On Variable Selection in SAT-LP-based Bounded Model Checking of - - PowerPoint PPT Presentation
On Variable Selection in SAT-LP-based Bounded Model Checking of Linear Hybrid Automata Marc Herbstritt (joint work with Bernd Becker, Erika Abrah am, Christian Herde) Institute of Computer Science Albert-Ludwigs-University Freiburg im
(joint work with Bernd Becker, Erika ´ Abrah´ am, Christian Herde)
Institute of Computer Science Albert-Ludwigs-University Freiburg im Breisgau, Germany
Presentation at DDECS 2007 April 13 2007
www.avacs.org
⇒ Automated analysis of complex systems is mandatory ⇒ Especially safety-critical ones ⇒ Real-world scenarios embed discrete control in continuous environments ⇒ Modeling relies on hybrid automata ⇒ Bounded Model Checking for correctness analysis
1
Bounded Model Checking for Linear Hybrid Automata
2
Variable counters for SAT-LP / Solvability Estimation
3
Train Example
4
Evaluation
5
Final LP-time Heuristics
6
Conclusions
˙ x ≤ 0 x ≥ xmin
x=xmin
˙ x ≥ 0 x ≤ xmax
x=xmax
c Marc Herbstritt (University Freiburg) Variable Selection in SAT-LP-based BMC of LHA DDECS’07 5 / 16
Counterexamples of length k described by first-order logic formulas over (R, +, <, 0, 1): ϕk(s0, . . . , sk) : Init(s0) ∧ Trans(s0, s1) ∧ . . . ∧ Trans(sk−1, sk) ∧ Bad(sk) ϕk is satisfiable ⇐ ⇒ exists run of length k leading to an unsafe state ⇒ Check ϕk incrementally for k = 0, 1, . . . using a suitable solver
[BMC for discrete systems: Biere et al. (TACAS 1999)]
c Marc Herbstritt (University Freiburg) Variable Selection in SAT-LP-based BMC of LHA DDECS’07 6 / 16
UNSAT SAT SAT−solver LP−solver ψ
Boolean abstraction
(In)equation set
unsat inconsistent consistent
Explanation
complete LP−consistent solution
[Fr¨ anzle/Herde, FMICS 2004]
c Marc Herbstritt (University Freiburg) Variable Selection in SAT-LP-based BMC of LHA DDECS’07 7 / 16
1: procedure HYSAT(Ψ) 2:
(ϕ, π) ←ABSTRACTION(Ψ)
3:
while ( true ) do
4:
if ( DECIDENEXTBRANCH(α) ) then
5:
while ( DEDUCE() == CONFLICT ) do
6:
(blevel, learnedClause) ←ANALYZECONFLICT()
7:
if ( blevel ≤ 0 ) then return UNSAT
8: 9:
// partial SAT solution
10:
ψ ←
11:
if ( LPSOLVE(ψ) == INCONSISTENT ) then
12:
(blevel, µ) ←ANALYZECONFLICT(ψ) //µ is MIS
13:
if (blevel ≤ 0) then return UNSAT
14:
else
15:
// all variables are assigned
16:
return SAT
c Marc Herbstritt (University Freiburg) Variable Selection in SAT-LP-based BMC of LHA DDECS’07 7 / 16
DECIDENEXTBRANCH(α)
GP (General Purpose): VSIDS-like, i.e., counter for activity in conflicts (increased for literals in conflict clauses) FWD (Forward): Forward computations [init → bad] preferenced BWD (Backward): Backward computations [bad → init] preferenced
HySAT Performance
HYSAT Benchmark GP FWD BWD CPU 77.78 117.85 28.30 train5 #LP 15593 29962 3824 CPU 498.22 75.57 >1000 car #LP 168023 18592 —
c Marc Herbstritt (University Freiburg) Variable Selection in SAT-LP-based BMC of LHA DDECS’07 7 / 16
What is the focus within SAT-LP?
conflict-free partial assignments: ⇒ counter BS(l) for literals in partial assignments (typically not considered in pure SAT) boolean conflicts in abstraction ϕ: ⇒ counter BC(l) for boolean conflicts (as in VSIDS) conflicts in real-valued domain: ⇒ counter LPC(l) for literals in minimal infeasible subsets
c Marc Herbstritt (University Freiburg) Variable Selection in SAT-LP-based BMC of LHA DDECS’07 8 / 16
1: procedure HYSAT(Ψ) 2: (ϕ, π) ←ABSTRACTION(Ψ) 3: while ( true ) do 4: if ( DECIDENEXTBRANCH(α) ) then 5: while ( DEDUCE() == CONFLICT ) do 6: (blevel, learnedClause) ←ANALYZECONFLICT() 7: for all (l ∈ learnedClause) do BC(l)+ = cBC 8: if ( blevel ≤ 0 ) then return UNSAT 9: 10: // partial SAT solution 11: for all (l ∈ Λ(α)) do BS(l)+ = cBS 12: ψ ← “V
v∈VA∧α(v)=1 π(v)
” // activated constraints 13: if ( LPSOLVE(ψ) == INCONSISTENT ) then 14: (blevel, µ) ←ANALYZECONFLICT(ψ) //µ is MIS 15: for all (l ∈ µ) do LPC(l)+ = cLPC 16: if (blevel ≤ 0) then return UNSAT 17: else 18: // all variables are assigned 19: return SAT
c Marc Herbstritt (University Freiburg) Variable Selection in SAT-LP-based BMC of LHA DDECS’07 8 / 16
Solvability estimation as heuristics to prevent local minima (Herbstritt/Becker, SAT’03).
Mass of (Un)Satisfiability
Compute mass of satisfiability (MS) and unsatisfiability (MU), resp.: MU ←
(BC(l) + LPC(l)) MS ←
BS(l)
Belief switching
When MU ≥ MS then Belief-UNSAT, else Belief-SAT.
Belief-dependent Variable Selection
l′ ← arg max(BS(l) − BC(l) − LPC(l)) : Belief − SAT arg min(BS(l) − BC(l) − LPC(l)) : Belief − UNSAT
c Marc Herbstritt (University Freiburg) Variable Selection in SAT-LP-based BMC of LHA DDECS’07 9 / 16
Solvability estimation as heuristics to prevent local minima (Herbstritt/Becker, SAT’03).
Mass of (Un)Satisfiability
Compute mass of satisfiability (MS) and unsatisfiability (MU), resp.: MU ←
(BC(l) + LPC(l)) MS ←
BS(l)
Belief switching
When MU ≥ MS then Belief-UNSAT, else Belief-SAT.
Belief-dependent Variable Selection
l′ ← arg max(BS(l) − BC(l) − LPC(l)) : Belief − SAT arg min(BS(l) − BC(l) − LPC(l)) : Belief − UNSAT
c Marc Herbstritt (University Freiburg) Variable Selection in SAT-LP-based BMC of LHA DDECS’07 9 / 16
Solvability estimation as heuristics to prevent local minima (Herbstritt/Becker, SAT’03).
Mass of (Un)Satisfiability
Compute mass of satisfiability (MS) and unsatisfiability (MU), resp.: MU ←
(BC(l) + LPC(l)) MS ←
BS(l)
Belief switching
When MU ≥ MS then Belief-UNSAT, else Belief-SAT.
Belief-dependent Variable Selection
l′ ← arg max(BS(l) − BC(l) − LPC(l)) : Belief − SAT arg min(BS(l) − BC(l) − LPC(l)) : Belief − UNSAT
c Marc Herbstritt (University Freiburg) Variable Selection in SAT-LP-based BMC of LHA DDECS’07 9 / 16
c Fr¨ anzle/Herde, FMICS 2004
System overview
n trains running on the same track trains cannot overtake each train has a collision avoidance controller controller has four control modes
c Marc Herbstritt (University Freiburg) Variable Selection in SAT-LP-based BMC of LHA DDECS’07 10 / 16
c Fr¨ anzle/Herde, FMICS 2004
Control Modes
1
Mode 1 (Free Run): No neighbouring train ⇒ de-/increase velocity
2
Mode 2 (Forward Proximity): Distance below 500m ⇒ decrease velocity proportional to intrusion depth
3
Mode 3 (Backward Proximity): Train approach from behind ⇒ increase velocity
4
Mode 4 (Two-sided Proximity): Acceleration according to sum of control forces
c Marc Herbstritt (University Freiburg) Variable Selection in SAT-LP-based BMC of LHA DDECS’07 10 / 16
F
c Fr¨ anzle/Herde, FMICS 2004
Control Modes
1
Mode 1 (Free Run): No neighbouring train ⇒ de-/increase velocity
2
Mode 2 (Forward Proximity): Distance below 500m ⇒ decrease velocity proportional to intrusion depth
3
Mode 3 (Backward Proximity): Train approach from behind ⇒ increase velocity
4
Mode 4 (Two-sided Proximity): Acceleration according to sum of control forces
c Marc Herbstritt (University Freiburg) Variable Selection in SAT-LP-based BMC of LHA DDECS’07 10 / 16
F
c Fr¨ anzle/Herde, FMICS 2004
Control Modes
1
Mode 1 (Free Run): No neighbouring train ⇒ de-/increase velocity
2
Mode 2 (Forward Proximity): Distance below 500m ⇒ decrease velocity proportional to intrusion depth
3
Mode 3 (Backward Proximity): Train approach from behind ⇒ increase velocity
4
Mode 4 (Two-sided Proximity): Acceleration according to sum of control forces
c Marc Herbstritt (University Freiburg) Variable Selection in SAT-LP-based BMC of LHA DDECS’07 10 / 16
F F
c Fr¨ anzle/Herde, FMICS 2004
Control Modes
1
Mode 1 (Free Run): No neighbouring train ⇒ de-/increase velocity
2
Mode 2 (Forward Proximity): Distance below 500m ⇒ decrease velocity proportional to intrusion depth
3
Mode 3 (Backward Proximity): Train approach from behind ⇒ increase velocity
4
Mode 4 (Two-sided Proximity): Acceleration according to sum of control forces
c Marc Herbstritt (University Freiburg) Variable Selection in SAT-LP-based BMC of LHA DDECS’07 10 / 16
Evaluation for different counter increments cBS, cBC, cLPC for the train example:
CPU times LPC BS 11500 11700 11900 12100 12300 12500 12700 12900 13100 13300 13500 BC LP Calls
c Marc Herbstritt (University Freiburg) Variable Selection in SAT-LP-based BMC of LHA DDECS’07 11 / 16
Evaluation for different counter increments cBS, cBC, cLPC for the train example:
CPU times LPC BS 11500 11700 11900 12100 12300 12500 12700 12900 13100 13300 13500 BC LP Calls
BC
c Marc Herbstritt (University Freiburg) Variable Selection in SAT-LP-based BMC of LHA DDECS’07 11 / 16
Evaluation for different counter increments cBS, cBC, cLPC for the train example:
20000 40000 60000 80000 100000 100 200 300 400 500
number of LP-calls CPU time
min-CPU(cat4) max-CPU(cat4) min-LP(cat4) max-LP(cat4)
c Marc Herbstritt (University Freiburg) Variable Selection in SAT-LP-based BMC of LHA DDECS’07 11 / 16
Lessons learned
guard variables of real-valued constraints have to be noticeably weighted ambiguity between time used by the LP-solver and number of LP-calls:
1
solving of a large number of LPs with small time resources
2
solving of a small number of LPs with huge time resources
c Marc Herbstritt (University Freiburg) Variable Selection in SAT-LP-based BMC of LHA DDECS’07 11 / 16
LPEasy and LPHard
Differentation between easily solvable LPs and hard-to-solve LPs Use different counter increments for LPC(l):ab
LPC = 2 · cLPC
LPC = cLPC
LPC = cLPC c′ LPC = 2 · cLPC
aγLP := time used for the last LP-call bφLP := average time used by all previous LP-calls c Marc Herbstritt (University Freiburg) Variable Selection in SAT-LP-based BMC of LHA DDECS’07 12 / 16
Performance for train example.
100 200 300 400 500 600 700 800 900 1000 5 10 15 20 25 30
Time [s]
Unfolding depth k
Train Example
GP FWD BWD
c Marc Herbstritt (University Freiburg) Variable Selection in SAT-LP-based BMC of LHA DDECS’07 12 / 16
Performance for train example.
100 200 300 400 500 600 700 800 900 1000 5 10 15 20 25 30
Time [s]
Unfolding depth k
Train Example
GP FWD BWD LPEasy(250) LPHard(250)
c Marc Herbstritt (University Freiburg) Variable Selection in SAT-LP-based BMC of LHA DDECS’07 12 / 16
Performance for car example.
200 400 600 800 1000 5 10 15 20 25 30 35
Time [s]
Unfolding depth k
Car Example
GP FWD BWD LPEasy(300) LPHard(250)
c Marc Herbstritt (University Freiburg) Variable Selection in SAT-LP-based BMC of LHA DDECS’07 12 / 16
Conclusions
Introduction of variable counters that focus on semantics of SAT-LP Evaluation for several counter increments using solvability estimation Derivation of heuristics that take LP-time into account LP-time heuristics perform very well
Future Work
Investigation on difference between LPEasy and LPHard ...
c Marc Herbstritt (University Freiburg) Variable Selection in SAT-LP-based BMC of LHA DDECS’07 13 / 16
c Marc Herbstritt (University Freiburg) Variable Selection in SAT-LP-based BMC of LHA DDECS’07 14 / 16
Alur et al., “The Algorithmic Analysis of Hybrid Systems”, TCS’95. Herbstritt, Becker, “Conflict-Based Selection of Branching Rules”, SAT’03. Fr¨ anzle, Herde, “Efficient Proof Engines for Bounded Model Checking
´ Abrah´ am et al., “Optimizing Bounded Model Checking for Linear Hybrid Systems”, VMCAI’05.
c Marc Herbstritt (University Freiburg) Variable Selection in SAT-LP-based BMC of LHA DDECS’07 15 / 16