We investigate a version of linear temporal logic whose propositional fragment is Gödel–Dummett logic (which is well known both as a superintuitionistic logic and a t-norm fuzzy logic). We define the logic using two natural semantics: first a real-valued semantics, where statements have a degree of truth in the real unit interval, and second a ‘bi-relational’ semantics. We then show that these two semantics indeed define one and the same logic: the statements that are valid for the real-valued semantics are the same as those that are valid for the bi-relational semantics. This Gödel temporal logic does not have any form of the finite model property for these two semantics: there are non-valid statements that can only be falsified on an infinite model. However, by using the technical notion of a quasimodel, we show that every falsifiable statement is falsifiable on a finite quasimodel, yielding an algorithm for deciding if a statement is valid or not. Later, we strengthen this decidability result by giving an algorithm that uses only a polynomial amount of memory, proving that Gödel temporal logic is PSPACE-complete. We also provide a deductive calculus for Gödel temporal logic, and show this calculus to be sound and complete for the above-mentioned semantics, so that all (and only) the valid statements can be proved with this calculus.

中文翻译：

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Gödel-Dummett 线性时间逻辑


我们研究了线性时间逻辑的一个版本，其命题片段是哥德尔-杜米特逻辑（众所周知的超直觉逻辑和 t-norm 模糊逻辑）。我们使用两种自然语义来定义逻辑：第一种是实值语义，其中语句在实单位区间中具有一定程度的真理，第二种是“双关系”语义。然后我们表明，这两种语义确实定义了一个相同的逻辑：对实值语义有效的语句与对双关系语义有效的语句相同。这个哥德尔时间逻辑没有这两个语义的任何形式的有限模型属性：有无效的陈述只能在无限模型上被证伪。然而，通过使用准模型的技术概念，我们证明了每个可证伪的陈述在有限准模型上都是可证伪的，从而产生了一种决定陈述是否有效的算法。后来，我们通过给出一个仅使用多项式内存量的算法来加强这个可判定性结果，证明哥德尔时间逻辑是 PSPACE 完备的。我们还为哥德尔时间逻辑提供了一个演绎演算，并证明这个演算对于上述语义是合理和完整的，因此所有（且唯一）有效陈述都可以用这个演算来证明。