Complexity theory for splicing systems
This paper proposes a notion of time complexity in splicing systems. The time complexity of a splicing system at length n is defined to be the smallest integer t such that all the words of the system having length n are produced within t rounds. For a function t from the set of natural numbers to itself, the class of languages with splicing system time complexity t(n) is denoted by SPLTIME[f(n)]. This paper presents fundamental properties of SPLTIME and explores its relation to classes based on standard computational models, both in terms of upper bounds and in terms of lower bounds. As to upper bounds, it is shown that for any function t(n)SPLTIME[t(n)] is included in l-NSPACE[t(n)]; i.e., the class of languages accepted by a t(n)-space-bounded non-deterministic Turing machine with one-way input head. Expanding on this result, it is shown that 1-NSPACE[t(n)] is characterized in terms of splicing systems: it is the class of languages accepted by a t(n)-space uniform family of extended splicing systems having production time O(t(n)) with the additional property that each finite automaton appearing in the family of splicing systems has at most a constant number of states. As to lower bounds, it is shown that for all functions t (n) >= log n, all languages accepted by a pushdown automaton with maximal stack height t(vertical bar x vertical bar) for a word x are in SPLTIME[t(n)]. From this result, it follows that the regular languages are in SPLTIME[O(log it)] and that the context-free languages are in SPLTIME[O(n)]. (c) 2007 Elsevier B.V. All rights reserved.
splicing systems dna computing computational complexity