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1 DCP 1172 Introduction to Artificial Intelligence Lecture notes for Ch.8 [AIAM-2nd Ed.] First-order Logic (FOL) Chang-Sheng Chen

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DCP1172, Ch.8 2 Midterm format Date: 11/19/2004 from 10:15 am – 11:55 am Location: EC016 Credits: 25% of overall grade Approx. 5 problems, several questions in each. Material: everything so far (or, up to Sec. 8.3). No books (or other material) are allowed. Duration will be 100 minutes.

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DCP1172, Ch.8 3 Question 1 – problems with short answer Problem 1-1: Forward search vs. backward search. Please explain with your example. Problem 1-2: Translate the following English sentences to FOL. … Problem 1-3: Please explain the phrase " Truth depends on interpretation.” by showing with an example. …

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DCP1172, Ch.8 4 Question Type 2: multiple-choices Please choose the topics that are candidates for the midterm exam ? (a)Informed search (b) Logical programming (c) Constraint Searching Program (d) Decision making under uncertainty (e) Probability Reasoning System Answer: (a), (c)

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DCP1172, Ch.8 5 Question Type 3- Matching Here are a few things that are related to NCTU and NTHU, please match the correct ones to each of them. (a) (b) (c) IPv4 address range, *.* (d) The current administrator of TANet Hsinchu-Miaoli regional center (e) The university with the largest number of students in Taiwan Answer:,

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DCP1172, Ch.8 6 Question type 4 – calculation or inference (with much more detailed involved) Problem 4-1: Please find an optimal solution to go from NCTU to the Hsinchu Railway Station. Here is the related information … the most economic way (e.g., less money) the most efficient way (e.g., less time)...

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DCP1172, Ch.8 7 Last time: Logic and Reasoning Knowledge Base (KB): contains a set of sentences expressed using a knowledge representation language TELL: operator to add a sentence to the KB ASK: operator to query the KB Logics are KRLs where conclusions can be drawn Syntax Semantics Entailment: KB |= α iff a is true in all worlds where KB is true Inference: KB |– i α, sentence α can be derived from KB using procedure i Sound: whenever KB |– i α then KB |= α is true Complete: whenever KB |= a then KB |– i a

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DCP1172, Ch.8 8 Last Time: Syntax of propositional logic

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DCP1172, Ch.8 9 Last Time: Semantics of Propositional logic

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DCP1172, Ch.8 10 Last Time: Inference rules for propositional logic

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DCP1172, Ch.8 11 This time First-order logic Syntax Semantics Wumpus world example Ontology (ont = ‘to be’; logica = ‘word’): kinds of things one can talk about in the language

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DCP1172, Ch.8 12 Why first-order logic? We saw that propositional logic is limited because it only makes the ontological commitment that the world consists of facts. Difficult to represent even simple worlds like the Wumpus world; e.g., “don’t go forward if the Wumpus is in front of you” takes 64 rules

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DCP1172, Ch.8 13 First-order logic (FOL) Ontological commitments: Objects: wheel, door, body, engine, seat, car, passenger, driver Relations: Inside(car, passenger), Beside(driver, passenger) Functions: ColorOf(car) Properties: Color(car), IsOpen(door), IsOn(engine) Functions are relations with single value for each object

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DCP1172, Ch.8 14 Semantics there is a correspondence between functions, which return values predicates, which are true or false Function: father_of(Mary) = Bill Predicate: father_of(Mary, Bill)

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DCP1172, Ch.8 15 Examples: “One plus two equals three” Objects:one, two, three, one plus two Relations:equals Properties:-- Functions:plus (“one plus two” is the name of the object obtained by applying function plus to one and two; three is another name for this object) “Squares neighboring the Wumpus are smelly” Objects:Wumpus, square Relations:neighboring Properties:smelly Functions:--

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DCP1172, Ch.8 16 FOL: Syntax of basic elements Constant symbols: 1, 5, A, B, Alex, Manos, … Predicate symbols: >, Friend, Student, Colleague, … Function symbols: +, sqrt, SchoolOf, TeacherOf, ClassOf, … Variables: x, y, z, next, first, last, … Connectives: , , , Quantifiers: , Equality: =

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DCP1172, Ch.8 17 Syntax of Predicate Logic Symbol set constants Boolean connectives variables functions predicates (relations) quantifiers

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DCP1172, Ch.8 18 Syntax of Predicate Logic Terms: a reference to an object variables, constants, functional expressions (can be arguments to predicates) Examples: first([a,b,c]), sq_root(9), sq_root(n), tail([a,b,c])

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DCP1172, Ch.8 19 Syntax of Predicate Logic Sentences: make claims about objects Well-formed formulas, (wffs) Atomic Sentences (predicate expressions): Loves(John,Mary), Brother(John,Ted) Complex Sentences (Atomic Sentences connected by booleans): ¬ Loves(John,Mary) Brother(John,Ted) ^ Brother(Ted,John) Mother(Alice, John) ⇒ Mother(Alice,Ted)

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DCP1172, Ch.8 20 Examples of Terms: Constants, Variables and Functions Constants: object constants refer to individuals Alan, Sam, R225, R216 Variables PersonX, PersonY, RoomS, RoomT Functions father_of(PersonX) product_of(Number1,Number2)

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DCP1172, Ch.8 21 Examples of Predicates and Quantifiers Predicates In(Alan,R225) PartOf(R225,BuildingEC3) FatherOf(PersonX,PersonY) Quantifiers All dogs are mammals. Some birds can’t fly. 3 birds can’t fly.

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DCP1172, Ch.8 22 Semantics Referring to individuals Jackie son-of(Jackie), Sam Referring to states of the world person(Jackie), female(Jackie) mother(Sam, Jackie)

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DCP1172, Ch.8 23 FOL: Atomic sentences AtomicSentence Predicate(Term, …) | Term = Term Term Function(Term, …) | Constant | Variable Examples: SchoolOf(Manos) Colleague(TeacherOf(Alex), TeacherOf(Manos)) >((+ x y), x)

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DCP1172, Ch.8 24 FOL: Complex sentences Sentence AtomicSentence | Sentence Connective Sentence | Quantifier Variable, … Sentence | Sentence | (Sentence) Examples: S1 S2, S1 S2, (S1 S2) S3, S1 S2, S1 S3 Colleague(Paolo, Maja) Colleague(Maja, Paolo) Student(Alex, Paolo) Teacher(Paolo, Alex)

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DCP1172, Ch.8 25 Semantics of atomic sentences Sentences in FOL are interpreted with respect to a model Model contains objects and relations among them Terms: refer to objects (e.g., Door, Alex, StudentOf(Paolo)) Constant symbols: refer to objects Predicate symbols: refer to relations Function symbols: refer to functional Relations An atomic sentence predicate(term 1, …, term n ) is true iff the relation referred to by predicate holds between the objects referred to by term 1, …, term n

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DCP1172, Ch.8 26 Example model Objects: John, James, Marry, Alex, Dan, Joe, Anne, Rich Relation: sets of tuples of objects {,,, …} {,,, …} E.g.: Parent relation -- {,, } then Parent(John, James) is true Parent(John, Marry) is false

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DCP1172, Ch.8 27 Quantifiers Expressing sentences about collections of objects without enumeration (naming individuals) E.g., All Trojans are clever Someone in the class is sleeping Universal quantification (for all): Existential quantification (three exists):

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DCP1172, Ch.8 28 Universal quantification (for all): “Every one in the DCP1172 class is smart”: x In(DCP1172, x) Smart(x) P corresponds to the conjunction of instantiations of P In(DCP1172, Manos) Smart(Manos) In(DCP1172, Dan) Smart(Dan) … In(DCP1172, Clinton) Smart(Clinton)

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DCP1172, Ch.8 29 Universal quantification (for all): is a natural connective to use with Common mistake: to use in conjunction with e.g: x In(DCP1172, x) Smart(x) means “every one is in DCP1172 and everyone is smart”

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DCP1172, Ch.8 30 Existential quantification (there exists): “Someone in the dcp1172 class is smart”: x In(dcp1172, x) Smart(x) P corresponds to the disjunction of instantiations of P In(dcp1172, Manos) Smart(Manos) In(dcp1172, Dan) Smart(Dan) … In(dcp1172, Clinton) Smart(Clinton)

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DCP1172, Ch.8 31 Existential quantification (there exists): is a natural connective to use with Common mistake: to use in conjunction with e.g: x In(dcp1172, x) Smart(x) is true if there is anyone that is not in dcp1172! (remember, false true is valid).

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DCP1172, Ch.8 32 Properties of quantifiers Proof? Not all by one person but each one at least by one

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DCP1172, Ch.8 33 Proof In general we want to prove: x P(x) ¬ x ¬ P(x) x P(x) = ¬(¬( x P(x))) = ¬(¬(P(x1) ^ P(x2) ^ … ^ P(xn)) ) = ¬(¬P(x1) v ¬P(x2) v … v ¬P(xn)) ) x ¬P(x) = ¬P(x1) v ¬P(x2) v … v ¬P(xn) ¬ x ¬P(x) = ¬(¬P(x1) v ¬P(x2) v … v ¬P(xn))

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DCP1172, Ch.8 34 Example sentences Brothers are siblings x, y Brother(x, y) Sibling(x, y) Sibling is transitive x, y, z Sibling(x, y) Sibling(y, z) Sibling(x, z) One’s mother is one’s sibling’s mother m, c,d Mother(m, c) Sibling(c, d) Mother(m, d) A first cousin is a child of a parent’s sibling c, d FirstCousin(c, d) p, ps Parent(p, d) Sibling(p, ps) Parent(ps, c)

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DCP1172, Ch.8 35 Example sentences One’s mother is one’s sibling’s mother m, c,d Mother(m, c) Sibling(c, d) Mother(m, d) c,d m Mother(m, c) Sibling(c, d) Mother(m, d) cd m Mother of sibling

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DCP1172, Ch.8 36 Translating English to FOL Every gardener likes the sun. x gardener(x) => likes(x,Sun) You can fool some of the people all of the time. x t (person(x) ^ time(t)) => can-fool(x,t)

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DCP1172, Ch.8 37 Translating English to FOL You can fool all of the people some of the time. x t (person(x) ^ time(t) => can-fool(x,t) All purple mushrooms are poisonous. x (mushroom(x) ^ purple(x)) => poisonous(x)

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DCP1172, Ch.8 38 Translating English to FOL… No purple mushroom is poisonous. ¬( x) purple(x) ^ mushroom(x) ^ poisonous(x) or, equivalently, ( x) (mushroom(x) ^ purple(x)) => ¬poisonous(x)

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DCP1172, Ch.8 39 Translating English to FOL… There are exactly two purple mushrooms. ( x)( y) mushroom(x) ^ purple(x) ^ mushroom(y) ^ purple(y) ^ ¬(x=y) ^ ( z) (mushroom(z) ^ purple(z)) => ((x=z) v (y=z)) Deb is not tall. ¬tall(Deb)

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DCP1172, Ch.8 40 Translating English to FOL… X is above Y if X is on directly on top of Y or else there is a pile of one or more other objects directly on top of one another starting with X and ending with Y. ( x)( y) above(x,y) (on(x,y) v ( z) (on(x,z) ^ above(z,y)))

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DCP1172, Ch.8 41 Equality

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DCP1172, Ch.8 42 Higher-order logic? First-order logic allows us to quantify over objects (= the first-order entities that exist in the world). Higher-order logic also allows quantification over relations and functions. e.g., “two objects are equal iff all properties applied to them are equivalent”: x,y (x=y) ( p, p(x) p(y)) Higher-order logics are more expressive than first-order; however, so far we have little understanding on how to effectively reason with sentences in higher-order logic.

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DCP1172, Ch.8 43 Logical agents for the Wumpus world 1.TELL KB what was perceived Uses a KRL to insert new sentences, representations of facts, into KB 2.ASK KB what to do. Uses logical reasoning to examine actions and select best. Remember: generic knowledge-based agent:

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DCP1172, Ch.8 44 Using the FOL Knowledge Base Set of solutions

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DCP1172, Ch.8 45 Wumpus world, FOL Knowledge Base

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DCP1172, Ch.8 46 Deducing hidden properties

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DCP1172, Ch.8 47 Situation calculus

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DCP1172, Ch.8 48 Describing actions May result in too many frame axioms

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DCP1172, Ch.8 49 Describing actions (cont’d)

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DCP1172, Ch.8 50 Planning

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DCP1172, Ch.8 51 Generating action sequences [ ] = empty plan Recursively continue until it gets to empty plan [ ]

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DCP1172, Ch.8 52 Summary

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