Living without Beth and Craig: Definitions and Interpolants in the Guarded and Two-Variable Fragments

J. C. Jung, F. Wolter
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引用次数: 11

Abstract

In logics with the Craig interpolation property (CIP) the existence of an interpolant for an implication follows from the validity of the implication. In logics with the projective Beth definability property (PBDP), the existence of an explicit definition of a relation follows from the validity of a formula expressing its implicit definability. The two-variable fragment, FO2, and the guarded fragment, GF, of first-order logic both fail to have the CIP and the PBDP. We show that nevertheless in both fragments the existence of interpolants and explicit definitions is decidable. In GF, both problems are 3EXPTIME-complete in general, and 2EXPTIME-complete if the arity of relation symbols is bounded by a constant c ≥ 3. In FO2, we prove a CON2EXPTIME upper bound and a 2EXPTIME lower bound for both problems. Thus, both for GF and FO2 existence of interpolants and explicit definitions are decidable but harder than validity (in case of FO2 under standard complexity assumptions).
没有贝丝和克雷格的生活:守卫和双变量片段中的定义和插入
在具有克雷格插值性质(CIP)的逻辑中,一个蕴涵的插值存在性由蕴涵的有效性推导出来。在具有投影贝丝可定义性的逻辑中,关系的显式定义的存在性来源于表达其隐式可定义性的公式的有效性。一阶逻辑的双变量片段(FO2)和保护片段(GF)都不具有CIP和PBDP。我们表明,尽管如此,在这两个片段的存在插值和明确的定义是可确定的。在GF中,一般情况下这两个问题都是3EXPTIME-complete,当关系符号的密度以常数c≥3为界时,这两个问题都是2EXPTIME-complete。在FO2中,我们证明了两个问题的CON2EXPTIME上界和2EXPTIME下界。因此,对于GF和FO2,内插和显式定义的存在性都是可确定的,但比有效性更难(在标准复杂性假设下的FO2情况下)。
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