Space-Bounded Complexity Classes and Iterated Deterministic Substitution

We investigate the effect on the space complexity when a language family K is extended by means of iterated λ-free deterministic substitution to the family η(K). If each language in K is accepted by a one-way nondeterministic multi-tape Turing machine within space S(n) for some monotonic space bound S(n) ⩾ log n, then η(K) is included in NSPACE(S(n)). Thus for each monotonic space bound S(n) ⩾ n, the inclusion K ⊆ NSPACE(S(n)) implies that η(K) is also included in NSPACE(S(n)). An implication similar to the latter one also holds for DSPACE(S(n)).

Consequently, some well-known space-bounded complexity classes such as the families of (non)deterministic context-sensitive languages, of two-way (non)deterministic nonerasing stack automaton languages, and are AFL's closed under intersection and iterated λ-free deterministic substitution. On the other hand no complexity class which includes DSPACE(log n) is closed under controlled iterated λ-free (non) deterministic substitution.