Knowledge Representation using First-Order Logic(Part II)

Reading: R&N Chapters 8, 9

Outline

• Review: KB |= S is equivalent to |= (KB S)– So what does {} |= S mean?

• Review: Follows, Entails, Derives– Follows: “Is it the case?”– Entails: “Is it true?”– Derives: “Is it provable?”

• Review: FOL syntax

• Finish FOL Semantics, FOL examples

• Inference in FOL

• Next Tuesday (1 June) is Review/Catch-up; Homework#7 is due• Next Thursday (3 June) NO CLASS; I’m reviewing grants for NIH• Next Friday (4 June) PROJECT REPORTS & CODE is due• Following Thursday (10 June) FINAL EXAM; 1:30-3:30pm

Review: KB |= S means |= (KB S)

• KB |= S is read “KB entails S.”– Means “S is true in every world (model) in which KB is true.”– Means “In the world, S follows from KB.”

• KB |= S is equivalent to |= (KB S)– Means “(KB S) is true in every world (i.e., is valid).”

• And so: {} |= S is equivalent to |= ({} S)

• So what does ({} S) mean?– Means “True implies S.”– Means “S is valid.”– In Horn form, means “S is a fact.” p. 256 (3rd ed.; p. 281, 2nd ed.)

• Why does {} mean True here, but False in resolution proofs?

Review: (True S) means “S is a fact.”

• By convention,– The null conjunct is “syntactic sugar” for True.– The null disjunct is “syntactic sugar” for False.– Each is assigned the truth value of its identity element.

• For conjuncts, True is the identity: (A True) A• For disjuncts, False is the identity: (A False) A

• A KB is the conjunction of all of its sentences.– So in the expression: {} |= S

• We see that {} is the null conjunct and means True.– The expression means “S is true in every world where True is true.”

• I.e., “S is valid.”– Better way to think of it: {} does not exclude any worlds (models).

• In Conjunctive Normal Form each clause is a disjunct.– So in, say, KB = { {P Q} {Q R} {} {X Y Z} }

• We see that {} is the null disjunct and means False.

Side Trip: Functions AND, OR, and null values(Note: These are “syntactic sugar” in logic.)

function AND(arglist) returns a truth-value return ANDOR(arglist, True)

function OR(arglist) returns a truth-value return ANDOR(arglist, False)

function ANDOR(arglist, nullvalue) returns a truth-value /* nullvalue is the identity element for the caller. */ if (arglist = {}) then return nullvalue if ( FIRST(arglist) = NOT(nullvalue) ) then return NOT(nullvalue) return ANDOR( REST(arglist) )

Review: Schematic for Follows, Entails, and Derives

If KB is true in the real world,then any sentence entailed by KBand any sentence derived from KB by a sound inference procedureis also true in the real world.

Sentences SentenceDerives

Inference

Schematic Example: Follows, Entails, and Derives

Inference

“Mary is Sue’s sister and Amy is Sue’s daughter.” “Mary is

Amy’s aunt.”Representation

Derives

Entails

FollowsWorld

Mary Sue

Amy

“Mary is Sue’s sister and Amy is Sue’s daughter.”

“An aunt is a sister of a parent.”

“An aunt is a sister of a parent.”

Sister

Daughter

Mary

Amy

Aunt

“Mary is Amy’s aunt.”

Is it provable?

Is it true?

Is it the case?

Review: Models (and in FOL, Interpretations)

• Models are formal worlds in which truth can be evaluated

• We say m is a model of a sentence α if α is true in m

• M(α) is the set of all models of α

• Then KB ╞ α iff M(KB) M(α)– E.g. KB, = “Mary is Sue’s sister

and Amy is Sue’s daughter.”– α = “Mary is Amy’s aunt.”

• Think of KB and α as constraints, and of models m as possible states.

• M(KB) are the solutions to KB and M(α) the solutions to α.

• Then, KB ╞ α, i.e., ╞ (KB a) , when all solutions to KB are also solutions to α.

Review: Wumpus models

• KB = all possible wumpus-worlds consistent with the observations and the “physics” of the Wumpus world.

Review: Wumpus models

α1 = "[1,2] is safe", KB ╞ α1, proved by model checking.

Every model that makes KB true also makes α1 true.

Review: Syntax of FOL: Basic elements

• Constants KingJohn, 2, UCI,...

• Predicates Brother, >,...

• Functions Sqrt, LeftLegOf,...

• Variables x, y, a, b,...

• Connectives , , , ,

• Equality =

• Quantifiers ,

Syntax of FOL: Basic syntax elements are symbols

• Constant Symbols:– Stand for objects in the world.

• E.g., KingJohn, 2, UCI, ...

• Predicate Symbols– Stand for relations (maps a tuple of objects to a truth-value)

• E.g., Brother(Richard, John), greater_than(3,2), ...– P(x, y) is usually read as “x is P of y.”

• E.g., Mother(Ann, Sue) is usually “Ann is Mother of Sue.”

• Function Symbols– Stand for functions (maps a tuple of objects to an object)

• E.g., Sqrt(3), LeftLegOf(John), ...

• Model (world) = set of domain objects, relations, functions• Interpretation maps symbols onto the model (world)

– Very many interpretations are possible for each KB and world!– Job of the KB is to rule out models inconsistent with our knowledge.

Syntax of FOL: Terms

• Term = logical expression that refers to an object

• There are two kinds of terms:

– Constant Symbols stand for (or name) objects:• E.g., KingJohn, 2, UCI, Wumpus, ...

– Function Symbols map tuples of objects to an object:• E.g., LeftLeg(KingJohn), Mother(Mary), Sqrt(x)• This is nothing but a complicated kind of name

– No “subroutine” call, no “return value”

Syntax of FOL: Atomic Sentences

• Atomic Sentences state facts (logical truth values).– An atomic sentence is a Predicate symbol, optionally

followed by a parenthesized list of any argument terms– E.g., Married( Father(Richard), Mother(John) )– An atomic sentence asserts that some relationship (some

predicate) holds among the objects that are its arguments.

• An Atomic Sentence is true in a given model if the relation referred to by the predicate symbol holds among the objects (terms) referred to by the arguments.

Syntax of FOL: Connectives & Complex Sentences

• Complex Sentences are formed in the same way, and are formed using the same logical connectives, as we already know from propositional logic

• The Logical Connectives: biconditional implication and or negation

• Semantics for these logical connectives are the same as we already know from propositional logic.

Syntax of FOL: Variables

• Variables range over objects in the world.

• A variable is like a term because it represents an object.

• A variable may be used wherever a term may be used.– Variables may be arguments to functions and predicates.

• (A term with NO variables is called a ground term.)• (A variable not bound by a quantifier is called free.)

Syntax of FOL: Logical Quantifiers

• There are two Logical Quantifiers:– Universal: x P(x) means “For all x, P(x).”

• The “upside-down A” reminds you of “ALL.”– Existential: x P(x) means “There exists x such that, P(x).”

• The “upside-down E” reminds you of “EXISTS.”

• Syntactic “sugar” --- we really only need one quantifier. x P(x) x P(x) x P(x) x P(x)– You can ALWAYS convert one quantifier to the other.

• RULES: and

• RULE: To move negation “in” across a quantifier,change the quantifier to “the other quantifier”and negate the predicate on “the other side.”

x P(x) x P(x) x P(x) x P(x)

Existential Quantification

• Existential quantification is equivalent to:– Disjunction of all sentences obtained by substitution of an object for the

quantified variable.

• Spot has a sister who is a cat. x Sister(x, Spot) Cat(x)

• Disjunction of all sentences obtained by substitution of an object for the quantified variable:Sister(Spot, Spot) Cat(Spot) Sister(Rick, Spot) Cat(Rick) Sister(LAX, Spot) Cat(LAX) Sister(Shayama, Spot) Cat(Shayama) Sister(France, Spot) Cat(France) Sister(Felix, Spot) Cat(Felix) …

Combining Quantifiers --- Order (Scope)

The order of “unlike” quantifiers is important. x y Loves(x,y)

– For everyone (“all x”) there is someone (“exists y”) whom they love

y x Loves(x,y) - there is someone (“exists y”) whom everyone loves (“all x”)

Clearer with parentheses: y ( x Loves(x,y) )

The order of “like” quantifiers does not matter.x y P(x, y) y x P(x, y)x y P(x, y) y x P(x, y)

De Morgan’s Law for Quantifiers

( )( )( )( )

x P x Px P x P

x P x Px P x P

( )( )

( )( )

P Q P QP Q P Q

P Q P QP Q P Q

De Morgan’s Rule Generalized De Morgan’s Rule

Rule is simple: if you bring a negation inside a disjunction or a conjunction,always switch between them (or and, and or).

FOL (or FOPC) Ontology:What kind of things exist in the world?What do we need to describe and reason about?Objects --- with their relations, functions, predicates, properties, and general rules.

Reasoning

Representation-------------------A Formal Symbol System

Inference---------------------Formal Pattern Matching

Syntax---------What is said

Semantics-------------What it means

Schema-------------Rules of Inference

Execution-------------Search Strategy

This lecture Next lecture

Semantics: Worlds

• The world consists of objects that have properties.– There are relations and functions between these objects– Objects in the world, individuals: people, houses, numbers,

colors, baseball games, wars, centuries• Clock A, John, 7, the-house in the corner, Tel-Aviv

– Functions on individuals:• father-of, best friend, third inning of, one more than

– Relations:• brother-of, bigger than, inside, part-of, has color, occurred

after– Properties (a relation of arity 1):

• red, round, bogus, prime, multistoried, beautiful

Semantics: Interpretation

• An interpretation of a sentence (wff) is an assignment that maps – Object constant symbols to objects in the world, – n-ary function symbols to n-ary functions in the world,– n-ary relation symbols to n-ary relations in the world

• Given an interpretation, an atomic sentence has the value “true” if it denotes a relation that holds for those individuals denoted in the terms. Otherwise it has the value “false.”– Example: Kinship world:

• Symbols = Ann, Bill, Sue, Married, Parent, Child, Sibling, …– World consists of individuals in relations:

• Married(Ann,Bill) is false, Parent(Bill,Sue) is true, …

Truth in first-order logic

• Sentences are true with respect to a model and an interpretation

• Model contains objects (domain elements) and relations among them

• Interpretation specifies referents for

constant symbols → objectspredicate symbols → relationsfunction symbols → functional relations

• An atomic sentence predicate(term1,...,termn) is trueiff the objects referred to by term1,...,termnare in the relation referred to by predicate

–

Semantics: Models

• An interpretation satisfies a wff (sentence) if the wff has the value “true” under the interpretation.

• Model: A domain and an interpretation that satisfies a wff is a model of that wff

• Validity: Any wff that has the value “true” under all interpretations is valid

• Any wff that does not have a model is inconsistent or unsatisfiable

• If a wff w has a value true under all the models of a set of sentences KB then KB logically entails w

Models for FOL: Example

Using FOL

• We want to TELL things to the KB, e.g. TELL(KB, ) TELL(KB, King(John) )

These sentences are assertions

• We also want to ASK things to the KB, ASK(KB, )

these are queries or goals

The KB should return the list of x’s for which Person(x) is true: {x/John,x/Richard,...}

, ( ) ( )x King x Person x

, ( )x Person x

FOL Version of Wumpus World

• Typical percept sentence:Percept([Stench,Breeze,Glitter,None,None],5)

• Actions:Turn(Right), Turn(Left), Forward, Shoot, Grab, Release, Climb

• To determine best action, construct query: a BestAction(a,5)

• ASK solves this and returns {a/Grab}– And TELL about the action.

Knowledge Base for Wumpus World

• Perception s,b,g,x,y,t Percept([s,Breeze,g,x,y],t) Breeze(t) s,b,x,y,t Percept([s,b,Glitter,x,y],t) Glitter(t)

• Reflex action t Glitter(t) BestAction(Grab,t)

• Reflex action with internal state t Glitter(t) Holding(Gold,t) BestAction(Grab,t)

Holding(Gold,t) can not be observed: keep track of change.

Deducing hidden properties

Environment definition:x,y,a,b Adjacent([x,y],[a,b])

[a,b] {[x+1,y], [x-1,y],[x,y+1],[x,y-1]}

Properties of locations:s,t At(Agent,s,t) Breeze(t) Breezy(s)

Squares are breezy near a pit:– Diagnostic rule---infer cause from effect

s Breezy(s) r Adjacent(r,s) Pit(r)

– Causal rule---infer effect from cause (model based reasoning)r Pit(r) [s Adjacent(r,s) Breezy(s)]

Set Theory in First-Order Logic

Can we define set theory using FOL? - individual sets, union, intersection, etc

Answer is yes.

Basics: - empty set = constant = { }

- unary predicate Set( ), true for sets

- binary predicates: x s (true if x is a member of the set x) s1 s2 (true if s1 is a subset of s2)

- binary functions: intersection s1 s2, union s1 s2 , adjoining {x|s}

A Possible Set of FOL Axioms for Set Theory

The only sets are the empty set and sets made by adjoining an element to a set

s Set(s) (s = {} ) (x,s2 Set(s2) s = {x|s2})

The empty set has no elements adjoined to it x,s {x|s} = {}

Adjoining an element already in the set has no effect x,s x s s = {x|s}

The only elements of a set are those that were adjoined into it. Expressed recursively:

x,s x s [ y,s2 (s = {y|s2} (x = y x s2))]

A Possible Set of FOL Axioms for Set Theory

A set is a subset of another set iff all the first set’s members are members of the 2nd set

s1,s2 s1 s2 (x x s1 x s2)

Two sets are equal iff each is a subset of the others1,s2 (s1 = s2) (s1 s2 s2 s1)

An object is in the intersection of 2 sets only if a member of bothx,s1,s2 x (s1 s2) (x s1 x s2)

An object is in the union of 2 sets only if a member of eitherx,s1,s2 x (s1 s2) (x s1 x s2)

Knowledge engineering in FOL

1. Identify the task

2. Assemble the relevant knowledge

3. Decide on a vocabulary of predicates, functions, and constants

4. Encode general knowledge about the domain

5. Encode a description of the specific problem instance

6. Pose queries to the inference procedure and get answers

7. Debug the knowledge base

The electronic circuits domain

One-bit full adder

Possible queries:- does the circuit function properly?

- what gates are connected to the first input terminal? - what would happen if one of the gates is broken? and so on

The electronic circuits domain

1. Identify the task– Does the circuit actually add properly?

2. Assemble the relevant knowledge– Composed of wires and gates; Types of gates (AND, OR, XOR, NOT)– Irrelevant: size, shape, color, cost of gates

3. Decide on a vocabulary– Alternatives:

Type(X1) = XOR (function)Type(X1, XOR) (binary predicate)XOR(X1) (unary predicate)

–––

The electronic circuits domain

4. Encode general knowledge of the domain t1,t2 Connected(t1, t2) Signal(t1) = Signal(t2)

t Signal(t) = 1 Signal(t) = 0

– 1 ≠ 0

t1,t2 Connected(t1, t2) Connected(t2, t1)

g Type(g) = OR Signal(Out(1,g)) = 1 n Signal(In(n,g)) = 1

g Type(g) = AND Signal(Out(1,g)) = 0 n Signal(In(n,g)) = 0

g Type(g) = XOR Signal(Out(1,g)) = 1 Signal(In(1,g)) ≠ Signal(In(2,g))

g Type(g) = NOT Signal(Out(1,g)) ≠ Signal(In(1,g))

The electronic circuits domain

5. Encode the specific problem instanceType(X1) = XOR Type(X2) = XORType(A1) = AND Type(A2) = ANDType(O1) = OR

Connected(Out(1,X1),In(1,X2)) Connected(In(1,C1),In(1,X1))Connected(Out(1,X1),In(2,A2)) Connected(In(1,C1),In(1,A1))Connected(Out(1,A2),In(1,O1)) Connected(In(2,C1),In(2,X1))Connected(Out(1,A1),In(2,O1)) Connected(In(2,C1),In(2,A1))Connected(Out(1,X2),Out(1,C1)) Connected(In(3,C1),In(2,X2))Connected(Out(1,O1),Out(2,C1)) Connected(In(3,C1),In(1,A2))

The electronic circuits domain

6. Pose queries to the inference procedureWhat are the possible sets of values of all the terminals for the adder

circuit? i1,i2,i3,o1,o2 Signal(In(1,C_1)) = i1 Signal(In(2,C1)) = i2 Signal(In(3,C1)) = i3 Signal(Out(1,C1)) = o1 Signal(Out(2,C1)) = o2

7. Debug the knowledge baseMay have omitted assertions like 1 ≠ 0–•–

Syntactic Ambiguity

• FOPC provides many ways to represent the same thing.• E.g., “Ball-5 is red.”

– HasColor(Ball-5, Red)• Ball-5 and Red are objects related by HasColor.

– Red(Ball-5)• Red is a unary predicate applied to the Ball-5 object.

– HasProperty(Ball-5, Color, Red)• Ball-5, Color, and Red are objects related by HasProperty.

– ColorOf(Ball-5) = Red• Ball-5 and Red are objects, and ColorOf() is a function.

– HasColor(Ball-5(), Red())• Ball-5() and Red() are functions of zero arguments that both

return an object, which objects are related by HasColor.– …

• This can GREATLY confuse a pattern-matching reasoner.– Especially if multiple people collaborate to build the KB, and they

all have different representational conventions.

Summary

• First-order logic:– Much more expressive than propositional logic– Allows objects and relations as semantic primitives– Universal and existential quantifiers– syntax: constants, functions, predicates, equality, quantifiers

• Knowledge engineering using FOL– Capturing domain knowledge in logical form

• Inference and reasoning in FOL– Next lecture

• Required Reading:– All of Chapter 8– Next lecture: Chapter 9

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