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l16bwpropsatis - Propositional Satisfiability Brian C...

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10/6/2005 1 Propositional Satisfiability th , 2005 copyright Brian Williams Brian C. Williams 16.410-13 November 9 10/6/2005 2 copyright Brian Williams Reading Assignment: Propositional Satisfiability AIMA Ch. 6 – Propositional Logic 10/6/2005 3 Main 1 Pressure 1 Pressure 2 ingle l i i copyright Brian Williams How Do We Reason About Complex Systems at a Commonsense Level? Helium tank Fuel tank Oxidizer tank Engines Flow = zero = nominal = nominal Acceleration = zero Model using propositional logic. Reason from s model to operate, diagnose and repair. Is the diagnosis --- “the red va ve is stuck” closed --- consistent w th the observat ons? a 1
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10/6/2005 copyright Brian Williams 4 Reduction to Clausal Form: Engine Example (mode(E1) = ok implies (thrust(E1) = on iff (flow(V1) = on and flow(V2) = on))) and (mode(E1) = ok or mode(E1) = unknown) and not (mode(E1) = ok and mode(E1) = unknown) not (mode(E1) = ok) or not (thrust(E1) = on) or flow(V1) = on; not (mode(E1) = ok) or not (thrust(E1) = on) or flow(V2) = on; not (mode(E1) = ok) or not (flow(V1) = on) or not (flow(V2) = on) or thrust(E1) = on; mode(E1) = ok or mode(E1) = unknown; not (mode(E1) = ok) or not (mode(E1) = unknown); E1 V1 V2 10/6/2005 copyright Brian Williams 5 Propositional Satisfiability Find a truth assignment that satisfies logical sentence T: Reduce sentence T to clausal form. Perform search similar to MAC = (BT+CP) [Davis, Logmann & Loveland, 1962] Propositional satisfiability testing : 1990: 100 variables / 200 clauses (constraints) 1998: 10,000 - 100,000 vars / 10^6 clauses Novel applications : e.g. diagnosis , planning, software / circuit testing, machine learning, and protein folding 10/6/2005 copyright Brian Williams 6 Outline Propositional Satisfiability Backtrack Search Unit Propagation DPLL: Unit Propagation + Backtrack Search Appendices a 2
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10/6/2005 copyright Brian Williams 7 Propositional Satisfiability Input: A Propositional Satisfiability Problem is a pair <P, Φ >, where: P is a finite set of propositions . Φ is a propositional sentence on P May be reduced to a set of clauses . Output: True iff there exists a model of Φ . Is an instance of a CSP: Variables: Propositions Domain: {True, False} Constraints: Clauses that must be true 10/6/2005 copyright Brian Williams 8 Propositional Satisfiability An interpretation (truth assignment to all propositions) such that all clauses are satisfied : A clause is satisfied if and only if at least one literal is true . A clause is violated if and only if all literals are false . C1: Not A or B C2: Not C or A C3: Not B or C 10/6/2005 copyright Brian Williams 9 Satisfiability Testing Procedures Satisfiability Testing Procedures Reduce to CNF (Clausal Form) then: 1. Apply systematic, complete procedure Depth-first backtrack search [Davis, Logmann, & Loveland 1962] unit propagation, shortest clause heuristic 2. Apply stochastic, incomplete procedure MinConflict [Minton et a. 90], GSAT [Selman et. al 1993)]– see Appendix 3. Apply exhaustive clausal resolution [Davis, Putnam, 1960] a 3
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10/6/2005 10 Outline ility Backtrack Search copyright Brian Williams Propositional Satisfiab Unit Propagation
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