Interpolation and Amalgamation for Arrays with MaxDiff (Extended Version)

TL;DR

This paper extends the theory of McCarthy's arrays with a maxdiff operation, analyzing their models and amalgamation properties to enable interpolation algorithms, thus enhancing expressivity in array theories.

Contribution

It introduces a maxdiff operation into array theory, providing a semantic analysis and hierarchical approach for interpolation, surpassing previous formal-only diff operations.

Findings

Maxdiff increases array theory expressivity.

Semantic analysis enables interpolation results.

Hierarchical approach facilitates concrete algorithms.

Abstract

In this paper, the theory of McCarthy's extensional arrays enriched with a maxdiff operation (this operation returns the biggest index where two given arrays differ) is proposed. It is known from the literature that a diff operation is required for the theory of arrays in order to enjoy the Craig interpolation property at the quantifier-free level. However, the diff operation introduced in the literature is merely instrumental to this purpose and has only a purely formal meaning (it is obtained from the Skolemization of the extensionality axiom). Our maxdiff operation significantly increases the level of expressivity; however, obtaining interpolation results for the resulting theory becomes a surprisingly hard task. We obtain such results via a thorough semantic analysis of the models of the theory and of their amalgamation properties. The results are modular with respect to the index theory and it is shown how to convert them into concrete interpolation algorithms via a hierarchical approach.