APPLICATION OF THE BANACH FIXED POINT THEOREM ON
FUZZY QUASI-METRIC SPACES TO STUDY THE COST OF
ALGORITHMS WITH TWO RECURRENCE EQUATIONS
Francisco Castro-Company
Departamento de Matem´atica Aplicada, Universidad Polit´ecnica de Valencia, Camino de Vera, Valencia, Spain
Salvador Romaguera, Pedro Tirado
Instituto Universitario de Matem´atica Pura y Aplicada, Universidad Polit´ecnica de Valencia
Camino de Vera, Valencia, Spain
Keywords:
Algorithm, Banach fixed point theorem, Fuzzy quasi-metric, Recurrence equations.
Abstract:
Considering recursiveness as a unifying theory for algorithm related problems, we take advantage of algo-
rithms formulation in terms of recurrence equations to show the existence and uniqueness of solution for the
two recurrence equations associated to a kind of algorithms defined as two procedures depending the one on
the other by applying the Banach contraction principle in a suitable product of fuzzy quasi-metrics on the
domain of words.
1 INTRODUCTION
Our study is motivated, in part, by the following al-
gorithm, considered by M.D. Atkinson in (Atkinson,
1996, p. 16-17), which is defined as two procedures
depending the one on the other P and Q, such that, for
n ∈ ω:
function P(n)
if n > 0 then
Q(n-1); C;
P(n-1); C;
Q(n-1)
function Q(n)
if n > 0 then
P(n-1); C;
Q(n-1); C;
P(n-1); C;
Q(n-1)
The algorithm is shown as a pair of recurrence
equations expressed in terms of P and Q procedures.
Concrete examples of this class of algorithms
could be extracted from language theory scenarios;
such a system of equations may represent a couple
of mutually dependent rules of a grammar. Another
scenario where many cases can be found is object-
oriented design. An algorithm like this one in the
object-oriented context might express a situation of
highly coupled design; a pair of objects from the sys-
tem with methods that rely non-interactively the one
on the other to fulfill a given and more general task.
In order to demonstrate the existence and unique-
ness of solution for these recurrence equations we
will apply the Banach fixed point theorem in a suit-
able product of fuzzy quasi-metrics on the domain
of words. This technique was already used in (Ro-
maguera et al., 2007) to prove on the existence and
uniqueness of solution for the recurrence equations
associated with Quicksort, and Divide and Conquer
algorithms.
2 PRELIMINARY CONCEPTS
AND RESULTS
Our approach is based on the notion of a fuzzy
quasi-metric space, which constitutes a nonsymmet-
ric generalization of the Kramosil-Michalek defini-
tion (Kramosil and Michalek, 1975) of a fuzzy met-
ric space. A different approach to this study, based
on the theory of complexity spaces of M. Schelleck-
ens (Schellekens, 1995), may be found in (Castro-
105
Castro-Company F., Romaguera S. and Tirado P..
APPLICATION OF THE BANACH FIXED POINT THEOREM ON FUZZY QUASI-METRIC SPACES TO STUDY THE COST OF ALGORITHMS WITH
TWO RECURRENCE EQUATIONS.
DOI: 10.5220/0003081301050109
In Proceedings of the International Conference on Fuzzy Computation and 2nd International Conference on Neural Computation (ICFC-2010), pages
105-109
ISBN: 978-989-8425-32-4
Copyright
c
 2010 SCITEPRESS (Science and Technology Publications, Lda.)

Company et al., 2010).
Definition 1. (Cho et al., 2006; Gregori and Roma-
guera, 2004). A fuzzy quasi-metric on a set X is a pair
(M,∗) such that ∗ is a continuous t-norm and M is a
fuzzy set in X × X × [0,∞) such that for all x,y,z ∈ X:
(KM1) M(x,y, 0) = 0.
(KM2) x = y if and only if M(x,y,t) = M(y, x,t) = 1
for all t > 0.
(KM3) M(x,z,t + s) ≥ M(x,y,t) ∗ M(y,z, s) for all
t,s ≥ 0.
(KM4) M(x,y, ) : [0,∞) → [0,1] is left continuous.
Definition 2. (Kramosil and Michalek, 1975). A
fuzzy metric on a set X is a fuzzy quasi-metric (M, ∗)
on X such that for each x,y ∈ X:
(KM5) M(x,y,t) = M(y,x,t) for all t > 0.
Definition 3. (Cho et al., 2006; Gregori and Roma-
guera, 2004). A fuzzy (quasi-)metric space is a triple
(X,M,∗) such that X is a set and (M, ∗) is a fuzzy
(quasi-)metric on X.
Each fuzzy (quasi-)metric (M,∗) on a set X in-
duces a topology τ
M
on X which has as a base the
family of open balls {B
M
(x,ε,t) : x ∈ X, 0 < ε < 1,
t > 0}, where B
M
(x,ε,t) = {y ∈ X :M(x,y,t) > 1−ε}.
If (M,∗) is a fuzzy quasi-metric on a set X, it is
obvious that (M
−1
,∗) is also a fuzzy quasi-metric on
X, where M
−1
is the fuzzy set in X × X × [0,∞) de-
fined by
M
−1
(x,y,t) = M(y, x,t).
Moreover, if we denote by M
i
the fuzzy set in X×X ×
[0,∞) given by
M
i
(x,y,t) = min{M(x,y,t),M
−1
(x,y,t)},
then (M
i
,∗) is, clearly, a fuzzy metric on X.
A fuzzy (quasi-)metric space (X,M,∗) such that
M(x,z,t) ≥ min{M(x,y,t),M(y,z,t)},
for all x, z ∈ X and t > 0, is said to be a non-
Archimedean fuzzy (quasi-)metric space.
In (Grabiec, 1988), M. Grabiec introduced the fol-
lowing notions in order to obtain a fuzzy version of
the classical Banach fixed point theorem (an exhaus-
tive study of fixed point theory on fuzzy metric spaces
and related structures may be found in (Hadzic and
Pap, 2001)):
A sequence (x
n
)
n
in a fuzzy metric space (X,M,∗)
is Cauchy provided that lim
n→∞
M(x
n
,x
n+p
,t) = 1 for
each t > 0 and p ∈ N.
A fuzzy metric space (X,M,∗) is complete pro-
vided that every Cauchy sequence in X is convergent.
In this case, (M, ∗) is called a complete fuzzy metric
on X.
In the sequel, and according to (Gregori and
Sapena, 2002) and (Vasuki and Veeramani, 2003), a
Cauchy sequence in Grabiec’s sense will be called G-
Cauchy and a complete fuzzy metric space in Gra-
biec’s sense will be called G-complete.
On the other hand, following (Sehgal and
Bharucha-Reid, 1972), a B-contraction on a fuzzy
metric space (X,M,∗) is a self-map f on X such that
there is a constant k ∈ (0,1) satisfying
M( f(x), f(y),kt) ≥ M(x, y,t)
for all x,y ∈ X, t > 0.
Thus, Grabiec’s fixed point theorem can be formu-
lated as follows.
Theorem 1. (Grabiec, 1988). Let (X,M,∗)
be a G-complete fuzzy metric space such that
lim
t→∞
M(x,y,t) = 1 for all x, y ∈ X. Then every B-
contraction on X has a unique fixed point.
The following quasi-metric generalizations of the
notions of B-contraction and G-completeness were
introduced in (Romaguera et al., 2007).
Definition 4. A B-contraction on a fuzzy quasi-metric
space (X,M, ∗) is a self-map f on X such that there is
a constant k ∈ (0, 1) satisfying
M( f(x), f(y),kt) ≥ M(x, y,t)
for all x,y ∈ X, t > 0. The number k is then called a
contraction constant of f.
Definition 5. A sequence (x
n
)
n
in a fuzzy quasi-
metric space (X, M, ∗) is called G-Cauchy if it is a G-
Cauchy sequence in the fuzzy metric space (X, M
i
,∗).
Definition 6. A fuzzy quasi-metric space (X,M, ∗)
is called G-bicomplete if the fuzzy metric space
(X,M
i
,∗) is G-complete.
Then, Grabiec’s theorem was generalized to fuzzy
quasi-metric spaces in (Romaguera et al., 2007) as
follows.
Theorem 2. (Romaguera et al., 2007). Let (X,M,∗)
be a G-bicomplete fuzzy quasi-metric space such
that lim
t→∞
M(x,y,t) = 1 for all x,y ∈ X. Then every B-
contraction on X has a unique fixed point.
Since G-(bi)completeness is a very strong kind of
completeness (see (George and Veeramani, 1994; Va-
suki and Veeramani, 2003)), George and Veeramani
introduced the following notions:
A sequence (x
n
)
n
in a fuzzy metric space (X,N, ∗)
is a Cauchy sequence (George and Veeramani, 1994)
if for each ε ∈ (0, 1), t > 0 there exists n
0
∈ N such
that M(x
n
,x
m
,t) > 1− ε for all n,m ≥ n
0
.
ICFC 2010 - International Conference on Fuzzy Computation
106

A fuzzy metric space is complete provided that ev-
ery Cauchy sequence in X is convergent.
Definition 7. A fuzzy quasi-metric space (X,M,∗) is
called bicomplete if the fuzzy metric space (X,M
i
,∗)
is bicomplete.
Then we have the following nice and useful fact
for our approach.
Theorem 3. (Romaguera et al., 2007). Each bicom-
plete non-Archimedeanfuzzy quasi-metric space is G-
bicomplete.
Let us recall (Gregori and Romaguera, 2004)
that if (X,d) is a (quasi-)metric space, then the pair
(M
d
,∧) is a fuzzy (quasi-)metric on X where M
d
is
the fuzzy set in X × X × [0,∞) given by M
d
(x,y,0) =
0, and, for t > 0, by
M
d
(x,y,t) =
t
t + d(x, y)
.
The triple (X,M
d
,∧) is called the standard fuzzy
(quasi-)metric space.
Furthermore, we have that (M
d
)
−1
= M
d
−1
and
(M
d
)
i
= M
d
s
. In addition, topology τ
d
, induced by d,
coincides with the topology τ
M
d
induced by the fuzzy
(quasi-)metric (M
d
,∧).
Next we recall several pertinent facts and results
on the domain of words and some non-Archimedean
quasi-metric that one can construct on it because they
will be used in section 3.
The domain of words Σ
∞
(Kunzi, 1995;
Matthews, 1994; Romaguera and Schellekens,
2005; Schellekens, 2004; Smyth, 1988, etc) consists
of all finite and infinite sequences (“words”) over a
nonempty set (“alphabet”) Σ, ordered by the so-called
information order ⊑ on Σ
∞
, i.e., x ⊑ y ⇔ x is a prefix
of y, where we assume that the empty sequence φ is
an element of Σ
∞
.
For each x, y ∈ Σ
∞
denote by x⊓y the longest com-
mon prefix of xand y, and for each x ∈ Σ
∞
denote
by ℓ(x) the length of x. Thus ℓ(x) ∈ [1,∞] whenever
x 6= φ, and ℓ(φ) = 0.
Given a nonempty alphabet Σ, Smyth introduced
in (Smyth, 1988) a non-Archimedeanquasi-metric d
⊑
on Σ
∞
given by d
⊑
(x,y) = 0 if x ⊑ y, and d
⊑
(x,y) =
2
−ℓ(x⊓y)
otherwise (see also (Kunzi, 1995; Rodr´ıguez-
L´opez et al., 2008; Romaguera et al., 2007, etc)).
This quasi-metric has the advantage that its spe-
cialization order coincides with the order ⊑, and thus
the quasi-metric space (Σ
∞
,d
⊑
) preserves the infor-
mation provided by ⊑. Moreover, the metric (d
⊑
)
s
is
given by (d
⊑
)
s
(x,y) = 0 if x = y, and (d
⊑
)
s
(x,y) =
2
−ℓ(x⊓y)
otherwise; so that (d
⊑
)
s
is exactly the cel-
ebrated Baire metric on Σ
∞
. Since the Baire metric
is complete, it follows that d
⊑
is a bicomplete non-
Archimedean quasi-metric on Σ
∞
.
3 THE BANACH FIXED POINT
THEOREM ON FUZZY
QUASI-METRIC SPACES
APPLIED TO ALGORITHMS
COST ANALYSIS
In order to apply techniques of fixed point for ob-
taining the existence and uniqueness of solution for
the two recurrence equations associated to algorithms
with two recurrence procedures, we shall combine the
above results with some facts on the product of (non-
Archimedean) fuzzy quasi-metrics that we present in
the sequel.
Similarly to (Cho et al., 2009) the product
(fuzzy quasi-metric) space of two fuzzy quasi-metric
spaces (X
1
,M
1
,∗) and (X
2
,M
2
,∗) is the fuzzy quasi-
metric space (X
1
× X
2
,M
1
× M
2
,∗) such that for each
(x
1
,x
2
),(y
1
,y
2
) ∈ X
1
× X
2
and each t ≥ 0,
(M
1
× M
2
)((x
1
,x
2
),(y
1
,y
2
),t) =
M
1
(x
1
,y
1
,t) ∗ M
2
(x
2
,y
2
,t).
In particular, if (X
1
,M
1
,∧) and (X
2
,M
2
,∧) are
non-Archimedean, then (X
1
× X
2
,M
1
× M
2
,∧) is non-
Archimedean.
Furthermore, it is clear that if (X
1
,M
1
,∗) and
(X
2
,M
2
,∗) are bicomplete, then (X
1
×X
2
,M
1
×M
2
,∗)
is bicomplete.
By applying the above results to the standard
fuzzy quasi-metric space of (Σ
∞
,d
⊑
) when ∗ = ∧, we
immediately deduce from Theorems 2 and 3 the fol-
lowing.
Theorem 4. (Σ
∞
× Σ
∞
,M
d
⊑
× M
d
⊑
,∧) is a bicom-
plete non-Archimedean fuzzy quasi-metric space such
that lim
t→∞
(M
d
⊑
× M
d
⊑
)((x
1
,x
2
),(y
1
,y
2
),t) = 1 for all
(x
1
,x
2
),(y
1
,y
2
) ∈ Σ
∞
× Σ
∞
. Therefore, every B-
contraction on this space has a unique fixed point.
As mentioned in Section 1, following Atkinson
(Atkinson, 1996, p. 16-17), consider the two recur-
sive procedure algorithm defined, for two procedures
P and Q, and n ∈ ω, by:
function P(n)
if n > 0 then
Q(n-1); C;
P(n-1); C;
Q(n-1)
function Q(n)
APPLICATION OF THE BANACH FIXED POINT THEOREM ON FUZZY QUASI-METRIC SPACES TO STUDY
THE COST OF ALGORITHMS WITH TWO RECURRENCE EQUATIONS
107

if n > 0 then
P(n-1); C;
Q(n-1); C;
P(n-1); C;
Q(n-1)
where C denotes any statements taking time indepen-
dent of n.
Then, the execution times S(n) and T(n) of P(n)
and Q(n), satisfy, at least approximately, the recur-
rences
S(n) = S(n− 1) + 2T(n− 1) + K
1
,
and
T(n) = 2S(n− 1) + 2T(n − 1) + K
2
,
for n ∈ N, and with K
1
,K
2
, nonnegative constants.
(We assume that S(0) > 0 and T(0) > 0).
We shall deduce the existence and uniqueness of
solution for the recurrences S and T by means of a
version of the Banach fixed point theorem on a suit-
able (product) fuzzy quasi-metric space constructed
on a certain product of domain of words.
To this end, consider the recurrences A and B
given by A(0) > 0, B(0) > 0, and
A(n) = pA(n − 1) + qB(n− 1) + K
1
,
and
B(n) = rA(n − 1) + sB(n− 1) + K
2
,
for all n ∈ N, where p,q,r,s,K
1
,K
2
, are nonnegative
constants with p, q, r,s > 0.
Note that recurrences S and T are a particular case
of A and B for p = 1, q = r = s = 2.
In the rest of this section by Σ
∞
we shall denote
the domain of words where the alphabet Σ is the set
of nonnegative real numbers.
Recurrences A and B suggest the construction of
the functional
Φ : Σ
∞
× Σ
∞
→ Σ
∞
× Σ
∞
,
given for each pair x
1
,x
2
∈ Σ
∞
, by
Φ(x
1
,x
2
) = (u
1
,u
2
),
where
(u
1
)
0
= A(0), (u
2
)
0
= B(0),
and
(u
1
)
n
= p(x
1
)
n−1
+ q(x
2
)
n−1
+ K
1
,
(u
2
)
n
= r(x
1
)
n−1
+ s(x
2
)
n−1
+ K
2
,
for all n ∈ N such that n ≤ (ℓ(x
1
) ∧ ℓ(x
2
)) + 1.
Note that then ℓ(u
j
) ≥ (ℓ(x
1
) ∧ ℓ(x
2
)) + 1, for j =
1,2.
Next we prove that for each (x
1
,x
2
),(y
1
,y
2
) ∈
Σ
∞
× Σ
∞
and each t > 0, one has
(M
d
⊑
× M
d
⊑
)(Φ((x
1
,x
2
)),Φ((y
1
,y
2
)),t/2) ≥
M
d
⊑
(x
1
,y
1
,t) ∧ M
d
⊑
(x
2
,y
2
,t).
Indeed, put Φ(x
1
,x
2
) = (u
1
,u
2
) and Φ(y
1
,y
2
) =
(v
1
,v
2
) and let t > 0.
First observe that if u
1
⊑ u
2
and v
1
⊑ v
2
, we obtain
(M
d
⊑
× M
d
⊑
)(Φ((x
1
,x
2
)),Φ((y
1
,y
2
)),t/2) =
M
d
⊑
(u
1
,v
1
,t/2) ∧ M
d
⊑
(u
2
,v
2
,t/2) = 1.
Otherwise, we will take into account that, by the
construction of u
1
,u
2
,v
1
and v
2
, we have
ℓ(u
k
⊓ v
k
) ≥ (ℓ(x
1
⊓ y
1
) ∧ ℓ(x
2
⊓ y
2
)) + 1,
for k = 1, 2.
Consequently
(M
d
⊑
× M
d
⊑
)(Φ((x
1
,x
2
)),Φ((y
1
,y
2
)),t/2)
= M
d
⊑
(u
1
,v
1
,t/2) ∧ M
d
⊑
(u
2
,v
2
,t/2)
=
t/2
t/2+ d
⊑
(u
1
,v
1
)
∧
t/2
t/2+ d
⊑
(u
2
,v
2
)
=
t
t + 2
−ℓ(u
1
⊓v
1
)+1
∧
t
t + 2
−ℓ(u
2
⊓v
2
)+1
≥
t
t + 2
−(ℓ(x
1
⊓y
1
)∧ℓ(x
2
⊓y
2
))
=
t
t + 2
−ℓ(x
1
⊓y
1
)
∧
t
t + 2
−ℓ(x
2
⊓y
2
)
= M
d
⊑
(x
1
,y
1
,t) ∧ M
d
⊑
(x
2
,y
2
,t)
= (M
d
⊑
× M
d
⊑
)((x
1
,x
2
),(y
1
,y
2
),t).
4 CONCLUSIONS
We have shown that there exists a B-contraction Φ
on the (non-Archimedean) G-bicomplete fuzzy quasi-
metric space (Σ
∞
× Σ
∞
,M
d
⊑
× M
d
⊑
,∧). By Theorem
4, Φ has a unique fixed point which is obviously the
solution of the recurrences A and B.
Finally, we observe that, in practice, one actu-
ally works on the set Σ
F
of all finite words (over
the alphabet [0,∞)), that endowed with the restric-
tion of (M
d
⊑
,∧) provides a non-Archimedean fuzzy
quasi-metric space which, obviously, is not bicom-
plete. In fact the product space (Σ
F
× Σ
F
,M
d
⊑
×
M
d
⊑
,∧) is also a non-bicomplete non-Archimedean
fuzzy quasi-metric space. However, for each pair
x
1
,x
2
∈ Σ
F
, the sequence of iterations (Φ
k
(x
1
,x
2
))
k
,
is a Cauchy sequence in the complete fuzzy metric
ICFC 2010 - International Conference on Fuzzy Computation
108

space (Σ
∞
× Σ
∞
,(M
d
⊑
× M
d
⊑
)
i
,∧) by the property of
B-contraction of Φ stated above, and thus it converges
to an element (y
1
,y
2
), with ℓ(y
1
) = ℓ(y
2
) = ∞, which
is, in fact, the solution for the pair of recurrence equa-
tions A and B.
ACKNOWLEDGEMENTS
The second and third authors acknowledge the sup-
port of the Spanish Ministry of Science and Innova-
tion, under grant MTM2009-12872-C02-01.
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APPLICATION OF THE BANACH FIXED POINT THEOREM ON FUZZY QUASI-METRIC SPACES TO STUDY
THE COST OF ALGORITHMS WITH TWO RECURRENCE EQUATIONS
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