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. 2 Announcements  Final Exam Date  Dec. 19 th  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. 3 Types of Inference  Resolution Theorem proving  Model Checking  Forward chaining with modus ponens  Backward chaining with modus ponens

4. 4 One Problem done all ways

5. 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. 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 enumerate every entailed sentence

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11. 11 Example

12. 12 Characteristics of FOL  Declarative  Expressive  Partial information  Negation  Compositionality

13. 13 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

14. 14 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)

15. 15 Problems with propositional logic

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17. 17 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

18. 18 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: ,

19. 19 Atomic Sentences  Atomic sentence = predicate (term 1 …term m ) or term 1 =term 2  Term = function (term 1, …, term m ) or constant or variable  E.g. know(Charles,Ken), Adjacent (x,y), father-of(Kathy) = Michael, Victor, x

20. 20 Complex Sentences  Complex sentences are made from atomic sentences using connectives ⌐S, S 1 ΛS 2, S 1 VS 2, S 1  S 2, S 1  S 2  E.g., adjacent(x,y)  adjacent (y,x), ⌐knows(Charles, Michael),

21. 21 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 (term 1,…,term n ) is true iff the objects referred to by term 1,…, term n are in the relation referred to by predicate.

22. 22 Universal quantification    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) …..

23. 23 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)

24. 24 Existential Quantification    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)

25. 25 Another Common Mistake  Typically, Λ is the main connective with   Common mistake: using  as the main connective  x At(x,Columbia)  Smart(x)

26. 26 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)

27. 27 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)

28. 28 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)

29. 29 Translation from English to FOL  A mother is a female parent  Andrew likes one of the homework problems  ?