Abstract
We consider functions of natural numbers which allow a combinatorial interpretation as counting functions (speed) of classes of relational structures, such as Fibonacci numbers, Bell numbers, Catalan numbers and the like. Many of these functions satisfy a linear recurrence relation over \(\mathbb Z\) or \({\mathbb Z}_m\) and allow an interpretation as counting the number of relations satisfying a property expressible in Monadic Second Order Logic (MSOL).
C. Blatter and E. Specker (1981) showed that if such a function f counts the number of binary relations satisfying a property expressible in MSOL then f satisfies for every m ∈ ℕ a linear recurrence relation over ℤ m .
In this paper we give a complete characterization in terms of definability in MSOL of the combinatorial functions which satisfy a linear recurrence relation over ℤ, and discuss various extensions and limitations of the Specker-Blatter theorem.
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Kotek, T., Makowsky, J.A. (2010). Definability of Combinatorial Functions and Their Linear Recurrence Relations. In: Blass, A., Dershowitz, N., Reisig, W. (eds) Fields of Logic and Computation. Lecture Notes in Computer Science, vol 6300. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15025-8_21
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DOI: https://doi.org/10.1007/978-3-642-15025-8_21
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