# 程序代写代做代考 data structure flex chain database First-Order Logic

First-Order Logic

CISC 6525

Artificial Intelligence

First-Order Logic

Russell & Norvig, Chapter 8

Outline

Why FOL?

Syntax and semantics of FOL

Using FOL

Wumpus world in FOL

Knowledge engineering in FOL

Last Week: Propositional Logic

Propositions; Syntax, Semantics

Entailment: α ╞ iff M(α) M()

Model checking for Wumpus world

KB ╞ g iff M(KB) M(g)

Inference: α ├i = sentence can be derived from α (sound?, complete?)

Resolution: show KBα unsatisfiable

Horn clauses; Back/Forward Chaining

Propositional Logic: Pros & Cons

Propositional logic is declarative

Propositional logic allows partial/disjunctive/negated information

(unlike many data structures and databases)

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 very limited expressive power

(unlike natural language)

E.g., cannot say “pits cause breezes in adjacent squares“

except by writing one sentence for each square

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, baseball games, wars, …

Relations: red, round, prime, brother of, bigger than, part of, comes between, …

Functions: father of, best friend, one more than, plus, …

Syntax of FOL: Basic elements

Constants KingJohn, 2, NUS,…

Predicates Brother, >,…

Functions Sqrt, LeftLegOf,…

Variables x, y, a, b,…

Connectives , , , ,

Equality =

Quantifiers ,

Atomic sentences

Atomic sentence predicate (term1,…,termn) or term1 = term2

Term function (term1,…,termn) or constant

or variable

E.g.,

Brother( KingJohn, RichardTheLionheart )

E.g.,

>( Length( LeftLegOf( Richard) ),

Length( LeftLegOf( KingJohn) ) )

Complex sentences

Complex sentences are made from atomic sentences using connectives

S, S1 S2, S1 S2, S1 S2, S1 S2,

E.g.,

Sibling( KingJohn, Richard )

Sibling( Richard, KingJohn )

E.g.,

>( 1, 2 ) ≤ ( 1, 2)

>( 1,2 ) >( 1, 2)

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 → objects

predicate symbols → relations

function symbols → functional relations

An atomic sentence predicate(term1,…,termn) is true

iff the objects referred to by term1,…,termn

are in the relation referred to by predicate

Models for FOL: Example

Universal quantification

Everyone at Fordham is smart:

x At(x, Fordham ) Smart(x)

x P is true in a model m iff P is true with x being each possible object in the model

Roughly speaking, equivalent to the conjunction of instantiations of P

At(KingJohn,Fordham) Smart(KingJohn)

At(Richard, Fordham) Smart(Richard)

At(Fordham,Fordham) Smart(Fordham)

…

A common mistake to avoid

Typically, is the main connective with

Common mistake: using as the main connective with –

x At(x,Fordham) Smart(x)

means “Everyone is at Fordham and everyone is smart”

Existential quantification

Someone at Fordham is smart:

x At(x,Fordham) Smart(x)

x P is true in a model m iff P is true with x being some possible object in the model

Roughly speaking, equivalent to the disjunction of instantiations of P

At(KingJohn,Fordham) Smart(KingJohn)

At(Richard, Fordham) Smart(Richard)

At(Fordham,Fordham) Smart(Fordham)

…

Another common mistake to avoid

Typically, is the main connective with

Common mistake: using as the main connective with –

x At(x,Fordham) Smart(x)

is true if there is anyone who is not at Fordham!

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)

Equality

term1 = term2 is true under a given interpretation if and only if term1 and term2 refer to the same object

E.g.,

definition of Sibling in terms of Parent:

x,y Sibling(x,y) [(x = y) m, f (m = f)

Parent( m, x ) Parent( f, x )

Parent( m, y ) Parent( f, y ) ]

Using FOL

The kinship domain:

Brothers are siblings

x,y Brother(x,y) Sibling(x,y)

One’s mother is one’s female parent

m,c Mother(c) = m (Female(m) Parent(m,c))

“Sibling” is symmetric

x,y Sibling(x,y) Sibling(y,x)

Using FOL

The set domain:

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

x,s {x|s} = {}

x,s x s s = {x|s}

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

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

s1,s2 (s1 = s2) (s1 s2 s2 s1)

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

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

Suppose a wumpus-world agent is using an FOL KB and perceives a smell and a breeze (but no glitter) at t=5:

Tell( KB, Percept([Smell,Breeze,None],5) )

Ask( KB, a BestAction(a,5) )

I.e., does the KB entail some best action at t=5?

Answer: Yes, {a/Shoot} ← substitution (binding list)

Given a sentence S and a substitution σ,

Sσ denotes the result of plugging σ into S; e.g.,

S = Smarter(x,y)

σ = {x/Hillary,y/Bill}

Sσ = Smarter(Hillary,Bill)

=> Ask(KB,S) returns some/all σ such that KB╞ σ

Interacting with FOL KBs

Knowledge base for the wumpus world

Perception

t,s,b Percept([s,b,Glitter],t) Glitter(t)

Reflex

t Glitter(t) BestAction(Grab,t)

Deducing hidden properties

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 squares:

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

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

Knowledge engineering in FOL

Identify the task

Assemble the relevant knowledge

Decide on a vocabulary of predicates,

functions, and constants

Encode general knowledge about the domain

Encode a description of the specific problem instance

Pose queries to the inference procedure and get answers

Debug the knowledge base

Summary

First-order logic:

objects and relations are semantic primitives

syntax: constants, functions, predicates, equality, quantifiers

Increased expressive power: sufficient to define wumpus world

The electronic circuits domain

One-bit full adder

The electronic circuits domain

Identify the task

Does the circuit actually add properly? (circuit verification)

Assemble the relevant knowledge

Composed of wires and gates; Types of gates (AND, OR, XOR, NOT)

Irrelevant: size, shape, color, cost of gates

Decide on a vocabulary

Alternatives:

Type(X1) = XOR

Type(X1, XOR)

XOR(X1)

The electronic circuits domain

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

Encode the specific problem instance

Type(X1) = XOR Type(X2) = XOR

Type(A1) = AND Type(A2) = AND

Type(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

Pose queries to the inference procedure

What 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

Debug the knowledge base

May have omitted assertions like 1 ≠ 0

Summary

First-order logic:

objects and relations are semantic primitives

syntax: constants, functions, predicates, equality, quantifiers

Increased expressive power: sufficient to define wumpus world