Forbidding Sets and Normal Forms for Language Forbidding-Enforcing Systems

This paper investigates ways to reduce redundancy in forbidding sets for language forbidding-enforcing systems. A language forbidding set disallows combinations of subwords in a word, while permitting the presence of some parts of these combinations. Since a forbidding set is a potentially infinite set of finite sets of words, finding normal forms for forbidding sets is interesting from a combinatorics on words perspective and important for the theoretical investigation of language fe-systems, the connection between variants of fe-systems, and their applications to molecular computation. This paper shows that the minimal normal forms for forbidding sets defining classes of languages (fe-families) are also normal forms for forbidding sets defining single languages (fe-languages), but not necessarily minimal. Thus, an investigation of minimality and sufficient conditions for fe-languages are presented and it is shown that in special cases they coincide with a minimal normal form for fe-families.