Turing machines and the spectra of first-order formulas

Introduction. H. Scholz [11] defined the spectrum of a formula s of first-order logic with equality to be the set of all natural numbers n for which (p 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 E* and are properly included in 43. However, no progress has been made toward establishing whether spectra properly include e*, or whether spectra are closed under complementation. A possible connection with automata theory arises from the fact that e2 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 62 ) 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], Ritchwe [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. A further characterization of spectra is by means of the "spectrum automaton" described in ?2 below. This model is of interest because it possesses almost no computational abilities, being limited to several read-only heads which scan the vertices of a multi-dimensional cube given to it by an "oracle." It is well known that a nondeterministic Turing machine can be regarded as a