First-order spectra with one variable

  • E. Grandjean
Part of the Lecture Notes in Computer Science book series (LNCS, volume 270)


Define a (∀1, unary)-sentence to be a prenex first-order sentence of unary type 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 iff 〈B,S〉 satisfies ∃U1...∃Up ϕ. It implies that
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 characterized by sentences with only one variable.


Linear Order Turing Machine Hamiltonian Cycle Unary Type Successor Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 1987

Authors and Affiliations

  • E. Grandjean
    • 1
  1. 1.Laboratoire d'InformatiqueUniversité de CaenCaen CedexFrance

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