Constant Time Factors

The constant speedup theorem, so well known from Tur-ing machine based complexity theory, is shown false for a natural imperative programming language I that manipulates tree-structured data. This relieves a tension between general programming practice, where linear factors are essential, and complexity theory, where linear time changes are traditionally regarded as trivial. Speciically, there is a constant b such that for any a > 0 there is a set X recognizable in time 1 a b n but not in time a n. Thus LIN, the collection of all sets recognizable in linear time by deterministic I-programs, contains an innnite hierarchy ordered by constant coef-cients. Constant hierarchies also exist for larger time bounds T(n), provided they are time-constructable. Second, a problem is exhibited which is complete for the nondeterministic linear time sets NLIN with respect to a natural notion of deterministic linear-time reduction. Third, Kleene's Second Recursion Theorem in essence shows that for any program p deened with self-reference, there is an equivalent nonreeexive program q. This is proven for an extension I " of I. Further, q can be simulated by an I program at most constantly slower than p. Language I " allows calls to the language's own interpretation function, and even to its running time function (without the usual high costs for nested levels of interpretation). The results all hold as well for a stronger language I su allowing selective updating of tree-structured data. The results are robust in that classes LIN su and NLIN su are identical for I su , I su" , Schh onhage's Storage Modii-cation Machines, Knuth/Tarjan's pointer machines, and successor RAMs 2 13,14,11]. 1 where n is the size of the input. 2 If the \more realistic and precise measure" of SMM computation time is used 13], and similarly for the other models.