Space Bounded Computations: Review and New Separation Results

In this paper we review the key results about space bounded complexity classes, discuss the central open problems and outline the relevant proof techniques. We show that, for a slightly modified Turing machine model, the low level deterministic and nondeterministic space bounded complexity classes are different. Furthermore, for this computation model, we show that Savitch and Immerman-Szelepcsenyi theorems do not hold in the range lg lg n to lg n. We also discuss some other computation models to bring out and clarify the importance of space constructibility and establish some results about these models. We conclude by enumerating a few open problems which arise out of the discussion.

[1]  Christopher B. Wilson.  Parallel Computation and the NC Hierarchy Relativized , 1986, Computational Complexity Conference.

[2]  Róbert Szelepcsényi.  The moethod of focing for nondeterministic automata , 1987, Bull. EATCS.

[3]  Walter J. Savitch,et al.  Relationships Between Nondeterministic and Deterministic Tape Complexities , 1970, J. Comput. Syst. Sci..

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

[5]  Michael Sipser,et al.  Halting space-bounded computations , 1978, 19th Annual Symposium on Foundations of Computer Science (sfcs 1978).

[6]  S.-Y. Kuroda,et al.  Classes of Languages and Linear-Bounded Automata , 1964, Inf. Control..

[7]  John Gill,et al.  Relativizations of the P =? NP Question , 1975, SIAM J. Comput..

[8]  Viliam Geffert,et al.  Nondeterministic Computations in Sublogarithmic Space and Space Constructibility , 1990, SIAM J. Comput..

[9]  Ravi Kannan,et al.  Alternation and the power of nondeterminism , 1983, STOC.

[10]  Andrzej Szepietowski.  Some Notes on Strong and Weak log log n Space Complexity , 1989, Inf. Process. Lett..

[11]  Richard Edwin Stearns,et al.  Memory bounds for recognition of context-free and context-sensitive languages , 1965, SWCT.

[12]  R. Solovay,et al.  Relativizations of the $\mathcal{P} = ?\mathcal{NP}$ Question , 1975 .

[13]  Viliam Geffert.  Nondeterministic Computations in Sublogarithmic Space and Space Constructibility , 1991, SIAM J. Comput..

[14]  Christopher B. Wilson.  Relativized circuit complexity , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[15]  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 .

[16]  Richard E. Ladner,et al.  Space Bounds for Processing Contentless Inputs , 1975, J. Comput. Syst. Sci..

[17]  A. Selman.  Structure in Complexity Theory , 1986, Lecture Notes in Computer Science.

[18]  Neil Immerman.  Nondeterministic Space is Closed Under Complementation , 1988, SIAM J. Comput..

[19]  Andrzej Szepietowski.  If Deterministic and Nondeterministic Space Complexities are Equal for log log n then they are also Equal for log n , 1989, STACS.

[20]  Richard Edwin Stearns,et al.  Hierarchies of memory limited computations , 1965, SWCT.