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Assignment
-
2 due now

CPSC 322, Lecture 20

Slide
2

Propositional Definite Clause
Logic:
Syntax, Semantics and
Bottom
-
up Proofs

Computer Science cpsc322, Lecture 20

(Textbook
Chpt

5.1.2
-

5.2.2 )


Oct, 23, 2010

CPSC 322, Lecture 20

Slide
3

Lecture Overview

•
Recap: Logic intro

•
Propositional Definite Clause Logic:
Semantics

•
PDCL: Bottom
-
up Proof


CPSC 322, Lecture 20

Slide
4

Logics as a R&R system

•
formalize a domain





•
reason about it


CPSC 322, Lecture 18

Slide
5

Logics in AI: Similar slide to the one for planning

Propositional
Logics

First
-
Order
Logics

Propositional Definite
Clause Logics

Semantics and Proof
Theory

Satisfiability

Testing
(SAT)

Description
Logics

Cognitive Architectures

Video Games

Hardware Verification

Product Configuration

Ontologies

Semantic Web

Information
Extraction

Summarization

Production Systems

Tutoring Systems

CPSC 322, Lecture 20

Slide
6

Propositional (Definite Clauses) Logic:
Syntax

We start from a restricted form of Prop. Logic:


Only two kinds of statements

•
that a proposition is true

•
that a proposition is true if one or more other propositions
are true

CPSC 322, Lecture 20

Slide
7

Lecture Overview


•
Recap: Logic intro

•
Propositional Definite Clause Logic:
Semantics

•
PDCL: Bottom
-
up Proof


CPSC 322, Lecture 20

Slide
8

Propositional Definite Clauses Semantics:
Interpretation

Definition (interpretation)

An
interpretation
I


assigns a truth value to each atom.

Semantics allows you to relate the symbols in the logic to the
domain you're trying to model
. An
atom

can be…..

If your domain can be represented by four atoms (propositions):

So an interpretation is just a…………………………..

CPSC 322, Lecture 20

Slide
9

PDC Semantics: Body

Definition (
truth values of statements
):
A
body
b
1

∧

b
2

is true in
I

if and only if
b
1

is true in
I

and
b
2

is true in
I
.

We can use the
interpretation

to determine the truth value of
clauses

and
knowledge bases
:

p

q

r

s

I
1

true

true

true

true

I
2

false

false

false

false

I
3

true

true

false

false

I
4

true

true

true

false

I
5

true

true

false

true

CPSC 322, Lecture 20

Slide
10

PDC Semantics: definite clause

Definition (
truth values of statements

cont’
):
A
rule
h

←

b

is
false in
I

if and only if
b

is true in
I

and
h

is false in
I
.

In other words:
”if
b is true
I am claiming that
h must be true
,
otherwise I am not making any claim”

p

q

r

s

I
1

true

true

true

true

I
2

false

false

false

false

I
3

true

true

false

false

I
4

true

true

true

false

…..

….

…..

….

....

PDC Semantics: Knowledge Base (KB)

p

q

r

s

I
1

true

true

false

false

p

r

s
←

q
∧

p

p

q

s
←

q

p

q
←

r
∧

s

KB
1

KB
2

KB
3

Which of the three KB above are True in I
1


•
A
knowledge base KB

is true in I if and only if
every clause in KB is true in I.

PDC Semantics: Knowledge Base (KB)

p

q

r

s

I
1

true

true

false

false

p

r

s
←

q
∧

p

p

q

s
←

q

p

q
←

r
∧

s

KB
1

KB
2

KB
3

Which of the three KB above are True in I
1
?


KB
3

•
A
knowledge base KB

is true in I if and only if
every clause in KB is true in I.

CPSC 322, Lecture 20

Slide
13

PDC Semantics: Knowledge Base

Definition (
truth values of statements cont’
):
A
knowledge base
KB

is true in
I

if and only if every clause in
KB

is true in
I
.

CPSC 322, Lecture 20

Slide
14

Models

Definition (model)

A
model

of a set of clauses (a KB) is an interpretation in which
all the clauses are
true
.

CPSC 322, Lecture 20

Slide
15

Example: Models

Which interpretations are
models?

p

q

r

s

I
1

true

true

true

true

I
2

false

false

false

false

I
3

true

true

false

false

I
4

true

true

true

false

I
5

true

true

false

true

CPSC 322, Lecture 20

Slide
16

Logical Consequence

Definition (logical consequence)

If
KB

is a set of clauses and
G

is a conjunction of atoms,
G

is
a
logical consequence
of
KB
, written
KB
⊧

d
Ⱐ楦i
䜠

楳i
瑲略u

楮
敶敲礠e潤o氠潦o
䭂
.

•

we also say that
G

logically follows
from
KB
, or that
KB

entails

G
.

•

In other words,
KB
⊧

d

楦⁴桥牥h楳漠楮瑥牰牥瑡i楯渠楮 睨楣i
䭂h

楳i
瑲略u

慮搠
䜠

楳i
晡汳f
.

CPSC 322, Lecture 20

Slide
17

Example: Logical Consequences

p

q

r

s

I
1

true

true

true

true

I
2

true

true

true

false

I
3

true

true

false

false

I
4

true

true

false

true

I
5

false

true

true

true

I
6

false

true

true

false

I
7

false

true

false

false

I
8

false

true

false

true

…

….

…

…

…

Which of the following is true?

•

KB
⊧

焬⁋䈠
⊧

瀬⁋䈠
⊧

猬s䭂
⊧

r


CPSC 322, Lecture 20

Slide
18

Lecture Overview


•
Recap: Logic intro

•
Propositional Definite Clause Logic:
Semantics

•
PDCL: Bottom
-
up Proof


CPSC 322, Lecture 20

Slide
19

One simple way to prove that G logically
follows from a KB

•
Collect all the models of the KB

•
Verify that G is true in all those models

Any problem with this approach?

•
The goal of proof theory is to find
proof
procedures

that allow us to prove that a logical
formula follows form a KB avoiding the above

CPSC 322, Lecture 20

Slide
20

Soundness and Completeness

•
If I tell you I have a
proof procedure for PDCL

•
What do I need to show you in order for you to
trust my procedure?

Definition (soundness)

A proof procedure is
sound

if
KB

⊦

d

業灬i敳e
䭂h
⊧

d
.

䑥fin楴楯渠(c潭ol整e湥獳s

䄠灲潯映灲潣敤畲攠楳i
捯浰汥瑥

楦i
䭂h
⊧

d

業灬i敳e
䭂

⊦

d
.

•
KB
⊦

䜠
浥m湳
d

捡渠扥n摥物r敤 批 浹⁰牯潦o
灲潣敤畲攠晲潭
䭂
.

•
Recall
KB

⊧

d

浥慮猠
d

楳i瑲t攠楮⁡e氠浯l敬e 潦o
䭂
.

CPSC 322, Lecture 20

Slide
21

Bottom
-
up Ground Proof Procedure

One
rule of derivation
, a generalized form of
modus
ponens
:


If “
h
←

b
1

∧

…
∧

b
m
” is a clause in the knowledge
base, and each
b
i

has been derived, then
h

can
be derived.


You are
forward chaining
on this clause.

(This rule also covers the case when
m=0
. )

CPSC 322, Lecture 20

Slide
22

Bottom
-
up proof procedure


KB

⊦

䜠

楦i
d

⊆

C

at the end of this procedure:


C
:={};

repeat


select
clause “
h
←

b
1

∧

…
∧

b
m
” in
KB

such



that
b
i

∈

C

景爠慬a
i
Ⱐ慮I
栠
∉

C
;


C

㨽:
䌠
∪

{ h }

until

no more clauses can be selected.



CPSC 322, Lecture 20

Slide
23

Bottom
-
up proof procedure: Example

z
←

f
∧

e

q
←

f
∧

g
∧

z

e
←

a
∧

b

a

b

r

f


C
:={};

repeat


select
clause “
h
←

b
1

∧

…
∧

b
m
” in
KB

such



that
b
i

∈

C

景f⁡汬
i
Ⱐ慮搠
栠
∉

C
;


C

㨽
䌠
∪

{ h }

until

no more clauses can be selected.



CPSC 322, Lecture 20

Slide
24

Bottom
-
up proof procedure: Example

z
←

f
∧

e

q
←

f
∧

g
∧

z

e
←

a
∧

b

a

b

r

f


C
:={};

repeat


select
clause “
h
←

b
1

∧

…
∧

b
m
” in
KB

such



that
b
i

∈

C

景f⁡汬
i
Ⱐ慮搠
栠
∉

C
;


C

㨽
䌠
∪

{ h }

until

no more clauses can be selected.



CPSC 322, Lecture 20

Slide
25

Bottom
-
up proof procedure: Example

z
←

f
∧

e

q
←

f
∧

g
∧

z

e
←

a
∧

b

a

b

r

f


r? q? z?


C
:={};

repeat


select
clause “
h
←

b
1

∧

…
∧

b
m
” in
KB

such



that
b
i

∈

C

景f⁡汬
i
Ⱐ慮搠
栠
∉

C
;


C

㨽
䌠
∪

{ h }

until

no more clauses can be selected.



CPSC 322, Lecture 4

Slide
26

Learning Goals for today’s class

You can:


•
Verify whether an
interpretation

is a
model

of
a PDCL KB.

•
Verify when a conjunction of atoms is a
logical consequence
of a knowledge base.

•
Define/read/write/trace/debug the
bottom
-
up
proof procedure
.

CPSC 322, Lecture 20

Slide
27

Next class




(still section 5.2)

•
Soundness and Completeness of Bottom
-
up
Proof Procedure

•
Using PDC Logic to model the electrical domain

•
Reasoning in the electrical domain


CPSC 322, Lecture
20

Slide
28

Study for midterm (Mon Oct 29 )




Midterm
:
~6
short
questions
(
10pts each
)
+ 2
problems
(
20pts each)

•
Study: textbook and
inked

slides

•
Work on
all
practice
exercises and
revise assignments
!

•
While you revise the

learning goals
, work on
review questions

(will
post today)

I
may even reuse some verbatim


•
Will post a
couple of problems
from previous offering
(maybe
slightly more
difficult)

… but I’ll give you the solutions
