First-Order Logic

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

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. 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. 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

11. 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),

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 (term1,…,termn) is true iff the objects referred to by term1,…, termn are in the relation referred to by predicate.

13. 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) …..

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. 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)

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

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

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

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

23. Unification for example

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

26. Example inference using forward chaining (production systems)

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

29. Example inference using backward chaining

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