index.md (2116B)
1 +++ 2 title = 'Lecture 7' 3 +++ 4 # Lecture 7 5 ## Standard translation 6 Aim to map formulas of basic modal logic to first-order predicate logic such that: 7 - φ in BML is *valid* iff its translation in first-order predicate logic is *valid* 8 - φ in BML is *satisfiable* iff its translation in first-order predicate logic is *satisfiable* 9 10 Translation: 11 - translate "p is true in world x" as "predicate P holds for x" 12 - translate "accessibility relation R" as "binary predicate R" 13 - translation is relative to some state 14 - notation: Px instead of P(x), Rxy instead of R(x,y) 15 16 Rules for standard translation: 17 18  19 20 Examples of translation: 21 22  23 24 first-order predicate logic is decidable _if it uses at most 2 variables_. 25 therefore, adapt standard translation to only use 2 variables. 26 27 In a formula, check the situation with bound variables, and rename where possible. 28 When you are in state _a_ and start a quantification, use the variable _b_, and vice versa. 29 30 ## Finite model property 31 If φ is satisfiable, then φ is satisfiable on a finite model. 32 33 Effective finite model property: if φ is satisfiable, then φ is satisfiable in a model of size ≤ f(φ) 34 35 Via selection: 36 - suppose there is a model that makes φ true 37 - unravel the model at x to a tree model 38 - φ has finite modal depth n, so restrict tree model to height n 39 - rewrite φ to a conjunction of first-order propositional logic formulas and diamonds 40 - take for every diamond formula a successor 41 42 Via filtration: 43 - suppose there is a model that makes φ true 44 - consider set S of all subformulas of φ 45 - define equivalence relation on S: u ~ v iff u and v are modally equivalent 46 - i.e., if they agree on letters/formulas 47 - define W' to consist of equivalence classes [u] of states from W 48 - define V' using u ∈ V(p) iff [u] ∈ V'(p) for every p in S 49 - define a R' using R'[u][v] if Ruv, and requiring: 50 - if R'[u][v] and ◇ ψ ∈ S and [v] ⊨ ψ, then [u] ⊨ ◇ φ 51 - then (W', R'), V' is finite and (W', R'), V', [x] ⊨ φ 52 53