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Unformatted text preview: 10/6/2005 1 Modeling Using Session 15 copyright Brian Wil iams Propositional Logic Brian C. Williams 16.41013 10/6/2005 2 copyright Brian Wil iams Reading Assignment: Propositional Logic & Satisfiability AIMA Ch. 6 Propositional Logic 10/6/2005 3 Main 1 Pressure 1 Pressure 2 ingle l i i copyright Brian Wil iams 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 10/6/2005 copyright Brian Wil iams 4 Propositional Sentences: Engine Example An Engine E1 can either be okay or broken, in some unknown way. When E1 is okay, it will thrust when there is a flow through V1 and v2. E1 V1 V2 (mode(E1) = ok or mode(E1) = unknown) and not (mode(E1) = ok and mode(E1) = unknown) and (mode(E1) = ok implies (thrust(E1) = on if and only if flow(V1) = on and flow(V2) = on)) 10/6/2005 copyright Brian Wil iams 5 Propositional Satisfiability 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 Wil iams 6 Outline Propositional Logic Syntax Semantics Clausal Reduction Appendices a 2 10/6/2005 7 Logic: truth of facts time, . Algebra copyright Brian Wil iams What formal languages exist for describing constraints? Propositional logic First order logic facts,objects,relations Temporal logic Modal logics knowledge, belief Probability degree of belief values of variables 10/6/2005 8 Logic in General Logics l i i l Semantics i copyright Brian Wil iams formal anguages for representing information such that conclusions can be drawn. Syntax def nes the sentences n the anguage. def nes the meaning of sentences; truth of a sentence in a world. 10/6/2005 9 Logic Example: Arithmetic legal i i l l copyright Brian Wil iams Syntax sentences X + 2 > Y is a legal sentence. X 2 + Y > is not a legal sentence. Semantics  truth in world X + 2 > Y is true ff the number x + 2 is not less than or equal to the number y X + 2 > Y is true in a world where X = 7, Y = 1 X + 2 > Y s fa se in a wor d where X = 0, Y = 6 a 3 10/6/2005 10 Propositional Logic: Syntax Propositions ( ) i )  )  ) (i ) => (( ) ) ( ) ( ( )( )) copyright Brian Wil iams A statement that is true or false (valve v1) = voltage high Propositional Sentences (S)...
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 Fall '05
 BrianWilliams

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