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Jim Pattison Leadership Centre
CSSS BBQ
Fri. Oct 26
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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
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