Filtrations and Basic Proof Theory Notes for Lecture 5
Note that !Σ is an equivalence relation. Let |w|Σ = {v | w !Σ v} denote the equivalence class of w under !Σ. Definition 1.1 (Filtration) Let M = 〈W,R, V 〉 be a Kripke model. Given a set of formulas closed under subformulas, a model Mf = 〈W f , Rf , V f 〉 is said to be a filtration of M through Σ provided • W f = {|w|Σ | w ∈W} • If wRv then |w|ΣR |v|Σ • If |w|ΣR |v|Σ then for each 3φ ∈ Σ, if M, v |= φ then M, w |= 3φ • V f (p) = {|w|Σ | w ∈ V (p)} /
FINITE MODELS CONSTRUCTED FROM CANONICAL FORMULAS
This paper obtains the weak completeness and decidability results for standard systems of modal logic using models built from formulas themselves. This line of work began with Fine (Notre Dame J. Form. Log. 16:229–237, 1975). There are two ways in which our work advances on that paper: First, the definition of our models is mainly based on the relation Kozen and Parikh used in their proof of the completeness of PDL, see (Theor. Comp. Sci. 113–118, 1981). The point is to develop a general model-construction method based on this definition. We do this and thereby obtain the completeness of most of the standard modal systems, and in addition apply the method to some other systems of interest. None of the results use filtration, but in our final section we explore the connection.
Rotational Analysis of the CrO Bands
The (0, 1), (0, 0) and (1, 0) bands of the CrO, which appear in the red and yellow region, are analysed rotationally, and we have been able to show that these bands owe to the transition 5 Π- 5 Π. We have shown the theoretical formulae for the quintet state, which can be applied in the case of nearly (a), and obtained the molecular constants as follows,