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The termination problem for affine programs over the integers was left openin\cite{Braverman}. For more that a decade, it has been considered and cited asa challenging open problem. To the best of our knowledge, we present here themost complete response to this issue: we show that termination for affineprograms over Z is decidable under an assumption holding for almost all affineprograms, except for an extremely small class of zero Lesbegue measure. We usethe notion of asymptotically non-terminating initial variable values} (ANT, forshort) for linear loop programs over Z. Those values are directly associated toinitial variable values for which the corresponding program does not terminate.We reduce the termination problem of linear affine programs over the integersto the emptiness check of a specific ANT set of initial variable values. Forthis class of linear or affine programs, we prove that the corresponding ANTset is a semi-linear space and we provide a powerful computational methodsallowing the automatic generation of these $ANT$ sets. Moreover, we are able toaddress the conditional termination problem too. In other words, by taking ANTset complements, we obtain a precise under-approximation of the set of inputsfor which the program does terminate.

On the Termination of Linear and Affine Programs over the Integers