Contributions Differential Refinement Logic � Maintains a modular and hierarchical proof structure � Abstracts implementation-specific designs � Leverages iterative system design � Prove time-triggered model refines event-triggered � Encouraging evidence of reduced user interaction and computation time 67
Appendix 68
Comparing dRL and dL We have proved that the refinement relation can be embedded in dL. As a result, dL and dRL are equivalent in terms of expressibility and provability . However, we can analyze dRL on familiar (challenging) case studies. We can consider: • Number of proof steps • Computation time • Qualitative difficulty to complete proof • Proof structure 69
Semantics of hybrid programs iff except for ρ ( x := θ ) = v = w the value of v w ρ ( x ρ ( x := θ ) = ) = { ( v, w ) : w = v except [[ x ]] w = [[ θ ]] v } ? ψ | Iff holds in state = ψ v v ρ (? ψ ) = ) = { ( v, v ) : v | = ψ } x 0 = θ If solves x 0 = θ y ( t ) v w x := y ( t ) ρ ( x 0 = θ ) = { ( ϕ (0) , ϕ ( t )) : ϕ ( s ) | = x 0 = θ for all 0 ≤ s ≤ t } [Platzer08] 70
Semantics of hybrid programs α ; β v u w β α ρ ( α ; β ) = { ( v, w ) : ( v, u ) ∈ ρ ( α ) , ( ) , ( u, w ) ∈ ρ ( β ) } [Platzer08] 71
Combining refinement and diamond modality 72
Nondeterministic Assignment 73
Nondeterministic Assignment ρ ( x := θ ) = v J θ K v v J x x 74
Nondeterministic Assignment v d 1 ∗ x = : x i h ρ ( x := θ ) = v J θ K v v J v J v d 2 x := ∗ x x x x x := ∗ v d 3 hi x 75
Nondeterministic Assignment v d 1 ∗ x = : x i h ρ ( x := θ ) = v J θ K v v J v J v d 2 x := ∗ x x x x x := ∗ v d 3 hi x 76
Nondeterministic Repetition α ∗ … α α α v w 77
Nondeterministic Repetition α ∗ … α α α v w β 78
Nondeterministic Repetition α ∗ … α α α v w β ? β 79
Nondeterministic Repetition α ∗ … α α α v w β ? β 80
Nondeterministic Repetition α ∗ … α α α v w β β 81
Nondeterministic Repetition α ∗ … α α α v w β ? β β 82
Nondeterministic Repetition α ∗ … α α α v w β ? β β 83
Nondeterministic Repetition α ∗ … α α α v w β β β 84
Nondeterministic Repetition α ∗ … α α α v w β β β β ∗ 85
Nondeterministic Repetition (KAT style) 86
Nondeterministic Repetition (KAT style) … β α α w 1 w 2 w 3 w 4 v 87
Nondeterministic Repetition (KAT style) … β α α w 1 w 2 w 3 w 4 v 88
Nondeterministic Repetition (KAT style) … β α α w 1 w 2 w 3 w 4 v 89
Nondeterministic Repetition (KAT style) … β α α w 1 w 2 w 3 w 4 v γ ? 90
Nondeterministic Repetition (KAT style) … β α α w 1 w 2 w 3 w 4 v γ ? 91
Nondeterministic Repetition (KAT style) … β α α w 1 w 2 w 3 w 4 v γ 92
Nondeterministic Repetition (KAT style) … β α α w 1 w 2 w 3 w 4 v γ ? γ 93
Nondeterministic Repetition (KAT style) … β α α w 1 w 2 w 3 w 4 v γ ? γ 94
Nondeterministic Repetition (KAT style) … β α α w 1 w 2 w 3 w 4 v γ ? γ 95
Nondeterministic Repetition (KAT style) … β α α w 1 w 2 w 3 w 4 v γ γ 96
Nondeterministic Repetition (KAT style) … β α α w 1 w 2 w 3 w 4 v γ … γ γ 97
Nondeterministic Repetition (KAT style) … β α α w 1 w 2 w 3 w 4 v γ … γ γ γ 98
Nondeterministic Repetition (KAT style) … β α α w 1 w 2 w 3 w 4 v 99
Nondeterministic Repetition (KAT style) … β α α w 1 w 2 w 3 w 4 v … γ γ γ γ 100
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