Abstract
The spectrum, Sp(ϑ), of a sentence ϑ is the set of cardinalities of finite structures which satisfy ϑ. 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 BIN 1, 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 ∪ 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 BIN 1. We also give some consequences for the image of spectra under polynomials of Q[x] .
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