Weakly Aggregative Modal logic (WAML) is a collection of disguised polyadic modal logics with n-ary modalities whose arguments are all the same. WAML has some interesting applications on epistemic logic and logic of g...
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Qualitative and quantitative approaches to reasoning about uncertainty can lead to different logical systems for formalizing such reasoning, even when the language for expressing uncertainty is the same. In the case o...
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It is well-known that natural axiomatic theories are well-ordered by consistency strength. However, it is possible to construct descending chains of artificial theories with respect to consistency strength. We provide...
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Epistemic logic with non-standard knowledge operators, especially the "knowing-value" operator, has recently gathered much attention. With the "knowing-value" operator, we can express knowledge of ...
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The class of abelian p-groups are an example of some very interesting phenomena in computable structure theory. We will give an elementary first-order theory Tp whose models are each bi-interpretable with the disjoint...
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A computable structure A is decidable if, given a formula (x¯) of elementary first-order logic, and a tuple ā ∈ A, we have a decision procedure to decide whether holds of ā. We show that there is no reasonable...
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We examine recursive monotonic functions on the Lindenbaum algebra of EA. We prove that no such function sends every consistent φ to a sentence with deductive strength strictly between φ and (φ∧Con(φ)). We genera...
Scott showed that for every countable structure A, there is a sentence of the infinitary logic Lω1ω, called a Scott sentence for A, whose models are exactly the isomorphic copies of A. Thus, the least quantifier com...
We formalize and analyze a new problem in formal language theory termed control improvisation. Given a specification language, the problem is to produce an improviser, a probabilistic algorithm that randomly generates...
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It is a classic result in lattice theory that a poset is a complete lattice iff it can be realized as fixpoints of a closure operator on a powerset. Dragalin [9,10] observed that a poset is a locale (complete Heyting ...
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ISBN:
(纸本)9781848902015
It is a classic result in lattice theory that a poset is a complete lattice iff it can be realized as fixpoints of a closure operator on a powerset. Dragalin [9,10] observed that a poset is a locale (complete Heyting algebra) iff it can be realized as fixpoints of a nucleus on the locale of upsets of a poset. He also showed how to generate a nucleus on upsets by adding a structure of "paths" to a poset, forming what we call a Dragalin frame. This allowed Dragalin to introduce a semantics for intuitionistic logic that generalizes Beth and Kripke semantics. He proved that every spatial locale (locale of open sets of a topological space) can be realized as fixpoints of the nucleus generated by a Dragalin frame. In this paper, we strengthen Dragalin's result and prove that every locale-not only spatial locales-can be realized as fixpoints of the nucleus generated by a Dragalin frame. In fact, we prove the stronger result that for every nucleus on the upsets of a poset, there is a Dragalin frame based on that poset that generates the given nucleus. We then compare Dragalin's approach to generating nuclei with the relational approach of Fairtlough and Mendler [11], based on what we call FM-frames. Surprisingly, every Dragalin frame can be turned into an equivalent FM-frame, albeit on a different poset. Thus, every locale can be realized as fixpoints of the nucleus generated by an FM-frame. Finally, we consider the relational approach of Goldblatt [13] and characterize the locales that can be realized using Goldblatt frames.
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