1 first order logic. 2 knowledge representation & reasoning introduction propositional logic is...

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1 First Order Logic

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Page 1: 1 First Order Logic. 2 Knowledge Representation & Reasoning  Introduction Propositional logic is declarative Propositional logic is compositional: meaning

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First Order Logic

Page 2: 1 First Order Logic. 2 Knowledge Representation & Reasoning  Introduction Propositional logic is declarative Propositional logic is compositional: meaning

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Knowledge Representation & Reasoning

Introduction

Propositional logic is declarative

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 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)

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Knowledge Representation & Reasoning

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))

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Knowledge Representation & Reasoning

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: red, round, prime, brother of, bigger

than, part of, …• Functions: Sqrt, Plus, …

Can express the following:– Squares neighboring the Wumpus are smelly;– Squares neighboring a pit are breezy.

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Knowledge Representation & Reasoning

Syntax Order LogicUser defines these primitives:1. Constant symbols (i.e., the

"individuals" in the world) e.g., Mary, 32. Function symbols (mapping

individuals to individuals) e.g., father-of(Mary) = John, colorof(Sky) = Blue

3. Predicate/relation symbols (mapping from individuals to truth values) e.g., greater(5,3),

green(apple), color(apple, Green)

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Knowledge Representation & Reasoning

Syntax Order LogicFOL supplies these primitives:1. Variable symbols. e.g., x,y2. Connectives. Same as in PL: ⇔,∧,∨, ⇒

3. Equality =

4. Quantifiers: Universal (∀) and Existential (∃)

A legitimate expression of predicate calculus is called a well-formed formula (wff) or, simply, a sentence.

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Knowledge Representation & Reasoning

Syntax Order Logic

Quantifiers: Universal (∀) and Existential (∃)

Allow us to express properties of collections of objects instead of enumerating objects by name

Universal: “for all”: ∀<variables> <sentence>

Existential: “there exists” ∃<variables> <sentence>

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Knowledge Representation & Reasoning

Syntax Order Logic: Constant Symbols

• A symbol, e.g. Wumpus, Ali.

• 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.

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Knowledge Representation & Reasoning

Syntax Order Logic: Variables• Used to represent objects or properties

of objects without explicitly naming the object.

• Usually lower case.

• For example:– x– father– square

.

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Knowledge Representation & Reasoning

Syntax 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:Instructor (CAP492, Souham)

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Knowledge Representation & Reasoning

Syntax 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

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Knowledge Representation & Reasoning

Syntax 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)

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Knowledge Representation & Reasoning

Syntax Order Logic: Atomic sentence

Atomic sentence = predicate (term1,...,termn)

or term1 = term2Term = function (term1,...,termn) or constant or variableExample terms:Brother(Ali , Mohamed)Greater(Length(x), Length(y))

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Knowledge Representation & Reasoning

Syntax 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,y Sibling(x,y) ⇒ Sibling(y,x)

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Knowledge Representation & Reasoning

While constant symbols, variables and connectives are like propositional logic, “What are functions and predicates?”

The language of logic is based on set theory:

– Sets;– Relations;– Functions.

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Knowledge Representation & Reasoning

SETS: The set of objects defines a “Universe of Discourse.” [Objects are represented by Constant Symbols.]

a

bc

de

Interpretation:A aB bC cD dE e

For example, in this blocks world, the universe of discourse is {a,b,c,d,e}.

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Knowledge Representation & Reasoning

• RELATIONS: Def: A binary relation is a set of ordered pairs

• Example: Consider set of blocks {a,b,c,d,e}

a

bc

de

The “on” relation:on = {<a,b>, <b,c>, <d,e>}.The predicate On(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.

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Knowledge Representation & Reasoning

• 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=zIf <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).

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Knowledge Representation & Reasoning

• FUNCTIONS: a

bc

de

hat = {<c,b>, <b,a>,<e,d>}hat (c) = bhat(b) = ahat(d) is not defined.

Hat(E) can be interpreted as d

Using FOL

On(A,B)On(B,C)On(D,E)On(A,Hat(C))

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Knowledge Representation & Reasoning

Syntax of First Order Logic

Sentence → Atomic Sentence |(sentence connective Sentence)

| Quantifier variable,… Sentence | ¬ Sentence

Atomic Sentence → Predicate (Term,…) |Term=TermTerm → Function(Term,…) | Constant |variable

Connective → ⇔ | ∧ | ∨ | Quantifier → ∀ | ∃Constant → A |X1…Variable → a | x | s | …Predicate → Before | hascolor | ….Function → Mother | Leftleg |…

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Knowledge Representation & Reasoning

Inference in First Order Logic

Inference in FOL can be performed by:

Reducing first-order inference to propositional inference

Unification Generalized Modus Ponens Resolution Forward chaining Backward chaining

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Knowledge Representation & Reasoning

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.

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Knowledge Representation & Reasoning

Inference in First Order LogicSubstitutionGiven a sentence α and 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/Sam, y/Pam}, Likes(x,y)) =

Likes(Ali, fatima)

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Knowledge Representation & ReasoningInference in First Order Logic

Universal instantiation (UI)Every instantiation of a universally quantifiedsentence is entailed by it:

∀v αSubst({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))

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Knowledge Representation & ReasoningInference 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 αSubst({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

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Knowledge Representation & Reasoning

Inference in First Order LogicReduction to propositional inferenceSuppose 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)• The new KB is propositionalized

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Knowledge Representation & ReasoningInference in First Order Logic

Reduction to PL• CLAIM: A ground sentence is entailed by a new KB iff entailed by the original KB.

• CLAIM: 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)))

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Knowledge Representation & Reasoning

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)∀ y Greedy(y)

It is intuitively clear that we can substitute{x/John,y/John} and obtain that Evil(John)

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Knowledge Representation & ReasoningInference in First Order Logic

2. 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)

• α β SubstKnows(John,x) Knows(John,Jane) {x/Jane}Knows(John,x) Knows(y,OJ)

{x/OJ,y/John}Knows(John,x) Knows(y,Mother(y))

{y/John,x/Mother(John)}}Knows(John,x) Knows(x,OJ) {fail}

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Knowledge Representation & Reasoning

Inference in First Order Logic3. 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.

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Knowledge Representation & Reasoning

Inference in First Order Logic3. ResolutionFull 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

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Knowledge Representation & ReasoningInference in First Order Logic

4. 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 known then add the conclusion to the KB

and continue chaining.

Forward chaining is data-driven,e.g., inferring properties and categories from percepts.

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Knowledge Representation & ReasoningInference in First Order Logic

4. Forward chaining example.• Rules

1. Buffalo(x) Pig(y) Faster(x, y) 2. Pig(y) Slug(z) Faster(y, z) 3. Faster(x, y) Faster(y, z) Faster(x, z)

Facts1. Buffalo(Bob)2. Pig(Pat)3. Slug(Steve)

New facts4. Faster(Bob,Pat)5. Faster(Pat, Steve)6. Faster (Bob, Steve)Buffalo(Bob) Pig (Pat)

{y/Pat}{x/Bob}

Faster (Bob,Pat)

Slug(Steve)

Faster(Pat,Steve)

Faster(Bob,Steve)

{z/Steve}

{x/Bob, y/Pat} {y/Pat, z/Steve}

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Knowledge Representation & ReasoningInference in First Order Logic

4. Backward chainingBackward chaining starts with a hypothesis and work

backwards , according to the rules in the knowledge base until reaching confirmed findings or facts.

Pig(y)∧Slug(z) ⇒ Faster (y, z)

Slimy(a)∧ Creeps(a) ⇒Slug(a)

Pig(Pat)Slimy(Steve)Creeps(Steve)

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Knowledge Representation & Reasoning

Other knowledge representation schemes

1. Semantic Networks A semantic network consists of entities and

relations between the entities. Generally it is represented as a graph.

• A bird is a kind of animal• Flying is the normal moving method of birds• An albatross is a bird• Albert is an albatross, and so is Ross

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Knowledge Representation & Reasoning

Other knowledge representation schemes

1. Semantic Networks

Birddaylight fly

animal

albatross

Black and whiteAlbert

Ross

colourisa isa

isa

Moving-methodActive-at

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Knowledge Representation & Reasoning

Other knowledge representation schemes

2. Frames A frame is a data structure whose components

are called slots. Slots have names and accomodate information of

various kinds.

Example:Frame: Bird A_kind_of: animal Moving_method: fly Active_at:daylight

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Knowledge Representation & Reasoning Expert Systems (ES)

Definition: an ES is a program that behaves like an expert for some problem domain.

Should be capable of explaining its decisions and the underlying reasoning.

Often, it is expected to be able to deal with uncertain and incomplete information.

Application domains: Medical diagnosis (MYCIN), locating equipment failures,…

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Knowledge Representation & Reasoning Expert Systems (ES)

Functions and structure: Expert systems are designed to lve

problems that require expert knowledge in a particular domain → possessing knowledge in some form.

ES are known as knowledge based systems.

Expert systems have to have a friendly user interaction capability that will make the system’s reasoning transparent to the user.

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Knowledge Representation & Reasoning Expert Systems (ES)

Functions and structure: The structure of an ES includes

three main modules: 1. A knowledge base2. An inference engine3. A user interface

Knowledge base

Inference engine

User interfaceuser

Shell

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Knowledge Representation & Reasoning Expert Systems (ES)

Functions and structure:1. The knowledge base comprises the

knowledge that is specific to the domain of application: simple facts, rules, constraints and possibly also methods, heuristics and ideas for solving problems.

2. Inference engine: designed to use actively the knowledge in the base deriving new knoweldge to help decision making.

3. User interface: caters for smooth communication between the user and the system.

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Knowledge Representation & Reasoning

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 sevaral 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 knoweldge representation and reasoning.