Towards a Hybrid Dynamic Logic for Hybrid Dynamic Systems e Platzer - PowerPoint PPT Presentation

Towards a Hybrid Dynamic Logic for Hybrid Dynamic Systems e Platzer 1 , 2 Andr´ 1 Carnegie Mellon University, Pittsburgh, PA, USA 2 University of Oldenburg, Department of Computing Science, Germany aplatzer@cs.cmu.edu LICS International Workshop on Hybrid Logic 2006 Andr´ e Platzer (CMU) Hybrid Dynamic Logic LICS - HyLo 2006 1 / 10

Towards a Hybrid Dynamic Logic for Hybrid Dynamic Systems e Platzer 1 , 2 Andr´ 1 Carnegie Mellon University, Pittsburgh, PA, USA 2 University of Oldenburg, Department of Computing Science, Germany aplatzer@cs.cmu.edu LICS International Workshop on Hybrid Logic 2006 Andr´ e Platzer (CMU) Hybrid Dynamic Logic LICS - HyLo 2006 1 / 10

Hybrid Dynamic Systems Hybrid Dynamic Logic Logic with state-references and program-modalities Hybrid Dynamic Systems Hybrid dynamic systems are subject to both continuous evolution along differential equations and discrete change. x t +0 . 5 Andr´ e Platzer (CMU) Hybrid Dynamic Logic LICS - HyLo 2006 2 / 10

Hybrid Dynamic Systems Hybrid Dynamic Systems Hybrid dynamic systems are subject to both continuous evolution along differential equations and discrete change. Example (Safety-Critical) Car / train / aircraft / chemical process / artificial pancreas discrete: digital controller of plant continuous: physical model of plant Andr´ e Platzer (CMU) Hybrid Dynamic Logic LICS - HyLo 2006 2 / 10

Hybrid Dynamic Systems: Verification Hybrid Dynamic Systems Hybrid dynamic systems are subject to both continuous evolution along differential equations and discrete change. Challenges (Compositional Verification) 1 Verify intricate dynamics in isolation 2 Integrability of local correctness Andr´ e Platzer (CMU) Hybrid Dynamic Logic LICS - HyLo 2006 2 / 10

Hybrid Dynamic Systems: Verification Hybrid Dynamic Systems Hybrid dynamic systems are subject to both continuous evolution along differential equations and discrete change. Challenges (Compositional Verification) 1 Verify intricate dynamics in isolation 2 Integrability of local correctness state-based reasoning: (transition to abstract state i ) 1 introspection: (statement about other state @ i φ ) 2 Andr´ e Platzer (CMU) Hybrid Dynamic Logic LICS - HyLo 2006 2 / 10

Outline Motivation 1 The Logic d L h 2 Syntax Semantics Compositional Introspection The d L h Calculus 3 Sequent Calculus State-based Reasoning Soundness & Co Conclusions & Future Work 4 Andr´ e Platzer (CMU) Hybrid Dynamic Logic LICS - HyLo 2006 2 / 10

Outline Motivation 1 The Logic d L h 2 Syntax Semantics Compositional Introspection The d L h Calculus 3 Sequent Calculus State-based Reasoning Soundness & Co Conclusions & Future Work 4 Andr´ e Platzer (CMU) Hybrid Dynamic Logic LICS - HyLo 2006 2 / 10

The Logic d L h : Syntax d L h formulas = first-order logic + dynamic logic + hybrid logic � �� � [ α ] φ, � α � φ Definition (System actions α ) x = f ( x ) ˙ (continuous evolution) x := θ (discrete mode switch) φ ? (conditional execution) α ; γ (seq. composition) α ∪ γ (nondet. choice) α ∗ (nondet. repetition) Details Andr´ e Platzer (CMU) Hybrid Dynamic Logic LICS - HyLo 2006 3 / 10

d L h Semantics: Hybrid System Evolution x e − t t x = − x +0 . 5 ˙ x = f ( x ) ˙ x > 1 → � ˙ x = − x ; x := x + 0 . 5; ˙ x = f ( x ) � safe Details Andr´ e Platzer (CMU) Hybrid Dynamic Logic LICS - HyLo 2006 4 / 10

d L h Semantics: Hybrid System Evolution x e − t t +0 . 5 ˙ x = − x ˙ x = f ( x ) x > 1 → � ˙ x = − x ; x := x + 0 . 5; ˙ x = f ( x ) � safe Details Andr´ e Platzer (CMU) Hybrid Dynamic Logic LICS - HyLo 2006 4 / 10

Compositional Introspection in ETCS Braking [poll-sensor; a := accel-sys; ¨ z = a ]( z ≥ m → @ i slope ) Andr´ e Platzer (CMU) Hybrid Dynamic Logic LICS - HyLo 2006 5 / 10

Compositional Introspection in ETCS Braking [poll-sensor; a := accel-sys; i ?; ¨ z = a ]( z ≥ m → @ i slope ) Andr´ e Platzer (CMU) Hybrid Dynamic Logic LICS - HyLo 2006 5 / 10

Outline Motivation 1 The Logic d L h 2 Syntax Semantics Compositional Introspection The d L h Calculus 3 Sequent Calculus State-based Reasoning Soundness & Co Conclusions & Future Work 4 Andr´ e Platzer (CMU) Hybrid Dynamic Logic LICS - HyLo 2006 5 / 10

Sequent Calculus (excerpt) @ i � x := θ � j ⊢ @ i F θ x (R1) @ i � x := θ � j ⊢ @ j F @ i � α � a, @ a φ ⊢ (R2) @ i � α � φ ⊢ @ i ∃ t ≥ 0 � x := y x ( t ) � φ ⊢ (R3) @ i � ˙ x = f ( x ) � φ ⊢ � � x = ˙ f ( x ) where y x solution of IVP x (0) = x Priority: R3 > R2 > R1 Andr´ e Platzer (CMU) Hybrid Dynamic Logic LICS - HyLo 2006 6 / 10

State-based Reasoning for Compositional Verification ∗ @ t � a := - b � r, @ t � ¨ z = - b � cr ⊢ @ t � ¨ z = - b � z ≥ m ... @ t � a := - b � r, @ r � ¨ z = a � cr ⊢ @ t � ¨ z = - b � z ≥ m @ t � c 2 ?; . . . � r ⊢ . . . @ t ( � a := - b � r ∨ � c 2 ?; a := 0 . 1 � r ) , @ r � ¨ z = a � cr ⊢ @ t � ¨ z = - b � z ≥ m @ t � a := - b ∪ ( c 2 ?; a := 0 . 1) � r, @ r � ¨ z = a � cr ⊢ @ t � ¨ z = - b � z ≥ m @ t � a := - b ∪ ( c 2 ?; a := 0 . 1) �� ¨ z = a � cr ⊢ @ t � ¨ z = - b � z ≥ m @ t � accel � cr ⊢ @ t � ¨ z = - b � z ≥ m @ t ¬� ¨ z = - b � z ≥ m , @ s � tctl � t, @ t � accel � cr ⊢ @ s [tctl] ¬� ¨ z = - b � z ≥ m , @ s � tctl � t, @ t � accel � cr ⊢ @ s [tctl] ¬� ¨ z = - b � z ≥ m , @ s � tctl �� accel � cr ⊢ @ s [tctl] ¬� ¨ z = - b � z ≥ m , @ s � tctl; accel � cr ⊢ @ s [tctl] ¬� ¨ z = - b � z ≥ m ⊢ @ s ¬� tctl; accel � cr � � Abbreviations: c 2 ≡ ( m − z ≥ 2 e ) and accel ≡ a := - b ∪ ( c 2 ?; a := 0 . 1) ; ¨ z = a Andr´ e Platzer (CMU) Hybrid Dynamic Logic LICS - HyLo 2006 7 / 10

State-based Reasoning for Compositional Verification ∗ @ t � a := - b � r, @ t � ¨ z = - b � cr ⊢ @ t � ¨ z = - b � z ≥ m ... @ t � a := - b � r, @ r � ¨ z = a � cr ⊢ @ t � ¨ z = - b � z ≥ m @ t � c 2 ?; . . . � r ⊢ . . . @ t ( � a := - b � r ∨ � c 2 ?; a := 0 . 1 � r ) , @ r � ¨ z = a � cr ⊢ @ t � ¨ z = - b � z ≥ m @ t � a := - b ∪ ( c 2 ?; a := 0 . 1) � r, @ r � ¨ z = a � cr ⊢ @ t � ¨ z = - b � z ≥ m @ t � a := - b ∪ ( c 2 ?; a := 0 . 1) �� ¨ z = a � cr ⊢ @ t � ¨ z = - b � z ≥ m @ t � accel � cr ⊢ @ t � ¨ z = - b � z ≥ m @ t ¬� ¨ z = - b � z ≥ m , @ s � tctl � t, @ t � accel � cr ⊢ @ s [tctl] ¬� ¨ z = - b � z ≥ m , @ s � tctl � t, @ t � accel � cr ⊢ @ s [tctl] ¬� ¨ z = - b � z ≥ m , @ s � tctl �� accel � cr ⊢ @ s [tctl] ¬� ¨ z = - b � z ≥ m , @ s � tctl; accel � cr ⊢ @ s [tctl] ¬� ¨ z = - b � z ≥ m ⊢ @ s ¬� tctl; accel � cr � � Abbreviations: c 2 ≡ ( m − z ≥ 2 e ) and accel ≡ a := - b ∪ ( c 2 ?; a := 0 . 1) ; ¨ z = a Andr´ e Platzer (CMU) Hybrid Dynamic Logic LICS - HyLo 2006 7 / 10

State-based Reasoning for Compositional Verification ∗ @ t � ¨ z = - b � s, @ s crash ⊢ @ s z ≥ m @ t � ¨ z = - b � s, @ s crash ⊢ @ t � ¨ z = - b � z ≥ m @ t � a := - b � r, @ t � ¨ z = - b � crash ⊢ @ t � ¨ z = - b � z ≥ m @ t � a := - b � r, @ r � ¨ z = a � crash ⊢ @ t � ¨ z = - b � z ≥ m Andr´ e Platzer (CMU) Hybrid Dynamic Logic LICS - HyLo 2006 7 / 10

State-based Reasoning for Compositional Verification ∗ @ t � ¨ z = - b � s, @ s crash ⊢ @ s z ≥ m @ t � ¨ z = - b � s, @ s crash ⊢ @ t � ¨ z = - b � z ≥ m @ t � a := - b � r, @ t � ¨ z = - b � crash ⊢ @ t � ¨ z = - b � z ≥ m @ t � a := - b � r, @ r � ¨ z = a � crash ⊢ @ t � ¨ z = - b � z ≥ m Andr´ e Platzer (CMU) Hybrid Dynamic Logic LICS - HyLo 2006 7 / 10

State-based Reasoning for Compositional Verification ∗ @ t � ¨ z = - b � s, @ s crash ⊢ @ s z ≥ m @ t � ¨ z = - b � s, @ s crash ⊢ @ t � ¨ z = - b � z ≥ m @ t � a := - b � r, @ t � ¨ z = - b � crash ⊢ @ t � ¨ z = - b � z ≥ m @ t � a := - b � r, @ r � ¨ z = a � crash ⊢ @ t � ¨ z = - b � z ≥ m Andr´ e Platzer (CMU) Hybrid Dynamic Logic LICS - HyLo 2006 7 / 10

State-based Reasoning for Compositional Verification ∗ @ t � ¨ z = - b � s, @ s crash ⊢ @ s z ≥ m @ t � ¨ z = - b � s, @ s crash ⊢ @ t � ¨ z = - b � z ≥ m @ t � a := - b � r, @ t � ¨ z = - b � crash ⊢ @ t � ¨ z = - b � z ≥ m @ t � a := - b � r, @ r � ¨ z = a � crash ⊢ @ t � ¨ z = - b � z ≥ m Andr´ e Platzer (CMU) Hybrid Dynamic Logic LICS - HyLo 2006 7 / 10

State-based Reasoning for Compositional Verification ∗ @ t � ¨ z = - b � s, @ s crash ⊢ @ s z ≥ m @ t � ¨ z = - b � s, @ s crash ⊢ @ t � ¨ z = - b � z ≥ m @ t � a := - b � r, @ t � ¨ z = - b � crash ⊢ @ t � ¨ z = - b � z ≥ m @ t � a := - b � r, @ r � ¨ z = a � crash ⊢ @ t � ¨ z = - b � z ≥ m Andr´ e Platzer (CMU) Hybrid Dynamic Logic LICS - HyLo 2006 7 / 10