lecture-4.md (1661B)
1 +++ 2 title = 'Lecture 4' 3 template = 'page-math.html' 4 +++ 5 6 # Lecture 4 7 ## Bisimulations 8 A non-empty relation Z ⊆ W × W' is bisimulation ($Z : M \underline{\leftrightarrow} M'$) if for all pairs (w, w') ∈ Z we have: 9 - w ∈ V(p) iff w' ∈ V'(p) 10 - if Rwv then for some v' ∈ W' we have R'w'v' and vZv' 11 - if R'w'v' then for some v ∈ W we have Rwv and vZv' 12 13 Two models are bisimilar ($M \underline{\leftrightarrow} M'$) if there exists a bisimulation Z ∈ W × W'. 14 15 Two pointed models are bisimilar if there exists a bisimulation such that (w,w') ∈ Z 16 17 Two states are modally equivalent if they satisfy exactly the same formulas. 18 So if M,w and M',w' are bisimilar, then they are modally equivalent. 19 20 ## Transforming and constructing models 21 Disjoint union of models: combine models by union of states, relations, and valuations. 22 A state in one of the models is modally equivalent with the state in the union. 23 24 Generated submodel: starting from state w, only take its future. 25 26 Tree unravelling: unravelling of world s in (W,R,V) is: 27 - $W' : (s_{1} \dots s_{n})$ with $s_{1} = s$ and $Rs_{i} s_{i+1}$ 28 - $R'$ relates ($s_{1} \dots s_{n}$) to ($s_{1} \dots s_{n+1}$) if $Rs_{n} s_{n+1}$ 29 - $V'(p) = \{ (s_{1} \dots s_{n} | s_{n} \in V(p) \}$ 30 - a state in (W',R',V') is bisimilar to $s_{n}$ in (W,R,V) 31 - if φ is satisfiable in M,w it is satisfiable in tree unravelling of s in M 32 33 Bisimulation contraction 34 - W' consists of equivalence classes |s| = { t such that $s \underline{\leftrightarrow} t$ } 35 - R' relates |s| to |t| if Ruv for some u ∈ |s| and some v ∈ |t| 36 - V'(p) = { |s| | s ∈ V(p) } 37 38 If two states are modally equivalent, then they are bisimilar.