CSE PredLogic 1 Knowledge Representation with Logic: First Order Predicate Calculus Outline –Introduction to First Order Predicate Calculus (FOPC) syntax semantics –Entailment –Soundness and Completeness

CSE PredLogic 2 Modeling Our World with Propositional Logic Limited. –Quickly gets explosive and cumbersome, can’t express generalizations –Can’t distinguish between objects and relations

CSE PredLogic 3 Comparing Logics Ontology (ont = ‘to be’; logica = ‘word’): kinds of things one can talk about in the language Examples: Propositional Logic Facts Predicate Logic Objects, Relationships among Objects Temporal Logics Time Points or Intervals (divorce)

CSE PredLogic 4 Syntax of Predicate Logic Symbol set – give examples for: –constants –Boolean connectives –variables –functions –predicates (aka relations) –Quantifiers Terms: variables, constants, functional expressions (can be arguments to predicates)

CSE PredLogic 5 Syntax of Predicate Logic Sentences: –atomic sentences (predicate expressions, literals) Ground literal? –complex sentences (atomic sentences connected by Booleans) –quantified sentences

CSE PredLogic 6 Examples of Terms: Constants, Variables and Functions Constants –Alan, Sam, R225, R216 Variables –PersonX, PersonY, RoomS, RoomT Functions –father_of(PersonX) –product_of(Number1,Number2)

CSE PredLogic 7 Examples of Predicates and Quantifiers Predicates –in(Alan,R225) –partOf(R225,Pender) –fatherOf(PersonX,PersonY) Quantifiers –All dogs are mammals. –Some birds can’t fly. –3 birds can’t fly.

CSE PredLogic 8 Semantics of Predicate Logic A term is a reference to an object –constants –variables –functional expressions Sentences make claims about objects –Well-formed formulas, (wffs)

CSE PredLogic 9 Semantics, part 2 Object constants refer to individuals There is a correspondence between –functions, which return values –Predicates (or relations), which are true or false Function: father_of(Mary) = Bill Predicate: father_of(Mary, Bill)

CSE PredLogic 10 Semantics, part 3 Referring to individuals –Jackie –son-of(Jackie), Sam Referring to states of the world –person(Jackie), female(Jackie) –mother(Sam, Jackie)

CSE PredLogic 11 Combining Logical Symbols Terms: logical expressions referring to objects –first([a,b,c]), sq_root(9), sq_root(n), tail([a,b,c]) Atomic Sentences: –loves(John,Mary), brother_of(John,Ted) Complex Sentences: –loves(John,Mary) brother_of(John,Ted) teases(Ted, John)

CSE PredLogic 12 Encoding Facts pass(John, courses,40) => graduate(John) cavity(molar) => x-ray_shadow(molar) leak(pipe, kitchen) /\ full(pipe,water) => location(water, kitchen_floor)

CSE PredLogic 13 KB Design Choices Design choice: red(block1) color(block1, red) val(color,block1,red) 2 nd order predicate calculus.... Implication of choice: ????? nice(red) property(color)

CSE PredLogic 14 Quantifiers Universal Quantifiers. –All cats are mammals. Existential Quantifiers. –There is a cat owned by John.

CSE PredLogic 15 Restricting Quantifiers To say “All lawyers ___.” (  x Lawyer(x)  _____ ) To say “a lawyer ___.” “some lawyers ___.” (  x Lawyer(x)  _____ ) To say “no lawyers ___.” (  x Lawyer(x)   _____ )  (  x Lawyer(x)  _____ ) All lawyers do not ___. There does not exist a lawyer who ___.

CSE PredLogic 16 Negation and Quantification When you move negation inside or outside of a quantifier, then  and  get flipped. “All lawyers are not naïve.” is equivalent to “There is no lawyer that is naïve.” (Conditional Law) (DeMorgan’s Law) (Flip Quantifier)

CSE PredLogic 17 Nested Quantification There is some cat that has some owner. All cats have an owner (but that owner could be different). There is some person who owns all the cats in the world. For all the owners and ownees in the world, the owner loves the ownee. Order matters when some are  and some .

CSE PredLogic 18 The Power of Expressivity Indirect knowledge: Tall(MotherOf(john)) Counterfactuals: ¬Tall(john) Partial knowledge (disjunction): IsSisterOf(b,a)  IsSisterOf(c,a) Partial knowledge (indefiniteness):  xIsSisterOf(X,a)

CSE PredLogic 19 Inference Procedures Mechanical rules that compute (derive) a new sentence from a set of sentences. Proof theory: set of rules for deducing the entailments of a set of sentences. Terminology: –proof = sequence of inference rule applications –derived wff = result of proof, theorem

CSE PredLogic 20 General Resolution Unit resolution with variables (requires unification) and more complex sentences Complete – if something can be proven from KB, general resolution will prove it

CSE PredLogic 21 Entailment is a Strong Requirement Q is a sentence; KB is a set of sentences If whenever the sentences in KB are true, Q is true, then KB Q (KB “entails” Q)

CSE PredLogic 22 Logic as a representation of the World entails Representation: Sentences Sentence Refers to (semantics) follows World: Facts Fact

CSE PredLogic 23 Soundness & Completeness Soundness: anything derived is entailed; inference procedure (rule) only produces entailments. Completeness: anything entailed is derived; inference procedure (rules) produces all entailments.

CSE PredLogic 24 General Resolution Complete, but only semi-decidable If its true it can be proven If its not true, theorem prover might not halt, may not be able to prove that its not true Closed World Assumption – no missing information, if P can’t be proven, assume P is false.