7 research outputs found
A note on first-order spectra with binary relations
The spectrum of a first-order sentence is the set of the cardinalities of its
finite models. In this paper, we consider the spectra of sentences over binary
relations that use at least three variables. We show that for every such
sentence , there is a sentence that uses the same number of
variables, but only one symmetric binary relation, such that its spectrum is
linearly proportional to the spectrum of . Moreover, the models of
are all bipartite graphs. As a corollary, we obtain that to settle
Asser's conjecture, i.e., whether the class of spectra is closed under
complement, it is sufficient to consider only sentences using only three
variables whose models are restricted to undirected bipartite graphs
First-order spectra with one variable
AbstractDefine a (∀1 unary)-sentence to be a prenex first-order sentence of unary type (i.e., a type which only contains unary relation and function symbols and constant symbols) with only one (universal) quantifier. A successor structure is a structure 〈B, S〉 such that S is a function which is a permutation of the basis B with only one cycle. We exhibit a (∀1, unary)-sentence φ of type {S, U1, …, Up} such that if B is finite then 〈B, S〉 is a successor structure if 〈B, S〉 satisfies ∃U1, …, ∃Upϕ. It implies that ⋃ NRAM(cn)=SPECTRA(∀1, unary), c⩾1 where NRAM(cn) denotes the class of sets of positive integers accepted by a nondeterministic random access machine in time cn (where n is the input integer) and SPECTRA(∀1, unary) is the class of finite spectra of (∀1, unary)-sentences. Another consequence is that some graph properties (hamiltonicity, connectedness) can be characterised by sentences with unary function symbols and constant symbols and only one variable. This contrasts with the result (by Fagin and De Rougemont) that these two graph properties are not definable by monadic generalized spectra (without function symbols) even in the presence of an underlying successor relation