SlideServe
Télécharger Obtenir Pro
Menu
Accueil Pro
Préférences
ApparenceLight Aide et commentaires
© SlideServe · v2026

First-Order Logic

deacon-holmes avatar
deacon-holmes
23 diapositives · 384 Views · Nov 20, 2014

This chapter outlines the foundational aspects of First-Order Logic (FOL) in artificial intelligence, detailing its syntax, semantics, and practical applications, such as the Wumpus world. It covers the advantages of FOL over propositional logic, emphasizing its ability to express complex relationships and statements about objects, functions, and quantifiers. The discussion includes atomic and complex sentences, truth in models, and common pitfalls in quantification. Understanding FOL is crucial for representing knowledge and reasoning in AI systems.

First-Order Logic - slide 1
01 / 23First-Order Logic
1 / 17

Télécharger la présentation

First-Order Logic
Une image / un lien ci-dessous est fourni (tel quel) pour télécharger la présentation

Politique de téléchargement : le contenu du site vous est fourni TEL QUEL pour votre information et votre usage personnel ; il ne peut être vendu, concédé sous licence ni partagé sur d’autres sites sans l’accord de son auteur.Le contenu vous est fourni TEL QUEL, pour votre information et votre usage personnel uniquement. Téléchargez la présentation en cliquant sur ce lien. Lors du téléchargement, si vous ne parvenez pas à télécharger une présentation, l’éditeur a peut-être supprimé le fichier de son serveur.Pendant le téléchargement, si vous ne parvenez pas à obtenir une présentation, le fichier a peut-être été supprimé par l’éditeur.

F I N

Plus de contenu associé

Transcription de la présentation

  1. First-Order Logic CIS 391 – Introduction to Artificial Intelligence AIMA, Chapter 8

  2. Outline • Why FOL? • Syntax and semantics of FOL • Using FOL • Wumpus world in FOL

  3. Pros and cons of propositional logic  Propositional logic is declarative  Propositional logic allows partial/disjunctive/negated information • (unlike most data structures and databases) • Propositional logic is compositional: • meaning of B1,1 P1,2 is derived from meaning of B1,1and of P1,2  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. First-order logic • 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, centuries… • Relations: red, round, prime, brother of, bigger than, part of, comes between, bogus … • Functions: father of, best friend, third inning of, one more than, plus, …

  5. Logics in general

  6. Syntax of FOL: Basic elements • Constants KingJohn, 2, Penn,... • Predicates Brother, >,... • Functions Sqrt, LeftLegOf,... • Variables x, y, a, b,... • Connectives , , , ,  • Equality = • Quantifiers , 

  7. Atomic sentences Term = function (term1,...,termn) or constant or variable Atomic sentence = predicate (term1,...,termn) orterm1 = term2 • For example: • Brother(KingJohn, RichardTheLionheart) • > (Length(LeftLegOf(Richard)),Length(LeftLegOf(KingJohn)))

  8. Complex sentences • Complex sentences are made from atomic sentences using connectives S, S1  S2, S1  S2, S1  S2, S1  S2, For example Sibling(KingJohn,Richard) Sibling(Richard,KingJohn) >(1,2)  ≤ (1,2) >(1,2)  >(1,2)

  9. 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 constantsymbols→objects predicatesymbols→relations functionsymbols→ functional relations • An atomic sentence predicate(term1,...,termn)is true iff the objects referred to by term1,...,termn are in the relation referred to by predicate

  10. Models for FOL: Example

  11. Truth example Consider the interpretation in which Richard→Richard the Lionheart John→the evil King John Brother→the brotherhood relation Under this interpretation Brother(Richard,John)is true just in case Richard the Lionheart and the evil King John are in the brotherhood relation in the model.

  12. Universal quantification • <variables> <sentence> Everyone at Penn is smart:  x At(x,Penn)  Smart(x) • 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 At(KingJohn,Penn)  Smart(KingJohn)  At(Richard,Penn)  Smart(Richard)  At(Penn,Penn)  Smart(Penn)  ...

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

  14. Existential quantification •  <variables> <sentence> • Someone at Penn is smart: x At(x,Penn)  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: At(KingJohn,Penn)  Smart(KingJohn)  At(Richard,Penn)  Smart(Richard)  At(Penn,Penn)  Smart(Penn)  ...

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

  16. Properties of quantifiers • x y is the same as yx • 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” • 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. Equality • term1 = term2is true under a given interpretation if and only if term1and term2refer to the same object • E.g., What’s wrong with the following definition?  x,ySibling(x,y)  [  m,f ( Parent(m,x)  Parent(f,x)  Parent(m,y)  Parent(f,y))]

  18. Equality • term1 = term2is true under a given interpretation if and only if term1and term2refer to the same object • E.g., definition of Sibling in terms of Parent:  x,ySibling(x,y)  [(x = y)   m,f ( (m = f)  Parent(m,x)  Parent(f,x)  Parent(m,y)  Parent(f,y))]

  19. Using FOL The kinship domain: • Brothers are siblings  x,y Brother(x,y)  Sibling(x,y) • One's mother is one's female parent  m,c Mother(c) = m  (Female(m)  Parent(m,c)) • “Sibling” is symmetric  x,y Sibling(x,y)  Sibling(y,x) • A first cousin is a child of a parent’s sibling  x,y FirstCousin(x,y)  p,ps Parent(p,x) Sibling(ps,p)  Parent(ps,y)

  20. Interacting with FOL KBs • Suppose a wumpus-world agent is using an FOL KB and perceives a smell and a breeze (but no glitter) at t = 5: Tell(KB,Percept([Smell,Breeze,None],5)) Ask(KB,a BestAction(a,5)) • I.e., does the KB entail some best action at t = 5? • Answer: Yes, {a/Shoot} ← substitution (binding list) • Given a sentence Sand a substitution σ,Sσ denotes the result of plugging σinto S; e.g., S = Smarter(x,y) σ = {x/Hillary,y/Bill} Sσ = Smarter(Hillary,Bill) • Ask(KB,S) returns some/all σ such that KB╞ Sσ

  21. Knowledge base for the wumpus world “Perception” •  s, b, t Percept([s,b,Glitter],t)  AtGold(t) •  b, g, t Percept([Smell,b,g],t)  Smelt(t) Reflex •  t AtGold (t)  Action(Grab,t) Reflex with internal state: do we have gold already? •  t AtGold (t)  Holding(Gold,t)  Action(Grab,t) Holding(Gold,t)cannot be observed • keeping track of change is essential

  22. Deducing hidden properties  x, y, a, b Adjacent([x,y],[a,b])  [a,b]  {[x+1,y], [x-1,y],[x,y+1],[x,y-1]} Properties of squares:  s, t At(Agent,s,t)  Breezy(t)  Breezy(s) Squares are breezy near a pit: • Diagnostic rule---infer cause from effect  s Breezy(s)  r Adjacent(r,s)  Pit(r) • Causal rule---infer effect from cause  r Pit(r)  [ s Adjacent(r,s)  Breezy(s) ]

  23. Summary • First-order logic: • objects and relations are semantic primitives • syntax: constants, functions, predicates, equality, quantifiers • Increased expressive power: sufficient to define wumpus world