Abstract
Smullyan [61] introduced the class R of rudimentary relations as the smallest class which contains the concatenation relation and which is closed under the boolean operations, explicit transformations and linearly bounded quantification. RUD, the class of rudimentary languages, consists of the sequential encodings of rudimentary relations. Wrathall [75] has shown that RUD can be described as the union LH of a linear time analogue of the polynomial time hierarchy of PH of Meyer,Stockmeyer [72].
Chandra,Kozen and Stockmeyer [76] introduced the concept of alternating Turing machines (=ATM). An ATM is a nondeterministic Turing machine with two disjoint sets of states,existential and universal states,which play dual roles in the definition of acceptance. A language L belongs to the alternation class STA(s,t,a) if there exists an ATM M such that for each word w in L there exists a finite accepting computation tree of M for w of depth ≤t(n), alternation depth ≤a(n) and space ≤s(n), where n=|w|.
There is a close connection between quantification and linear alternation. Chandra,Kozen and Stockmeyer noted that PH may be described as the union of a hierarchy of bounded alternation. An analogous result will be shown for RUD = LH:
-
(1)
RUD = U<STA (−,O(n) , k) :k ε N> ⊑ ATIME (O(n)) = STA(−,O(n),O(n)) Extending a result of Nepomnjascii [70] for NLOGSPACE we are able to prove that the alternating LOGSPACE hierarchy of Chandra, Kozen and Stockmeyer is contained in RUD:
-
(2)
U<STA (O (nα), Oβ),k):k ε N> ⊑ RUD for α<1≤β, and hence
-
(3)
U<STA(log(n),−,k):k ∈N>⊑RUD However, the question whether the inclusion RUD⊑ATIME (O(n)) is proper remains open. A negative answer would solve important open problems in complexity theory.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Preview
Unable to display preview. Download preview PDF.
References
Bennett, J.H. [62]: On Spectra, Ph.D.Thesis, Princeton Univ., Princeton N.J., 1962, 135pp.
Berman, L. [80]: The Complexity of Logical Theories, Theoret.Comp.Sci. 11 (1980), 71–77
Book, R.,Greibach, S. [70]: Quasirealtime Languages, Math.Systems Theory 4 (1970),97–111
Chandra,A.K.,Stockmeyer,L.J. [76]: Alternation, in: Proc. 17th IEEE Symp. on Found. of Comp.Sci. (1976),98–108.
Chandra, A.K.,Kozen, D.C.,Stockmeyer, L.J. [81]: Alternation,J.ACM 28 (1981), 114–133
Harrow, K. [78]: The Bounded Arithmetic Hierarchy, Information and Control 36 (1978),102–117
Jones, N.D. [69]: Context-free Languages and Rudimentary Attributes, Math. Systems Theory 3 (1969) 102–109, 11 (1977/8),379–380
Jones, N.D. [75]: Space-Bounded Reducibility among Combinatorial Problems, J.Comp.System Sci. 11 (1975),68–85, 15 (1977),241
King, K.N.,Wrathall, C. [78]: Stack Languages and Log n Space, J.Comp.System Sci. 17 (1978),281–299
Kozen,D.C.[76]: On Parallelism in Turing Machines, in: Proc. 17th IEEE Symp. on Found.of Comp.Sci. (1976), 89–97
Meloul, J. [79]: Rudimentary Predicates, Low Level Complexity Classes and Related Automata, Ph.D.Thesis, Oxford Univ., Oxford 1979, 210pp.
Meyer,A.R.,Stockmeyer,L.J. [72]: The Equivalence Problem for Regular Expressions with Squaring requires Exponential Space, in: Proc. 13th IEEE on Switching and Automata Theory (1972), 125–129
Nepomnjascii, V.A. [70a]: Rudimentary Predicates and Turing Computations, Soviet Math.Dokl. 11 (1970), 1462–1465
Nepomnjascii, V.A. [70b]: Rudimentary Interpretation of Two-Tape Turing Computation, Kibernetika 6 (1970),29–35
Nepomnjascii,V.A. [78]: Examples of Predicates not expressible by S-RUD Formulae, Kibernetika 12 (1978)
Ruzzo, W.L. [80]: Tree-Size Bounded Alternation, J.Comp.System Sci. 21 (1980),218–235
Ruzzo, W.L.,Simon, J.,Tompa, M. [82]: Space-Bounded Hierarchies and Probabilistic Computations, in: Proc. 14th ACM Symp. on Theory of Computing, San Francisco 1982,215–223
Smullyan,R. [61]: Theory of Formal Systems, Annals of Math.Studies 47, Princeton Univ.Press 1961, 147pp.
Stockmeyer,L.J. [75]: The Polynomial-Time Hierarchy, IBM Fesearch Report RC 5379 (1975)
Stockmeyer, L.J. [77]: The Polynomial-Time Hierarchy, Theoret.Comp.Sci. 3 (1977),1–22
Volger, H. [83]: Turing Machines with Linear Alternation, Theories of Bounded Concatenation and the Decision Problem of First Order Theories, Theoret. Comp.Sci. 23 (1983), 333–338
Wrathall, C. [75]: Subrecursive Predicates and Automata, Ph.D.Thesis, Harvard Univ., Cambridge MA, 1975, 156pp.
Wrathall, C. [77]: Complete Sets and the Polynomial-Time Hierarchy, Theoret. Comp.Sci. 3 (1977),23–33
Wrathall, C. [78]: Rudimentary Predicates and Relative Computation, SIAM J. Computing 7 (1978),194–209
Yu, Y.Y. [70]: Rudimentary Relations and Formal Languages, Ph.D.Thesis, Univ. of California, Berkeley 1970, 47pp.
Yu, Y.Y. [77]: Rudimentary Relations and Stack Languages, Math.Systems Theory 10 (1977),337–343
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1984 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Volger, H. (1984). Rudimentary relations and Turing machines with linear alternation. In: Börger, E., Hasenjaeger, G., Rödding, D. (eds) Logic and Machines: Decision Problems and Complexity. LaM 1983. Lecture Notes in Computer Science, vol 171. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-13331-3_38
Download citation
DOI: https://doi.org/10.1007/3-540-13331-3_38
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-13331-5
Online ISBN: 978-3-540-38856-2
eBook Packages: Springer Book Archive