A Characterisation of the Relations Definable in Presburger Arithmetic

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.