CS 416 Artificial Intelligence Lecture Lecture 17 17 First Order First Order Logic Logic Chapter Chapter 99 Guest Speaker Topics Topics in in Optimal Optimal Control Control Minimax Minimax Control Control and and Game Game Theory Theory March March 28 28thth 22 p m p m OLS OLS 005 005 Onesimo Onesimo Hernandez Lerma Hernandez Lerma Department Department of of Mathematics Mathematics CINVESTAV IPN CINVESTAV IPN Mexico Mexico City City This This is is aa nontechnical nontechnical introduction introduction mainly mainly thru thru examples examples to to some some recent recent topics topics in in control control and and game game theory theory including including adaptive adaptive control control minimax minimax control control a k a a k a worst case worst case control control or or games games against against nature nature partially partially observable observable systems systems a k a a k a controlled controlled hidden hidden Markov Markov models models cooperative cooperative and and noncooperative noncooperative game game equilibria equilibria etc etc Inference in first order logic Our Our goal goal is is to to prove prove that that KB KB entails entails aa fact fact We We use use logical logical inference inference Forward Forward chaining chaining Backward Backward chaining chaining Resolution Resolution All All three three logical logical inference inference systems systems rely rely on on search search to to find find aa sequence sequence of of actions actions that that derive derive the the empty empty clause clause Search and forward chaining Start Start with with KB KB full full of of first order first order definite definite clauses clauses Disjunction Disjunction of of literals literals with with exactly exactly one one positive positive Equivalent Equivalent to to implication implication with with conjunction conjunction of of positive positive literals literals on on left left antecedent antecedent body body premise premise and and one one positive positive literal literal on on right right consequent consequent head head conclusion conclusion Propositional Propositional logic logic used used Horn Horn clauses clauses which which permit permit zero zero or or one one to to be be positive positive Look Look for for rules rules with with premises premises that that are are satisfied satisfied use use substitution substitution to to make make matches matches and and add add conclusions conclusions to to KB KB Search and forward chaining Breadth Breadth First First A A B B D D G G H H A A E E C C Which Which rules rules have have premises premises that that are are satisfied satisfied modus modus ponens ponens B B D D E E A A E E C C nope nope E E C C G G H H II B B D D E E yes yes E E C C G G H H I I nope nope A A E E C C yes yes E E C C G G H H I I nope nope one one more more try try yes yes Search and forward chaining Would Would other other search search methods methods work work Yes Yes this this technique technique falls falls in in standard standard domain domain of of all all searches searches Search and backward chaining Start Start with with KB KB full full of of implications implications Find Find all all implications implications with with conclusion conclusion matching matching the the query query Add Add to to fringe fringe list list the the unknown unknown premises premises Adding Adding could could be be to to front front or or rear rear of of fringe fringe depth depth or or breadth breadth Search and backward chaining Depth Depth First First A A B B D D G G H H A A E E C C B B D D E E C C E E G G H H II Are Are all all the the premises premises of of II satisfied satisfied No No For For each each C C E EG G H H are are each each of of their their premises premises satisfied satisfied C C no no put put its its premises premises on on fringe fringe For For each each A A and and E E are are their their premises premises satisfied satisfied A A yes yes E E no no add add premises premises for for each each B B and and D D B B yes yes D D yes yes E E G G H H yes yes Search and backward chaining Breadth Breadth First First A A B B D D G G H H A A E E C C B B D D E E C C E E G G H H II Are Are all all the the premises premises of of II satisfied satisfied No No For For each each C C E EG G H H are are each each of of their their premises premises satisfied satisfied C C no no put put its its premises premises on on fringe fringe end end E E no no put put its its premises premises on on fringe fringe end end G G H H yes yes Are Are C s C s premises premises A A E E satisfied satisfied A A yes yes E E no no add add premises premises Are Are E s E s premises premises B B D D satisfied satisfied Yes Yes Return Return to to C C and and II Backward forward chaining Don t Don t explicitly explicitly tie tie search search method method to to chaining chaining direction direction Inference with resolution We We put put each each first order first order sentence sentence into into conjunctive conjunctive normal normal form form We We remove remove quantifiers quantifiers We We make make each each sentence sentence aa disjunction disjunction of of literals literals each each literal literal is is universally universally quantified quantified We We show show KB KB is is unsatisfiable unsatisfiable by by deriving deriving the the empty empty clause clause Resolution Resolution inference inference rule rule is is our our method method Keep Keep resolving resolving until until the the empty empty clause clause is is reached reached Resolution Look Look for for matching matching sentences sentences Shared Shared literal literal with with opposite opposite sign sign Substitution Substitution may may be be required required Animal Animal F x F x V V Loves Loves G x G x x x and and Loves Loves u v u v V V Kills Kills u u v v F x F x animal animal unloved unloved by by xx G x G x someone someone who who loves loves xx Resolution What What does does this this mean mean in in English English Animal Animal F x F x V V Loves Loves G x G x x x F x F x animal animal unloved unloved by by xx G x G x someone someone who who loves loves xx Loves Loves u v u v V V Kills Kills u u v v For For all all people people either either aa person person doesn t doesn t …