Algebraic Semantics for Coalgebraic Logics
Algebraic Semantics for Coalgebraic Logics
复制标题
代数逻辑的代数语义
DOI：
10.1016/j.entcs.2004.02.037
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发表时间：
2004
期刊：
影响因子：
--
通讯作者：
D. Pattinson
中科院分区：
文献类型：
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作者：
C. Kupke;A. Kurz;D. Pattinson
余代数通常用集合上的闭函子T来定义，本文证明了T-余代数的模态逻辑可以自然地描述为布尔代数上的函子L。在此基础上，我们从对偶理论的角度研究了余代数逻辑的可靠性、完备性和可表达性。也就是说，给定一个内函子T的余代数的逻辑L，我们构造一个内函子L，使得L-代数提供逻辑的一个可靠的和完备的（代数）语义。我们表明，如果L是对偶T，然后健全和完备的代数语义立即产生相应的属性的coalgebraic语义。最后，我们用L的公理刻画了L和T之间的对偶性。这提供了一个标准，证明具体给定的逻辑是健全的，完整的和表达。
With coalgebras usually being defined in terms of an endofunctor T on sets, this paper shows that modal logics for T-coalgebras can be naturally described as functors L on boolean algebras. Building on this idea, we study soundness, completeness and expressiveness of coalgebraic logics from the perspective of duality theory. That is, given a logic L for coalgebras of an endofunctor T, we construct an endofunctor L such that L-algebras provide a sound and complete (algebraic) semantics of the logic. We show that if L is dual to T, then soundness and completeness of the algebraic semantics immediately yield the corresponding property of the coalgebraic semantics. We conclude by characterising duality between L and T in terms of the axioms of L. This provides a criterion for proving concretely given logics to be sound, complete and expressive.