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doi:10.1016/0304-3975(95)00112-3 
Copyright © 1996 Published by Elsevier Science B.V.
Regular paper
First-order spectra with one binary predicate
Communicated by M. Nivat
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Abstract
The spectrum, Sp(
), of a sentence is the set of cardinalities of finite structures which satisfy . We prove that any set of integers which is in 
Func1∞, 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 
cNTIMERAM(cn) is included in Func1∞ for unary languages (n denotes the input integer), we deduce that the set of primes and many “natural” sets belong to BIN1.
We also give some consequences for the image of spectra under polynomials of
.
