Title: First-Order Logi B.Y. ✫ ✬ AIMA: Chapter 8 (Se tions 8.1 and 8.2) Choueiry Se tion 8.3, dis ussed brie�y , is also required reading In tro du tion to Arti� ial In telligen e CSCE 476-876, Spring 2016 URL: www. se.unl.edu/� ho uei ry/ S1 6-4 76- 87 6 1 Berthe Y. Choueiry (Sh u-w e-ri) (402)472-5444 Instru tor's April notes 8, 2016 #13 ✪ ✩
B.Y. ✫ ✬ Choueiry Outline First-order logi : � basi elemen ts � syn tax • � seman ti s Examples 2 • Instru tor's April notes 8, 2016 #13 ✪ ✩
Pros and ons of prop ositional logi Prop ositional logi is de larativ e : pie es of syn tax orresp ond to fa ts B.Y. ✫ ✬ Choueiry Prop ositional logi allo ws partial/disjun tiv e/negated information • (unlik e most data stru tures and databases) Prop ositional logi is omp ositional : • meaning of B 1 , 1 ∧ P 1 , 2 is deriv ed from meaning of B 1 , 1 and of Meaning in prop ositional logi is on text-indep enden t • 3 (unlik e natural language, where meaning dep ends on on text) but... P 1 , 2 Prop ositional logi has v ery limited expressiv e p o w er • E.g., annot sa y �pits ause breezes in adja en t squares� Instru tor's ex ept b y writing one sen ten e for ea h square • April notes 8, 2016 #13 ✪ ✩
Prop ositional Logi B.Y. is simple ✫ ✬ Choueiry illustrates imp ortan t p oin ts: mo del, inferen e, v alidit y , satis�abilit y , .. • is restri tiv e: w orld is a set of fa ts • la ks expressiv eness: In PL, w orld on tains fa ts • 4 First-Order Logi • → more sym b ols (ob je ts, prop erties, relations) more onne tiv es (quan ti�er) Instru tor's • April • notes 8, 2016 #13 ✪ ✩
First Order Logi B.Y. ✫ F OL pro vides more "primitiv es" to express kno wledge: ✬ Choueiry � ob je ts (iden tit y & prop erties) � relations among ob je ts (in luding fun tions) → Ob je ts: p eople, houses, n um b ers, Einstein, Husk ers, ev en t, .. Prop erties : smart, ni e, large, in telligen t, lo v ed, o urred, .. Relations : brother-of, bigger-than, part-of, o urred-after, .. F un tions : father-of, b est-friend, double-of, .. 5 Examples : (ob je ts? fun tion? relation? prop ert y?) � one plus t w o equals four [si ℄ � squares neigh b oring the wumpus are smelly Instru tor's April notes 8, 2016 #13 ✪ ✩
Logi A ttra ts : mathemati ians, philosophers and AI p eople B.Y. A dv an tages: ✫ ✬ Choueiry � allo ws to represen t the w orld and reason ab out it � expresses an ything that an b e programmed Non- ommittal to : � sym b ols ould b e ob je ts or relations ( e.g. , King(Gusta v e), King(Sw eden, Gusta v e), Mer iless(King)) � lasses, ategories, time, ev en ts, un ertain t y .. but amenable to extensions: OO F OL, temp oral logi s, 6 situation/ev en t al ulus, mo dal logi , et . Some p eople think F OL *is* the language of AI true/false? donno :�( but it will remain around for some time.. Instru tor's April − → notes 8, 2016 #13 ✪ ✩
T yp es of logi Logi s are hara terized b y what they ommit to as �primitiv es� B.Y. On tologi al ommitmen t : ✫ ✬ Choueiry what exists�fa ts? ob je ts? time? b eliefs? Epistemologi al ommitmen t : what states of kno wledge? 7 Language Ontological Commitment Epistemological Commitment (What exists in the world) (What an agent believes about facts) Propositional logic facts true/false/unknown First-order logic facts, objects, relations true/false/unknown Higher-Order Logi : views relations and fun tions of F OL as Temporal logic facts, objects, relations, times true/false/unknown ob je ts Instru tor's Probability theory facts degree of belief 0…1 Fuzzy logic degree of truth degree of belief 0…1 April notes 8, 2016 #13 ✪ ✩
Syn tax of F OL : w ords and grammar The w ords: sym b ols B.Y. ✫ ✬ Constan t sym b ols stand for ob je ts: QueenMary , 2, UNL, et . Choueiry V ariable sym b ols stand for ob je ts: x , y , et . Predi ate sym b ols stand for relations: Odd, Ev en, Brother, • Sibling, et . • F un tion sym b ols stand for fun tions (viz. relation) F ather-of, Square-ro ot, LeftLeg, et . • Quan ti�y ers ∀ , ∃ 8 • Conne tiv es: ∧ , ∨ , ¬ , ⇒ , ⇔ , (Sometimes) equalit y = • Predi ates and fun tions an ha v e an y arit y (n um b er of argumen ts) Instru tor's • • April notes 8, 2016 #13 ✪ ✩
Basi elemen ts in F OL (i.e., the grammar) B.Y. ✫ In prop ositional logi , ev ery expression is a sen ten e ✬ Choueiry In F OL , T erms Sen ten es: � atomi sen ten es � omplex sen ten es • 9 Quan ti�ers: • � Univ ersal quan ti�er � Existen tial quan ti�er • Instru tor's April notes 8, 2016 #13 ✪ ✩
B.Y. ✫ ✬ T erm Choueiry logi al expression that refers to an ob je t � built with: onstan t sym b ols, v ariables, fun tion sym b ols T erm = or onstan t or v ariable 10 � ground term : term with no v ariable function ( term 1 , . . . , term n ) Instru tor's April notes 8, 2016 #13 ✪ ✩
A tomi sen ten es B.Y. ✫ ✬ Choueiry state fa ts built with terms and predi ate sym b ols A tomi sen ten e = or term 1 = term 2 Examples : 11 Brother (Ri hard, John) predicate ( term 1 , . . . , term n ) Married (F atherOf(Ri hard), MotherOf(John)) Instru tor's April notes 8, 2016 #13 ✪ ✩
Complex Sen ten es B.Y. ✫ ✬ built with atomi sen ten es and logi al onne tiv es Choueiry ¬ S S 1 ∧ S 2 S 1 ∨ S 2 12 Examples : S 1 ⇒ S 2 Sibling(KingJohn,Ri hard) ⇒ Sibling(Ri hard,KingJohn) S 1 ⇔ S 2 Instru tor's > (1 , 2) ∨ ≤ (1 , 2) April > (1 , 2) ∧ ¬ > (1 , 2) notes 8, 2016 #13 ✪ ✩
B.Y. T ruth in �rst-order logi : Seman ti ✫ ✬ Choueiry Sen ten es are true with resp e t to a mo del and an in terpretation Mo del on tains ob je ts and relations among them In terpretation sp e i�es referen ts for onstant symb ols → ob je ts pr e di ate symb ols → relations fun tion symb ols → fun tional relations An atomi sen ten e predicate ( term 1 , . . . , term n ) is true 13 i� the ob je ts referred to b y term 1 , . . . , term n are in the relation referred to b y predicate Instru tor's April notes 8, 2016 #13 ✪ ✩
Mo del in F OL : example B.Y. ✫ ✬ Choueiry crown on head brother person person king brother R J The domain of a mo del is the set of ob je ts it on tains: $ left leg left leg �v e ob je ts 14 In tended in terpretation: Ri hard refers Ri hard the Lion Heart, John refers to Evil King John, Brother refers to brotherho o d relation, et . Instru tor's April notes 8, 2016 #13 ✪ ✩
Mo dels for F OL: Lots! B.Y. W e an en umerate the mo dels for a giv en KB v o abulary: ✫ ✬ Choueiry F or ea h n um b er of domain elemen ts n from 1 to ∞ F or ea h k -ary predi ate P k in the v o abulary F or ea h p ossible k -ary relation on n ob je ts F or ea h onstan t sym b ol C in the v o abulary F or ea h hoi e of referen t for C from n ob je ts . . . Computing en tailmen t b y en umerating mo dels is not going to b e easy! 15 There are man y p ossible in terpretations, also some mo del domain are not b ounded Che king en tailmen t b y en umerating is not an option Instru tor's April − → notes 8, 2016 #13 ✪ ✩
B.Y. ✫ ✬ Choueiry Quan ti�ers allo w to mak e statemen ts ab out en tire olle tions of ob je ts univ ersal quan ti�er: mak e statemen ts ab out ev erything existen tial quan ti�er: mak e statemen ts ab out some things 16 • • Instru tor's April notes 8, 2016 #13 ✪ ✩
Univ ersal quan ti� ation B.Y. Example : all dogs lik e b ones ∀ xDog ( x ) ⇒ LikeBones ( x ) ✫ ✬ Choueiry x = Indy is a dog x = Indiana Jones is a p erson ∀ � variables � � sentence � is equiv alen t to the onjun tion of instan tiations of P ∀ x P 17 Dog ( Indy ) ⇒ LikeBones ( Indy ) T ypi ally : ⇒ is the main onne tiv e with ∀ Dog ( Rebel ) ⇒ LikeBones ( Rebel ) ∧ Common mistak e : using ∧ as the main onne tiv e with ∀ Dog ( KingJohn ) ⇒ LikeBones ( KingJohn ) ∧ Example: ∀ x Dog ( x ) ∧ LikeBones ( x ) ∧ . . . all ob je ts in the w orld are dogs, and all lik e b ones Instru tor's April notes 8, 2016 #13 ✪ ✩
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