The Non-Self-Embedding Property for Generalized Fuzzy Context-Free Grammars

A fuzzy context-free $K$-grammar is a fuzzy context-free grammar with a countable rather than a finite number of rules satisfying the following condition: for each symbol $\alpha$, the set containing all right-hand sides of rules with left-hand side equal to $\alpha$ forms a fuzzy language that belongs to a given family $K$ of fuzzy languages. In this paper we study the effect of the non-self-embedding restriction on the generating power of fuzzy context-free $K$-grammars. Our main result shows that under weak assumptions on the family $K$, a fuzzy language is generated by a non-self-embedding fuzzy context-free $K$-grammar if and only if either it is a fuzzy regular language or it belongs to the substitution closure $K_\infty$ of the family $K$. The proof heavily relies on the closure properties of the families $K$ and $K_\infty$.