Skolem functions play a central role in logic, from eliminating quantifiers in first order logic formulas to providing functional implementations of relational specifications. While classical results in logic are only interested in their existence, the question of how to effectively compute them is also interesting, important and useful for several applications. In the restricted case of Boolean propositional logic formula, this problem of synthesizing Boolean Skolem functions has been addressed in depth, with various recent work focussing on both theoretical and practical aspects of the problem. However, there are few existing results for the general case, and the focus has been on heuristical algorithms. In this article, we undertake an investigation into the computational hardness of the problem of synthesizing Skolem functions for first order logic formula. We show that even under reasonable assumptions on the signature of the formula, it is impossible to compute or synthesize Skolem functions. Then we determine conditions on theories of first order logic which would render the problem computable. Finally, we show that several natural theories satisfy these conditions and hence do admit effective synthesis of Skolem functions.

中文翻译：

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一阶逻辑公式的Skolem函数合成

从消除一阶逻辑公式中的量词到提供相关规范的功能实现，Skolem函数在逻辑中都起着核心作用。尽管逻辑中的经典结果仅对它们的存在感兴趣，但如何有效地计算它们的问题对于一些应用程序也很有趣，重要且有用。在布尔命题逻辑公式的受限情况下，已深入解决了合成布尔Skolem函数的问题，最近的各种工作都集中在该问题的理论和实践方面。但是，对于一般情况，现有的结果很少，并且重点一直放在启发式算法上。在本文中，我们对一阶逻辑公式合成Skolem函数问题的计算难度进行了研究。我们证明，即使在公式签名的合理假设下，也无法计算或合成Skolem函数。然后，我们确定一阶逻辑理论的条件，这些条件可使问题可计算。最后，我们证明了几种自然理论都满足这些条件，因此确实接受了Skolem函数的有效合成。