By Lev Beklemishev (auth.), Nick Bezhanishvili, Sebastian Löbner, Kerstin Schwabe, Luca Spada (eds.)
Edited in collaboration with FoLLI, the organization of good judgment, Language and data, this e-book constitutes the refereed complaints of the eighth foreign Tbilisi Symposium on common sense, Language, and Computation, TbiLLC 2009, held in Bakuriani, Georgia, in September 2009. The 20 revised complete papers incorporated within the booklet have been rigorously reviewed and chosen from quite a few shows given on the symposium. the focal point of the papers is at the following issues: traditional language syntax, semantics, and pragmatics; confident, modal and algebraic common sense; linguistic typology and semantic universals; logics for synthetic intelligence; info retrieval, question resolution platforms; good judgment, video games, and formal pragmatics; language evolution and learnability; computational social selection; historic linguistics, heritage of logic.
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Additional info for Logic, Language, and Computation: 8th International Tbilisi Symposium on Logic, Language, and Computation, TbiLLC 2009, Bakuriani, Georgia, September 21-25, 2009. Revised Selected Papers
Example text
Let X be a topological space. We recall that a nonempty subset A of X is irreducible if from A = B ∪ C, with B and C closed, it follows that A = B or A = C, and that X is sober if each closed irreducible subset of X is the closure of a point. We also recall that X is coherent if the compact open subsets of X are closed under finite intersections and form a basis for the topology. Definition 9 ([10]). A topological space X is a spectral space if X is compact, T0 , sober, and coherent. Spectral spaces play an important role in the theory of distributive lattices and commutative rings with identity.
By the induction hypothesis, ξ ∈ s. Let t ∈ c ∈ V and s ∈ d ∈ V . – If cSd, then as ξ ∈ d, by the definition of S, we have ♦ξ ∈ t. – If c = d and t = s, then Definition 7 yields ♦ξ ∈ t. – If c = d and t = s, then two cases are possible: either cSd, in which case we use the argument of the first considered case, or t = s is regular. In the latter case, by the definition of a regular type, we immediately obtain ♦ξ ∈ t. Thus, in all possible cases we have ♦ξ ∈ t, which finishes the proof of the claim. Now, since r |=ν ϕ and ϕ ∈ Σ, we have ϕ ∈ t(r) ∈ c(r) ∈ V .
For this it is sufficient to show that each open subset U of X⊕ is compact. Since Fs is finite, each open subset U of X⊕ is a finite union of open sets Vi such that π(Vi ) = {c} ∪ Rs (c) for some c ∈ Ws . Therefore, we may assume that π(U ) = {c} ∪ Rs (c) for some c ∈ Ws . Let (U i )i∈I be an open cover of U . By condition (a) of Definition 8, Uc , Uci ∈ τc for each i ∈ I. Clearly Uc is compact in Xc and (Uci )i∈I is an open cover of Uc . Thus, there exists a finite subcover Uci1 , . . , Ucin of Uc .