1 / 29

Propositional Logic

Propositional Logic. Agenda: Other forms of inference in propositional logic Basics of First Order Logic (FOL) Vision Final Homework now posted on web site. Announcements. Final Exam Date Dec. 19 th 1:10-4pm 833 Mudd Getting homeworks back Game playing will be returned 12/10 in class

keaira
Download Presentation

Propositional 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. Propositional Logic Agenda: Other forms of inference in propositional logic Basics of First Order Logic (FOL) Vision Final Homework now posted on web site

  2. Announcements • Final Exam Date • Dec. 19th • 1:10-4pm • 833 Mudd • Getting homeworks back • Game playing will be returned 12/10 in class • Machine learning will be returned in final exam • Class participation grade will be posted by 12/10 • Midterm curve will be given in class 12/10 • Final class will wrap up vision and do what’s next and review

  3. Types of Inference • Resolution Theorem proving • Model Checking • Forward chaining with modus ponens • Backward chaining with modus ponens

  4. One Problem done all ways

  5. Model Checking • Enumerate all possible worlds • Restrict to possible worlds in which the KB is true • Check whether the goal is true in those worlds or not

  6. Inference as Search • State: current set of sentences • Operator: sound inference rules to derive new entailed sentences from a set of sentences • Can be goal directed if there is a particular goal sentence we have in mind • Can also try to enumerateevery entailed sentence

  7. Example

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

  9. 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

  10. 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)

  11. Problems with propositional logic

  12. 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

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

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

  15. 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(Charles, Michael),

  16. 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.

  17. 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) …..

  18. 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)

  19. 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)

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

  21. 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) •  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)

  22. 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) • 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)

  23. 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)

  24. Translation from English to FOL • A mother is a female parent • Andrew likes one of the homework problems • ?

More Related