ver al recent papers, [3, 4, 5,17,19, 26, 27] to characterize a variety of classes of languages arising naturally in different situations — machines, grammars, string relations, complexity classes, etc. For example, the class of finite reversai checking automaton languages is the closure of the regular sets under homomorphic replication and also the closure of the regular sets under itération of controls on linear context-free grammars [17, 19]. The class of recursively enumerable languages is the smallest class of languages containing the regular sets and closed under intersection and homomorphic replication [3], The opération of substitution and some of its extensions and restrictions (such as nested iterated substitution [20], iterated substitution [2, 31], and insertion [21]) have also proved usefuL For example, the class of dérivation bounded languages is the substitution closure of the class of linear context-free languages [14]. Both substitution and homomorphic replication are syntactic operators in the sense that, if jSf is a full semiAFL not closed under the operator, a proper hierarchy is obtained by iterating applications of the operator and if the closure (*) Received 24 January 1978. () Oepartment of System Science, University of California, Los Angeles. This work was supported in part by the National Science Foundation under Grants No. DCR7415091, and MCS78-04725. R.A.I.R.O. Informatique théorique/Theoretical Computer Science