# Rudimentary relations and Turing machines with linear alternation

## 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|.

- (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 (n

^{α}), O^{β}),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

Turing Machine Universal State Important Open Problem Polynomial Time Hierarchy Bound Alternation## Preview

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## References

- Bennett, J.H. [62]: On Spectra, Ph.D.Thesis, Princeton Univ., Princeton N.J., 1962, 135pp.Google Scholar
- Berman, L. [80]: The Complexity of Logical Theories, Theoret.Comp.Sci. 11 (1980), 71–77Google Scholar
- Book, R.,Greibach, S. [70]: Quasirealtime Languages, Math.Systems Theory 4 (1970),97–111Google Scholar
- Chandra,A.K.,Stockmeyer,L.J. [76]: Alternation, in: Proc. 17th IEEE Symp. on Found. of Comp.Sci. (1976),98–108.Google Scholar
- Chandra, A.K.,Kozen, D.C.,Stockmeyer, L.J. [81]: Alternation,J.ACM 28 (1981), 114–133Google Scholar
- Harrow, K. [78]: The Bounded Arithmetic Hierarchy, Information and Control 36 (1978),102–117Google Scholar
- Jones, N.D. [69]: Context-free Languages and Rudimentary Attributes, Math. Systems Theory 3 (1969) 102–109, 11 (1977/8),379–380Google Scholar
- Jones, N.D. [75]: Space-Bounded Reducibility among Combinatorial Problems, J.Comp.System Sci. 11 (1975),68–85, 15 (1977),241Google Scholar
- King, K.N.,Wrathall, C. [78]: Stack Languages and Log n Space, J.Comp.System Sci. 17 (1978),281–299Google Scholar
- Kozen,D.C.[76]: On Parallelism in Turing Machines, in: Proc. 17th IEEE Symp. on Found.of Comp.Sci. (1976), 89–97Google Scholar
- Meloul, J. [79]: Rudimentary Predicates, Low Level Complexity Classes and Related Automata, Ph.D.Thesis, Oxford Univ., Oxford 1979, 210pp.Google Scholar
- 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), 125–129Google Scholar
- Nepomnjascii, V.A. [70a]: Rudimentary Predicates and Turing Computations, Soviet Math.Dokl. 11 (1970), 1462–1465Google Scholar
- Nepomnjascii, V.A. [70b]: Rudimentary Interpretation of Two-Tape Turing Computation, Kibernetika 6 (1970),29–35Google Scholar
- Nepomnjascii,V.A. [78]: Examples of Predicates not expressible by S-RUD Formulae, Kibernetika 12 (1978)Google Scholar
- Ruzzo, W.L. [80]: Tree-Size Bounded Alternation, J.Comp.System Sci. 21 (1980),218–235Google Scholar
- 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 1982,215–223Google Scholar
- Smullyan,R. [61]: Theory of Formal Systems, Annals of Math.Studies 47, Princeton Univ.Press 1961, 147pp.Google Scholar
- Stockmeyer,L.J. [75]: The Polynomial-Time Hierarchy, IBM Fesearch Report RC 5379 (1975)Google Scholar
- Stockmeyer, L.J. [77]: The Polynomial-Time Hierarchy, Theoret.Comp.Sci. 3 (1977),1–22Google Scholar
- 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), 333–338Google Scholar
- Wrathall, C. [75]: Subrecursive Predicates and Automata, Ph.D.Thesis, Harvard Univ., Cambridge MA, 1975, 156pp.Google Scholar
- Wrathall, C. [77]: Complete Sets and the Polynomial-Time Hierarchy, Theoret. Comp.Sci. 3 (1977),23–33Google Scholar
- Wrathall, C. [78]: Rudimentary Predicates and Relative Computation, SIAM J. Computing 7 (1978),194–209Google Scholar
- Yu, Y.Y. [70]: Rudimentary Relations and Formal Languages, Ph.D.Thesis, Univ. of California, Berkeley 1970, 47pp.Google Scholar
- Yu, Y.Y. [77]: Rudimentary Relations and Stack Languages, Math.Systems Theory 10 (1977),337–343Google Scholar