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MIT16_410F10_lec13

# MIT16_410F10_lec13 - Optimal Satisfiability and...

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10/27/10 1 Optimal Satisfiability and Conflict-directed A* Brian C. Williams 16.410 / 16.413 October 27 th , 2010 Brian C. Williams, copyright 2000 Assignment • Remember: Problem Set #6 Propositional Logic, due Today. 16:413 Project Part 1: Sat-based Activity Planner, due Wednesday, November 3 rd . Problem Set #7 Diagnosis, Conflict-directed A* and RRTs, due Wednesday, November 10 th . • Reading Today: Brian C. Williams, and Robert Ragno, "Conflict-directed A* and its Role in Model-based Embedded Systems," Special Issue on Theory and Applications of Satisfiability Testing, Journal of Discrete Applied Math , January 2003. 11/02/09 copyright Brian Williams, 2000-10 2 Image credit: NASA.

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10/27/10 2 When you have eliminated the impossible, whatever remains, however improbable, must be the truth. - Sherlock Holmes. The Sign of the Four . Model-based Diagnosis as Conflict-directed Best First Search 1. Generate most likely Hypothesis. 2. Test Hypothesis. 3. If Inconsistent, learn reason for inconsistency (a Conflict ). 4. Use conflicts to leap over similarly infeasible options to next best hypothesis. Compare Most Likely Hypothesis to Observations Helium tank Fuel tank Oxidizer tank Main Engines Flow 1 = zero Pressure 1 = nominal Pressure 2 = nominal Acceleration = zero It is most likely that all components are okay.
10/27/10 3 Isolate Conflicting Information Helium tank Fuel tank Oxidizer tank Main Engines Flow 1 = zero The red component modes conflict with the model and observations. Helium tank Fuel tank Oxidizer tank Main Engines Flow 1 = zero Leap to the Next Most Likely Hypothesis that Resolves the Conflict The next hypothesis must remove the conflict.

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10/27/10 4 New Hypothesis Exposes Additional Conflicts Pressure 1 = nominal Pressure 2 = nominal Acceleration = zero Helium tank Fuel tank Oxidizer tank Main Engines Another conflict, try removing both. Final Hypothesis Resolves all Conflicts Helium tank Fuel tank Oxidizer tank Main Engines Pressure 1 = nominal Flow 1 = zero Pressure 2 = nominal Flow 2 = positive Acceleration = zero Implementation: Conflict-directed A* search.
10/27/10 5 Outline Model-based Diagnosis Optimal CSPs Informed Search Conflict-directed A* 10/27/10 10 Constraint Satisfaction Problem CSP = <X, D X ,C> – variables X with domain D X . – Constraint C(X): D X {True, False}. Problem: Find X in D X s.t. C(X) is True. R , G, B G R , G Different-color constraint V 1 V 2 V 3

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10/27/10 6 10/27/10 11 Optimal CSP OCSP= <Y, g , CSP> – Decision variables Y with domain D Y . – Utility function g(Y): D Y . – CSP over variables <X;Y>. Find leading arg max g (Y) Y D y s.t. X D X s.t. C(X,Y) is True. g: multi-attribute utility with preferential independence, value constraint, … CSP: propositional state logic, simple temporal problem, mixed logic-linear program, … 10/27/10 12 CSPs Are Frequently Encoded in Propositional State Logic (mode(E1) = ok implies (thrust(E1) = on if and only if flow(V1) = on and flow(V2) = on)) and (mode(E1) = ok or mode(E1) = unknown) and not (mode(E1) = ok and mode(E1) = unknown) E1 V1 V2
10/27/10 7 10/27/10 13 Multi Attribute Utility Functions g(Y) = G(g 1 (y 1 ), g 2 (y 2 ), . . .) where G(u 1 , u

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• Fall '10
• Prof.BrianWilliams
• A* search algorithm, Best-first search, Admissible heuristic, Consistent heuristic, Brian C. Williams, conflict-directed a*