index.md (3707B)
1 +++ 2 title = 'Lecture 2' 3 +++ 4 # Lecture 2 5 ## Semantics: local truth 6 Valuation notation: 7 - V : Var → P(W) means Var → W → {0,1} 8 - V(p,w) = 1 is the same as w ∈ V(p) 9 10 A formula φ _characterizes_ a state x in model M if φ is true in x but not in other states of M. 11 12 A formula φ _distinguishes_ state x from state y in a model M if φ is true in x but not in y. 13 14  15 16 Above: 17 - the formula 3 ⊨ □ ⊥ characterizes state 3 18 - the formula 2 ⊨ ◇ □ ⊥ characterizes state 2 19 20 ## Game semantics 21 This is an approach to determine if a formula φ holds in a pointed model M, w. 22 23 We have: 24 - model M = ((W,R), V), world w ∈ W, formula φ 25 - two players: 26 - Verifier V claims that φ is true in w (sort of like ∀) 27 - Falsifier F claims that φ is false in w (sort of like ∃, try to find a _witness_) 28 - position: pair (w, φ) with w ∈ W a world and φ a formula 29 - move: from position (w, φ), determined by main operator of φ 30 31 Assume that negation only applied to prop. variables. 32 Transform formulas from ¬ □ p to ◇ ¬ p, ¬(p ∧ q) to ¬p ∨ ¬q. 33 34 The position determines the move, e.g. in a position (t, ◇ φ), V chooses a successor _u_ of _t_, and play continues with (u, φ). 35 For (t,p), if p is true in t then V wins, otherwise F wins. 36 For (t, ¬p), if p is false in t then V wins, otherwise F wins. 37 If a player should but cannot choose a successor, they lose. 38 39 Who starts: 40 - V: ∨, ◇ 41 - F: ∧, □ 42 43 A complete game tree for φ and (M,w) starts with (w,φ) and contains all possible moves. 44 A strategy for player P is subset of steps for P, and it's a winning strategy if it ensures that P wins the game. 45 φ is true in M in s ⇔ V has a winning strategy for M,s,φ. 46 47 ### Example 48 Diagram: 49 50  51 52 <details> 53 <summary>Graphviz code</summary> 54 55 <!-- :Tangle(dot) states.dot --> 56 ```dot 57 digraph states { 58 1 -> 2 59 1 -> 3 60 2 -> 3 61 3 -> 2 62 3 -> 4 63 4 -> 2 64 4 -> 3 65 } 66 ``` 67 68 </details> 69 70 Given: 71 - formula ◇ p ∨ □ ◇ p, in state 2. 72 - p is true in state 3. 73 74 Complete game tree: 75 76  77 78 <details> 79 <summary>Graphviz code</summary> 80 81 <!-- :Tangle(dot) tree.dot --> 82 ```dot 83 digraph gametree { 84 top [label="[V] ◇ p ∨ □ ◇ p, 2"] 85 l11 [label="[V] ◇ p, 2"] 86 l12 [label="[F] □ ◇ p, 2"] 87 top -> l11 88 top -> l12 89 l21 [label="p 3"] 90 l22 [label="[V] ◇ p, 3"] 91 l11 -> l21 92 l12 -> l22 93 l31 [label="Verifier wins"] 94 l32 [label="p, 2"] 95 l33 [label="p, 4"] 96 l21 -> l31 97 l22 -> l32 98 l22 -> l33 99 l41 [label="Falsifier wins."] 100 l42 [label="Falsifier wins."] 101 l32 -> l41 102 l33 -> l42 103 } 104 ``` 105 106 </details> 107 108 ## Truth and validity 109 (((W,R),V),w) ⊨ φ means φ is valid in a point w. 110 (W,R) ⊨ φ means φ is valid in a frame (W,R). 111 ⊨ φ means φ is valid/tautology. 112 If a part is omitted, it's implicitly universally quantified. 113 114 Satisfiability: 115 - φ _satisfiable in model M_ if there's a world w ∈ M such that M,w ⊨ φ 116 - φ _satisfiable_ if there's a model M and a world w ∈ M such that M,w ⊨ φ 117 - φ and ψ _semantically equivalent_ if ∀ M,w: M,w ⊨ φ ⇔ M,w ⊨ ψ 118 - φ valid iff ¬ φ not satisfiable 119 120 ### Example 121 Show universal validity of □ (φ → ψ) → (□ φ → □ ψ) 122 123 1. let F = (W,R) be frame, V valuation on F, let x ∈ W. 124 2. Assume a1: F,V,x ⊨ □ (φ → ψ) 125 3. Assume a2: F,V,x ⊨ □ φ 126 4. Aim to show F,V,x ⊨ □ ψ. 127 5. □ is universal quantification, so take an arbitrary successor y ∈ W. 128 6. If Rxy, aim to show y ⊨ ψ. If not, □ ψ holds. 129 7. Have y ⊨ φ → ψ and y ⊨ φ. 130 8. From a2, have y ⊨ φ. 131 9. From a1, have have y ⊨ ψ. 132 10. Hence x ⊨ □ ψ, hence x ⊨ □ φ → □ ψ. Hence formula is valid.