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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

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Propositional Logic

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  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 • ?

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