1 / 31

First-Order Logic

First-Order Logic. Reading: C. 8 and C. 9 Pente specifications handed back at end of class. First-Order Logic: Outline. Expressing Information in first-order logic An example Inference in FOL Resolution theorem proving Production systems (forward chaining)

ivans
Télécharger la présentation

First-Order Logic

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. First-Order Logic Reading: C. 8 and C. 9 Pente specifications handed back at end of class

  2. First-Order Logic: Outline • Expressing Information in first-order logic • An example • Inference in FOL • Resolution theorem proving • Production systems (forward chaining) • Logic-based programming (backward chaining)

  3. Characteristics of FOL • Declarative • Expressive • Partial information • Negation • Compositionality

  4. Ontological Commitment • Propositional logic: • There are facts that either hold or do not hold in the world • Logic constrains facts • First-order logic: • The world consists of objects and relations between objects • Logic constrains allowable objects, properties of objects, relations between objects

  5. Ontological commitments of higher order logics • Temporal logic • Facts hold at particular times and those times are ordered • Epistemological • Agents hold beliefs about facts • Three possible states of knowledge • The agent believes a fact • The agent does not believe it • The agent has no opinion • Probabilistic • Facts are true to different degrees (Truth value from 0 to 1)

  6. Problems with propositional logic

  7. Propositional Logic is lacking in expressiveness • Cannot represent knowledge of complex environments in a concise way • E.g., Squares adjacent to pits are breezy • Need objects • Squares, pits, Kathy • Need relations • Adjacent, breezy, smelly, know • Need functions • Father-of, mother-of

  8. Syntax of FOL: basic elements • Constants: Vijay, Andrew, Sowmya • Predicates: knows, adjacent, > • Functions: Sqrt, father-of • Variables: x,y,a,b • Connectives: Λ,V,⌐,→,↔ • Equality: = • Quantifiers: ,

  9. Atomic Sentences • Atomic sentence = predicate (term1…termm) or term1=term2 • Term = function (term1, …, termm) or constant or variable • E.g. know(Kathy,Sowmya), Adjacent (x,y), father-of(Kathy) = Michael, Andrew, x

  10. Complex Sentences • Complex sentences are made from atomic sentences using connectives⌐S, S1ΛS2, S1VS2, S1S2, S1S2 • E.g., adjacent(x,y)  adjacent (y,x), ⌐knows(Nunzio, Michael),

  11. Truth in First-order Logic • Sentences are true with respect to a model and an interpretation • Model contains  1 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 (term1,…,termn) is true iff the objects referred to by term1,…, termn are in the relation referred to by predicate.

  12. Universal quantification • <variables> <sentence> • Everyone at Columbia is smart:x At(x,Columbia)  Smart(x) • x P is true in a model m iff P with x being each possible object in the model At (Leia, Columbia)  Smart(Leia) At (Ryan, Columbia)  Smart (Ryan) At (Archana, Columbia)  Smart (Archana) At (Stanley, Columbia)  Smart (Stanley) …..

  13. A common mistake • Typically,  is the main connective used with  • Common mistake: using as the main connective Λx At(x,Columbia) Λ Smart(x)

  14. Existential Quantification • <variables> <sentence> • Someone at Columbia is smartx At(x,Columbia) Smart(x) •  x P is true in a model m iff P with x being each possible object in the model • Equivalent to the disjunction of instantiations of P At (Leia, Columbia) Λ Smart(Leia) V At (Ryan, Columbia) Λ  Smart (Ryan) V At (Archana, Columbia) Λ  Smart (Archana) V At (Stanley, Columbia) Λ  Smart (Stanley)

  15. Another Common Mistake • Typically, Λ is the main connective with  • Common mistake: using  as the main connective x At(x,Columbia)  Smart(x)

  16. Properties of Quantifiers • xy is the same 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 is loved by someone. • 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. Translation from English to FOL • A mother is a female parent • Andrew likes the problem of one of the book exercises • ?

  18. Example • Family trees • What does the model look like? • Father-of • Mother-of • Sibling • What can we infer? • Cousin • Ancestors

  19. To Make Inferences in FOL • Method 1 • Unification of variables with literals (in the KB) • Generalized Modus Ponens • Forward-chaining or Backward-chaining • Method 2 • Resolution

  20. Unification • We want to find a substitution  such that x and y match literals • Unify (,) =  if  =  • Some examples

  21. Standardizing apart eliminates overlap of variables, e.g., Knows(z17,Michel)

  22. Unification for example

  23. P`1= father-of(Kathy)=Michael P1= father-of(x)=y ={x/Kathy,y/Michael} q=ancestor(x,y) q`=ancestor(Kathy,Michael)

  24. Example inference using forward chaining (production systems)

  25. Properties of forward-chaining • Sound and complete for first-order definite clauses • Datalog is first-order definite clauses and no functions • May not terminate in general if is not entailed • This is unavoidable: entailment with definite clauses is semi-decidable • Forward chaining is widely used in deductive databases

  26. Example inference using backward chaining

  27. Properties of backward-chaining • Depth-first recursive proof search: space is linear in size of proof • Incomplete due to infinite loops • Fix by checking current goal against every goal on stack • Inefficient due to repeated subgoals (both success and failure) • Fix using cache of previous results (extra space!) • Widely used (without improvements!) for logic programming (e.g., Prolog)

  28. Midterm results • Exams will only be given back to person the owner of the exam

More Related