Polite theory combination is a method for obtaining a solver for a combination of two (or more) theories using the solvers of each individual theory as black boxes. Unlike the earlier Nelson–Oppen method, which is usable only when both theories are stably infinite, only one of the theories needs to be strongly polite in order to use the polite combination method. In its original presentation, politeness was required from one of the theories rather than strong politeness, which was later proven to be insufficient. The first contribution of this paper is a proof that indeed these two notions are different, obtained by presenting a polite theory that is not strongly polite. We also study several variants of this question.

The cost of the generality afforded by the polite combination method, compared to the Nelson–Oppen method, is a larger space of arrangements to consider, involving variables that are not necessarily shared between the purified parts of the input formula. The second contribution of this paper is a hybrid method (building on both polite and Nelson–Oppen combination), which aims to reduce the number of considered variables when a theory is stably infinite with respect to some of its sorts but not all of them. The time required to reason about arrangements is exponential in the worst case, so reducing the number of variables considered has the potential to improve performance significantly. We show preliminary evidence for this by demonstrating significant speed-up on a smart contract verification benchmark.

礼貌理论组合是一种使用每个单独理论的求解器作为黑匣子来获得两个（或多个）理论组合的求解器的方法。与早期的 Nelson-Oppen 方法不同，该方法仅在两种理论都稳定无限时才可用，只有其中一种理论需要具有强礼貌才能使用礼貌组合方法。在最初的表述中，礼貌是一种理论所要求的，而不是强烈的礼貌，后来被证明是不够的。本文的第一个贡献是通过提出一种非强礼貌的礼貌理论来证明这两个概念确实不同。我们还研究了这个问题的几种变体。

与 Nelson-Oppen 方法相比，礼貌组合方法提供的通用性的代价是要考虑更大的安排空间，涉及输入公式的纯化部分之间不一定共享的变量。本文的第二个贡献是一种混合方法（建立在礼貌和纳尔逊-奥本组合的基础上），其目的是当一个理论对于某些种类但不是全部种类而言稳定无限时，减少考虑变量的数量。在最坏的情况下，推理安排所需的时间呈指数级增长，因此减少考虑的变量数量有可能显着提高性能。我们通过展示智能合约验证基准的显着加速来证明这一点的初步证据。