dTL 2 : Differential Temporal Dynamic Logic with Nested Modalities - PowerPoint PPT Presentation

dTL 2 : Differential Temporal Dynamic Logic � with Nested Modalities for Hybrid Systems Jean-Baptiste Jeannin and André Platzer 
 Carnegie Mellon University � � IJCAR, July 21 st , 2014 � Jeannin & Platzer � dTL 2 : Differential Temporal Dynamic Logic with Nested Modalities for Hybrid Systems � 1 �

Hybrid Systems � ¢ Continuous Evolutions 
 (differential equations, 
 e.g. flight dynamics) � ¢ Discrete Jumps 
 (control decisions, 
 e.g. pilot actions) � Jeannin & Platzer � dTL 2 : Differential Temporal Dynamic Logic with Nested Modalities for Hybrid Systems � 2 �

Differential Dynamic Logic � ¢ used to reason about (nondeterministic) hybrid systems � ¢ comes with a (relatively) complete axiomatization � ¢ proves properties about the end state of the execution � all behaviors of � hybrid system � α φ satisfy at the end � [ α ] φ x φ No guarantee on intermediate states � φ No guarantee on infinite executions � … � t Jeannin & Platzer � dTL 2 : Differential Temporal Dynamic Logic with Nested Modalities for Hybrid Systems � 3 �

Differential Temporal Dynamic Logic � ¢ What about property 
 “these airplanes never collide”? � ¢ We need some temporal reasoning � [ α ] ⇤ φ x φ φ φ Guarantees on φ φ φ φ intermediate states � φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ Guarantees on φ φ φ φ φ φ φ φ infinite executions � φ … � t Jeannin & Platzer � dTL 2 : Differential Temporal Dynamic Logic with Nested Modalities for Hybrid Systems � 4 �

Nested Alternating Modalities � ¢ What about property “this satellite can 
 reach its orbit and then stay there”? � ¢ We need nested alternating modalities � ¢ A step towards dTL*, handling temporal formulas of CTL* � x h α i ⌃⇤ φ φ φ φ φ φ φ … � t Jeannin & Platzer � dTL 2 : Differential Temporal Dynamic Logic with Nested Modalities for Hybrid Systems � 5 �

Temporal Properties of Hybrid Systems � φ , ψ State Property � Trace property � π φ ≤ , ¬ , ∧ , ∨ , ∀ , ∃ ¢ � ¢ � [ α ] π ⇤ π ¢ for all traces of � ¢ for all suffix of � α σ h α i π ♦ π ¢ there is a trace of � ¢ there is a suffix of � α σ φ φ [ α ] ⇤ φ x x φ h α i ⌃⇤ φ φ φ φ φ φ φφ φ φ φ … � … � φ φ φ φ φ t t x x φ [ α ] ♦ φ [ α ] ⇤⌃ φ φ φ … � … � φ φ φ t t Jeannin & Platzer � dTL 2 : Differential Temporal Dynamic Logic with Nested Modalities for Hybrid Systems � 6 �

Hybrid Programs � They model systems and are non deterministic. They are: � x := θ ¢ Discrete variable assignment � ? χ ¢ Test � x 0 = θ & χ ¢ Differential Equation � α ∪ β ¢ Nondeterministic choice � α ; β ¢ Sequential composition � ¢ Nondeterministic repetition � α ∗ Jeannin & Platzer � dTL 2 : Differential Temporal Dynamic Logic with Nested Modalities for Hybrid Systems � 7 �

Trace Semantics of Hybrid Programs � A trace represents the evolution of the variable over time, σ consisting of continuous evolutions and discrete jumps � x t The trace semantics of a hybrid program is a set of traces � Jeannin & Platzer � dTL 2 : Differential Temporal Dynamic Logic with Nested Modalities for Hybrid Systems � 8 �

Trace Semantics of Hybrid Programs � x := θ Variable assignment � x val ( x 1 , θ ) x 1 x 2 val ( x 2 , θ ) t Jeannin & Platzer � dTL 2 : Differential Temporal Dynamic Logic with Nested Modalities for Hybrid Systems � 9 �

Trace Semantics of Hybrid Programs � ? χ Test � x no state change � x 1 χ error � x 2 t Jeannin & Platzer � dTL 2 : Differential Temporal Dynamic Logic with Nested Modalities for Hybrid Systems � 10 �

Trace Semantics of Hybrid Programs � x 0 = θ & χ Differential equation � x x 1 can continue forever � x 2 χ x 3 t Jeannin & Platzer � dTL 2 : Differential Temporal Dynamic Logic with Nested Modalities for Hybrid Systems � 11 �

Trace Semantics of Hybrid Programs � Nondeterministic choice � α ∪ β x in α in β t Jeannin & Platzer � dTL 2 : Differential Temporal Dynamic Logic with Nested Modalities for Hybrid Systems � 12 �

Trace Semantics of Hybrid Programs � � α ; β Sequential composition The intermediate x state has to match � in β in α … in α t Jeannin & Platzer � dTL 2 : Differential Temporal Dynamic Logic with Nested Modalities for Hybrid Systems � 13 �

Trace Semantics of Hybrid Programs � α ∗ Nondeterministic repetition � x in α in α in α t Jeannin & Platzer � dTL 2 : Differential Temporal Dynamic Logic with Nested Modalities for Hybrid Systems � 14 �

Simplification of Trace Formulas � ⌃⌃⌃⇤⌃⇤⇤ φ ⇤⇤ π ≡ ⇤ π ♦♦ π ≡ ♦ π ≡ ⌃⇤⌃⇤ φ ⇤⌃⇤ φ ≡ ⌃⇤ φ ≡ ⌃⌃⇤ φ ⌃⇤⌃ φ ≡ ⇤⌃ φ ≡ ⌃⇤ φ x x ⇤⇤ π ≡ ⇤ π ⇤⌃⇤ φ ≡ ⌃⇤ φ π π π φ π φ π π π π φ π π π φ π t t Jeannin & Platzer � dTL 2 : Differential Temporal Dynamic Logic with Nested Modalities for Hybrid Systems � 15 �

Simplification of Trace Formulas � ⌃⌃⌃⇤⌃⇤⇤ φ ⇤⇤ π ≡ ⇤ π ♦♦ π ≡ ♦ π ≡ ⌃⇤⌃⇤ φ ⇤⌃⇤ φ ≡ ⌃⇤ φ ≡ ⌃⌃⇤ φ ⌃⇤⌃ φ ≡ ⇤⌃ φ ≡ ⌃⇤ φ The only interesting temporal properties thus are � ⇤ φ ⌃ φ ⌃⇤ φ ⇤⌃ φ and this corresponds to modal system S4.2 � We focus on the study of and particularly on � h α i ⇤ φ ⇤ φ Jeannin & Platzer � dTL 2 : Differential Temporal Dynamic Logic with Nested Modalities for Hybrid Systems � 16 �

A Technical Issue: the Composition � h α i ⇤ φ ^ h α ih β i ⇤ φ ( unsound ) h α ; β i ⇤ φ h α i ( ⇤ φ ^ h β i ⇤ φ ) ( OK if the trace of terminates ) α (if the trace of does not terminate ) h α i ⇤ φ α x x counterexample � h α ; β i ⇤ φ in β in α ⇤ φ ⇤ φ in β infinite trace in , � α α ; β thus in � ⇤ φ in α t t Jeannin & Platzer � dTL 2 : Differential Temporal Dynamic Logic with Nested Modalities for Hybrid Systems � 17 �

Solution: Introducing � φ u ⇤ ψ σ ✏ φ u ⇤ ψ if and only if � last σ ✏ φ σ ✏ ⇤ ψ ¢ and � � � if terminates � σ σ ✏ ⇤ ψ ¢ � � � � � otherwise (infinite or error) � ⇤ φ ⌘ true u ⇤ φ and � h α i ( h β i ⇤ φ u ⇤ φ ) x h ; i ⇤ h α ; β i ⇤ φ in β in α ⇤ φ ⇤ φ in β … in α ⇤ φ in α t Jeannin & Platzer � dTL 2 : Differential Temporal Dynamic Logic with Nested Modalities for Hybrid Systems � 18 �

Solution: Introducing � φ u ⇤ ψ σ ✏ φ u ⇤ ψ if and only if � last σ ✏ φ σ ✏ ⇤ ψ ¢ and � � � if terminates � σ σ ✏ ⇤ ψ ¢ � � � � � otherwise (infinite or error) � ⇤ φ ⌘ true u ⇤ φ and � h α i ( h β i ⇤ φ u ⇤ φ ) x h ; i ⇤ h α ; β i ⇤ φ φ in β h α i ( h β i ( φ u ⇤ ψ ) u ⇤ ψ ) in α h ; iu h α ; β i ( φ u ⇤ ψ ) ⇤ ψ ⇤ ψ in β … in α ⇤ ψ in α t Jeannin & Platzer � dTL 2 : Differential Temporal Dynamic Logic with Nested Modalities for Hybrid Systems � 19 �

New Rules for � φ u ⇤ ψ ψ ^ h x := θ i ( φ ^ ψ ) h := iu h x := θ i ( φ u ⇤ ψ ) x φ ∧ ψ ψ t Jeannin & Platzer � dTL 2 : Differential Temporal Dynamic Logic with Nested Modalities for Hybrid Systems � 20 �

New Rules for � φ u ⇤ ψ ( ¬ χ ^ ψ ) _ h x 0 = θ & ( χ ^ ψ ) i φ _ [ x 0 = θ ]( χ ^ ψ ) h x 0 = θ & χ i ( φ u ⇤ ψ ) x ψ ψ ψ ψ ψ ψ … φ ψ ψ ψ χ ψ t Jeannin & Platzer � dTL 2 : Differential Temporal Dynamic Logic with Nested Modalities for Hybrid Systems � 21 �

New Rules for � φ u ⇤ ψ 8 α 8 r > 0 ( ϕ ( r ) ! h α i ( ϕ ( r � 1) u ⇤ ψ )) ( 9 r ϕ ( r )) ^ ψ ! h α ∗ i (( 9 r  0 ϕ ( r )) u ⇤ ψ ) x ϕ (2) ⇤ ψ ϕ ( − 1) in α in α ϕ ( − 2) in α ⇤ ψ ϕ (1) ϕ (0) in α ⇤ ψ ⇤ ψ t Jeannin & Platzer � dTL 2 : Differential Temporal Dynamic Logic with Nested Modalities for Hybrid Systems � 22 �

φ t ♦ ψ φ ⇣ ⇤⌃ ψ φ ⇣ ⌃⇤ ψ Similarly , , � σ ✏ φ u ⇤ ψ Remember: if and only if � last σ ✏ φ σ ✏ ⇤ ψ ¢ and � � � if terminates � σ σ ✏ ⇤ ψ ¢ � � � � � otherwise (infinite or error) � σ ✏ φ t ⌃ ψ � � if and only if � σ ✏ ⌃ ψ last σ ✏ φ ¢ or � � � if terminates � σ σ ✏ ⌃ ψ ¢ � � � � � otherwise (infinite or error) � σ ✏ φ ⇣ ⇤⌃ ψ if and only if � last σ ✏ φ ¢ � � � � � if terminates � σ σ ✏ ⇤⌃ ψ ¢ � � � � � otherwise (infinite or error) � σ ✏ φ ⇣ ⌃⇤ ψ is defined similarly � Jeannin & Platzer � dTL 2 : Differential Temporal Dynamic Logic with Nested Modalities for Hybrid Systems � 23 �