In this article, we enrich McCarthy’s theory of extensional arrays with a length and a maxdiff operation. As is well-known, some diff operation (i.e., some kind of difference function showing where two unequal arrays differ) is needed to keep interpolants quantifier free in array theories. Our maxdiff operation returns the max index where two arrays differ; thus, it has a univocally determined semantics.

The length function is a natural complement of such a maxdiff operation and is needed to handle real arrays. Obtaining interpolation results for such a rich theory is a surprisingly hard task. We get such results via a thorough semantic analysis of the models of the theory and of their amalgamation and strong amalgamation properties. The results are modular with respect to the index theory; we show how to convert them into concrete interpolation algorithms via a hierarchical approach realizing a polynomial reduction to interpolation in linear arithmetics endowed with free function symbols.

在本文中，我们用长度和 maxdiff 操作丰富了 McCarthy 的扩展数组理论。众所周知，需要一些差异运算（即某种差异函数显示两个不相等数组的不同之处）以保持数组理论中的插值量词自由。我们的 maxdiff 操作返回两个数组不同的最大索引；因此，它具有明确确定的语义。

长度函数是这种 maxdiff 操作的自然补充，需要处理实际数组。为如此丰富的理论获得插值结果是一项异常艰巨的任务。我们通过对理论模型及其融合和强融合特性的彻底语义分析得到这样的结果。结果是关于指数理论的模块化；我们展示了如何通过分层方法将它们转换为具体的插值算法，实现了赋予自由函数符号的线性算术中插值的多项式约简。