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Craig interpolation has emerged as an effective means of generating candidateprogram invariants. We present interpolation procedures for the theories ofPresburger arithmetic combined with (i) uninterpreted predicates (QPA+UP), (ii)uninterpreted functions (QPA+UF) and (iii) extensional arrays (QPA+AR). Weprove that none of these combinations can be effectively interpolated withoutthe use of quantifiers, even if the input formulae are quantifier-free. We goon to identify fragments of QPA+UP and QPA+UF with restricted forms of guardedquantification that are closed under interpolation. Formulae in these fragmentscan easily be mapped to quantifier-free expressions with integer division. ForQPA+AR, we formulate a sound interpolation procedure that potentially producesinterpolants with unrestricted quantifiers.

Beyond Quantifier-Free Interpolation in Extensions of Presburger Arithmetic (Extended Technical Report)