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Logic and Natural Language

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Colorless green ideas sleep furiously - Noam Chomsky. Sentences ('well formed formulas' ... all cars require oil to run. some houses have decks. FOL variables ... – PowerPoint PPT presentation

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Title: Logic and Natural Language

1
Logic and Natural Language

By Willie Milnor
LSDIS Lab
University of Georgia

2
Outline

Logic
Propositional Logic
First-order Logic
Logic Programming
Natural Language

3
Logic is a Formal Language

Syntax
what is allowed - expressions
Semantics
what expressions mean
Proof system
derive new expressions/semantics
reveal something new about the word

4
Inferences from Proofs

Agent reasoning
given multiple percepts
wet umbrella, muddy tracks, pitter-patter on roof
make conclusions about the world
Effect of an action
current state of the world
knowledge about the action
next state of the world?

5
Propositional Logic

Syntax
Colorless green ideas sleep furiously
- Noam Chomsky
Sentences (well formed formulas)
true and false
propositional variables P, Q
sentential operators
F ? ?, F ? ?, F, (F), F ? ?, F ? ?

6
Propositional Logic

Semantics
truth value t, f
true t
false f
Interpretation
assignment of truth values
holds(F, i)
fails(F, i)

7
Propositional Logic

Semantics
Rules
holds(true, i) for all i
fails(false, i) for all i
holds(P, i) iff i(P) t
fails(P, i) iff i(P) f
holds(P and Q, i) iff holds(P, i) and holds(Q, i)
...

8
Propositional Logic

valid
t in all interpretations
ex raining or not raining
satisfiable
t in at least one interpretation
ex raining implies thunder
Unsatisfiable
f in all interpretations
ex raining and not raining

9
Satisfiability

find an interpretation where the sentence is true
- holds(S,i)
constraint satisfaction
assignment of variables where constraints hold
methods
brute force
hueristics
constraint propagation

10
Satisfiability

Example
given KB
P ? Q
Q ? S
P
S?
n propositional variables 2n interpretations
8 interpretations in the example
4 variables 16 interpretations

11
Entailment

A KB entails S iff S is true in every
interpretations that makes KB true
S follows from KB

1
12
Entailment

Computation
enumerate all interpretations
find those in which KB is true
check against S
O(2n) time
Use a proof

13
Proof

Given a knowledge base
Sequence of premises (known facts)
Apply inference rules to premises
new sentences
Finished when S is inferred
sound KB can prove any S it entails
complete KB entails any S that it can prove

14
Natural Deductions

modus
ponens
a ? b
a
_______________________
b

And-
introduction
a
b
_______________________
a ? b

modus tolens a ? b b _______________________
a
And- elimination a ? b _______________________ a

15
Natural Deductions

proof checkers for Natural Deduction
what conclusions the system should find from what
premises
sound but not complete
many inference rules large branching factor

16
Conjunctive Normal Form

sentence conjunction of clauses
clause disjunction of literals
literal variable or negation thereof
example (P ? Q) ? (R ? S) ? (Q ? S)
conversion
eliminate ?, ?
move into clauses
distribute ? over ?
O(2n) space

17
Propositional Resolution

resolution rule (sound and complete)
a ? b
b ? y
_____________________
a ? y
resolution refutation (sound and complete)
convert all sentences to CNF
negate desired conclusion
apply resolution rule
conclusion is entailed if the contradiction is
derived

18
First-Order Logic

PL has facts true or false
what about groups of things?
all cars require oil to run
some houses have decks
FOL variables refer to quantifiable things

19
First-Order Logic

Syntax
terms
constant symbols square, dog
variables x,y
function symbol with 1 terms
sentence
predicate symbol
t1 t
quantifiers for all, there exists
closure over sentential operators

20
First-Order Logic Interpretations

set of objects
universe or domain of discourse
constants map to elements
predicates map to relations
functions map to functions
binary relation

21
First-Order Logic Interpretations

Holds
relation P
terms t1,,tn
interpretation I
holds (P(t1,,tn), I) iff ltI(t1),, I(tn)gt
belongs to I(P)
Equality
holds(t1 t2, I) iff I(t1) is the same object as
I(t2)

22
Entailment

A KB entails S iff holds(KB, I) implies holds(S,
I)
Computing entailment is impossible
infinitely many interpretations
holds(S, I) may be impossible because of
quantifiers
for all in an infinite universe
Proofs
S is entailed by KB implies that there is a
finite proof of S from KB

23
Proofs

Axiomitization
Want to prove holds(S, I1) for some
interpretations I1
Set of sentences, axioms, as a KB
holds(KB, I1) and holds(S, I1)
axioms may be too weak
holds(KB, I2) and fails(S, I2),
KB does not entail S
Add an axiom to KB so that
holds(KB, I2) and holds(S, I2)
but, holds(KB, I3) and fails(S, I3)

24
First-Order Resolution

for all x, P(x) ? Q(x)
P(A)
________________________________________________
Q(A)
for all x, P(x) ? Q(x)
P(A)
________________________________________________
Q(A)
for all x, P(A) ? Q(A)
P(A)
_________________________________________________
Q(A)

Syllogism
Equivalence by
definition of
implication
Substitute A for x

25
First-Order Resolution

Convert to clausal form
Substitutions
v1/t1,,vn/tn
substitute each variable vi with term ti
Unification
find substitution to make two expressions match
expressions e1 and e2 are unifiable if there
exists a substitution s such that e1s e2s

26
Substitution

P(x, f(y), B) an atomic sentence

1
27
Unification

Let e1 x and e2 y, unifier s

1
28
Most General Unifier (MGU)

G is a MGU iff for all unifiers S, there exists a
unifier s such that the result of applying G
followed by s to e1 is the same result as
applying S to e1, and the same for e2
All other unifiers can be expressed as an extra
substitution added on to the MGU
MGU for P(x) and P(A) is A

29
First-Order Resolution

Given a ? F, ? ? ß, and MGU(F, ?) ?
Conclude (a ? ß) ?

P(x) ? Q(x,y)
P(A) ? R(B,z)
___________________________________________
(Q(x,y) ? R(B,z))?
Q(A,y) ? R(B,Z)

T x/A
30
Proving Validity

Valid sentence is always true
derivable from nothing
entailed by the empty set of sentences
Proof by resolution refutation
Negate the sentence
derive a contradiction
Example
syllogism (P(x) ? Q(x)) ? (P(A) ? Q(A))
negated clausal form ( P(x) ? Q(x)) ? P(A) ?
Q(A)

31
Proving Validity

Proof

1
32
Completeness and Decidability

complete
If KB entails S, then KB can prove S
Godels Completeness Theorem
There exists a complete proof system for FOL
Robinsons Completeness Theorem
Resolution refutation is a complete proof system
for FOL
FOL is semi-decidable
If there is a proof, we will halt with it
If not, we may never halt

1
33
Rules and Programming

How to compute what is true
(Parent(x,y) ? Parent(y,z)) ? Grandparent (x,z)
Find grandparents given parents
Find parents given grandparents
Rule-Based systems
Logic Programming
x ? y ? z
z is consequent
x and y are antecedents
z - x, y

34
Horn Clauses

A clause with at most one positive literal
Rule
exactly one positive literal
one or more negative literals
Fact
exactly one positive literal
no negative literals
Consistency constraint
no positive literals

35
Inference Backchaining

Prove C
Push C and Ans literal on stack
Repeat
Pop literal L
Choose a consequent to unify with L
Push antecedents onto stack
Apply unifier to entire stack
If no match, fail and backup to last choice
Just like resolution

36
Example KB

Father(A,B)
Mother(B,C)
GrandP(?x,?z) - Parent(?x,?y), Parent(?y,?z)
Parent(?x,?y) - Father(?x,?y)
Parent(?x,?y) - Mother(?x,?y)
Prove
GrandP(A,?w), Ans(?w)

1
37
Sentence Order

parent(A,B)
parent(B,C)
ancestor(?x,?z) - ancestor(?x,?y), parent(?x,?z)
ancestor(?x,?y) - parent(?x,?y)
Prove ancestor(?x,C), Ans(?x)
Infinite loop!

1
38
Negation

No!
Illegal
P ? Q ? R
same as P ? Q ? R
not Horn
Closed world assumption
We know everything there is to know about our
world
Anything we dont know is false
Use failure to prove as a negation

39
Natural Language

Applications
DB interfaces
Automated customer service
Grammar checking
Web search
Document translation

40
Natural Language

Speech recognition
sound frequency
Syntactic analysis
grammar
Semantics analysis
meaning
Pragmatics
context of utterance
connect meaning with general knowledge

41
Syntax

Grammar the legal structures
Parsing finding legal structures of a sentence
Parse tree

S
NP
VP
V
V
N
Art
N
threw
Haines
1
the
frisbee
42
Grammar

Rewrite rules
S ? NP VP
NP ? Name
NP ? Art N
Name ? Haines
Art ? the
N ? frisbee
Legal sentence
A sentence is legal if, starting at S, we can
find a sequence of rewrite rules that generate
the sentence

43
Types of Grammar