First-order spectra with one binary predicate
The spectrum, Sp(ϕ), of a sentence ϕ is the set of cardinalities of finite structures which satisfy ϕ. We prove that any set of integers which is in Func 1 ∞ i.e. in the class of spectra of first-order sentences of type containing only unary function symbols is also in BIN1 i.e. in the class of spectra of first-order sentences of type involving only a single binary relation.
We give similar results for generalized spectra and some corollaries: in particular, from the fact that the large complexity class \(\mathop \cup \limits_c \)NTIME RAM (cn) is included in Func 1 ∞ for unary languages (n denotes the input integer), we deduce that the set of primes and many “natural” sets belong to BIN1
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- 2.S.A Cook. A hierarchy for nondeterministic time complexity. J. Comput. Systems Sci., vol.7:pp.343–353, 1973.Google Scholar
- 3.R. Fagin. Generalized first-order spectra and polynomial-time recognizable sets. Complexity of computations, vol.7:pp.43–73, 1974.Google Scholar
- 4.R. Fagin. Monadic generalized spectra. Z. Math. Logik. Grundlag. Math., 21:pp.89–96, 1975.Google Scholar
- 5.R. Fagin. A spectrum hierarchy. Z. Math. Logik Grundlag. Math., (21):pp.123–134, 1975.Google Scholar
- 7.R. Fagin, L.J. Stockmeyer, and M.Y. Vardi. On monadic np vs monadic co — np. IBM Research Report, 1993.Google Scholar
- 11.N.D Jones and A.L Selman. Turing machines and the spectra of first-order formulas with equality. J. Symb. Logic, vol.39:pp.139–150, 1974.Google Scholar
- 13.F. Olive. Personal communication.Google Scholar
- 14.P. Pudlak. The observational predicate calculus and complexity of computations. Comment. Math. Univ. Carolin., vol.16, 1975.Google Scholar
- 15.M. De Rougemont. Second-order and inductive definability on finite structures. Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik, vol.33:pp.47–63, 1987.Google Scholar
- 16.A.R. Woods. Some problems in logic and number theory and their connections. PhD thesis, University of Manchester, 1981.Google Scholar