Knowledge Representation with Logic: First Order Predicate Calculus

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

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

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) PredLogic

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) PredLogic

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

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) PredLogic

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. PredLogic

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) PredLogic

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) PredLogic

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

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) PredLogic

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

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

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

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 ___. PredLogic

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) PredLogic

Nested Quantification Order matterswhen some are and some . 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. PredLogic

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) PredLogic

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 PredLogic

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 PredLogic

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) PredLogic

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

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

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. PredLogic

Knowledge Representation with Logic: First Order Predicate Calculus