一句话:提出直觉主义时态逻辑 ITL^c,证明其可判定,并表明它能表达极小性与庞加莱回归。
Kremer 曾指出,命题线性时态逻辑的语言可在动力系统类上解释,从而得到一种自然的直觉主义时态逻辑。本文引入该逻辑的一个变体 ITL^c,研究其表达力与判定问题。作者证明 ITL^c 是可判定的,并进一步证明动力系统中的极小性(minimality)与庞加莱回归(Poincaré recurrence)都可在 ITL^c 中表达。因此,该逻辑既能刻画非平凡的渐近行为,又保持可判定性,为动力系统提供了一种有表达力的逻辑推理工具。
A dynamical system is a pair $(X,f)$, where $X$ is a topological space and $f\colon X\to X$ is continuous. Kremer observed that the language of propositional linear temporal logic can be interpreted over the class of dynamical systems, giving rise to a natural intuitionistic temporal logic. We introduce a variant of Kremer's logic, which we denote ${\sf ITL^c}$, and show that it is decidable. We also show that minimality and Poincar\'e recurrence are both expressible in the language of ${\sf ITL^c}$, thus providing a decidable logic expressive enough to reason about non-trivial asymptotic behavior in dynamical systems.