Download Chapter 2-3 - Dr. Djamel Bouchaffra

Chapter 2 & 3 (Part 2):
A Representation & Reasoning
System & Using Definite Knowledge
 Proofs (2.7)
(How to compute logical consequences of a KB rather that
testing whether a given formula is a KB logical consequence?)

Functions Symbols: Language Extension (2.8)

Representing Abstract Concepts (3.5)
D. Poole, A. Mackworth, and R. Goebel, Computational Intelligence: A Logical Approach,
Oxford University Press, January 1998
Proofs (2.7)
2

A proof is a mechanically derivable demonstration that a
formula logically follows from a knowledge base.

Given a proof procedure, KB
from knowledge base KB.

Recall KB
g means g can be derived
g means g is true in all models of KB.

A proof procedure is sound (w.r.t. a semantics) if KB
g
implies KB
g (every thing that can be derived from KB is a
logical consequence of KB).

A proof procedure is complete (w.r.t a semantics) if there is a
proof of each logical consequence of KB: That is,
if KB
g implies KB
g (there is a proof of g).
Dr. Djamel Bouchaffra
CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge
Proof (2.7) (cont.)
 Bottom-up
3
Ground Proof Procedure
 One
rule of derivation, a generalized form of
modus ponens:
“h  b1  …  bm” is a clause in the knowledge
base, and each bi has been derived from KB, then h
can be derived from KB.
 If
 You
are forward chaining on this clause (going
forward from what is known but not backward
from the query!).
 (This
Dr. Djamel Bouchaffra
rule also covers the case when m = 0.
CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge
4
Proof (2.7) (cont.)
 Bottom-up
proof procedure
If g C then g is
proven
(C: consequence set)
Dr. Djamel Bouchaffra
CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge
Proof (2.7) (cont.)
5
 Example
a b
c.
^
b d e.
^
b g e.
^
c e.
d.
e.
fa
Dr. Djamel Bouchaffra
^
g.
CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge
Proof (2.7) (cont.)
 Example
6
(cont.)
{}
{d}
{e,d}
{c,e,d}
{b,c,e,d}
{a,b,c,e,d}
Dr. Djamel Bouchaffra
CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge
7
Proof (2.7) (cont.)

Soundness of bottom-up proof procedure
If KB
g then KB
g.
(every atom in C is a logical consequence of KB)
Suppose there is a g (atom) such that KB
g and KB
g.
Let g be the first atom added to C that’s not true in every model of
KB. Suppose g isn’t true in model I of KB.There must be a clause in
KB of form: g  b1  …  bm where:
Each bi is true in I (since g is the first) and g is false in I. So this
clause is false in I.
Therefore I isn’t a model of KB.
Contradiction: thus no such g exists !
Dr. Djamel Bouchaffra
CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge
Proof (2.7) (cont.)

8
Fixed Point

The C generated at the end of the bottom-up algorithm is
called a fixed point (any further application of the rule
won’t change C).

Let I be the interpretation in which every element of the
fixed point is true and every other atom not in C is false
Then I is a model of KB.
Proof: suppose h  b1  …  bm in KB is false in I. The
only way this could happen is if b1,b2…,bm  C and h C.
Thus h can be added to C.
Contradiction to C being the fixed point. In conclusion,
there can be no clause in KB that is false in the
interpretation I (e.g., I is a model)

I is called a Minimal Model.
Dr. Djamel Bouchaffra
CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge
Proof (2.7) (cont.)

9
Completeness of the bottom-up proof

If KB
g then KB
g.
Suppose KB
g. Then g is true in all models of KB.
Thus g is true in the minimal model (defined by the fixed
point).
Thus g is generated by the bottom up algorithm.
Thus KB g.

Complexity of the bottom-up proof:

The bottom-up proof algorithm halts and is linear w.r.t. to
the size of the knowledge base.
Dr. Djamel Bouchaffra
CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge
Proof (2.7) (cont.)

10
Top-down Proof Procedure
Idea: search backward from a query to determine if it is a
logical consequence of KB.
 An answer clause is of the form:
yes  a1  a2  …  am
 The SLD (Selector Function on Definite Clauses)
Resolution of this answer clause on atom ai with
the clause:
ai  b1 … bp
is the answer clause
yes  a1 … ai-1  b1 … bp  ai+1 … am

Dr. Djamel Bouchaffra
CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge
Proof (2.7) (cont.)
 Top-down
11
Ground Proof Procedure (cont.)
 Derivations
 An
answer is an answer clause with m = 0. That is, it is
the answer clause yes  .
derivation of query “?q1  … qk” from KB is a
sequence of answer clauses y0; y1, …, yn such that
A
y0 is the answer clause yes  q1  …  qk, (query)
 yi is obtained by resolving yi-1 with a clause in KB,
and
 yn is an answer.

Dr. Djamel Bouchaffra
CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge
Proof (2.7) (cont.)
 Top-down
12
ground proof procedure (cont.)
 Search
backward from a query to determine if it
is a logical consequence of the given clauses
Solve(q1
…
qk)
^^ ^
ac:=“yes q1
… qk”
^
^
^
repeat
select a conjunct ai from the body of ac
choose clause C from KB with ai as head
replace ai in the body of ac by the body of C
until ac is an answer
Dr. Djamel Bouchaffra
CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge
Proof (2.7) (cont.)

13
Example
a b
c.
^
bd e.
^
bg e.
^
ce.
d.
e.
fa
g.
^
?a. (query)
Dr. Djamel Bouchaffra
CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge
14
Proof (2.7) (cont.)

Example (cont.)
ac:=yes a.
yesb
c. (select b and choose b d
^
yesd e c.
^ ^
yese c.
^
(or g
e
)
^
^e: failure: no rule chosen headed by g!)
yesc.
yese.
yes.
“The two operations involved are “select” (What to Do ?) and
“choose” (How to Do ?).
Dr. Djamel Bouchaffra
CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge
Proof (2.7) (cont.)
 Nondeterministic
15
Choice
 “Don’t-care
nondeterminism”: If one selection
doesn’t lead to a solution, there is no point in trying
other alternatives. Select [no resources available:
requests should not be kept waiting! No backtracking]

“Don’t-know nondeterminism”: If one choice
doesn’t lead to a solution, other choices may, but
this search may be prohibitive. Choose (NP≠ P: no
polynomial oracle does exist!)
Dr. Djamel Bouchaffra
CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge
Proof (2.7) (cont.)

16
Reasoning with Variables

An instance of an atom or a clause is obtained by
uniformly substituting terms for variables.

A substitution is a finite set of the form
{V1/ t1, …, Vn/ tn},
where each Vi is a distinct variable and each ti is a term.

The application of a substitution  ={V1/ t1, …, Vn / tn} to
an atom or clause e, written e, is the instance of e with
every occurrence of Vi replaced by ti.
Dr. Djamel Bouchaffra
CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge
17
Proof (2.7) (cont.)
Application
Examples
Z
Dr. Djamel Bouchaffra
CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge
Proof (2.7) (cont.)
18
 Unifiers
 Substitution

 is a unifier of e1 and e2 if e1 = e2.
Substitution  is a most general unifier (mgu) of
e1 and e2 if

is a unifier of e1 and e2; and
 if substitution ’ also unifies e1 and e2, then e’ is an
instance of e for all atoms e.

If two atoms have a unifier, they have a most
general unifier.
Dr. Djamel Bouchaffra
CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge
Proof (2.7) (cont.)
 Unification
Dr. Djamel Bouchaffra
19
Example
CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge
Proof (2.7) (cont.)

20
Most General Unifiers

Example:
Atoms e1= p(X,Y) and e2=p(Z,Z) have as unifiers:
1={X/b, Y/b, Z/b}, 2={X/c, Y/c, Z/c} and
3={X/Z, Y/Z}. The 3 unifier is mgu, since:
(i)
(ii)
3 unifies e1 and e2
If 1= ’ then e1’ =p(b,b) is an instance of
e13=p(Z,Z)
Similarly, e2’ =p(b,b) is an instance of e23=p(Z,Z)
If 2= ’ then e1’ =p(c,c) is an instance of
e13=p(Z,Z)
Similarly, e2’ =p(c,c) is an instance of e23=p(Z,Z)
Dr. Djamel Bouchaffra
CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge
Proof (2.7) (cont.)

21
Proofs using Bottom-up Procedure with Variables

A ground instance of a clause is obtained by uniformly
substituting constants appearing in a KB or in any
query for the variables in the clause
Dr. Djamel Bouchaffra
CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge
Proof (2.7) (cont.)
22
 Example
q(a).
q(b).
r(a).
s(W) r(W).
p(X,Y) q(X) ^ s(Y).
The set of all ground instances is:
q(a).
q(b).
r(a).
s(a) r(a)
s(b) r(b)
Dr. Djamel Bouchaffra
CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge
Proof (2.7) (cont.)

23
Example (cont’d)
p(a,a)  q(a) ^ s(a).
p(a,b)  q(a) ^ s(b).
p(b,a)  q(b) ^ s(a).
p(b,b)  q(b) ^ s(b)
the bottom-up proof procedure will derive:
q(a), q(b), r(a), s(a), p(a,a) and p(b,a).
Dr. Djamel Bouchaffra
CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge
Proof (2.7) (cont.)

24
Definite Resolution with Variables: Top-down
Derivation
A generalized answer clause is of the form
yes(t1,…,tk) a1  a2  …  am
where t1, …, tk are terms and a1, …, am are atoms.
 The SLD resolution of this generalized answer clause on
ai with the clause
a  b1  …  bp
 where ai and a have most general unifier , is the clause:
(yes (t1, …, tk) 
a1  …  a i-1  b1  …  bp  a i+1  …  am

Dr. Djamel Bouchaffra
CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge
:

Proof (2.7) (cont.)
25
To solve query ?B with variables V1, …, Vk
Set ac to generalized answer clause yes(V1, …, Vk) B;
while ac is not an answer do \\ (yes .) is false
Suppose ac is yes(t1, …, tk)  a1  a2  …  am
Select atom ai in the body of ac;
Choose clause a  b1  …  bp in KB;
Rename all variables in a  b1  … bp to have new names;
Let  be the most general unifier of ai and a.
Fail if they don’t unify;
Set ac to
(yes(t1, …, tk)  a1  …  ai-1 b1  … bp  ai+1  … am)
end while.
Dr. Djamel Bouchaffra
CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge
Proof (2.7) (cont.)
26
 Example1
Consider the database of Figure 2.3 (page 44) and
the query: You want to know:
Which room is two doors east of r107 ?
You translate the query
? two_doors_east(R,r107)
two_doors_east(R,r107)
into: yes(R)
(class exercise).
Dr. Djamel Bouchaffra
CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge
27
Proof (2.7) (cont.)
Solution:
?two_doors_east(R, r107).
yes(R)two_doors_east(R, r107) % this is ac
% resolve with two_doors_east(E1,W1)imm_east(E1,M1)
% substitution: {E1/R, W1/r107}
^
imm_east(M1,W1)
yes(R)imm_east(R,M1) ^ imm_east(M1, r107)
% select leftmost conjunct and resolve with
imm_east(E2,W2)imm_west(W2,E2)
% substitution : {E2/R, W2/M1} ( mgu for ai and a)
yes(R)imm_west(M1, R) ^ imm_east(M1, r107)
% select leftmost conjunct and resolve with imm_west(r109,r111) *****
% substitution: {M1/r109, R/r111}
yes(r111)imm_east(r109, r107)
% resolve with imm_east(E3,W3)imm_west(W3,E3)
% substitution: {E3/r109, W3/r107}
yes(r111)imm_west(r107, r109)
% resolve with imm_west(r107, r109) and substitution: {}
yes(r111).
Dr. Djamel Bouchaffra
CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge
Example2 (Class Exercise)
28
Resolve and Substitute: {Y1/A}
Resolve with connected_to(w6,w5)
Dr. Djamel Bouchaffra
CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge
Function Symbols: Language Extension (2.8) 29

Often we want to refer to individuals in terms of components.

Examples:You may want to say that some course is held at
11:30am. You do not want to have to choose a constant for
each different time.

We extend the notion of term . So that a term can be
f (t1, …, tn) where f is a function symbol and the ti are terms.

In an interpretation and with a variable assignment, term
f (t1, …, tn) denotes an individual in the domain.

With one function symbol and one constant we can refer to
infinitely many individuals.
Example: before(T1,T2), true if T1 is before T2 in a day.
Dr. Djamel Bouchaffra
CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge
Representing Abstract Concept: Lists (3.5)

A list is an ordered sequence of elements.

Let’s use the constant nil to denote the empty list, and the function cons(H, T)
to denote the list with first element H and rest-of-list T. These are not built-in.

The list containing david, alan and randy is
Cons(david, cons(alan, cons(randy, nil)))

append(X, Y, Z) is true if list Z contains the elements of X followed by the
elements of Y
append(nil, Z, Z).
append(cons(A, X), Y, cons(A, Z))  append(X, Y, Z).
30
Example:
X=abc Y=123 Z=abc123 if append(X,Y,Z) is true
cons(A,X)=[A,abc]
cons(A,Z)=[A,abc123]
cons(A,Z)=cons(A,X)Y if Append(X,Y,Z) is true
which means:
[A,abc123]=[A,abc123]
Dr. Djamel Bouchaffra
CS 426 Artificial Intelligence, Ch. 2 & 3:A representation & reasoning system, Using definite knowledge