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Probabilistic CEGAR* Björn Wachter Joint work with Holger Hermanns, Lijun Zhang TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A A AAA A AVACS Supported by Uni Saar *To appear in CAV
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2 Introducing Probabilistic Model Checking CEGAR (counterexample-guided abstraction refinement) PASS does CEGAR for probabilistic models 1
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3 PRISM & PASS PRISM Very popular probabilistic model checker Finite-state PASS Supports PRISM models handles infinite-state as well Under the Hood: Predicate abstraction SMT Interpolation
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4 Comparison to PRISM Network protocols Wireless LAN, CSMA Bounded Retransmission Sliding Window Model (#)State reduction Speed-up WLAN (3) WLAN (1) 16x-152x ? 1,3x-7x TO->311s CSMA (4)41x-248x1x-2x BRP (3)1x1/2x - 1/3x PRISM vs PASS
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5 Basics Paths, Markov Chains, MDPs Counterexamples Probabilistic Programs Predicate Abstraction Abstraction Refinement Abstract Counterexamples Path Analysis Strongest Evidence CEGAR algorithm Experimental Results Conclusion Program e Probabilistic Reachability Problem Overview
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6 Paths, MCs, MDPs Weighted Path Markov Chain non-determinism … 2/3 1/3 2/31/3
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7 Paths, MCs, MDPs 2/3 1/3 2/3 1/3 1 1/2 1/3 2/31/3 Weighted Path Markov Chain Markov Decision Process
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8 Adversary Adversary resolves transition non-determinism 2/3 1/3 1 1/2
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9 Probabilistic Reachability Probability to get from green to red Weighted Path Markov Chain Markov Decision Process 2/3 1/3 2/3 1/3 1 1/2 1/3 2/31/3
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10 Guarded command language à la PRISM Variables: integer, real, bool Non-determinism: interleaving Example: Program = (variables, commands, initial condition) Probabilistic Programs x=1 0.2: (x‘:=x+1) x=2 Update #1 0.8: (x‘:=x+2) x=3 Update #2 Guard: x>0 guard Labels for CEX Analysis
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11 Predicates: partition the state space are boolean expressions x>0, x<y, x + y = 3 (variables x,y) Abstract MDP Probabilistic may-transitions Similar to Blast, SLAM, Magic … See our [Qest’07] paper Abstraction guarantees upper bound Predicate Abstraction actual 1 0 Probability: Abstract MDP
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12 May Transitions Hier ist‘s noch nicht verständlich genug! Besseres Beispiel wo #abs. trans < #conc. trans 0.2 0.8 1.0 0.2 0.8 1.0 abstract concrete
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13 CEGAR Loop p actual upper abstract check refine Probability CEX ? Real CEX Low enough
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14 Counterexamples (CEX) Resolution of non-determinism initial state adversary induces a Markov chain Counterexample: Resolution of non-det such that probability threshold exceeded Example: CEX for Witness of Reachability probability in MDP 2/3 1/3 1 1/2
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15 Path 1Path 2Path 3Path 4… Counterexample Analysis: Idea Idea: Enumerate paths of Markov chain Sort paths by probability [Han\Katoen2007]: visit paths with highest measure first Realizable Spurious Path 1Path 2Path 3Path 4… Probability of Abstract CEX / Markov Chain How much MEASURE is REALIZABLE? More than p?
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16 Path Analysis Abstract path: Two cases Realizable if there‘s a corresponding concrete path Spurious: no corresponding path Splitter predicate exists iff path spurious Interpolation: predicate from unsatisfiable path formula uu´ u´´ uu´ u´´ uu´ u´´ Reachable with prefix Can do postfix Path formula SAT UNSAT Logic (SMT)
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17 Path Analysis Abstract path: Two cases Realizable if there‘s a corresponding concrete path Spurious: no corresponding path Splitter predicate (interpolant): uu´ u´´ uu´ u´´ 0 1 x´:=x+1 2 10 9 x´:=x+1 Reachable with prefix Can do postfix Path formula SAT UNSAT Logic (SMT) x=0 x=1 X 10 x>1
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18 Example 1.0 concrete abstract 0.2 0.8 0.5 0 Probability: Upper: 1.0 0.80.2 ?
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19 Example(cont): after refinement 0.4 Concrete abstract 0.4 0 Probability: Upper: 0.4 0.8 0.5 lower
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20 Example 2 1.0 0.8 1.0 0.8 0.2 0.8 0.2 concrete abstract 0.8 0.2 0 lower 0.8 Upper 1.0 Multiple Initial states
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21 Example 2 1.0 0.8 1.0 0.8 0.2 0.8 0.2 concrete abstract 0.8 0.2 Maximum Find Maximal Combination by MAX-SMT ( paper) 0.8 0 Probability: lower 0.8 Upper 1.0
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22 CEX Analysis: Semi decision procedure Problem in general: undecidable Too many spurious paths abort counterexample analysis Output: collection of predicates Enough realizable probability Path 1Path 2Path 3Path 4…Path 1Path 2Path 3Path 4… > C Limit # of spurious paths to enforce termination Path 1Path 2Path 3Path 4…Path 1Path 2Path 3Path 4… Can take many paths To obtain enough realizable probability 0 lower = real
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23 Related Work Probabilistic Counterexamples: … however not in the context of abstraction Hermanns/Aljazzar (FORMATS’05), Han/Katoen (TACAS’07) Abstraction Refinement for Prob. Finite-state Models CEGAR for stochastic games, Chatterjee et al (UAI’05) Not based on counterexamples D‘Argenio (Papm-Probmiv02), Fecher & al (SPIN’06): simulation Magnifying-lens, de Alfaro et al (CAV’07): probability values
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24 Conclusion & Future Work Abstraction refinement … Counterexamples ~ Markov Chains Markov Chains have cycles Model Checking Infinite-state Probabilistic Models Speed-up for huge finite-state models Future Work Better Lower bounds
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25 References Tool website http://depend.cs.uni-sb.de/pass Literature Our work Hermanns, Wachter, Zhang: Probabilistic CEGAR (CAV’08) Wachter, Zhang, Hermanns: MC Modulo Theories (Qest’07) Counterexamples Hermanns, Aljazar: CEX for timed prob reachability, FORMATS‘05 Han, Katoen: CEX in probabilistic model checking, TACAS‘07 Probabilistic Abstraction Refinement De Alfaro, Magnifying-lens abstraction for MDPs, CAV‘07 Chatterjee, Henzinger, Majumdar: CEX-guided planning, UAI’05
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26 Questions?
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27 Is Counterexample analysis problem undecidable? Semi-decision algorithm heuristics If we only need finiteley many paths decidable if logic is If we need infinitely many undecidable
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