Bellerophon: Tactical Theorem Proving for Hybrid Systems. Nathan Fulton, Stefan Mitsch, Brandon Bohrer, André Platzer Carnegie Mellon University
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1 Bellerophon: Tactical Theorem Proving for Hybrid Systems Nathan Fulton, Stefan Mitsch, Brandon Bohrer, André Platzer Carnegie Mellon University
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3 Cyber-Physical Systems Cyber-Physical Systems combine computation and control. Hybrid Systems model combinations of discrete and continuous dynamics.
4 Bellerophon Verifying hybrid systems is hard.
5 Bellerophon Verifying hybrid systems is hard. Bellerophon demonstrates how to tackle hybrid systems with tactics:
6 Bellerophon Verifying hybrid systems is hard. Bellerophon demonstrates how to tackle hybrid systems with tactics: Build on a sound core.
7 Bellerophon Verifying hybrid systems is hard. Bellerophon demonstrates how to tackle hybrid systems with tactics: Build on a sound core. Implement high-level primitives for hybrid systems proofs.
8 Bellerophon Verifying hybrid systems is hard. Bellerophon demonstrates how to tackle hybrid systems with tactics: Build on a sound core. Implement high-level primitives for hybrid systems proofs. Automate common constructions (for ODEs and control software)
9 Bellerophon Theorem Bellerophon LOC Conceptual Proof Steps Hybrid Systems Axiom Applications Static Safety ,355 Passive-Friendly Safety ,620 Orientation Safety ,989 Pass Intersection Liveness ,878
10 KeYmaera X: Trustworthy Foundations Interactive Reachability Analysis Bellerophon combinator language Bellerophon standard library for hybrid systems Demonstration Bellerophon for Automation and Tooling Conclusions & Resources
11 Trustworthy Foundations KeYmaera X enables trustworthy automation for hybrid systems analysis: A well-defined logical foundations, implemented in a small trustworthy core that ensures correctness of automation and tooling.
12 Trustworthy Foundations Hybrid Programs a := t a=a 0 b=b 0 c=c 0... a=t b=b 0 c=c 0...
13 Trustworthy Foundations Hybrid Programs a := t a=a 0 b=b 0 c=c 0... a=t b=b 0 c=c 0... a;b a a;b b
14 Trustworthy Foundations Hybrid Programs a := t a=a 0 b=b 0 c=c 0... a=t b=b 0 c=c 0... a;b a a;b b?p If P is true: no change If P is false: terminate
15 Trustworthy Foundations Hybrid Programs a := t a=a 0 b=b 0 c=c 0... a=t b=b 0 c=c 0... a;b a a;b b?p If P is true: no change If P is false: terminate a b
16 Trustworthy Foundations Hybrid Programs a := t a=a 0 b=b 0 c=c 0... a=t b=b 0 c=c 0... a;b a a;b b?p If P is true: no change If P is false: terminate a b
17 Trustworthy Foundations Hybrid Programs a := t a=a 0 b=b 0 c=c 0... a=t b=b 0 c=c 0... a;b a a;b b?p If P is true: no change If P is false: terminate a b a* a...a...
18 Trustworthy Foundations Hybrid Programs a := t a=a 0 b=b 0 c=c 0... a=t b=b 0 c=c 0... a;b a a;b b?p If P is true: no change If P is false: terminate a b x=f(0)... a...a... x=x a* x =f 0... x=f(t)...
19 Trustworthy Foundations Reachability Specifications [a]p after every execution of a, P <a>p after some execution of a, P
20 Trustworthy Foundations Reachability Specifications [a]p after every execution of a, P <a>p after some execution of a, P init [{x := u(x); x = f(x)}*]safe
21 Trustworthy Foundations Hello, World { {?Dive r := r p }; t:=0; {x = v, V = f(v,g,r), t =1 & 0 x & t T} }* Control: Continue diving if safe, else open parachute. Plant: Downward velocity determined by gravity, air resistance. x v =f(v,g,r)
22 Trustworthy Foundations Hello, World { {?Dive r := r p }; t:=0; {x = v, V = f(v,g,r), t =1 & 0 x & t T} }* Control: Continue diving if safe, else open parachute. Plant: Downward velocity determined by gravity, air resistance. x v =f(v,g,r)
23 Trustworthy Foundations Hello, World { {?Dive r := r p }; t:=0; {x = v, V = f(v,g,r), t =1 & 0 x & t T} }* Control: Continue diving if safe, else open parachute. Plant: Downward velocity determined by gravity, air resistance. x v =f(v,g,r)
24 Trustworthy Foundations Hello, World { {?Dive r := r p }; t:=0; {x = v, V = f(v,g,r), t =1 & 0 x & t T} }* Control: Continue diving if safe, else open parachute. Plant: Downward velocity determined by gravity, air resistance. x v =f(v,g,r)
25 Trustworthy Foundations Hello, World { {?Dive r := r p }; t:=0; {x = v, V = f(v,g,r), t =1 & 0 x & t T} }* Control: Continue diving if safe, else open parachute. Plant: Downward velocity determined by gravity, air resistance. x v =f(v,g,r)
26 Trustworthy Foundations Hello, World { {?Dive r := r p }; t:=0; {x = v, V = f(v,g,r), t =1 & 0 x & t T} }* Control: Continue diving if safe, else open parachute. Plant: Downward velocity determined by gravity, air resistance. x v =f(v,g,r)
27 Trustworthy Foundations Reachability Specifications (Dive & g>0 & ) [{ {?Dive r := r p }; {x = v, V = f(v,g,r) & 0 x} }*](x=0 m v) x v =f(v,g,r)
28 Trustworthy Foundations Reachability Specifications (Dive & g>0 & ) [{ {?Dive r := r p }; {x = v, V = f(v,g,r) & 0 x} }*](x=0 m v) x v =f(v,g,r) If the parachuter is on the ground, their speed is safe (m v 0)
29 Introduction to Differential Dynamic Logic Dynamical Axioms [x:=t]f(x) f(t) [a;b]p [a b]p [a][b]p ([a]p & [b]p) [x =f&q]p (Q P)...
30 Introduction to Differential Dynamic Logic Trusted Core AXIOM BASE [x:=t]f(x) f(t) [a;b]p [a][b]p [a b]p ([a]p & [b]p) [x =f&q]p (Q P)... KeYmaera X Core Q.E.D.
31 Introduction to Differential Dynamic Logic Trustworthy Implementations Bellerophon Tooling Automated Analyses AXIOM BASE [x:=t]f(x) f(t) [a;b]p [a][b]p [a b]p ([a]p & [b]p) [x =f&q]p (Q P)... KeYmaera X Core Q.E.D.
32 Introduction to Differential Dynamic Logic Prover Core Comparison
33 Bellerophon Bellerophon enables interactive verification and tool development:
34 Bellerophon Bellerophon enables interactive verification and tool development: A standard library of common proof techniques.
35 Bellerophon Bellerophon enables interactive verification and tool development: A standard library of common proof techniques. A combinator language/library for decomposing theorems and composing proof strategies.
36 Bellerophon Standard Library Tactic Meaning prop unfold Applies propositional reasoning exhaustively. Symbolically executes discrete, loop-free programs. loop(j, i) Applies loop invariance axiom to position i. di,dg,dc,dw Reasoning principles for differential equations.
37 Bellerophon Standard Library prop Tactic Meaning Applies propositional reasoning exhaustively unfold Symbolically executes discrete, loop-free programs. loop(j, i) Applies loop invariance axiom to position i. di,dg,dc,dw Reasoning principles for differential equations.
38 Bellerophon Combinators prop Tactic Meaning Applies propositional reasoning exhaustively unfold loop(j, i) di,dg,dc,dw Symbolically executes discrete, loop-free programs. Applies loop invariance axiom to position i, extends J with constants. Reasoning principles for differential equations. A ; B A B Combinator Meaning Execute A on current goal, then execute B on the result. Try executing A on current goal. If A fails, execute B on current goal. A* Run A until it no longer applies. A<( B 1,B 2,,B N ) Execute A on current goal to create N subgoals. Run B i on subgoal i.
39 Bellerophon Isolating Interesting Questions (Dive & g>0 & ) [{ }*](x=0 m v)
40 Bellerophon Isolating Interesting Questions (Dive & g>0 & ) [{ (Dive & g>0 & ) J Loop invariant holds initially prop ; loop(j,1) J [ ]J Loop invariant is preserved }*](x=0 m v) J x=0 m v Loop invariant implies safety
41 Bellerophon Isolating Interesting Questions (Dive & g>0 & ) [{ (Dive & g>0 & ) J Loop invariant holds initially prop ; loop(j,1) J [ ]J Loop invariant is preserved }*](x=0 m v) J x=0 m v Loop invariant implies safety
42 Bellerophon Isolating Interesting Questions (Dive & g>0 & ) [{ (Dive & g>0 & ) J prop ; loop(j,1) J [ unfold J & Dive & r=r a [x =v,v =...]J ]J J & r=r p [x =v,v =...]J }*](x=0 m v) J x=0 m v
43 Bellerophon Isolating Interesting Questions (Dive & g>0 & ) [{ (Dive & g>0 & ) J prop ; loop(j,1) J [ unfold J & Dive & r=r a [x =v,v =...]J ]J J & r=r p [x =v,v =...]J }*](x=0 m v) J x=0 m v
44 Bellerophon Isolating Interesting Questions prop ; loop(j, 1) <( QE, /* Real arith. solver */ QE, unfold ; <( /* parachute open case */ /* parachute closed case */ ) )
45 Interactive Verification in Bellerophon Trustworthy Standard Library at High Abstraction Level J [{ctrl; plant}*]j J = v > -sqrt(g/pr) > m & Parachute Open Case: v v 0 - gt v 0 - gt > -sqrt(g/pr) Inductive invariants x v =rv 2 -g
46 Interactive Verification in Bellerophon From Axioms to Proof Steps DI Axiom: [{x'=f&q}]p ([?Q]P (Q [{x'=f&q}]p'))
47 Interactive Verification in Bellerophon From Axioms to Proof Steps DI Axiom: [{x'=f&q}]p ([?Q]P (Q [{x'=f&q}]p')) Example: [v =r p v 2 -g,t =1]v v 0 - gt
48 Interactive Verification in Bellerophon From Axioms to Proof Steps DI Axiom: [{x'=f&q}]p ([?Q]P (Q [{x'=f&q}]p')) Example: [v =r p v 2 -g,t =1]v v 0 - gt [v :=r p v 2 -g][t :=1]v -g*t r p v 2 -g -g r p 0
49 Interactive Verification in Bellerophon From Axioms to Proof Steps di Tactic: Side derivation: (v v 0 - gt) (v) (v 0 - gt) (v) (v 0 - gt) (v) (v 0 ) -(gt) (v) (v 0 ) - (t(g) +g(t )) V v 0 - (tg +gt ) H=r p 0 & r a 0 & g>0 &... DI Axiom: [{x'=f&q}]p ([?Q]P (Q [{x'=f&q}]p')) Example: [v =r p v 2 -g,t =1]v v 0 - gt [v :=r p v 2 -g][t :=1]v -g*t r p v 2 -g -g H r p 0
50 Automation and Tooling Hybrid Systems Analyses can be built on top of KeYmaera X. Examples: ODE Solver Runtime Monitoring
51 Automation and Tooling Solving Differential Equations 1. Use untrusted code to find a conjecture. Untrusted ODE Solver Axiomatic Solver (Bellerophon Program) 2. Prove the conjecture systematically, leveraging standard library. AXIOM BASE [x:=t]f(x) f(t) [a;b]p [a][b]p [a b]p ([a]p & [b]p) [a*]p (J P & J [b]j) [x =f&q]p (Q P)... KeYmaera X Core Q.E.D.
52 Automation and Tooling Solving Differential Equations 1. Use untrusted code to find a conjecture. Untrusted ODE Solver Axiomatic Solver (Bellerophon Program) 2. Prove the conjecture systematically, leveraging standard library. AXIOM BASE [x:=t]f(x) f(t) [a;b]p [a][b]p [a b]p ([a]p & [b]p) [a*]p (J P & J [b]j) [x =f&q]p (Q P)... KeYmaera X Core Q.E.D.
53 Automation and Tooling ModelPlex Tactic
54 Toward Automated Deduction Other Proof Automation & Tooling Taylor Series Bifurcations Limit Cycles Numerical tools... ODE & Controls Tooling Clever Bellerophon Programs AXIOM BASE [x:=t]f(x) f(t) [a;b]p [a][b]p [a b]p ([a]p & [b]p) [a*]p (J P & J [b]j) [x =f&q]p (Q P)... KeYmaera X Core Q.E.D.
55 Toward Automated Deduction Other Proof Automation & Tooling Taylor Series Bifurcations Limit Cycles Numerical tools... ODE & Controls Tooling Clever Bellerophon Programs Other Tooling: Component-based Verification Web UI AXIOM BASE [x:=t]f(x) f(t) [a;b]p [a][b]p [a b]p ([a]p & [b]p) [a*]p (J P & J [b]j) [x =f&q]p (Q P)... KeYmaera X Core Q.E.D.
56 Conclusion There is a wide gap between sound foundations for hybrid systems and practical interactive theorem proving technology for cyber-physical systems verification.
57 Conclusion There is a wide gap between sound foundations for hybrid systems and practical interactive theorem proving technology for cyber-physical systems verification. Bellerophon demonstrates how to verify hybrid systems using tactics.
58 Conclusion There is a wide gap between sound foundations for hybrid systems and practical interactive theorem proving technology for cyber-physical systems verification. Bellerophon demonstrates how to verify hybrid systems using tactics.
59 Conclusion There is a wide gap between sound foundations for hybrid systems and practical interactive theorem proving technology for cyber-physical systems verification. Bellerophon demonstrates how to verify hybrid systems using tactics. di Tactic: Side derivation: (v v 0 - gt) DI Axiom: [{x'=f&q}]p ([?Q]P (Q [{x'=f&q}]p')) Example: [v =r p v 2 -g,t =1]v v 0 - gt [v :=r p v 2 -g][t :=1]v -g*t r p v 2 -g -g H r p 0 H=r p 0 & r a 0 & g>0 &...
60 Conclusion There is a wide gap between sound foundations for hybrid systems and practical interactive theorem proving technology for cyber-physical systems verification. Bellerophon demonstrates how to verify hybrid systems using tactics. di Tactic: Side derivation: (v v 0 - gt) H=r p 0 & r a 0 & g>0 &... DI Axiom: [{x'=f&q}]p ([?Q]P (Q [{x'=f&q}]p')) Example: [v =r p v 2 -g,t =1]v v 0 - gt [v :=r p v 2 -g][t :=1]v -g*t r p v 2 -g -g H r p 0 ODE & Controls Tooling Axioms Clever Bellerophon Programs KyX qed
61 Conclusion There is a wide gap between sound foundations for hybrid systems and practical interactive theorem proving technology for cyber-physical systems verification. Bellerophon demonstrates how to verify hybrid systems using tactics. Project Website (start here) keymaerax.org Online Demo Open Source (GPL) web.keymaerax.org github.com/ls-lab/keymaerax-release Thanks: students, Jean-Baptiste Jeannin, Khalil Ghorbal, Yanni Kouskoulas et al., and many others!
62 Developers: Stefan Mitsch Nathan Fulton André Platzer Brandon Bohrer Jan-David Quesel Yong Kiam Tan Markus Völp
63
64 Interactive Reachability Analysis in KeYmaera X Differential Ghosts Parachute Closed: J & t=0 & r=r p [x =v,v =rv 2 -g & 0 x & t T]v>-sqrt(g/pr) > m Proof requires a differential ghost because the property is not inductive. x v =rv 2 -g
65 Interactive Reachability Analysis in KeYmaera X Differential Ghosts An example differential ghost. x>0 [x =-x]x>0
66 Interactive Reachability Analysis in KeYmaera X Differential Ghosts An example differential ghost. x>0 [x =-x]x>0 Ghost: y =y/2 Conserved: 1=xy 2
67 Interactive Reachability Analysis in KeYmaera X Differential Ghosts An example differential ghost. x>0 [x =-x]x>0 Ghost: y =y/2 Conserved: 1=xy 2 Notice: x>0 y.1=xy 2 Therefore, suffices to show: 1=xy 2 y.[x =-x,y =y/2]1=xy 2
68 Introduction to Differential Dynamic Logic Prover Core Comparison Tool Trusted LOC (approx.) KeYmaera X 1,682 (out of 100,000+) KeYmaera 65,989 Isabelle/Pure 8,113 Coq 20,000 HSolver 20,000 dreal 50,000 SpaceEx 100,000
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