Translational lemmas, polynomial time, and (log n)j-space

AbstractTranslational lemmas are stated in a general framework and then applied to specific complexity classes. Necessary and sufficient conditions are given for every set accepted by a Turing acceptor which operates in linear or polynomial time to be accepted by a Turing acceptor which operates in space (log n)j for some j ⩾ 1

Polynomial-time reducibilities and “almost all” oracle sets

AbstractIt is shown for every k>0 and for almost every tally setT, {A|A ⩽Pk−ttT} ≠ {A|A ⩽P(k+1)−ttT}. In contrast, it is shown that for every set A, the following holds: (a) for almost every set B,A ⩽ Pm B if and only if A ⩽ P(logn)−T B; and (b) for almost every set B, A ⩽Ptt B if and only ifA ⩽PTB

ALMOST-R: characterizations using different concepts of randomness

We study here the classes of the form ALMOST-R, for R a reducibility. This includes among other the classes BPP, P and PH. We give a characterization of this classes in terms of reducibility to n-random languages, a subclass of algorithmically random languages. We also give a characterization of classes of the form ALMOST-R in terms of resource bounded measure, for R reducibility of a restricted kind

On inefficient special cases of NP-complete problems

AbstractEvery intractable set A has a polynomial complexity core, a set H such that for any P-subset S of A or of Ā, S∩H is finite. A complexity core H of A is proper if H⊆A. It is shown here that if P≠NP, then every currently known (i.e., either invertibly paddable or k-creative) NP-complete set A and its complement Ā have proper polynomial complexity cores that are nonsparse and are accepted by deterministic machines in time 2cn for some constant c. Turning to the intractable class DEXT=∪c>0DTIME(2cn), it is shown that every set that is ⩽pm-complete for DEXT has an infinite proper polynomial complexity core that is nonsparse and recursive

Reset machines

AbstractA reset tape has one read-write head which moves only left-to-right except that the head can be reset once to the left end and the tape rescanned; a multiple-reset machine has reset tapes as auxiliary storage and a one-way input tape. Linear time is no more powerful than real time for nondeterministic multiple-reset machines and so the family MULTI-RESET of languages accepted in real time by nondeterministic multiple-reset machines is closed under linear erasing. MULTI-RESET is closed under Kleene. It can be characterized as the smallest family of languages containing the regular sets and closed under intersection and linear-erasing homomorphic duplication or as the smallest intersection-closed semiAFL containing COPY = {ww | w in {a, b}∗}. A circular tape is read full-sweep from left-to-right only and then reset to the left, any number of times; a nonwriting circular tape cannot be altered after the first sweep. For nondeterministic machines operating in real time, multiple reset tapes, circular tapes or nonwriting circular tapes have the same power. Languages in MULTI-RESET can be accepted in real time by nondeterministic machines using only three reset tapes or using only one reset tape and one nonwriting circular tape

On bounded query machines

AbstractSimple proofs are given for each of the following results: (a) P = Pspace if and only if, for every set A, P(A) = Pquery(A) (Selman et al., 1983): (b) NP = Pspace if and only if, for every set A, NP(A) = NPquery(S) (Book, 1981); (c) PH = Pspace if and only if, for every set A, PH(A) = PQH(A) (Book and Wrathall, 1981); (c) PH = Pspace if and only if, for every set set S, PH(S) = PQH(S) = Pspace(S) (Balcázar et al., 1986; Long and Selman, 1986)

The base of the intersection of two free submonoids

AbstractGiven two free submonoids of a free monoid, one wishes to find a specification for the base of the intersection. An algorithm to construct a graph-theoretic specification of the base is presented. From this specification it can easily be determined whether the base is finite. In addition, a a polynomial-time algorithm to determine if a regular set is a circular code is presented

Time-bounded grammars and their languages

Formal grammars and formal languages are studied from the viewpoint of time-bounded grammars. A time-bound on a grammar is a measure of the “derivational complexity” of the language generated. Based on results of Gladkii [7] on connectivity in grammars, it is shown that a “linear speedup” can be obtained and that one can construct Turing acceptors to simulate grammars without loss of time. Positive containment and closure properties are also studied