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First-Order Logic Reading: C. 8 and C. 9 Pente specifications handed back at end of class 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) 2 Characteristics of FOL Declarative Expressive Compositionality • Partial information • Negation 3 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 4 Ontological commitments of higher order logics Temporal logic • Facts hold at particular times and those times are ordered Epistemological • The agent believes a fact • The agent does not believe it • The agent has no opinion Probabilistic • Agents hold beliefs about facts • Three possible states of knowledge • Facts are true to different degrees (Truth value from 0 to 1) 5 Problems with propositional logic 6 7 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 Need relations Need functions • Squares, pits, Kathy • Adjacent, breezy, smelly, know • Father-of, mother-of 8 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: , 9 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 10 Complex Sentences Complex sentences are made from atomic sentences using connectives ⌐S, S1ΛS2, S1VS2, S1S2, S1S2 E.g., adjacent(x,y) adjacent (y,x), ⌐knows(Nunzio, Michael), 11 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. 12 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) ….. 13 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) 14 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) 15 Another Common Mistake Typically, Λ is the main connective with Common mistake: using as the main connective x At(x,Columbia) Smart(x) 16 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) 17 Translation from English to FOL A mother is a female parent Andrew likes the problem of one of the book exercises ? 18 Example Family trees What does the model look like? What can we infer? • Father-of • Mother-of • Sibling • Cousin • Ancestors 19 To Make Inferences in FOL Method 1 Method 2 • Unification of variables with literals (in the KB) • Generalized Modus Ponens • Forward-chaining or Backward-chaining • Resolution 20 Unification We want to find a substitution such that x and y match literals Unify (,) = if = Some examples 21 Knows(John,x) Knows(John, Jane) {x/Jane} Knows(John,x) Knows(y,Michel) {x/Michel,y/John} Knows(John,x) Knows(y,Motherof(y)) {y/John,x/Motherof(John) Knows(John,x) Knows(x,Michel) fail Standardizing apart eliminates overlap of variables, e.g., Knows(z17,Michel) Unification for example 23 P`1= father-of(Kathy)=Michael P1= father-of(x)=y ={x/Kathy,y/Michael} q=ancestor(x,y) q`=ancestor(Kathy,Michael) 24 25 Example inference using forward chaining (production systems) 26 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 27 28 Example inference using backward chaining 29 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) 30 Midterm results Exams will only be given back to person the owner of the exam 31