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Propositional Logic

Propositional Logic
Propositional Logic www.ThesisScientist.comLogic and AI • Would like our AI to have knowledge about the world, and logically draw conclusions from it • Search algorithms generate successors and evaluate them, but do not “understand” much about the setting • Example question: is it possible for a chess player to have 8 pawns and 2 queens – Search algorithm could search through tons of states to see if this ever happens, but… www.ThesisScientist.comA story • You roommate comes home; he/she is completely wet • You know the following things: – Your roommate is wet – If your roommate is wet, it is because of rain, sprinklers, or both – If your roommate is wet because of sprinklers, the sprinklers must be on – If your roommate is wet because of rain, your roommate must not be carrying the umbrella – The umbrella is not in the umbrella holder – If the umbrella is not in the umbrella holder, either you must be carrying the umbrella, or your roommate must be carrying the umbrella – You are not carrying the umbrella • Can you conclude that the sprinklers are on • Can AI conclude that the sprinklers are on www.ThesisScientist.comKnowledge base for the story • RoommateWet • RoommateWet = (RoommateWetBecauseOfRain OR RoommateWetBecauseOfSprinklers) • RoommateWetBecauseOfSprinklers = SprinklersOn • RoommateWetBecauseOfRain = NOT(RoommateCarryingUmbrella) • UmbrellaGone • UmbrellaGone = (YouCarryingUmbrella OR RoommateCarryingUmbrella) • NOT(YouCarryingUmbrella) www.ThesisScientist.comSyntax • What do wellformed sentences in the knowledge base look like • A BNF grammar: • Symbol → P, Q, R, …, RoommateWet, … • Sentence → True False Symbol NOT(Sentence) (Sentence AND Sentence) (Sentence OR Sentence) (Sentence = Sentence) • We will drop parentheses sometimes, but www.ThesisScientist.com formally they really should always be thereSemantics • A model specifies which of the proposition symbols are true and which are false • Given a model, I should be able to tell you whether a sentence is true or false • Truth table defines semantics of operators: a b NOT(a) a AND b a OR b a = b false false true false false true false true true false true true true false false false true false true true false true true true • Given a model, can compute truth of sentence www.ThesisScientist.com recursively with theseCaveats • TwoIsAnEvenNumber OR ThreeIsAnOddNumber is true (not exclusive OR) • TwoIsAnOddNumber = ThreeIsAnEvenNumber is true (if the left side is false it’s always true) All of this is assuming those symbols are assigned their natural values… www.ThesisScientist.comTautologies • A sentence is a tautology if it is true for any setting of its propositional symbols P Q P OR Q NOT(P) AND (P OR Q) NOT(Q) OR (NOT(P) AND NOT(Q)) false false false true true false true true false true true false true false true true true true false true www.ThesisScientist.com • (P OR Q) OR (NOT(P) AND NOT(Q)) is a tautologyIs this a tautology • (P = Q) OR (Q = P) www.ThesisScientist.comLogical equivalences • Two sentences are logically equivalent if they have the same truth value for every setting of their propositional variables P Q P OR Q NOT(NOT(P) AND NOT(Q)) false false false false false true true true true false true true true true true true • P OR Q and NOT(NOT(P) AND NOT(Q)) are logically equivalent www.ThesisScientist.com • Tautology = logically equivalent to TrueFamous logical equivalences • (a OR b) ≡ (b OR a) commutatitvity • (a AND b) ≡ (b AND a) commutatitvity • ((a AND b) AND c) ≡ (a AND (b AND c)) associativity • ((a OR b) OR c) ≡ (a OR (b OR c)) associativity • NOT(NOT(a)) ≡ a doublenegation elimination • (a = b) ≡ (NOT(b) = NOT(a)) contraposition • (a = b) ≡ (NOT(a) OR b) implication elimination • NOT(a AND b) ≡ (NOT(a) OR NOT(b)) De Morgan • NOT(a OR b) ≡ (NOT(a) AND NOT(b)) De Morgan • (a AND (b OR c)) ≡ ((a AND b) OR (a AND c)) distributitivity • (a OR (b AND c)) ≡ ((a OR b) AND (a OR c)) distributitivity www.ThesisScientist.comInference • We have a knowledge base of things that we know are true – RoommateWetBecauseOfSprinklers – RoommateWetBecauseOfSprinklers = SprinklersOn • Can we conclude that SprinklersOn • We say SprinklersOn is entailed by the knowledge base if, for every setting of the propositional variables for which the knowledge base is true, SprinklersOn is also true RWBOS SprinklersOn Knowledge base false false false false true false true false false true true true www.ThesisScientist.com • SprinklersOn is entailedSimple algorithm for inference • Want to find out if sentence a is entailed by knowledge base… • For every possible setting of the propositional variables, – If knowledge base is true and a is false, return false • Return true propositional variables • Not very efficient: 2 settings www.ThesisScientist.comInconsistent knowledge bases • Suppose we were careless in how we specified our knowledge base: • PetOfRoommateIsABird = PetOfRoommateCanFly • PetOfRoommateIsAPenguin = PetOfRoommateIsABird • PetOfRoommateIsAPenguin = NOT(PetOfRoommateCanFly) • PetOfRoommateIsAPenguin • No setting of the propositional variables makes all of these true • Therefore, technically, this knowledge base implies anything • TheMoonIsMadeOfCheese www.ThesisScientist.comReasoning patterns • Obtain new sentences directly from some other sentences in knowledge base according to reasoning patterns • If we have sentences a and a = b , we can correctly conclude the new sentence b – This is called modus ponens • If we have a AND b , we can correctly conclude a • All of the logical equivalences from before www.ThesisScientist.com also give reasoning patternsFormal proof that the sprinklers are on 1) RoommateWet 2) RoommateWet = (RoommateWetBecauseOfRain OR RoommateWetBecauseOfSprinklers) 3) RoommateWetBecauseOfSprinklers = SprinklersOn 4) RoommateWetBecauseOfRain = NOT(RoommateCarryingUmbrella) 5) UmbrellaGone 6) UmbrellaGone = (YouCarryingUmbrella OR RoommateCarryingUmbrella) 7) NOT(YouCarryingUmbrella) 8) YouCarryingUmbrella OR RoommateCarryingUmbrella (modus ponens on 5 and 6) 9) NOT(YouCarryingUmbrella) = RoommateCarryingUmbrella (equivalent to 8) 10) RoommateCarryingUmbrella (modus ponens on 7 and 9) 11) NOT(NOT(RoommateCarryingUmbrella) (equivalent to 10) 12) NOT(NOT(RoommateCarryingUmbrella)) = NOT(RoommateWetBecauseOfRain) (equivalent to 4 by contraposition) 13) NOT(RoommateWetBecauseOfRain) (modus ponens on 11 and 12) 14) RoommateWetBecauseOfRain OR RoommateWetBecauseOfSprinklers (modus ponens on 1 and 2) 15) NOT(RoommateWetBecauseOfRain) = RoommateWetBecauseOfSprinklers (equivalent to 14) 16) RoommateWetBecauseOfSprinklers (modus ponens on 13 and 15) www.ThesisScientist.com 17) SprinklersOn (modus ponens on 16 and 3)Reasoning about penguins 1) PetOfRoommateIsABird = PetOfRoommateCanFly 2) PetOfRoommateIsAPenguin = PetOfRoommateIsABird 3) PetOfRoommateIsAPenguin = NOT(PetOfRoommateCanFly) 4) PetOfRoommateIsAPenguin 5) PetOfRoommateIsABird (modus ponens on 4 and 2) 6) PetOfRoommateCanFly (modus ponens on 5 and 1) 7) NOT(PetOfRoommateCanFly) (modus ponens on 4 and 3) 8) NOT(PetOfRoommateCanFly) = FALSE (equivalent to 6) 9) FALSE (modus ponens on 7 and 8) 10) FALSE = TheMoonIsMadeOfCheese (tautology) 11) TheMoonIsMadeOfCheese (modus ponens on 9 and 10) www.ThesisScientist.comGetting more systematic • Any knowledge base can be written as a single formula in conjunctive normal form (CNF) – CNF formula: (… OR … OR …) AND (… OR …) AND … – … can be a symbol x, or NOT(x) (these are called literals) – Multiple facts in knowledge base are effectively ANDed together RoommateWet = (RoommateWetBecauseOfRain OR RoommateWetBecauseOfSprinklers) becomes (NOT(RoommateWet) OR RoommateWetBecauseOfRain OR RoommateWetBecauseOfSprinklers) www.ThesisScientist.comConverting story problem to conjunctive normal form • RoommateWet – RoommateWet • RoommateWet = (RoommateWetBecauseOfRain OR RoommateWetBecauseOfSprinklers) – NOT(RoommateWet) OR RoommateWetBecauseOfRain OR RoommateWetBecauseOfSprinklers • RoommateWetBecauseOfSprinklers = SprinklersOn – NOT(RoommateWetBecauseOfSprinklers) OR SprinklersOn • RoommateWetBecauseOfRain = NOT(RoommateCarryingUmbrella) – NOT(RoommateWetBecauseOfRain) OR NOT(RoommateCarryingUmbrella) • UmbrellaGone – UmbrellaGone • UmbrellaGone = (YouCarryingUmbrella OR RoommateCarryingUmbrella) – NOT(UmbrellaGone) OR YouCarryingUmbrella OR RoommateCarryingUmbrella • NOT(YouCarryingUmbrella) www.ThesisScientist.com – NOT(YouCarryingUmbrella)Unit resolution • Unit resolution: if we have • l OR l OR … OR l 1 2 k and • NOT(l ) i we can conclude • l OR l OR … l OR l OR … OR l 1 2 i1 i+1 k • Basically modus ponens www.ThesisScientist.comApplying resolution to story problem 1) RoommateWet 2) NOT(RoommateWet) OR RoommateWetBecauseOfRain OR RoommateWetBecauseOfSprinklers 3) NOT(RoommateWetBecauseOfSprinklers) OR SprinklersOn 4) NOT(RoommateWetBecauseOfRain) OR NOT(RoommateCarryingUmbrella) 5) UmbrellaGone 6) NOT(UmbrellaGone) OR YouCarryingUmbrella OR RoommateCarryingUmbrella 7) NOT(YouCarryingUmbrella) 8) NOT(UmbrellaGone) OR RoommateCarryingUmbrella (6,7) 9) RoommateCarryingUmbrella (5,8) 10) NOT(RoommateWetBecauseOfRain) (4,9) 11) NOT(RoommateWet) OR RoommateWetBecauseOfSprinklers (2,10) 12) RoommateWetBecauseOfSprinklers (1,11) www.ThesisScientist.com 13) SprinklersOn (3,12)Limitations of unit resolution • P OR Q • NOT(P) OR Q • Can we conclude Q www.ThesisScientist.com(General) resolution • General resolution: if we have • l OR l OR … OR l 1 2 k and • m OR m OR … OR m 1 2 n where for some i,j, l = NOT(m ) i j we can conclude • l OR l OR … l OR l OR … OR l OR m OR m 1 2 i1 i+1 k 1 2 OR … OR m OR m OR … OR m j1 j+1 n • Same literal may appear multiple times; remove those www.ThesisScientist.comApplying resolution to story problem (more clumsily) 1) RoommateWet 2) NOT(RoommateWet) OR RoommateWetBecauseOfRain OR RoommateWetBecauseOfSprinklers 3) NOT(RoommateWetBecauseOfSprinklers) OR SprinklersOn 4) NOT(RoommateWetBecauseOfRain) OR NOT(RoommateCarryingUmbrella) 5) UmbrellaGone 6) NOT(UmbrellaGone) OR YouCarryingUmbrella OR RoommateCarryingUmbrella 7) NOT(YouCarryingUmbrella) 8) NOT(RoommateWet) OR RoommateWetBecauseOfRain OR SprinklersOn (2,3) 9) NOT(RoommateCarryingUmbrella) OR NOT(RoommateWet) OR SprinklersOn (4,8) 10) NOT(UmbrellaGone) OR YouCarryingUmbrella OR NOT(RoommateWet) OR SprinklersOn (6,9) 11) YouCarryingUmbrella OR NOT(RoommateWet) OR SprinklersOn (5,10) 12) NOT(RoommateWet) OR SprinklersOn (7,11) 13) SprinklersOn (1,12) www.ThesisScientist.comSystematic inference • General strategy: if we want to see if sentence a is entailed, add NOT(a) to the knowledge base and see if it becomes inconsistent (we can derive a contradiction) • CNF formula for modified knowledge base is satisfiable if and only if sentence a is not entailed – Satisfiable = there exists a model that makes the modified knowledge base true = modified www.ThesisScientist.com knowledge base is consistentResolution algorithm • Given formula in conjunctive normal form, repeat: • Find two clauses with complementary literals, • Apply resolution, • Add resulting clause (if not already there) • If the empty clause results, formula is not satisfiable – Must have been obtained from P and NOT(P) • Otherwise, if we get stuck (and we will eventually), the formula is guaranteed to be satisfiable (proof in a couple of slides) www.ThesisScientist.comExample • Our knowledge base: – 1) RoommateWetBecauseOfSprinklers – 2) NOT(RoommateWetBecauseOfSprinklers) OR SprinklersOn • Can we infer SprinklersOn We add: – 3) NOT(SprinklersOn) • From 2) and 3), get – 4) NOT(RoommateWetBecauseOfSprinklers) www.ThesisScientist.com • From 4) and 1), get empty clauseIf we get stuck, why is the formula satisfiable • Consider the final set of clauses C • Construct satisfying assignment as follows: • Assign truth values to variables in order x , x , …, x 1 2 n • If x is the last chance to satisfy a clause (i.e., all the other variables in the j clause came earlier and were set the wrong way), then set x to satisfy it j – Otherwise, doesn’t matter how it’s set • Suppose this fails (for the first time) at some point, i.e., x must be j set to true for one lastchance clause and false for another • These two clauses would have resolved to something involving only up to x (not to the empty clause, of course), which must be j1 satisfied • But then one of the two clauses must also be satisfied www.ThesisScientist.com contradiction Special case: Horn clauses • Horn clauses are implications with only positive literals • x AND x AND x = x AND x 1 2 4 3 6 • TRUE = x 1 • Try to figure out whether some x is entailed j • Simply follow the implications (modus ponens) as far as you can, see if you can reach x j • x is entailed if and only if it can be reached (can set j everything that is not reached to false) • Can implement this more efficiently by maintaining, for each implication, a count of how many of the lefthand side variables have been reached www.ThesisScientist.com
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21-07-2017