GUHA - theory 2
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We will now study some special kinds of non-standard quantifiers.
Definition 4. Let (x), (x) be two fixed formulae of a language Ln such that x is the
only free variable in both of them and they don’t have common predicates. Let M and
N be two models. Then we have the following two four-fold tables:
We define: N is associational
M
non-
N
non-
better than M if a2a1, d2d1,
a1
b1
a2
b2
c1c2, b1b2. Moreover,
non-
c1
d1
non-
c2
d2
a binary quantifier is associational if, for all formulae (x) and (x), all models M, N:
if vM((x)(x)) = TRUE, N associational better than M, then vN((x)(x)) = TRUE.
Obviously, the quantifier of simple association is associational: this follows by the fact
that, under the given circumstances, a2d2a1d1>b1c1b2c2.
Church quantifier of implication is associational, too. Indeed, given a model M such that
vM((x) =>C(x)) = TRUE, the corresponding four-fold table has a form
M
non-
a1
c1
non-
0
d1
N
non-
a2
c2
non-
0
d2
Thus, any model N
that is associational
better than M has a
form
Thus,
vN((x) =>C(x))
= TRUE.
Quantifiers of founded p-implication
are associational: if a2a1 n, b1b2,
then a2b1a1b2 , therefore
a2a1+ a2b1 a2a1+ a1b2 and finally,
a2
a2 b2
a1
a 1 b1
p
(Today called : Basic implication)
I
Definition 5. Let (x) , (x) be two fixed formulae of a language Ln such that x is the
only free variable in both of them and they don’t have common predicates. Let M and
N be two models. Then we have the following two four-fold tables:
M
non-
a1
c1
non-
b1
d1
N
non-
a2
c2
non-
b2
d2
We define: N is implicational
better than M if a2a1, b1b2.
Moreover,
a binary quantifier is implicational if, for all formulae (x) and (x), all models M, N:
if vM((x)(x)) = TRUE, N implicational better than M, then vN((x)(x)) = TRUE.
Church quantifier of implication is implicational, quantifiers of founded p-implication
are implicational [proof: by a similar argument that they are associational]. However,
quantifier of simple association is NOT implicational: consider the following counter
example:
Clearly, N is implicational better than M, and a1d1c1b1
thus, vM((x)~(x)) = TRUE. However, a2d2<c2b2, thus
M
non-
vN((x)~(x)) = FALSE. Therefore ~ is not implicational.
non-
a1 = 1
c1 = 1
b1 = 1
d1 = 2
N
non-
a2 = 1
c2 = 2
non-
b2 = 1
d2 = 1
Lemma. Let be an implicational quantifier. Then is
associational.
Proof. Let be implicational and vM((x)(x)) = TRUE
If N is associational better than M, then N is clearly also
implicational better than M, so vN((x)(x)) = TRUE.
Therefore is associational, too.
Theorem 2. Let (x), (x), (x) be formulae, Proof. Let M be a model such that
and let be an implicational quantifier. Then vM((x) (x)) = TRUE and
(x) (x)
M
non-
(x) [(x)(x)]
a1
b1
is a sound rule of inference.
non-
c1
d1
In the proof we used an obvious fact:
for all implicational quantifiers , if We realise that
a1 = #{x | vM((x)) = vM((x)) = TRUE}
vM((x) (x)) = TRUE and
#{x | vM((x)(x)) = vM((x))=TRUE}
M
non-
= a2 and
a1
b1
b1 = #{x | vM((x)) = vM((x)) = TRUE }
#{x | vM((x) (x)) = vM((x)) = TRUE}
non-
c1
d1
Then, for any other formulae *(x), = #{x | vM(((x)(x))) = vM((x)) = TRUE }.
Thus, we have
*(x) such that
M
*
non-*
*
a2 >= a1
c2
non-*
b1 >= b2
d2
we have
vM(*(x) *(x)) = TRUE, too.
We will use this fact in the next
Theorem, too.
M
non-
-OR-
a2 >= a1
c2
non-(OR)
b1 >= b2
d2
Since is implicational we conclude that
vM((x) [(x)(x)]) = TRUE, too.
Theorem 3. Let (x), (x), (x) be formulae, Proof. Let M be a model such that
and let be an implicational quantifier. Then vM( ([(x) (x)] (x)) = TRUE and
[(x) (x)] (x)
M
non-
(x) [(x)(x)]
and non-
a1
b1
is a sound rule of inference.
non- or
Theorem 4. Let (x) and (x)
be formulae, and let ~ be the
simple association quantifier.
Then
(x) ~ (x)
SYM
(x) ~ (x)
and
(x) ~ (x) NEG
(x) ~ (x)
are sound rules of inference.
Exercises
13. Prove Theorem 4.
14. Prove that Theorem 4 does
not hold for founded p-implication
quantifiers.
c1
d1
We realise that
a1 = #{x | vM((x)(x))) = vM((x)) = TRUE}
#{x | vM((x)(x))) = vM((x)) = TRUE}
= a2 and
b1 = #{x | vM((x)(x))) = vM((x)) = TRUE}
= #{x | vM((x)) = vM((x) (x)))= TRUE }
= #{x | vM((x)) = vM(((x)(x)))= TRUE }
= b1. Thus, we have in the model M
M
non-
OR
a2 >= a1
c2
non-( OR )
b1 >= b2
d2
Since is implicational we conclude that
vM((x) [(x)(x)]) = TRUE, too.•
We have introduced deduction rules (i.e. sound rules of inference) mainly to minimise
the amount tautologies, called hypothesis i.e. outputs in practical GUHA data mining
tasks. For example, Theorem 1 says that if is an implicational quantifier and
(x) (x) is true in a given model M, so is (x) [(x)(x)] true.
Thus, we do not have to print (x) [(x)(x)] as a data mining result. Next we will
study some other useful deduction rules.
Consider elementary conjunctions EC and elementary disjunctions ED, i.e. open
formulae of a form P1(x) ... kPk(x) and P1(x)... kPk(x), where i:s are either
‘’ or empty sign. For example,
P1(x)P5(x)
and P1(x)P3(x) P5(x) are EC’s
P2(x)P3(x)P4(x) and P2(x)P4(x)
are ED’s.
Denote EC’s or T by symbols , 2, 3,… (maybe empty) and denote ED’s or by
symbols , 2, 3,… (maybe empty).
Definition 6. An elementary association is a sentence of the form , where is a
quantifier and , are disjoint, i.e. have no common predicates.
Let and 22 be elementary associations. We say that results from 22 by
specification if either and 22 are identical or there is an ED 0 disjoint from 1
such that 2 and 01 are logically equivalent (i.e. have always the same truth value)
and is logically equivalent to 20. [We say also: despecifies to 22]
Example. P1(x)P3(x) P5(x) P2(x)P4(x) results from
P1(x)P5(x) P2(x)P3(x)P4(x) by specification [indeed, 0 = P3(x)]
Moreover, we say that results from 22 by reduction [or dereduces to
22] if is 2 and 1 is a subdisjunction of 2
Example.[P1(x) P5(x)] [P2(x) P3(x)P4(x)] results from
[P1(x)P5(x)] [P2(x)P3(x)P4(x)P6(x)P7(x)] by reduction
[indeed, 2 = P6(x)P7(x)].
We introduce the despecifying-dereduction rules (SpRd-rules); they are of the form
where results from 22 by successive reduction and specification,
22
i.e. there is a ED 3 (a sub-ED of 2 ) such that 1 despecifies to 23
and 23 dereduces to 22.
Example. [P1P3 P5] [P2P4] despecifies to [P1 P5] [P2P3P4] and
[P1 P5] [P2P3P4] dereduces to [P1 P5] [P2P3P4P6P7] Thus, we
have an SpRd-rule
[P1P3 P5] [P2P4]
[P1 P5] [P2P3P4P6P7]
Theorem 5. For any implicational quantifier , SpRd-rules are sound rules of inference.
Proof. In a same manner than Theorem 4 and Theorem 5.ž
Remark. Theorem 5 can be reformulated in the following way: whenever
is a SpRd-rule, then ( ) (22 ) [i.e. is a tautology].
22
Theorem 5. SpRd-rules are transitive, that is, if and
22 then 1
22
33
33
Proof. The result is obvious as soon as we realise that the order of despecification and
dereduction can be reverted, i.e.
( 2 )
( 2 ) (2 )
dereduces to
despecifies to
despecifies to
( 2 )
dereduces to
We introduce two more types of quantifiers:
p- equivalence quantifiers, where 0 < p 1.
(today: Basic equivalence) For any model M,
v(x ((x) p (x))) = TRUE
iff (a+d) p(a+b+c+d),
I
in particular, in a model M such that
M
non-
0
c
non-
b
0
b+c > 0, v(x ((x) p (x))) = FALSE
(2 2 )
p- equivalence quantifiers, also called
-double quantifiers, where 0 < p 1.
(Basic double implication) For any model M,
v(x ((x) p(x))) = TRUE
iff a p(a+b+c),
I
in particular, in a model M such that
M
non-
0
0
non-
0
d
d > 0, v(x ((x) p (x))) = FALSE
Exercises. Prove that
15. p- equivalence quantifiers and 16. p - equivalence quantifiers are associational
