1/19 First-Order Logic Chapter 8 Modified by Vali Derhami

2/19 Outline Why FOL? Syntax and semantics of FOL Using FOL Wumpus world in FOL Knowledge engineering in FOL

3/19 propositional logic Propositional logic is declarative ((اظهاری جدا بودن دانش و استنتاج (کنترل) در منطق گزاره ای، در حالیکه در برنامه ریزی سنتی برای تازه سازی ساختار داده از یک رویه یِ وابسته به مساله و جزییات ان توسط برنامه نویس تعیین می شود انجام می پذیرد. Propositional logic allows partial/disjunctive/negated information –(unlike most data structures and databases) Propositional logic is compositional: –meaning of B 1,1  P 1,2 is derived from meaning of B 1,1 and of P 1,2 Meaning in propositional logic is context-independent –(unlike natural language, where meaning depends on context)  Propositional logic has very limited expressive power –(unlike natural language) –E.g., cannot say "pits cause breezes in adjacent squares“ except by writing one sentence for each square

4/19 First-order logic Whereas propositional logic assumes the world contains facts, first-order logic (like natural language) assumes the world contains –Objects: people, houses, numbers, colors, baseball games, wars, … –Relations: یک ارتباط یگانی یا خصوصیات verbs and verb phrases that refer to relations among objects Ex. : red, round, prime, brother of, bigger than, part of, comes between, … –Functions: father of, best friend, one more than, plus, … در توابع تنها یک مقدار برای یک ”ورودی“ مفروض وجود دارد

5/19 Some examples "One plus two equals three" –Objects: one, two, three, one plus two; Relation: equals; Function: plus. ("One plus –two" is a name for the object that is obtained by applying the function "plus" to the –objects "one" and "two." Three is another name for this object.) "Squares neighboring the wumpus are smelly." –Objects: wumpus, squares; Property: smelly; Relation: neighboring. "Evil King John ruled England in 1200." –Objects: John, England, 1200; Relation: ruled; Properties: evil, king.

6/19 Five different logic یک منطق می تواند با تعهدات هستی شناسی اش (حالات ممکن دانش که با توجه به واقعیت مجاز می باشد) مشخص شود

7/19 Syntax of FOL: Basic elements Constants (for objects):KingJohn, 2, NUS,... Predicates(for relation): Brother, >,... Functions(for functions): Sqrt, LeftLegOf,... سه مورد فوق با حرف بزرگ شروع میشوند. Variablesx, y, a, b,... Connectives , , , ,  Equality= Quantifiers , 

8/19 Atomic sentences Atomic sentence =predicate (Term 1,...) or Term 1 = Term 2 Term =function (Term 1,...) or constant or variable E.g., Brother(KingJohn,RichardTheLionheart) Length(LeftLegOf(Richard)) Length(LeftLegOf(KingJohn))

9/19 Complex sentences Complex sentences are made from atomic sentences using connectives  S, S 1  S 2, S 1  S 2, S 1  S 2, S 1  S 2, Sentence —> Atomic Sentence or ( Sentence Connective Sentence ) or Quantifier Variable,... Sentence or  Sentence E.g. Sibling(KingJohn,Richard)  Sibling(Richard,KingJohn) >(1,2)  ≤ (1,2) >(1,2)   >(1,2)

10/19 Truth in first-order logic Sentences are true with respect to a model and an interpretation Model contains objects (domain elements) and relations among them Interpretation specifies referents for constant symbols → objects predicate symbols → relations function symbols →functional relations An atomic sentence predicate(term 1,...,term n ) is true iff the objects referred to by term 1,...,term n are in the relation referred to by predicate

11/19 Models for FOL: Example Brother (Richard, John). LeftLeg(John).

12/19 Universal quantification: سورهای عمومی  Example: "All kings are persons,"  x King(x) => Person(x). Everyone at NUS is smart:  x At(x,NUS)  Smart(x) خوانده می شود : ”به ازای تمامی x ها... A term with no variables is called a ground term.  x P is true in a model m iff P is true with x being each possible object in the model Roughly speaking, equivalent to the conjunction of instantiations of P  x At(x,NUS)  Smart(x) باید همه مصداقهای زیر درست باشد At(KingJohn,NUS)  Smart(KingJohn)  At(Richard,NUS)  Smart(Richard)  At(NUS,NUS)  Smart(NUS) ...

13/19 A common mistake to avoid Typically,  is the main connective with  Common mistake: using  as the main connective with  :  x King(x)  Person(x)  x At(x,NUS)  Smart(x) means “Everyone is at NUS and everyone is smart”

14/19 Existential quantification  Someone at NUS is smart:  x At(x,NUS)  Smart(x)  x P is true in a model m iff P is true with x being some possible object in the model Roughly speaking, equivalent to the disjunction of instantiations of P کافی است یکی از مصداقهای زیر درست باشد. At(KingJohn,NUS)  Smart(KingJohn)  At(Richard,NUS)  Smart(Richard)  At(NUS,NUS)  Smart(NUS) ...

15/19 Another common mistake to avoid Typically,  is the main connective with  Common mistake: using  as the main connective with  :  x At(x,NUS)  Smart(x) is true if there is anyone who is not at NUS!

16/19 Properties of quantifiers  x  y is the same as  y  x –  x  y Brother(x,y) => Sibling(x,y)   x,y Brother(x,y) => Sibling(x,y)  x  y is the same as  y  x  x  y is not the same as  y  x  x  y Loves(x,y)  –“There is a person who loves everyone in the world”  y  x Loves(x,y) –“Everyone in the world is loved by at least one person” Example: Quantifier duality: each can be expressed using the other  x Likes(x,IceCream)  x  Likes(x,IceCream)  x Likes(x,Broccoli)  x  Likes(x,Broccoli)

17/19 The De Morgan rules for quantified and unquantified sentences

18/19 Equality term 1 = term 2 is true under a given interpretation if and only if term 1 and term 2 refer to the same object –Father (Ali)= Hasan E.g., definition of Sibling in terms of Parent:  x,y Sibling(x,y)  [  (x = y)   m,f  (m = f)  Parent(m,x)  Parent(f,x)  Parent(m,y)  Parent(f,y)] Richard has at least two brothers,  x, y Brother(x, Richard)  Brother(y, Richard),  x, y Brother(x, Richard)  Brother(y, Richard)   (x = y).

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