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First Order Logic

First Order Logic. Introduction to Artificial Intelligence CS440/ECE448 Lecture 10 SECOND HOMEWORK DUE MONDAY. Last lecture. Propositional logic Syntax and semantics (Sound) inference rules and procedures This Lecture First order logic Syntax and semantics Fun (?) with sentences

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First Order Logic

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  1. First Order Logic Introduction to Artificial Intelligence CS440/ECE448 Lecture 10 SECOND HOMEWORK DUE MONDAY

  2. Last lecture • Propositional logic • Syntax and semantics • (Sound) inference rules and procedures This Lecture • First order logic • Syntax and semantics • Fun (?) with sentences • The Wumpus world in first-order logic Reading • Chapter 8

  3. Propositional logic: Syntax • Propositional logic is the simplest logic. • Logical constants TRUE and FALSE are sentences. • Proposition symbols P1, P2 etc. are sentences. • Symbols P1 and negated symbols  P1 are called literals. • If S is a sentence,  S is a sentence (NOT). • If S1 and S2 is a sentence, S1  S2 is a sentence (AND). • If S1 and S2 is a sentence, S1  S2 is a sentence (OR). • If S1 and S2 is a sentence, S1  S2 is a sentence (Implies). • If S1 and S2 is a sentence, S1  S2 is a sentence (Equivalent).

  4. Model of P  Q Propositional logic: Semantics Most sentences are sometimes true (satisfiable). P  Q Some sentences are always true (valid). P  P Some sentences are never true (unsatisfiable). P  P

  5. Propositional Inference: Enumeration Method • Let    and KB = (  C)  B  C) • Is it the case that KB  ? • Check all possible models --  must be true whenever KB is true

  6. An Inference Rule: Modus Ponens • From an implication and the premise of the implication, you can infer the conclusion.    Premise ___________  Conclusion • An inference rule is sound if the conclusion is true in all cases where the premises are true.

  7. An Inference Rule: And - Elimination • From a conjunction, you can infer any of the conjuncts. 1 2 … n Premise _______________ i Conclusion • An inference rule is sound if the conclusion is true in all cases where the premises are true.

  8. And-Introduction & Double Negation • And-Introduction 1, 2, …, n Premise _______________ 1 2 … n Conclusion • Double Negation Premise _______  Conclusion

  9. Inference in Wumpus World Initial KB Some inferences: Apply Modus Ponens to R1 Add to KB W1,1Æ W2,1Æ W1,2 Apply to this AND-Elimination Add to KB W1,1 W2,1 W1,2 Percept Sentences S1,1 B1,1 S2,1 B2,1 S1,2 B1,2 … Environment Knowledge R1: S1,1 W1,1 W2,1 W1,2 R2: S2,1 W1,1 W2,1 W2,2 W3,1 R3: B1,1  P1,1 P2,1 P1,2 R4: B2,1  P1,1 P2,1  P2,2  P3,1 R5: B1,2  P1,1 P1,2  P2,2  P1,3 ...

  10. First Order Logic

  11. First Order Logic(First Order Predicate Calculus) • Propositional logic had limited ontology: • World consists of facts. • First order logic: • Objects (boxes, people, schools, …); • Relations (above, bigger than, …); • Properties (red, large, stinky, breeze, …); • Functions (father of, best friend, next class, …); • Quantification (every, there exists). • Can express the following: • Squares neighboring the Wumpus are smelly; • Squares neighboring a pit are breezy.

  12. Syntax of FOL: Basic elements Constants Wumpus, 2, UIUC, ... Predicates Brother, >, ... Functions Sqrt, LeftLegOf, ... Connectives     Variables x, y, a, b, ... Quantifiers  Equality = A legitimate expression of predicate calculus is called a well-formed formula (wff) or, simply, a sentence.

  13. Constant Symbols • A symbol, e.g. Jean, Wumpus, Arrow, GeorgeWBush, JacquesChirac. • Each constant symbol names exactly one object in a universe of discourse, but: • not all objects have symbol names; • some objects have several symbol names. • Usually denoted with upper-case first letter.

  14. Variables • Used to represent objects or properties of objects without explicitly naming the object. • Usually lower case. • For example: • x • father • square

  15. Relation (Predicate) Symbols • A predicate symbol is used to represent a relation in a universe of discourse. • The sentence Relation(Term1, Term2,…) is either TRUE or FALSE depending on whether Relation holds of Term1, Term2,… • To write “Voltaire wrote Candide” in a universe of discourse of names and written works: Wrote(Voltaire, Candide) This sentence is true in the intended interpretation. • Another example: Instructor (CS448, Ponce)

  16. Function Symbols • Functions talk about the binary relation of pairs of objects. • For example, the Father relation might represent all pairs of persons in father-daughter or father-son relationships: • Father(Jane) Refers to the father of Jane • Father(Joe) Refers to Joe’s father • Father(x) Refers to the father of variable x

  17. Atomic sentences Atomic Sentence: predicate(term1, …, termn) or term1 = term2 Term: function (term1, …, termn) or constant or variable For example: Intructor(CS448,Jean) Instructor(CS448, Son(JeanJacques)) Instructor(CS448, Son(x))

  18. Complex sentences Complex sentences are made from atomic sentences using connectives: S S1S2 S1S2 S1  S2 S1  S2 For example: Married (Bill, Hillary)  Married (Hillary, Bill) >(1,2)  (1,2) >(1,2)  >(1,2)

  19. While constant symbols, variables and connectives are like propositional logic, “What are functions and predicates?” The language of logic is based on set theory: • Sets; • Relations; • Functions.

  20. a d b c e Sets • The set of objects defines a “Universe of Discourse.” [Objects are represented by Constant Symbols.] • For example, in this blocks world, the universe of discourse is {a,b,c,d,e}. Interpretation: A  a B  b C  c D  d E  e

  21. a d b c e Relations • Def: A binary relation is a set of ordered pairs • Example: Consider set of blocks {a,b,c,d,e} • The “on” relation: • on = {<a,b>, <b,c>, <d,e>}. • The predicate On(A,B) can be interpreted as: <a,b>  on. • On(A,B) is TRUE, but On(A,C) and On(C,D) are FALSE (in this interpretation).

  22. Relations • Def: A binary relation is a set of ordered pairs. • Note: Binary relations are not necessarily finite sets. • Example: “less than” < < = {<x, y> | x, y N, z N, z  0, x+z = y }. The predicate: <(2, 3) is equivalent to: <2,3> <.

  23. Functions • A function is a binary relation such that no two distinct members have the same first element. • In other words, if F is a function <x,y>  F and <x,z>  F  y=z • If <x,y> F : • x is an argument of F ; • y is the value of F at x ; • y is the image of x under F. • F(x) designates the object y such that y=F(X).

  24. a d b c e Function Example • hat = {<c,b>, <b,a>,<e,d>} • hat (c) = b • hat(b) = a • hat(d) is not defined. • Hat(E) can be interpreted as d.

  25. a d b c e Formalizing Knowledge • The first step in formalizing knowledge is to construct a conceptualization, i.e., a representation of the objects in the world, and their relationships: • Objects – Universe of discourse; • Functions – Functional basis set; • Relations – Relational basis set. • A conceptualization is a triple < U of D, FBS, RBS >. • Blocks World: <{a,b,c,d,e}, {hat, …}, {on, …}>. • Second step is to express conceptualization using first order logic.

  26. a d b c e Formalizing Knowledge II • Second step is to express conceptualization using first order logic. • On(A,B) • On(B,C) • On(D,E) • On(A,Hat(C))

  27. Universal quantification •  <variables> <sentence> For all (every) instances of the variables, the <sentence> is true. • Everyone at UIUC is smart:  x At (x, UIUC)  Smart(x) •  x P is equivalent to the conjunction of instantiations of P (i.e., all x’s in universe of discourse). At (Jonathan, UIUC)  Smart(Jonathan) At (MarcSnir, UIUC)  Smart(MarcSnir)  At (BozoTheClown, UIUC)  Smart(BozoTheClown) … • Universe of discourse may be infinite, compare to propositional logic.

  28. Universal quantification cont. Typically,  is the main connective with  A common mistake is to use  as the main connective with  What does the following mean?  x At (x, UIUC)  Smart(x) “Everyone is at UIUC and everyone is smart.”

  29. x Bird (x)  Fly(x) predicate predicate • The block on top of any block lies on it x On(Hat(x),x) Some more examples • “Birds Fly”

  30. x Bird (x)  Fly(x) predicate predicate Some more examples • “Birds Fly” • But what about penguins? x Bird (x)Penguin(x)  Fly(x) • “Fish Swim.”  x Fish (x)  Swim(x)

  31. Existential quantification •  <variables> <sentence> There exist instances of the <variables> such that the <sentence> is true. • Someone at Ohio State is smart:  x At (x, OhioState)  Smart(x) •  x P is equivalent to the disjunction of instantiations of P (i.e., all x’s in universe of discourse). At (Rodrigo, OhioState)  Smart(Rodrigo)  At (Nincompoop, OhioState)  Smart(Nincompoop) …

  32. Existential quantification cont. Typically,  is the main connective with . A common mistake is to use  as the main connective with .  x At (x, UIUC)  Smart(x) is true if there is anyone who is not at UIUC.

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