SOF: A Semantic Restriction over Second-Order Logic and Its Polynomial-Time Hierarchy

  • Alejandro L. Grosso
  • José M. Turull Torres
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7260)


We introduce a restriction of second order logic, SO F , for finite structures. In this restriction the quantifiers range over relations closed by the equivalence relation ≡  FO . In this equivalence relation the equivalence classes are formed by k-tuples whose First Order type is the same, for some integer k ≥ 1. This logic is a proper extension of the logic SO ω defined by A. Dawar and further studied by F. Ferrarotti and the second author. In the existential fragment of SO F , \(\Sigma^{1,F}_1\), we can express rigidity, which cannot be expressed in SO ω . We define the complexity class NP F by using a variation of the relational machine of S. Abiteboul and V. Vianu (RMF) and we prove that this complexity class is captured by \(\Sigma^{1,F}_1\). Then we define an RMF k machine with a relational oracle and show the exact correspondence between prenex fragments of SO F and the levels of the PHF polynomial-time hierarchy.


Finite Model Theory Descriptive Complexity Relational Machines 


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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Alejandro L. Grosso
    • 1
  • José M. Turull Torres
    • 2
    • 3
  1. 1.Dpto. de InformáticaUniversidad Nacional de San LuisSan LuisArgentina
  2. 2.ICTICUniversidad de la Cuenca del PlataCorrientesArgentina
  3. 3.Dpto. de InformáticaUniversidad Nacional de San LuisArgentina

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