无限无限的无限与有限的无限域，用于Z3中的真实量的通用定量
这是一个理论上的问题。我想知道，求解普遍量化的公式对量化的无限无限域与无限无限的无限域上的量化时，Z3的性能是否存在差异。我正在研究一些有限的无限域在性能方面更好地工作的事情，并为其寻找简单的理论解释。我无法在文献中找到任何相关的东西，因此将不胜感激。
This is a bit of a theoretical question. I was wondering if there's a difference in performance of z3 when solving universally quantified formulas over reals where the quantification is over a bounded infinite domain vs an unbounded infinite domain. I am working on something where I see that bounded infinite domains work much better in terms of performance and looking for a simple theoretical explanation for it. I am not able to find anything related in the literature, so any guidance would be appreciated.
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大多数 SMT 求解器的内部工作原理都是黑匣子，其中有许多启发式/算法在起作用。然而，对于算术，有众所周知的技术来处理有趣的片段。我建议阅读 https://theory.stanford.edu/~ nikolaj/programmingz3.html#sec-arithmetic，其中列出了每个片段使用的算法。
量化是一个完全不同的问题。通常的方法是应用量词消除（https://en.wikipedia.org/wiki/Quantifier_elimination）。 z3 将对线性实数算术执行量词消除，但不会对非线性问题执行量词消除。请参阅 Z3 v4.3+ 是否支持量词消除对于非线性算术，因为后者是一个更困难的问题并且默认情况下不打开。
您设置界限的特殊情况是摆脱量词的简单方法，因此 z3 可能表现更好也就不足为奇了。然而，如果不研究内部结构，就不可能说是否有自定义算法可以利用它。您最好的选择可能是在 z3 讨论论坛上提问，我看到您已经这样做了： https ://github.com/Z3Prover/z3/discussions/5949
The inner workings of most SMT solvers is black-box, where many heuristics/algorithms are in play. For arithmetic however, there are well-known techniques to deal with interesting fragments. I'd recommend reading through https://theory.stanford.edu/~nikolaj/programmingz3.html#sec-arithmetic, where they list what algorithms they use for each fragment.
Quantification is a totally different question. The usual approach is to apply quantifier elimination (https://en.wikipedia.org/wiki/Quantifier_elimination). z3 will perform quantifier elimination for linear-real arithmetic, but not for non-linear problems. See Does Z3 v4.3+ support quantifier elimination for NON-linear arithmetic, as the latter is a much more difficult problem and is not turned on by default.
Your particular case of putting bounds is an easy way to get rid of quantifiers, so it's not surprising that z3 might perform better. However, it's impossible to say without studying the internals if there are custom algorithms to take advantage of that. Your best bet is probably to ask at the z3 discussion forum, which I see you already have done: https://github.com/Z3Prover/z3/discussions/5949