Logspace and Logtime Leaf Languages

The computation tree of a nondeterministic machineMwith inputxgives rise to aleaf stringformed by concatenating the outcomes of all the computations in the tree in lexicographical order. We may characterize problems by considering, for a particular “leaf language”Y, the set of allxfor which the leaf string ofMis contained inY. In this way, in the context of polynomial time computation, leaf languages were shown to capture many complexity classes. In this paper, we study the expressibility of the leaf language mechanism in the contexts of logarithmic space and of logarithmic time computation. We show that logspace leaf languages yield a much finer classification scheme for complexity classes than polynomial time leaf languages, capturing also many classes withinP. In contrast, logtime leaf languages basically behave like logtime reducibilities. Both cases are more subtle to handle than the polynomial time case. We also raise the issue of balanced versus nonbalanced computation trees underlying the leaf language. We indicate that it is a nontrivial problem to obtain information about the leaf string of a nonbalanced computation tree and present conditions under which it does not matter whether the computation tree is balanced or not.

[1]  David A. Mix Barrington,et al.  Bounded-width polynomial-size branching programs recognize exactly those languages in NC1 , 1986, STOC '86.

[2]  G. Buntrock,et al.  Finite monoids: From word to circuit evaluation , 1997 .

[3]  Bernd Borchert.  On the Acceptance Power of Regular Languages , 1994, STACS.

[4]  Liming Cai,et al.  Circuit bottom fan-in and computational power , 1997, Proceedings of Computational Complexity. Twelfth Annual IEEE Conference.

[5]  Liming Cai,et al.  On Input Read-Modes of Alternating Turing Machines , 1995, Theor. Comput. Sci..

[6]  Michael Sipser,et al.  Parity, circuits, and the polynomial-time hierarchy , 1981, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981).

[7]  Bernd Borchert,et al.  On the Acceptance Power of Regular Languages , 1994, Theor. Comput. Sci..

[8]  Andrew Chi-Chih Yao,et al.  ON ACC and threshold circuits , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[9]  Christoph Meinel,et al.  Structure and Importance of Logspace-MOD-Classes , 1991, STACS.

[10]  Larry J. Stockmeyer,et al.  The Polynomial-Time Hierarchy , 1976, Theor. Comput. Sci..

[11]  Neil Immerman,et al.  On Uniformity within NC¹ , 1990, J. Comput. Syst. Sci..

[12]  L. Hodes,et al.  Implementing the Algebra of Logic Functions in Terms of Bounded Depth Formulas in the Basis of , 1961 .

[13]  Denis Thérien,et al.  Finite Moniods: From Word to Circuit Evaluation , 1997, SIAM J. Comput..

[14]  Denis Thérien,et al.  Non-Uniform Automata Over Groups , 1987, Inf. Comput..

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

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

[17]  Andrew Chi-Chih Yao,et al.  Separating the Polynomial-Time Hierarchy by Oracles (Preliminary Version) , 1985, FOCS.

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

[19]  Denis Thérien,et al.  Finite monoids and the fine structure of NC1 , 1987, STOC.

[20]  Thomas Schwentick,et al.  On the power of polynomial time bit-reductions , 1993, [1993] Proceedings of the Eigth Annual Structure in Complexity Theory Conference.

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

[22]  Allan Borodin,et al.  A Time-Space Tradeoff for Sorting on Non-Oblivious Machines , 1981, J. Comput. Syst. Sci..

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

[24]  Theodore P. Baker,et al.  A Second Step toward the Polynomial Hierarchy , 1976, FOCS.

[25]  Pierre McKenzie,et al.  Extensions to Barrington's M-Program Model , 1993, Theor. Comput. Sci..

[26]  Janos Simon,et al.  Space-bounded hierarchies and probabilistic computations , 1982, STOC '82.

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

[28]  Helmut Veith,et al.  Succinct representation, leaf languages, and projection reductions , 1996, Proceedings of Computational Complexity (Formerly Structure in Complexity Theory).

[29]  Howard Straubing,et al.  Non-Uniform Automata Over Groups , 1990, Inf. Comput..

[30]  N. Vereshchagin.  RELATIVIZABLE AND NONRELATIVIZABLE THEOREMS IN THE POLYNOMIAL THEORY OF ALGORITHMS , 1994 .

[31]  Allan Borodin,et al.  A time-space tradeoff for sorting on non-oblivious machines , 1979, 20th Annual Symposium on Foundations of Computer Science (sfcs 1979).

[32]  N. Immerman,et al.  On uniformity within NC 1 . , 1988 .

[33]  Ulrich Hertrampf.  Complexity Classes with Finite Acceptance Types , 1993, STACS.

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

[35]  John Gill,et al.  Counting Classes: Thresholds, Parity, Mods, and Fewness , 1990, Theor. Comput. Sci..

[36]  Samuel R. Buss,et al.  An Optimal Parallel Algorithm for Formula Evaluation , 1992, SIAM J. Comput..

[37]  Pierluigi Crescenzi,et al.  Complexity classes and sparse oracles , 1991, [1991] Proceedings of the Sixth Annual Structure in Complexity Theory Conference.

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

[39]  Johan Håstad,et al.  Almost optimal lower bounds for small depth circuits , 1986, STOC '86.

[40]  Ivan Hal Sudborough,et al.  On the Tape Complexity of Deterministic Context-Free Languages , 1978, JACM.

[41]  Miklós Ajtai,et al.  ∑11-Formulae on finite structures , 1983, Ann. Pure Appl. Log..

[42]  Janusz A. Brzozowski,et al.  Dot-Depth of Star-Free Events , 1971, Journal of computer and system sciences (Print).

[43]  Janos Simon,et al.  Space-Bounded Hierarchies and Probabilistic Computations , 1984, J. Comput. Syst. Sci..

[44]  Dana Angluin,et al.  On Counting Problems and the Polynomial-Time Hierarchy , 1980, Theor. Comput. Sci..

[45]  Carme Àlvarez,et al.  A Very Hard log-Space Counting Class , 1993, Theor. Comput. Sci..

[46]  C. Papadimitriou,et al.  Two remarks on the power of counting , 1983 .

[47]  Theodore P. Baker,et al.  A second step toward the polynomial hierarchy , 1976, 17th Annual Symposium on Foundations of Computer Science (sfcs 1976).

[48]  Pierluigi Crescenzi,et al.  A Uniform Approach to Define Complexity Classes , 1992, Theor. Comput. Sci..