The guarded fragment of FO fails to have the Craig Interpolation Property (CIP) and the Projective Beth Definability Property (PBDP). Thus, not every valid implication between guarded formulas has a guarded interpolant, and not every implicitly definable relation has an explicit guarded definition. In this article, we show that nevertheless the existence of guarded interpolants and explicit definitions is decidable. Moreover, it is 3ExpTime-complete in general, and 2ExpTime-complete if the arity of relation symbols is bounded by a constant. Deciding the existence of guarded interpolants and explicit definitions is thus by one exponential harder than validity in the guarded fragment.

中文翻译：

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没有 Beth 和 Craig 的生活：受保护片段中的显式定义和插值

FO 的受保护片段没有 Craig Interpolation Property (CIP) 和 Projective Beth Definability Property (PBDP)。因此，并非受保护公式之间的每个有效含义都有受保护的插值，也不是每个可隐式定义的关系都有明确的受保护定义。在本文中，我们证明了受保护的插值和显式定义的存在是可判定的。此外，如果关系符号的元数以常数为界，则一般为 3ExpTime-complete 和 2ExpTime-complete。因此，确定受保护的插值和显式定义的存在比受保护片段的有效性要困难一个指数。