A Polite Non-Disjoint Combination Method: Theories with Bridging Functions Revisited

The Nelson-Oppen combination method is ubiquitous in Satisfiability Modulo Theories solvers. However, one of its major drawbacks is to be restricted to disjoint unions of theories. We investigate the problem of extending this combination method to particular non-disjoint unions of theories connected via bridging functions. The motivation is, e.g., to solve verification problems expressed in a combination of data structures connected to arithmetic with bridging functions such as the length of lists and the size of trees. We present a sound and complete combination procedure a la Nelson-Oppen for the theory of absolutely free data structures, including lists and trees. This combination procedure is then refined for standard interpretations. The resulting theory has a nice politeness property, enabling combinations with arbitrary decidable theories of elements.

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