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 Breisgau, Germany Presentation at DDECS 2007 April 13 2007 www.avacs.org

Relation to DDECS

Relation to DDECS Systems Diagnostics

Relation to DDECS Linear Hybrid Automata Systems Diagnostics

Relation to DDECS Linear Hybrid Automata Systems Diagnostics Bounded Model Checking

Context ⇒ 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

Overview Bounded Model Checking for Linear Hybrid Automata 1 Variable counters for SAT-LP / Solvability Estimation 2 Train Example 3 Evaluation 4 Final LP-time Heuristics 5 Conclusions 6

� Hybrid automaton (Thermostat controller) �� �� � �� �� x = x min off on x = x max � ˙ ˙ x ≤ 0 x ≥ 0 �� �� �� �� x ≤ x max x ≥ x min x = x max � Marc Herbstritt (University Freiburg) c Variable Selection in SAT-LP-based BMC of LHA DDECS’07 5 / 16

BMC of Linear Hybrid Automata Counterexamples of length k described by first-order logic formulas over ( R , + , <, 0 , 1 ) : ϕ k ( s 0 , . . . , s k ) : Init ( s 0 ) ∧ Trans ( s 0 , s 1 ) ∧ . . . ∧ Trans ( s k − 1 , s k ) ∧ Bad ( s k ) exists run of length k ϕ k is satisfiable ⇐ ⇒ leading to an unsafe state ⇒ Check ϕ k incrementally for k = 0 , 1 , . . . using a suitable solver [BMC for discrete systems: Biere et al. (TACAS 1999)] � Marc Herbstritt (University Freiburg) c Variable Selection in SAT-LP-based BMC of LHA DDECS’07 6 / 16

SAT-LP-Solver: HySAT (eager SMT approach) ψ UNSAT Boolean unsat abstraction complete LP−consistent SAT−solver SAT solution consistent (In)equation set Explanation inconsistent LP−solver [Fr¨ anzle/Herde, FMICS 2004] � Marc Herbstritt (University Freiburg) c Variable Selection in SAT-LP-based BMC of LHA DDECS’07 7 / 16

SAT-LP-Solver: HySAT (eager SMT approach) 1: procedure H Y S AT ( Ψ ) ( ϕ, π ) ← A BSTRACTION (Ψ) 2: while ( true ) do 3: if ( D ECIDE N EXT B RANCH ( α ) ) then 4: while ( D EDUCE () == CONFLICT ) do 5: ( blevel , learnedClause ) ← A NALYZE C ONFLICT () 6: if ( blevel ≤ 0 ) then return UNSAT 7: 8: // partial SAT solution 9: �� � // activated constraints ψ ← v ∈ V A ∧ α ( v )= 1 π ( v ) 10: if ( LPS OLVE ( ψ ) == INCONSISTENT ) then 11: // µ is MIS ( blevel , µ ) ← A NALYZE C ONFLICT ( ψ ) 12: if ( blevel ≤ 0) then return UNSAT 13: 14: else // all variables are assigned 15: return SAT 16: � Marc Herbstritt (University Freiburg) c Variable Selection in SAT-LP-based BMC of LHA DDECS’07 7 / 16

SAT-LP-Solver: HySAT (eager SMT approach) D ECIDE N EXT B RANCH ( α ) GP (General Purpose): VSIDS-like, i.e., counter for activity in conflicts (increased for literals in conflict clauses) F WD (Forward): Forward computations [ init → bad ] preferenced B WD (Backward): Backward computations [ bad → init ] preferenced HySAT Performance H Y S AT Benchmark G P F WD B WD CPU 77.78 117.85 28.30 train5 #LP 15593 29962 3824 CPU 498.22 75.57 > 1000 car #LP 168023 18592 — � Marc Herbstritt (University Freiburg) c Variable Selection in SAT-LP-based BMC of LHA DDECS’07 7 / 16

HySAT extended by counters 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 � Marc Herbstritt (University Freiburg) c Variable Selection in SAT-LP-based BMC of LHA DDECS’07 8 / 16

HySAT extended by counters 1: procedure H Y S AT ( Ψ ) 2: ( ϕ, π ) ← A BSTRACTION (Ψ) 3: while ( true ) do 4: if ( D ECIDE N EXT B RANCH ( α ) ) then while ( D EDUCE () == CONFLICT ) do 5: 6: ( blevel , learnedClause ) ← A NALYZE C ONFLICT () 7: for all ( l ∈ learnedClause ) do BC ( l )+ = c BC if ( blevel ≤ 0 ) then return UNSAT 8: 9: // partial SAT solution 10: 11: for all ( l ∈ Λ( α ) ) do BS ( l )+ = c BS “V ” // activated constraints 12: ψ ← v ∈ V A ∧ α ( v )= 1 π ( v ) if ( LPS OLVE ( ψ ) == INCONSISTENT ) then 13: // µ is MIS 14: ( blevel , µ ) ← A NALYZE C ONFLICT ( ψ ) 15: for all ( l ∈ µ ) do LPC ( l )+ = c LPC if ( blevel ≤ 0) then return UNSAT 16: 17: else // all variables are assigned 18: return SAT 19: � Marc Herbstritt (University Freiburg) c Variable Selection in SAT-LP-based BMC of LHA DDECS’07 8 / 16

Solvability Estimation 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.: � � ( BC ( l ) + LPC ( l )) BS ( l ) MU ← MS ← l ∈ L l ∈ L Belief switching When MU ≥ MS then Belief-UNSAT , else Belief-SAT . Belief-dependent Variable Selection � arg max ( BS ( l ) − BC ( l ) − LPC ( l )) Belief − SAT : l ′ ← Belief − UNSAT arg min ( BS ( l ) − BC ( l ) − LPC ( l )) : � Marc Herbstritt (University Freiburg) c Variable Selection in SAT-LP-based BMC of LHA DDECS’07 9 / 16

Solvability Estimation 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.: � � ( BC ( l ) + LPC ( l )) BS ( l ) MU ← MS ← l ∈ L l ∈ L Belief switching When MU ≥ MS then Belief-UNSAT , else Belief-SAT . Belief-dependent Variable Selection � arg max ( BS ( l ) − BC ( l ) − LPC ( l )) Belief − SAT : l ′ ← Belief − UNSAT arg min ( BS ( l ) − BC ( l ) − LPC ( l )) : � Marc Herbstritt (University Freiburg) c Variable Selection in SAT-LP-based BMC of LHA DDECS’07 9 / 16

Solvability Estimation 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.: � � ( BC ( l ) + LPC ( l )) BS ( l ) MU ← MS ← l ∈ L l ∈ L Belief switching When MU ≥ MS then Belief-UNSAT , else Belief-SAT . Belief-dependent Variable Selection � arg max ( BS ( l ) − BC ( l ) − LPC ( l )) Belief − SAT : l ′ ← Belief − UNSAT arg min ( BS ( l ) − BC ( l ) − LPC ( l )) : � Marc Herbstritt (University Freiburg) c Variable Selection in SAT-LP-based BMC of LHA DDECS’07 9 / 16

Train Distance Control � Fr¨ c 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 � Marc Herbstritt (University Freiburg) c Variable Selection in SAT-LP-based BMC of LHA DDECS’07 10 / 16

Train Distance Control � Fr¨ c anzle/Herde, FMICS 2004 Control Modes Mode 1 (Free Run): No neighbouring train ⇒ de-/increase 1 velocity Mode 2 (Forward Proximity): Distance below 500m ⇒ decrease 2 velocity proportional to intrusion depth Mode 3 (Backward Proximity): Train approach from behind ⇒ 3 increase velocity Mode 4 (Two-sided Proximity): Acceleration according to sum of 4 control forces � Marc Herbstritt (University Freiburg) c Variable Selection in SAT-LP-based BMC of LHA DDECS’07 10 / 16

Train Distance Control F � Fr¨ c anzle/Herde, FMICS 2004 Control Modes Mode 1 (Free Run): No neighbouring train ⇒ de-/increase 1 velocity Mode 2 (Forward Proximity): Distance below 500m ⇒ decrease 2 velocity proportional to intrusion depth Mode 3 (Backward Proximity): Train approach from behind ⇒ 3 increase velocity Mode 4 (Two-sided Proximity): Acceleration according to sum of 4 control forces � Marc Herbstritt (University Freiburg) c Variable Selection in SAT-LP-based BMC of LHA DDECS’07 10 / 16

Train Distance Control F � Fr¨ c anzle/Herde, FMICS 2004 Control Modes Mode 1 (Free Run): No neighbouring train ⇒ de-/increase 1 velocity Mode 2 (Forward Proximity): Distance below 500m ⇒ decrease 2 velocity proportional to intrusion depth Mode 3 (Backward Proximity): Train approach from behind ⇒ 3 increase velocity Mode 4 (Two-sided Proximity): Acceleration according to sum of 4 control forces � Marc Herbstritt (University Freiburg) c Variable Selection in SAT-LP-based BMC of LHA DDECS’07 10 / 16

Train Distance Control F F � Fr¨ c anzle/Herde, FMICS 2004 Control Modes Mode 1 (Free Run): No neighbouring train ⇒ de-/increase 1 velocity Mode 2 (Forward Proximity): Distance below 500m ⇒ decrease 2 velocity proportional to intrusion depth Mode 3 (Backward Proximity): Train approach from behind ⇒ 3 increase velocity Mode 4 (Two-sided Proximity): Acceleration according to sum of 4 control forces � Marc Herbstritt (University Freiburg) c Variable Selection in SAT-LP-based BMC of LHA DDECS’07 10 / 16