Abstract
We introduce a restriction of second order logic, SOF, 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 SOω defined by A. Dawar and further studied by F. Ferrarotti and the second author. In the existential fragment of SOF, \(\Sigma^{1,F}_1\), we can express rigidity, which cannot be expressed in SOω. We define the complexity class NPF 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 RMFk machine with a relational oracle and show the exact correspondence between prenex fragments of SOF and the levels of the PHF polynomial-time hierarchy.
Keywords
- Finite Model Theory
- Descriptive Complexity
- Relational Machines
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Grosso, A.L., Torres, J.M.T. (2012). SOF: A Semantic Restriction over Second-Order Logic and Its Polynomial-Time Hierarchy. In: Düsterhöft, A., Klettke, M., Schewe, KD. (eds) Conceptual Modelling and Its Theoretical Foundations. Lecture Notes in Computer Science, vol 7260. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28279-9_10
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DOI: https://doi.org/10.1007/978-3-642-28279-9_10
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