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Turing machines and the spectra of first-order formulas

  • Neil D. Jones (a1) and Alan L. Selman (a2)


H. Scholz [11] defined the spectrum of a formula φ of first-order logic with equality to be the set of all natural numbers n for which φ has a model of cardinality n. He then asked for a characterization of spectra. Only partial progress has been made. Computational aspects of this problem have been worked on by Gunter Asser [1], A. Mostowski [9], and J. H. Bennett [2]. It is known that spectra include the Grzegorczyk class and are properly included in . However, no progress has been made toward establishing whether spectra properly include , or whether spectra are closed under complementation.

A possible connection with automata theory arises from the fact that contains just those sets which are accepted by deterministic linear-bounded Turing machines (Ritchie [10]). Another resemblance lies in the fact that the same two problems (closure under complement, and proper inclusion of ) have remained open for the class of context sensitive languages for several years.

In this paper we show that these similarities are not accidental—that spectra and context sensitive languages are closely related, and that their open questions are merely special cases of a family of open questions which relate to the difference (if any) between deterministic and nondeterministic time or space bounded Turing machines.

In particular we show that spectra are just those sets which are acceptable by nondeterministic Turing machines in time 2cx, where c is constant and x is the length of the input. Combining this result with results of Bennett [2], Ritchie [10], Kuroda [7], and Cook [3], we obtain the “hierarchy” of classes of sets shown in Figure 1. It is of interest to note that in all of these cases the amount of unrestricted read/write memory appears to be too small to allow diagonalization within the larger classes.



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[1]Asser, G., Das Repräsentenproblem im Prädikatenkalkul der ersten Stufe mit Identität, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 1 (1955), pp. 252263.
[2]Bennett, J., On spectra, Doctoral Dissertation, Princeton University, Princeton, N.J., 1962.
[3]Cook, S. A., Characterization of push-down machines in terms of time-bounded computers, Journal of the Association for Computing Machinery, vol. 18 (1971), pp. 418.
[4]Cook, S. A., The complexity of theorem-proving procedures, Third ACM Symposium on Theory of Computing, 1971, pp. 151158.
[5]Grzegorczyk, A., Some classes of recursive functions, Rozprawy Mathematyczne, vol. 44 (1953), pp. 145.
[6]Hartmanis, J. and Stearns, R. E., On the computational complexity of algorithms, Transactions of the American Mathematical Society, vol. 117 (1965), pp. 285306.
[7]Kuroda, S. Y., Classes of languages and linear-bounded automata, Information and Control, vol. 7 (1964), pp. 207223.
[8]Mager, G., Writing pushdown acceptors, Journal of Computing and Systems Sciences, vol. 3 (1969), pp. 276319.
[9]Mostowski, A., Concerning a problem of H. Scholz, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 2 (1956), pp. 210214.
[10]Ritchie, R. W., Classes of predictably computable functions, Transactions of the American Mathematical Society, vol. 106 (1963), pp. 139173.
[11]Scholz, H., Problems, this Journal, vol. 17 (1952), p. 160.
[12]Tractenbrot, B. A., Impossibility of an algorithm for the decision problem in finite classes, Doklady Akademii Nauk SSSR, vol. 70 (1950), pp. 569572.

Turing machines and the spectra of first-order formulas

  • Neil D. Jones (a1) and Alan L. Selman (a2)


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