SAT Modulo ODE: A Direct SAT Approach to Hybrid Systems

Transcription

1 Welcome ATVA 2008 SAT Modulo ODE: A Direct SAT Approach to Hbrid Sstems Andreas Eggers, Martin Fränzle, and Christian Herde DFG Transregional Collaborative Research Center AVACS

2 Outline Motivation Bounded model checking of hbrid sstems SMT formulae and the isat algorithm ODE enclosures as propagators Safel enclosing ODEs Integrated algorithm First benchmark results Conclusions and future work 2/1

3 Motivation Analsis of comple embedded sstems Interaction of discrete and continous dnamics Continuous behavior described b ordinar differential equations (ODEs) 3/1

4 Bounded Model Checking of Hbrid Sstems ϑ i [19, 2] c = 0 Heat off d ϑ i /dt = 0.1 (ϑ i ϑ o ) d c / d t = 0.0 c ϑ i 19 c 0.04 ϑ i 19 c 0.04 ϑ i 21 Heat on d ϑ i /dt = 0.2 (3 ϑ i ) 0.1 (ϑ i ϑ o ) d c / d t = c ϑ i 21 ϑ i [1, 21] c = 0 Bounded Model Checking (BMC): Are there an trajectories leading from an inital to an unsafe state in k steps? 4a/1

5 Bounded Model Checking of Hbrid Sstems ϑ i [19, 2] c = 0 Heat off d ϑ i /dt = 0.1 (ϑ i ϑ o ) d c / d t = 0.0 c ϑ i 19 c 0.04 ϑ i 19 c 0.04 GUARDS & ACTIONS ϑ i 21 Heat on d ϑ i /dt = 0.2 (3 ϑ i ) 0.1 (ϑ i ϑ o ) d c / d t = c ϑ i 21 CONTINUOUS DYNAMICS FLOW FLOW FLOW JUMP JUMP error trace JUMP FLOW JUMP time ϑ i [1, 21] c = 0 Bounded Model Checking (BMC): Are there an trajectories leading from an inital to an unsafe state in k steps? 4b/1

6 Bounded Model Checking of Hbrid Sstems Heat off d ϑ i /dt = 0.1 (ϑ i ϑ o ) d c / d t = 0.0 c ϑ i 19 c 0.04 ϑ i 19 c 0.04 ϑ i [19, 2] c = 0 Heat on d ϑ i /dt = 0.2 (3 ϑ i ) 0.1 (ϑ i ϑ o ) d c / d t = c ϑ i 21 ϑ i [1, 21] c = 0 ϑ i 21 init = ( ϑ o 20 c = 0 ) 19 ϑ i 2 on 1 ϑ i 21 on trans = ( on on ϑ i 19 c 0.04 ϑ i = ϑ i ϑ o = ϑ o c = c) ( on on ϑ i 21 ϑ i = ϑ i ϑ o = ϑ o c = c) ( on on dϑ i dt = 0.1(ϑ i ϑ o ) dc dt = 0.0c (ϑ i 19 c 0.04) ϑ o = ϑ o ) ( on on dϑ i target = (c > 0.1) dt = ϑ i + 0.1ϑ o dc dt = c ϑ i 21 ϑ o = ϑ o ) Bounded Model Checking (BMC): Check satisfiabilit of SMT formula Φ k := init( 0 ) trans( 0, 1 ) trans( k 1, k ) target( k ) 4c/1

7 SAT Modulo Theories Formulae Bounded Model Checking (BMC): Check satisfiabilit of SMT formula Φ k := init( 0 ) trans( 0, 1 ) trans( k 1, k ) target( k ) Boolean combination of potentiall non-linear arithmetic atoms (including transcendental functions and ODEs) over the reals Goal: Determine whether or not there is a satisfing valuation for the variables of this formula Undecidable problem, i.e. method will necessaril be incomplete /1

8 Semantics (intuitive) A valuation satisfies an ODE constraint trajector (solution of the ODE) connecting valuations i and i+1 valuation of i+1 valuation of i 2 = 7 time d dt = 2 1 = 4 t 2 t 1 = 1. 6/1

9 Semantics (more formal) Given an ODE constraint c : d dt (t) = f( ), the valuation v of two successive BMC instances v( i ) and v( i+1 ) satisfies c iff there eists a function : t (t) such that (0) = v( i ) (correct starting point), there eists a point of time τ such that (τ) = v( i+1 ) (correct ending point), and for all t [0, τ] : d dt (t) = f( ) (correct slope in between). 7/1

10 The isat Algorithm in a Nutshell Generalization of DPLL solving manipulating interval bounds [3, 7], [ 2, 2] Deductions: prune off definite non-solutions - Unit propagation: ( > 8 = 2 ) - Interval constraint propagation: Decisions: Split interval (e.g. at its midpoint), propagate resulting bound Conflict-driven Learning: - Deduction can ield empt bo - Learn reasons from implication graph (conflict clause) - Jump back undoing decisions Termination: Stop search when - unresolvable conflict is found or - reasonabl small conflict-free bo found = = Use optimizations from propositional SAT (backjumps, two-watched literal scheme, isomorph inference, restarts,... ) 8a/1

11 The isat Algorithm in a Nutshell Generalization of DPLL solving manipulating interval bounds [3, 7], [ 2, 2] Deductions: prune off definite non-solutions - Unit propagation: ( > 8 = 2 ) - Interval constraint propagation: Decisions: Split interval (e.g. at its midpoint), propagate resulting bound Conflict-driven Learning: - Deduction can ield empt bo Add - Learn a similar reasons deduction from implication mechanism graph for(conflict ODEs that clause) - Jump - works backwith undoing intervals, decisions Goal: Handle ODEs directl Termination: - prunes off definite non-solutions, Stop search and when - unresolvable - safel keeps conflict all solutions. is found or - reasonabl Enclosuresmall mechanism conflict-free for ODEs bo found = = Use optimizations from propositional SAT (backjumps, two-watched literal scheme, isomorph inference, restarts,... ) 8b/1

12 ODE Enclosures as Propagators forward propagation backward propagation (2) postbo prebo (1) time of interest horizon /1

13 Safe Enclosures of ODEs Euler s method: Numerical approimation Euler s method: Numerical approimation t t a/1

14 Safe Enclosures of ODEs Euler s method: Numerical approimation Talor series Talor series Slope of unknown eact solution: d (t) = f((t)) dt Talor epansion of eact solution: (t0 + h) =(t0) + h1 d (t0) (EULER S METHOD) 1! dt + h2 d 2 (t0) ! dt2 + hn d n n! dt (t0) n (LAGRANGE REMAINDER) + hn+1 d n+1 (t0 + θh), with 0 < θ < 1 (n + 1)! dtn+1 t Slope of unknown eact solution: d (t) = f((t)) dt Talor epansion of eact solution: (t 0 + h) =(t 0 ) + h1 1! + h2 2! + hn n! + hn+1 (n + 1)! d dt (t 0) (EULER S METHOD) d 2 dt (t 0) d n dt (t 0) n (LAGRANGE REMAINDER) d n+1 dt (t n θh), with 0 < θ < 1 b/1

15 Safe Enclosures of ODEs Euler s method: Numerical approimation Error enclosures using bounding bo Talor series Slope of unknown eact solution: d (t) = f((t)) dt Talor epansion of eact solution: (t0 + h) =(t0) + h1 d (t0) (EULER S METHOD) 1! dt + h2 d 2 (t0) ! dt2 + hn d n n! dt (t0) n (LAGRANGE REMAINDER) + hn+1 d n+1 (t0 + θh), with 0 < θ < 1 (n + 1)! dtn+1 Error enclosures using bounding bo h n+1 d n+1 (t0 + θh) (n + 1)! dt, with 0 < θ < 1 n+1 }{{} - Value of (n + 1)-th derivative of unknown function at unknown position. - Need bounds for the (n + 1)-th derivative of an unknown function for a known interval. - Need bounds for the (n)-th derivative of f(1,...,n) for an unknown bo. Need bo which encloses eact solution. t h n+1 (n + 1)! d n+1 dt (t n θh), with 0 < θ < 1 }{{} - Value of (n + 1)-th derivative of unknown function at unknown position. - Need bounds for the (n + 1)-th derivative of an unknown function for a known interval. - Need bounds for the (n)-th derivative of f( 1,..., n ) for an unknown bo. Need bo which encloses eact solution. c/1

16 u2 u1 Safe Enclosures of ODEs Euler s method: Numerical approimation t Wrapping effect & coordinate transformation Talor series Slope of unknown eact solution: d (t) = f((t)) dt Talor epansion of eact solution: (t0 + h) =(t0) + h1 d (t0) (EULER S METHOD) 1! dt + h2 d 2 (t0) ! dt2 + hn d n n! dt (t0) n (LAGRANGE REMAINDER) + hn+1 d n+1 (t0 + θh), with 0 < θ < 1 (n + 1)! dtn+1 u 2 q [t, u] p Error enclosures using bounding bo h n+1 d n+1 (t0 + θh) (n + 1)! dt, with 0 < θ < 1 n+1 }{{} [c, d] - Value of (n + 1)-th derivative of unknown function at unknown position. - Need bounds for the (n + 1)-th derivative of an unknown function for a known interval. - Need bounds for the (n)-th derivative of f(1,...,n) for an unknown bo. Need bo which encloses eact solution. [r, s] Wrapping effect & coordinate transformation [c,d] q [t, u] p u 1 [a, b] [r, s] [a, b] d/1

17 u t u t 6 Safe Enclosures of ODEs Euler s method: Numerical approimation Talor series Slope of unknown eact solution: d (t) = f((t)) dt Talor epansion of eact solution: (t0 + h) =(t0) + h1 d (t0) (EULER S METHOD) 1! dt + h2 d 2 (t0) ! dt2 + hn d n n! dt (t0) n (LAGRANGE REMAINDER) + hn+1 d n+1 (t0 + θh), with 0 < θ < 1 (n + 1)! dtn+1 Error enclosures using bounding bo h n+1 d n+1 (t0 + θh) (n + 1)! dt, with 0 < θ < 1 n+1 }{{} - Value of (n + 1)-th derivative of unknown function at unknown position. - Need bounds for the (n + 1)-th derivative of an unknown function for a known interval. - Need bounds for the (n)-th derivative of f(1,...,n) for an unknown bo. Need bo which encloses eact solution. Wrapping effect & coordinate transformation q [t, u] t p Enclosure over intervals of time t t 6 [c,d] [r, s] [a, b] Enclosure over intervals of time 11 e/1

18 u t u t 6 Safe Enclosures of ODEs Euler s method: Numerical approimation Talor series Slope of unknown eact solution: d (t) = f((t)) dt Talor epansion of eact solution: (t0 + h) =(t0) + h1 d (t0) (EULER S METHOD) 1! dt + h2 d 2 (t0) ! dt2 + hn d n n! dt (t0) n (LAGRANGE REMAINDER) + hn+1 d n+1 (t0 + θh), with 0 < θ < 1 (n + 1)! dtn+1 Error enclosures using bounding bo h n+1 d n+1 (t0 + θh) (n + 1)! dt, with 0 < θ < 1 n+1 }{{} - Value of (n + 1)-th derivative of unknown function at unknown position. - Need bounds for the (n + 1)-th derivative of an unknown function for a known interval. - Need bounds for the (n)-th derivative of f(1,...,n) for an unknown bo. Need bo which encloses eact solution. Wrapping effect & coordinate transformation [c,d] [a, b] Enclosure over intervals of time 11 q [t, u] [r, s] t p Essentiall using method described b Moore, Lohner, Stauning: Determine rough first enclosure (bounding bo) Use Talor series and remainder term evaluation over bounding bo for tighter enclosure Use interval evaluation to enclose trajectories over intervals of time Use coordinate transformations to avoid wrapping effect f/1

19 Using the ODE Enclosures initial postbo t 14 11a/1

20 Using the ODE Enclosures tightened postbo and TOI initial postbo t 14 11b/1

21 Integrated Algorithm (Eample) ( > ) ( 28 a) ( a d dt = 3 20 a {0, 1}, 1 [, 20], 2 [, 7], [0, 30] (3 )) a/1

22 Integrated Algorithm (Eample) ( > ) ( 28 a) ( a d dt = 3 20 a {0, 1}, 1 [, 20], 2 [, 7], [0, 30] (3 )) b/1

23 Integrated Algorithm (Eample) ( > ) ( 28 a) ( a d dt = 3 20 a {0, 1}, 1 [, 20], 2 [, 7], [0, 27] (3 )) < 30 < > 1 [, 20] 2 [, 7] < c/1

24 Integrated Algorithm (Eample) ( > ) ( 28 a) ( a d dt = 3 20 a {0, 1}, 1 [, 20], 2 [, 7], [0, 27] (3 )) d/1

25 Integrated Algorithm (Eample) ( > ) ( 28 a ) ( a d dt = 3 20 a { 1}, 1 [, 20], 2 [, 7], [0, 27] (3 )) e/1

26 Integrated Algorithm (Eample) ( > ) ( 28 a) ( a d dt = 3 20 a { 1}, 1 [, 20], 2 [, 7], [0, 27] (3 )) f/1

27 Integrated Algorithm (Eample) ( > ) ( 28 a) ( a d dt = 3 20 a { 1}, 1 [, 20], 2 [3, 7], [0, 27] (3 ) ) g/1

28 Integrated Algorithm ODE deductions can ield new bounds for variables... which can be propagated b the other deduction mechanisms... which can cause conflicts... reasons for conflicts can be added as conflict clauses ODE deductions are wa more epensive Heuristics: Run ODE deductions after other deductions 13/1

29 First Benchmark Results Fehnker s room heating benchmark : Two heaters, three rooms, fied initial temperature, reach a temperature below threshold runtime in seconds number of conflicts 97% of this time spent with enclosing ODEs! BMC unwinding depth Throwing too man results awa on backtracking. 14/1

30 Conclusions & Future Work Directl use ODE enclosures as propagation method within DPLL-based arithmetic constraint solving Implemented first prototpe using Lohner s method as enclosure method (coordinate transformation in general onl effective for linear ODEs) Other enclosure methods possible: e.g. Talor models [Makino, Berz] (fight wrapping effect for non-linear cases) Benchmark results: Need to improve persistent learning of results Using cheaper knowledge (monotonicit, stabilit) as redundant encodings ma reduce number of necessar enclosures 1/1