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

2. 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. 3 Characteristics of FOL Declarative Expressive Partial information Negation Compositionality

4. 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. 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. 6 Problems with propositional logic

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8. 8 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

9. 9 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: , 

10. 10 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(Kathy,Sowmya), Adjacent (x,y), father-of(Kathy) = Michael, Andrew, x

11. 11 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(Nunzio, Michael),

12. 12 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.

13. 13 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) …..

14. 14 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)

15. 15 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)

16. 16 Another Common Mistake Typically, Λ is the main connective with  Common mistake: using  as the main connective  x At(x,Columbia)  Smart(x)

17. 17 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)

18. 18 Translation from English to FOL A mother is a female parent Andrew likes the problem of one of the book exercises ?

19. 19 Example Family trees What does the model look like? Father-of Mother-of Sibling What can we infer? Cousin Ancestors

20. 20 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

21. 21 Unification We want to find a substitution  such that x and y match literals Unify ( ,  ) =  if  =  Some examples

22.  Knows(John,x)Knows(John, Jane) {x/Jane} Knows(John,x)Knows(y,Michel){x/Michel,y/John} Knows(John,x)Knows(y,Mother- of(y)) {y/John,x/Mother- of(John) Knows(John,x)Knows(x,Michel)fail Standardizing apart eliminates overlap of variables, e.g., Knows(z 17,Michel)

23. 23 Unification for example

24. 24 P`1= father-of(Kathy)=Michael P 1 = father-of(x)=y  ={x/Kathy,y/Michael} q=ancestor(x,y) q`=ancestor(Kathy,Michael)

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26. 26 Example inference using forward chaining (production systems)

27. 27 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

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29. 29 Example inference using backward chaining

30. 30 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)

31. 31 Midterm results Exams will only be given back to person the owner of the exam