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Chapter 6

King Saud University College of Computer and Information Sciences Information Technology Department IT422 - Intelligent systems . Chapter 6. Knowledge Representation & Reasoning (part 2) First Order Logic. Introduction. Propositional logic pros:

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Chapter 6

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  1. King Saud University College of Computer and Information Sciences Information Technology Department IT422 - Intelligent systems Chapter 6 Knowledge Representation & Reasoning (part 2) First Order Logic

  2. Introduction • Propositional logic pros: • Propositional logic is declarative: Knowledge base and inference are separated • Propositional logic is compositional: meaning of B1,1∨ P1,2 is derived from meaning of B1,1 and of P1,2 • Meaning in propositional logic is context-independent unlike natural language, where meaning depends on context • Propositional logic cons: • Propositional logic has limited expressive power unlike natural language • e.g. cannot say "pits cause breezes in adjacent squares“ (except by writing one sentence for each square) Question : How can we write one sentence only that can be applied on a group of objects?

  3. From propositional logic (PL) to First order logic (FOL) • Examples of things we can say: • All men are mortal: • ∀x Man(x) ⇒ Mortal(x) • Everybody loves somebody • ∀x ∃y Loves(x, y) • The meaning of the word “above” • ∀ x ∀ y above(x,y) ⇔(on(x,y) ∨ ∃ z (on(x,z) ∧ above(z,y))

  4. First Order Logic • Whereas propositional logic assumes the world contains facts, first-order logic (like natural language) assumes the world contains: • Objects: people, houses, numbers, colors, … • Relations: • Unary(properties) red, round, prime, • N-ary: brother of, bigger than, part of, … • Functions:Sqrt, Plus, … • Can express the following: Squaresneighboring the Wumpus are smelly Squaresneighboring a pit are breezy

  5. Syntax of First Order Logic • User defines these primitives: • Constant symbols (i.e., the "individuals" in the world) e.g., Mary, 3, apple • Predicate/relation symbols (mapping from individuals to truth values) e.g., greater(5,3), green(apple), color(apple, Green) • Function symbols (mapping individuals to individuals) e.g., fatherOf(Mary) = John, colorOf(Sky) = Blue

  6. Syntax of First Order Logic • FOL supplies these primitives: • Variable symbols. x, y, … • Connectives. Same as in PL: ⇔,∧,∨, ⇒ • Equality = • Quantifiers: Universal (∀) : “for all”: and Existential (∃) : “thereexists” • A legitimate expression of FOL (also called first order predicate calculus) is called a well-formed formula (wff) or, simply, a sentence.

  7. Syntax of First Order Logic: Constant Symbols • Each constant symbol names exactly one object in a universe of discourse, but: • not all objects have symbol names; • some objects have several symbol names. • Usually denoted with upper-case first letter. • e.g. Wumpus, Ali.

  8. Syntax of First Order Logic: Relation (Predicate) Symbols • A predicate symbol is used to represent a relation in a universe of discourse. • The sentence: Relation(Term1, Term2,…) • is either TRUE or FALSE depending on whether Relation holds of Term1, Term2,… • To write “Malek wrote Muata” in a universe of discourse of names and written works: • Wrote(Malek, Muata)  This sentence is true in the intended interpretation. • Another example: - A proposition in predicate logic: Taking(IT422, Sarah)

  9. Syntax of First Order Logic: Function symbols • Functions talk about the binary relation of pairs of objects. • For example, the Father relation might represent all pairs of persons in father-daughter or father-son relationships: • Father(Ali)  Refers to the father of Ali • Father(x)  Refers to the father of variable x

  10. Syntax of First Order Logic: Variables • Used to represent objects or properties of objects without explicitly naming the object. • Usually lower case. • For example: • x • father • square

  11. Syntax of First Order Logic: Quantifiers • Quantifiers: Universal (∀) and Existential (∃) • Allow us to express properties of collections of objects instead of enumerating objects by name • Universal:“for all”: • ∀ <variables> <sentence> • ∀x At(x,KSU) ⇒ Smart(x) • Existential: “there exists” • ∃ <variables> <sentence> • ∃x At(x,PNU) ∧ Smart(x)

  12. Syntax of First Order Logic: Quantifiers • ⇒ is the main connective with (∀) • ∀x At(x,KSU) ⇒ Smart(x) • “everyone that is at KSU is smart” • ∧ is the main connective with (∃) • ∃x At(x,PNU) ∧ Smart(x) • “there are students at PNU that are smart”

  13. Syntax of First Order Logic: properties of quantifiers • ∀ x ∀ y is the same as ∀ 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 in the world is loved by at least one person” • 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)

  14. Syntax of First Order Logic: Atomic sentence • Atomic sentence = • predicate (term1,...,termn). • term1 = term2 • Term = • function (term1,...,termn). • constant. • variable • Example terms: • Brother(Ali , Mohamed) • Greater(Length(x), Length(y))

  15. Syntax of First Order Logic: Complex sentence • Complex sentences are made from atomic sentences using connectives and by applying quantifiers. • Examples: • Sibling(Ali,Mohamed) ⇒ Sibling(Mohamed,Ali) • greater(1,2) ∨ less-or-equal(1,2) • ∀ x,ySibling(x,y) ⇒ Sibling(y,x)

  16. Syntax of First Order Logic Sentence→ Atomic Sentence |(Sentence connective Sentence) | Quantifier variable,… Sentence | ¬ Sentence Atomic Sentence → Predicate (Term,…) |Term=Term Term → Function(Term,…) | Constant |variable Connective →⇔ | ∧ | ∨ |  Quantifier →∀ | ∃ Constant→ A |X1… Variable→ a | x | s | … Predicate→ Before |hascolor| …. Function→ Mother | Leftleg |…

  17. The language of logic • In first order logic, we use functions and predicates • The language of logic is based on set theory: • Sets • Relations • Functions

  18. a d b c e The language of logic • SETS: • The set of objects defines a “Universe of Discourse.” [Objects are represented by Constant Symbols.] Interpretation: A  a B  b C  c D  d E  e For example, in this blocks world, the universe of discourse is {a,b,c,d,e}.

  19. a d b c e The language of logic • RELATIONS: • A binary relation is a set of ordered pairs • Example: Consider set of blocks {a,b,c,d,e} The “on” relation: on = {<a,b>, <b,c>, <d,e>}. The predicateOn(A,B) can be interpreted as: <a,b>  on. On(A,B) is TRUE, but On(A,C) and On(C,D) are FALSE in this interpretation.

  20. The language of logic • FUNCTIONS: • A function is a binary relation such that no two distinct members have the same first element. • In other words, if F is a function • <x,y>  F and <x,z>  F  y=z • If <x,y>  F : • x is an argument of F ; • y is the value of F at x ; • y is the image of x under F. • F(x) designates the object y such that y=F(X).

  21. a d b c e The language of logic • FUNCTIONS: • hat = {<c,b>, <b,a>,<e,d>} • hat (c) = b • hat(b) = a • hat(d) is not defined. • Hat(E) can be interpreted as d • UsingFOL • On(A,B) • On(B,C) • On(D,E) • On(A, Hat(C))

  22. RECALL • Knowledge base is a set of sentences. Each sentence is expressed in a language called a knowledge representation language. • A sentence represents some assertion about the world. • Entailment α╞ β if and only if in every model in which α is true, β is also true • Inference is the process of deriving new sentences from old ones KB α • Sound inference algorithms derive only sentences that are entailed. • Complete inference algorithms derive all sentences that are entailed. • Model checking Enumerates all possible models to check that α is true in all models in which KB is true. • Inference rules (e.g. modus ponens, resolution rule)are patterns of sound inference that can be used to find proofs • Proof a sequence of applications of inference rules. • Resolution is a sound and complete method of inference • Forward chaining and backward chaining are natural reasoning algorithms for knowledge bases in Horn form. • Horn clause is a disjunction of literals of which at most one is positive. i

  23. Inference in First Order Logic Purpose of inference: KB ╞ α ? Inference in FOL can be performed by: • Reducing FOL to PL and then apply PL inference • Use inference rules of FOL • Unification • Generalized Modus Ponens • Resolution

  24. Inference in First Order Logic From FOL to PL • First order inference can be done by converting the knowledge base to PL and using propositional inference. • Two questions ? • How to convert universal quantifiers (∀)? • Replace variable by ground term. • How to convert existential quantifiers (∃)? • Skolemization.

  25. Inference in First Order Logic Substitution • Given a sentence αand binding list , the result of applying the substitution  to α is denoted by Subst(, α). Example:  = {x/Ali, y/Fatima} = Likes(x,y) Subst({x/Ali, y/Fatima}, Likes(x,y)) = Likes(Ali, Fatima)

  26. Inference in First Order Logic Universal instantiation (UI) • Every instantiation of a universally quantified sentence is entailed by it: ∀ v α Subst({v/g}, α) • for any variable v and ground term g e.g., ∀ x King(x) ∧ Greedy(x) Evil(x)yields: King(John) ∧ Greedy(John) Evil(John) King(Richard) ∧ Greedy(Richard) Evil(Richard) King(Father(John)) ∧ Greedy(Father(John))Evil(Father(John)) • UI can be applied several times to add new sentences

  27. Inference in First Order Logic Existential instantiation (EI) • For any sentence α, variable v, and constant symbol k that does not appear elsewhere in the knowledge base: ∃v α Subst({v/k}, α) • e.g., ∃x Crown(x) ∧OnHead(x,John) yields: Crown(C1) ∧OnHead(C1,John) provided C1 is a new constant symbol, called a Skolem constant • EI can be applied once to replace the existential sentence

  28. Inference in First Order Logic Reduction to propositional inference • Suppose the KB contains just the following: • ∀ x King(x) ∧ Greedy(x)Evil(x) • King(John) • Greedy(John) • Brother(Richard,John) • Instantiating the universal sentence in all possible ways, we have: • King(John) ∧ Greedy(John)Evil(John) • King(Richard) ∧ Greedy(Richard) Evil(Richard)  irrelevant substitution • King(John) • Greedy(John) • Brother(Richard,John) • The new KB is propositionalized

  29. Inference in First Order Logic Reduction to PL • A ground sentence is entailed by a new KB iff entailed by the original KB. • Every FOL KB can be propositionalized so as to preserve entailment • IDEA: propositionalize KB and query, apply resolution, return result • PROBLEM: with function symbols, there are infinitely many ground terms, e.g., Father(Father(Father(John)))

  30. Inference in First Order Logic • Instead of translating the knowledge base to PL, we can make the inference rules work in FOL. • For example, given: • ∀ x King(x) ∧ Greedy(x)  Evil(x) • King(John) • Greedy(John) • Can we prove Evil(John)? • The inference that John is evil works like this: • find some x such that x is a king and x is greedy, • and then infer that x is evil. • Generally: • It is intuitively clear that we can substitute {x/John} and obtain that Evil(John)

  31. Inference in First Order Logic • What if we have: • ∀ x King(x) ∧ Greedy(x)  Evil(x) • King(John) • ∀ y Greedy(y) • It is intuitively clear that we can substitute {x/John,y/John} and obtain that Evil(John)

  32. Inference in First Order Logic Unification • We can make the inference if we can find a substitution such that King(x) and Greedy(x) match King(John) and Greedy(y){x/John,y/John} works • Unify(α ,β) = θ if Subst(θ, α) = Subst(θ, β)

  33. Inference in First Order Logic 1) Generalized Modus Ponens (GMP) • Suppose that Subst( θ, pi’) = Subst( θ, pi) for all i then: p1', p2', … , pn', ( p1 ∧ p2 ∧ … ∧ pn q ) Subst(θ, q) • p1' is King(John) p1 is King(x) • p2' is Greedy(y) p2 is Greedy(x) • θ is {x/John,y/John} q is Evil(x) • Subst(θ, q) is Evil(John) • All variables assumed universally quantified.

  34. Inference in First Order Logic 2) Resolution • Full first-order version: l1 ∨ ··· ∨ lk, m1 ∨ ··· ∨ mn Subst( θ, l1 ∨ ··· ∨ li-1 ∨ li+1 ∨ ··· ∨ lk ∨ m1 ∨ ··· ∨ mj-1 ∨ mj+1 ∨ ··· ∨ mn) where θ = Unify( li, ¬mj) ¬Rich(x) ∨ Unhappy(x) , Rich(Ken) Unhappy(Ken) with θ = {x/Ken} • Apply resolution steps to CNF(KB ∧ ¬α); complete for FOL

  35. Inference in First Order Logic Forward chaining When a new fact P is added to the KB: For each rule such that P unifies with a premise,if the other premises are knownthen add the conclusion to the KB and continue chaining. • Forward chaining is data-driven,e.g., inferring conclusions from incoming percepts.

  36. Inference in First Order Logic Forward chaining example • Rules • Buffalo(x)  Pig(y)  Faster(x, y) • Pig(y)  Slug(z)  Faster(y, z) • Faster(x, y)  Faster(y, z)  Faster(x, z) • Facts • Buffalo(Bob) • Pig(Pat) • Slug(Steve) • New facts • Faster(Bob, Pat) • Faster(Pat, Steve) • Faster (Bob, Steve) Faster(Bob,Steve) {x/Bob, y/Pat} {y/Pat, z/Steve} Faster(Pat,Steve) Faster (Bob,Pat) {y/Pat} {x/Bob} {z/Steve} Buffalo(Bob) Pig (Pat) Slug(Steve)

  37. Inference in First Order Logic Backward chaining Backward chaining starts with a hypothesis (query) and work backwards, according to the rules in the knowledge base until reaching confirmed findings or facts. Example • Pig(y) ∧ Slug(z) ⇒ Faster (y, z) • Slimy(z) ∧ Creeps(z) ⇒Slug(z) • Pig(Pat) • Slimy(Steve) • Creeps(Steve)

  38. Other knowledge representation schemes Semantic Networks • A semantic network consists of entities and relations between the entities. • Generally it is represented as a graph. • Example: • A bird is a kind of animal • Flying is the normal moving method of birds • Birds are usually active at the day-light • An albatross is a bird • Albert is an albatross, and so is Ross

  39. Other knowledge representation schemes Semantic Networks • Example: animal Active-at Moving-method Bird fly daylight Is a albatross colour Is a Is a Black and white Ross Albert

  40. Other knowledge representation schemes Frames • A frame is a data structure whose components are called slots. • Slots have names and accommodate information of various kinds. • Example: Frame: Bird A_kind_of: animal Moving_method: fly Active_at: daylight

  41. Summary • FOL extends PL by adding new concepts such as sets, relations and functions and new primitives such as variables, equality and quantifiers. • There exist several alternatives to perform inference in FOL. • Logic is not the only one alternative to represent knowledge. • Inference algorithms depend on the way knowledge is represented. • Development of expert systems relies heavily on knowledge representation and reasoning.

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