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.
Effect of pretreatment with antibiotics on the hydrolysis of salicyluric acid in rabbit intestinal microorganisms.
The effect of pretreatment with antibiotics on the hydrolysis of salicyluric acid in rabbit intestinal microorganisms was investigated. Latamoxef sodium (LMOX, 25 mg/kg/d, intravenously) and cephalexin (CEX, 16.7 mg/kg/d, orally) were administered for 1 or 3 d. The blood concentration of salicyluric acid and salicylic acid following oral, intracecal and rectal administration of salicyluric acid was determined. By the pretreatment with LMOX for 1 or 3 d, the blood concentration of salicylic acid following oral administration of salicyluric acid was slightly decreased. In rabbits pretreated with CEX for 3 d, the blood concentration of salicylic acid was detected at low concentration. By the pretreatment with LMOX and CEX, however, the decrease in the blood concentration of salicylic acid following rectal administration of salicyluric acid was not observed. Although the examination of population of intestinal microorganisms induced by the pretreatment with antibiotics was not performed, the metabolic activity of intestinal microorganisms may be changed.