Abstract
Four sub-recursive classes of functions, \({\mathcal{B}}\), \({\mathcal{D}}\), \({\mathcal{BD}}\) and \({\mathcal{BDD}}\) are defined, and compared to the classes G 0, G 1 and G 2, originally defined by Grzegorczyk, based on bounded minimalisation, and characterised by Harrow in [5]. \({\mathcal{B}}\) is essentially G 0 with predecessor substituted for successor; \({\mathcal{BD}}\) is G 1 with (truncated) difference substituted for addition. We prove that the induced relational classes are preserved (\(G^0_\star={\mathcal{B}}_\star\) and \(G^1_\star={\mathcal{BD}}_\star\)). We also obtain \({\mathcal{D}}_\star={{{\mathfrak{PrA}}^{\text{\tiny qf}}}_\star}\) (the quantifier free fragment of Presburger Arithmetic), and \({\mathcal{BD}}_\star={\mathfrak{PrA}}_\star\), and \({\mathcal{BDD}}_\star=G^2_\star\), where \({\mathcal{BDD}}\) is G 2 with integer division and remainder substituted for multiplication, and where \(G^2_\star\) is known to be equal to the predicates definable by a bounded formula in Peano Arithmetic.
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Barra, M. (2008). A Characterisation of the Relations Definable in Presburger Arithmetic. In: Agrawal, M., Du, D., Duan, Z., Li, A. (eds) Theory and Applications of Models of Computation. TAMC 2008. Lecture Notes in Computer Science, vol 4978. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79228-4_23
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DOI: https://doi.org/10.1007/978-3-540-79228-4_23
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