The complexity of ranking simple languages

Ranking is the problem of computing for an input string its lexicographic index in a given (fixed) language. This paper concerns the complexity of ranking. We show that ranking languages accepted by 1-way unambiguous auxiliary pushdown automata operating in polynomial time is inNC(2). We also prove negative results about ranking for several classes of simple languages.C is rankable in deterministic polynomial time iffP=P#P, whereC is any of the following six classes of languages: (1) languages accepted by logtime-bounded nondeterministic Turing machines, (2) languages accepted by (uniform) families of unbounded fan-in circuits of constant depth and polynomial size, (3) languages accepted by 2-way deterministic pushdown automata, (4) languages accepted by multihead deterministic finite automata, (5) languages accepted by 1-way nondeterministic logspace-bounded Turing machines, and (6) finitely ambiguous linear context-free languages.

[1]  Juris Hartmanis.  Context-free languages and turing machine computations , 1967 .

[2]  Oscar H. Ibarra,et al.  On Pebble Automata , 1986, Theor. Comput. Sci..

[3]  Stephen A. Cook,et al.  A Taxonomy of Problems with Fast Parallel Algorithms , 1985, Inf. Control..

[4]  Alberto Bertoni,et al.  Computing the Counting Function of Context-Free Languages , 1987, STACS.

[5]  Uzi Vishkin,et al.  Simulation of Parallel Random Access Machines by Circuits , 1984, SIAM J. Comput..

[6]  Steven Fortune,et al.  Parallelism in random access machines , 1978, STOC.

[7]  Michael Sipser,et al.  Borel sets and circuit complexity , 1983, STOC.

[8]  Andrew V. Goldberg,et al.  Compression and ranking , 1985, STOC '85.

[9]  Eitan M. Gurari,et al.  Path Systems: Constructions, Solutions and Applications , 1980, SIAM J. Comput..

[10]  Stephen A. Cook,et al.  Characterizations of Pushdown Machines in Terms of Time-Bounded Computers , 1971, J. ACM.

[11]  A. Yao.  Separating the polynomial-time hierarchy by oracles , 1985 .

[12]  Walter L. Ruzzo.  On Uniform Circuit Complexity , 1981, J. Comput. Syst. Sci..

[13]  Leslie G. Valiant,et al.  The Complexity of Computing the Permanent , 1979, Theor. Comput. Sci..

[14]  Franz-Josef Brandenburg.  On one-way auxiliary pushdown automata , 1977, Theoretical Computer Science.

[15]  Walter L. Ruzzo,et al.  Tree-size bounded alternation(Extended Abstract) , 1979, J. Comput. Syst. Sci..

[16]  守屋 悦朗,et al.  J.E.Hopcroft, J.D. Ullman 著, "Introduction to Automata Theory, Languages, and Computation", Addison-Wesley, A5変形版, X+418, \6,670, 1979 , 1980 .

[17]  Stephen A. Cook.  Path systems and language recognition , 1970, STOC '70.

[18]  Jeffrey D. Ullman,et al.  Introduction to Automata Theory, Languages and Computation , 1979 .

[19]  S. Ginsburg,et al.  Finite-Turn Pushdown Automata , 1966 .

[20]  Andrew Chi-Chih Yao,et al.  K + 1 heads are better than K , 1976, 17th Annual Symposium on Foundations of Computer Science (sfcs 1976).

[21]  Samuel R. Buss,et al.  The Boolean formula value problem is in ALOGTIME , 1987, STOC.