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Fuzzy Sets and Systems 228 (2013) 88 – 104 www.elsevier.com/locate/fss
On quasi-metric aggregation functions and fixed point theorems J. Martín, G. Mayor, O. Valero∗ Department of Mathematics and Computer Science, University of the Balearic Islands, Ctra. de Valldemossa km. 7.5, 07122 Palma de Mallorca, Spain Available online 28 August 2012
Abstract The problem of how to merge, by means of a function, a family of metrics into a single one was studied deeply by J. Borsík and J. Doboš [On a product of metric spaces, Math. Slovaca 31 (1981) 193–205]. Motivated by the utility of quasi-metrics in Computer Science, the Borsík and Doboš study was extended to the quasi-metric context in such a way that a general description of how to combine through a function a family of quasi-metrics in order to obtain a single one as output was provided by G. Mayor and O. Valero [Aggregation of asymmetric distances in Computer Science, Inform. Sci. 180 (2010) 803–812]. In this paper, inspired by the fact that fixed point theory provides an efficient tool in many fields of applied sciences, we have proved fixed point theorems for a new type of contractions, that we have called projective -contractions, defined between quasi-metric spaces that have been obtained via the so-called quasi-metric aggregation functions. Moreover, we show that the new fixed point results are useful to discuss, on the one hand, the complexity of a collection of recursive programs whose running times of computing hold a coupled system of recurrence equations and, on the other hand, to analyze simultaneously the complexity and the correctness of recursive algorithms that perform a computation by means of a recursive denotational specification. © 2012 Elsevier B.V. All rights reserved. Keywords: Metric; Quasi-metric; Aggregation function; Homogeneous function; Projective contraction; Projective -contraction; Asymptotic complexity analysis; Denotational semantics
1. Introduction and mathematical preliminaries In applied sciences the mathematical theory of information aggregation plays a central role because of its wide range of applications to practical problems. In many situations it is necessary to process incoming data, symbolized via numerical values, which comes from sources of different nature in order to obtain a unique numerical value that allows to make a decision. In this sense aggregation functions become of great importance because they allow to develop efficient numerical fusion methods. A wide class of aggregation techniques impose a constraint in order to select the most suitable numerical aggregation function for the problem under study. In general this constraint consists of considering only those functions that provide the output data keeping the main properties of the inputs. An example of this type of situation arises when a collection of metrics are merged in order to obtain a new one. Since the notion of metric is crucial in the study of many processes in applied research, many authors have studied in depth which operators allow to aggregate a collection of metrics into a single one. A distinguished work is due to J. Borsík and J. Doboš, who studied intensely the general problem of merging a family of metrics into a single one in 1981 [6]. ∗ Corresponding author. Tel.: +34 971259817.
E-mail addresses:
[email protected] (J. Martín),
[email protected] (G. Mayor),
[email protected] (O. Valero). 0165-0114/$ - see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.fss.2012.08.009
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Among other things, they were able to connect the class of metric aggregation functions (functions that merge a family of metrics into a new one) with the class of monotone and subadditive ones. In order to present the aforesaid connection let us recall some pertinent notions. Throughout this paper we shall use the letters R, R+ , and N to denote the set of real numbers, the set of nonnegative real numbers, and the set of nonnegative integer numbers, respectively. Similarly, given n ∈ N, we set Rn+ = {(x1 , . . . , xn ) : x1 , . . . , xn ∈ R+ }. From now on, we will denote by 1i , where i = 1, . . . , n, the element of Rn+ given by i
1i = (0, . . . , 0, 1 , 0, . . . , 0). According to [6], a function : Rn+ −→ R+ is a metric aggregation function (metric preserving in [6]) if the function n , where X = n X and D : X × X −→ R+ is a metric for every arbitrary family of metric spaces (X i , di )i=1 i=1 i D (x, y) = (d1 (x1 , y1 ), . . . , dn (xn , yn )) for all x = (x1 , . . . , xn ), y = (y1 , . . . , yn ) ∈ X . Following [6], we will denote by O the set of all functions : Rn+ → R+ satisfying: (x) = 0 ⇔ x = 0, where 0 ∈ Rn+ with 0 = (0, . . . , 0). In addition, we will consider, as usual, the set Rn+ ordered by the pointwise order relation , i.e. x y ⇔ xi ≤ yi for all i = 1, . . . , n (where ≤ stands for the usual order on R+ ). Moreover, we will say that a function : Rn+ −→ R+ is monotone provided that (x) ≤ (y) for all x, y ∈ Rn+ with x y. Furthermore, a function : Rn+ −→ R+ is said to be subadditive if (x + y) ≤ (x) + (y) for all x, y ∈ Rn+ . According to [17], a function : Rn+ −→ R+ is called homogeneous provided that (x) = (x) for all x ∈ Rn+ and ∈ R+ . As we announced before, Borsík and Doboš gave a connection between subadditive and monotone functions and metric aggregation functions. In particular they proved the below useful result. Proposition 1. Let ∈ O. If is monotone and subadditive, then is a metric aggregation function. Unfortunately, the preceding proposition does not characterize metric aggregation functions because there are metric aggregation functions which are not monotone (see [26, Example 8]). Of course Borsík and Doboš proved a characterization of metric aggregation functions in terms of the so-called triplets. However we will not recall such a notion and result, since they are not useful for our purpose here. Following the original ideas of Borsík and Doboš, A. Pradera, E. Trillas and E. Castiñeira have provided a general solution to the problem of merging data represented by means of a family of generalized metrics and pseudometrics in [32,31,30]. Recently, several connections between information fusion theory and metric aggregation functions have been given in [38,7,27,43]. In the last years a generalization of the notion of metric space, the so-called quasi-metric spaces, has gained interest because they become useful in some fields of Computer Science. Metric tools based on quasi-metric spaces have been introduced and developed in order to provide an efficient framework to model processes, for instance, in complexity analysis of programs and algorithms [42,35,14,13,8,21,41,36], logic programming [18,39,40] and denotational semantics [2,3,22,23,42]. Next let us recall a few concepts about quasi-metric spaces. In our context by a quasi-metric [20] on a (nonempty) set X we mean a nonnegative real-valued function d on X×X such that for all x, y, z ∈ X : (i) d(x, y) = d(y, x) = 0 ⇔ x = y. (ii) d(x, z) ≤ d(x, y) + d(y, z). Note that a metric on a set X is a quasi-metric d on X satisfying, in addition, the following condition for all x, y ∈ X : (iii) d(x, y) = d(y, x). Of course, a quasi-metric space is a pair (X,d) such that X is a (nonempty) set and d is a quasi-metric on X. If d is a quasi-metric on X, then a metric d s can be defined on X×X by d s (x, y) = max{d(x, y), d(y, x)} for all x, y ∈ X .
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A quasi-metric space (X,d) is said to be bicomplete if the associated metric space (X, d s ) is complete. Of course, the notion of bicomplete quasi-metric space concurs with the notion of complete metric space when the quasi-metric is in fact a metric. Most of the aforesaid applications of quasi-metric spaces to model processes in Computer Science are obtained via the use of the celebrated Banach fixed point theorem for metric spaces and extensions of this result to the quasi-metric context (see, for instance, [42,39,3,35,40,13,34,14,18,36,37]. Let us recall, in order to help the reader, that a function F from a (quasi-)metric space (X,d) into itself is said to be a contraction if there exists c ∈ [0, 1[ such that d(F(x), F(y)) ≤ cd(x, y) for all x, y ∈ X . In the light of the preceding notion the Banach fixed point theorem can be stated as follows: Theorem 2. Let (X,d) be a complete metric space and let F : X −→ X . If F is a contraction from (X,d) into itself, then F has a unique fixed point x0 . Taking into account that every contraction from a quasi-metric space into itself is a contraction from the associated metric space into itself, Theorem 2 can be easily extended to the context of bicomplete quasi-metric spaces. Theorem 3. Let (X,d) be a bicomplete quasi-metric space and let F : X −→ X . If F is a contraction from (X,d) into itself, then F has a unique fixed point x 0 . Motivated by the applicability of quasi-metric spaces, Mayor and Valero [26] extended the notion of metric aggregation function of Borsík and Doboš to the quasi-metric context. Thus they introduced the so-called quasi-metric aggregation function in the following way. A function : Rn+ −→ R+ is a quasi-metric aggregation function if the n function n D : X × X −→ R+ is a quasi-metric for every arbitrary family of quasi-metric spaces (X i , di )i=1 , where X = i=1 X i and D (x, y) = (d1 (x1 , y1 ), . . . , dn (xn , yn )) for all x = (x1 , . . . , xn ), y = (y1 , . . . , yn ) ∈ X . In order to provide a general description of how to combine a family of quasi-metrics into a single one the next characterization of quasi-metric aggregation functions was obtained in [26] (see also [24]): Theorem 4. A function : Rn+ −→ R+ is a quasi-metric aggregation function if and only if ∈ O and is subadditive and monotone. The preceding theorem was crucial in [26,24] to connect the aggregation theory of quasi-metric spaces with the complexity analysis of programs and algorithms in Computer Science. Specifically, Mayor and Valero showed that, in many instances, the quasi-metrics used in order to quantify the efficiency gained when an algorithm is substituted by another one, in the spirit of Schellekens [42], can be retrieved from our exposed theory by means of a family of distinguished quasi-metric aggregation functions. Inspired by the aforementioned utility of Theorems 2 and 3 in Computer Science, a fixed point theorem for a kind of contractions between quasi-metric spaces obtained through quasi-metric aggregation functions was proved in [25] in such a way that Theorems 2 and 3 are retrieved as a particular case. To this end, given a family of quasi-metric n and a quasi-metric aggregation function , it was introduced the notion of projective contraction spaces (X, di )i=1 from the quasi-metric space (X, D ) into itself as a mapping such that each ith coordinate function is a contraction from the quasi-metric space (X i , di ) into itself (see Definition 6 in Section 2). Moreover, sufficient conditions on the quasi-metric space (X, D ) and the aggregation function were given to ensure the existence and uniqueness of fixed point for this new type of contractions in [25] (see Corollary 12 in Section 2). In this paper we focus our attention on providing a fixed point theorem for a new type of contraction on quasi-metric spaces which are obtained through quasi-metric aggregation functions. Thus we introduce the notion of projective -contraction and we show that every projective contraction is always a projective -contraction. Moreover, we prove that the existence and uniqueness of fixed point for this type of contractions are guaranteed whenever every member of
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the family of quasi-metric spaces to be merged is bicomplete and the quasi-metric aggregation function used to merge is homogeneous and holds the condition (1, . . . , 1) ≤ 1. Several examples elucidate that the last conditions cannot be deleted from the hypothesis of our main result in order to ensure the existence and uniqueness of fixed point and, in addition, a few connections between aggregation theory of quasi-metric spaces and the classical theory of aggregation functions are given. Moreover, we show that the obtained fixed point results are useful to discuss, on the one hand, the complexity of a collection of recursive programs whose running times of computing hold a coupled system of recurrence equations and, on the other hand, to analyze simultaneously the complexity and the correctness of recursive algorithms that perform a computation by means of a recursive denotational specification. 2. The fixed point theorem and aggregation In this section our aim is to prove a fixed point theorem for a new kind of contractions between quasi-metric spaces which have been obtained through the aggregation of a family of quasi-metric spaces in such a way that Theorems 2 and 3 can be retrieved as a particular case. n n Let (X i )i=1 be a family of nonempty sets and X = i=1 X i . In order to state our new fixed point result, we need to introduce the new contraction mapping type notion, what we call projective -contraction. To this end, let us recall that, given a mapping F : X −→ X , the coordinate functions of F are the functions Fi : X −→ X i , i = 1, . . . , n, such that F(x) = (F1 (x), . . . , Fn (x)) for all x ∈ X . n n be a family of arbitrary quasi-metric spaces, X = i=1 X i and : Rn+ −→ R+ a quasiDefinition 5. Let (X i , di )i=1 metric aggregation function. We will say that a mapping F : X −→ X is a projective -contraction from (X, D ) into itself, provided the existence of n constants c1 , . . . , cn ∈ [0, 1[ such that di (Fi (x), Fi (y)) ≤ ci (d1 (x1 , y1 ), . . . , dn (xn , yn )) for all x, y ∈ X and for all i = 1, . . . , n. Observe that whenever n=1 and is the identity function in Definition 5 we retrieve as a particular case of our new notion the classical one for quasi-metric spaces, i.e., the notion of a contraction from a quasi-metric space into itself. As we have explained in Section 1, in order to take the first steps in the study of fixed points and the aggregation of quasi-metrics the notion of projective contraction was introduced in [25]. The aforementioned notion can be formulated as follows: n be a family of arbitrary quasi-metric spaces and X = n X . We will say that a mapping Definition 6. Let (X i , di )i=1 i=1 i F : X −→ X is a projective contraction from (X, D ) into itself, whenever there exist n constants c1 , . . . , cn ∈ [0, 1[ such that
di (Fi (x), Fi (y)) ≤ ci di (xi , yi ) for all x, y ∈ X and for all i = 1, . . . , n. Note that a projective contraction is always a projective -contraction (for an appropriate function ) and, thus, n Definition 5 extends in some sense Definition 6. Indeed, n let (X i , di )i=1 be a family of arbitrary quasi-metric spaces and F : X −→ X a projective contraction, where X = i=1 X i . Next consider a homogeneous quasi-metric aggregation function : Rn+ −→ R+ such that (1i ) = 1 for all i = 1, . . . , n. From the fact that F is a projective contraction we obtain that di (Fi (x), Fi (y)) ≤ ci di (xi , yi ) ≤ cdi (xi , yi )
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for all i = 1, . . . , n, where c = max{c1 , . . . , cn }. Since is monotone, homogeneous and satisfies (1i ) = 1 we deduce that di (xi , yi ) = (1i di (xi , yi )) ≤ (d1 (x1 , y1 ), . . . , dn (xn , yn )) for all i = 1, . . . , n. Hence di (Fi (x), Fi (y)) ≤ cdi (xi , yi ) ≤ c(d1 (x1 , y1 ), . . . , dn (xn , yn )) for all i = 1, . . . , n. Notice that examples of homogeneous quasi-metric aggregation functions satisfying the condition (1i ) = 1 for all i = 1, . . . , n are the functions : Rn+ −→ R+ and : Rn+ −→ R+ given by (x) = max{x1 , . . . , xn }, (x) =
n
xi
i=1
for all x = (x1 , . . . , xn ) ∈ Rn+ . In fixed point theory the completeness of quasi-metric spaces plays a central role and, for this reason, the next lemma will be useful in the proof of our main result later on. n be a family of arbitrary quasi-metric spaces and X = Lemma 7. Let (X i , di )i=1 quasi-metric aggregation function, then the following assertions are equivalent:
n
i=1 X i .
If is a homogeneous
(1) For each i = 1, . . . , n, the quasi-metric space (X i , di ) is bicomplete. (2) The quasi-metric space (X, D ) is bicomplete. s ). Then, given > 0, there exists k ∈ N such that Proof. (1) ⇒ (2). Let (x k )k∈N be a Cauchy sequence in (X, D 0 s k j D (x , x ) < (1i ) for all k, j ≥ k0 (observe that (1i ) 0, since is a quasi-metric aggregation function). Then j
j
(d1 (x1k , x1 ), . . . , dn (xnk , xn )) < (1i ) · and j
j
(d1 (x1 , x1k ), . . . , dn (xn , xnk )) < (1i ) · for all k, j ≥ k0 . Moreover, from monotonicity of we obtain that j
j
j
(di (xik , xi ) · 1i )≤(d1 (x1k , x1 ), . . . , dn (xnk , xn )) < (1i ) · and j
j
j
(di (xi , xik ) · 1i ) ≤ (d1 (x1 , x1k ), . . . , dn (xn , xnk )) < (1i ) · for all i = 1, . . . , n. Since is homogeneous we deduce that j
j
j
j
max{di (xik , xi ), di (xi , xik )} · (1i ) = max{(di (xik , xi ) · 1i ), (di (xi , xik ) · 1i )} < (1i ) · . j
for all i = 1, . . . , n. It follows that dis (xik , xi ) < for all k, j ≥ k0 and for all i = 1, . . . , n. Whence we deduce that there exist xi ∈ X i such that limk→∞ xik = xi in (X i , dis ) for all i = 1, . . . , n, since (X i , di ) is a bicomplete quasi-metric space for all i = 1, . . . , n. s ) with x = (x , . . . , x ). Indeed, since is monotone and homogeneous Next we prove that limk→∞ x k = x in (X, D 1 n we have that D (x, x k ) = (d1 (x1 , x1k ), . . . , dn (xn , xnk ))≤(d1s (x1 , x1k ), . . . , dns (xn , xnk )) ≤ max{d1s (xi , xik ), . . . , dns (xi , xik )} · (1, . . . , 1). Hence we deduce that D (x, x k ) ≤ eventually. In the same manner we can see that D (x k , x) < eventually. s ) and, thus, the quasi-metric space (X, D ) is bicomplete. Consequently, we have that limk→∞ x k = x in (X, D
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(2) ⇒ (1). Let (xik )k∈N be a Cauchy sequence in the quasi-metric space (X i , di ) and fix x j ∈ X j for all i j. Then, given > 0, there exists k0 ∈ N such that di (xim , xil ) < /(1i ) for all m, l ≥ k0 . From the fact that is homogeneous we deduce that D (y m , y l ) = (0, . . . , 0, di (xim , xil ), 0, . . . , 0) = di (xim , xil ) · (1i ) < (1i ) = (1i ) for all m, l ≥ k0 , where the sequence (y k )k∈N is given by y kj = x j for all j i and yik = xik . It follows that the sequence (y k )k∈N is Cauchy in (X, D ). The bicompleteness of (X, D ) provides the existence of y ∈ X such that s ). Hence, by the monotonicity of and the fact that is homogeneous, we have y = limk→+∞ y k in (X, D s (y, x k ) < dis (yi , xik )(1i ) = max{(di (yi , xik ) · 1i ), (di (xik , yi ) · 1i )} ≤ D
eventually. Consequently, we obtain that yi = limk→+∞ xik in (X, dis ). So the quasi-metric space (X, di ) is bicomplete. The next result states, among other things, that every projective -contraction is a contraction from the quasi-metric space obtained via aggregation into itself. n n Theorem 8. Let (X i , di )i=1 be a family of arbitrary bicomplete quasi-metric spaces and X = i=1 X i . If is a homogeneous quasi-metric aggregation function such that (1, . . . , 1) ≤ 1 and F is a projective -contraction, then the following assertions hold: (1) F is a contraction from the quasi-metric space (X, D ) into itself. (2) F has a unique fixed point x 0 ∈ X . Proof. Let x, y ∈ X . Then the monotonicity of and the fact that F is a projective -contraction yields D (F(x), F(y)) = (d1 (F1 (x), F1 (y)), . . . , dn (Fn (x), Fn (y))) ≤ (c1 (d1 (x1 , y1 ), . . . , dn (xn , yn )), . . . , cn (d1 (x1 , y1 ), . . . , dn (xn , yn ))), (c(d1 (x1 , y1 ), . . . , dn (xn , yn )), . . . , c(d1 (x1 , y1 ), . . . , dn (xn , yn ))), where c = max{c1 , . . . , cn }. From the fact that is homogeneous we deduce that (c(d1 (x1 , y1 ), . . . , dn (xn , yn )), . . . , c(d1 (x1 , y1 ), . . . , dn (xn , yn ))) = c(1, . . . , 1)(d1 (x1 , y1 ), . . . , dn (xn , yn )) and, hence, D (F(x), F(y)) ≤ c(1, . . . , 1)D (x, y) ≤ cD (x, y). Therefore, the mapping F : X −→ X is a contraction from the quasi-metric space (X, D ) into itself. Since Lemma 7 yields that the quasi-metric space (X, D ) is bicomplete we obtain from Theorem 3 that F has a unique fixed point x0 ∈ X . The proof is complete. Observe that whenever n=1 and is the identity function in the statement of Theorem 8 we retrieve as a particular case Theorem 3. Next we show that, in the proof of Theorem 8 the assumption “ is homogeneous” and “(1, . . . , 1) ≤ 1” are crucial in order to prove assertion (1). In fact the next examples show that the aforementioned assumptions cannot be deleted from the statement of Theorem 8 in order to guarantee that a projective -contraction is, at the same time, a contraction from (X, D ) into itself. Example 9. Consider the (bicomplete) quasi-metric space ([0, 1], du ), where du (x, y) = max{y − x, 0} for all x, y ∈ [0, 1] and the family of (bicomplete) quasi-metric spaces ([0, 1], di )i=1,2 such that d1 = d2 = du .
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Define the function : R2+ −→ R+ by (x) =
0 if x1 = x2 = 0, 1 otherwise.
(1)
It is clear that the function is a quasi-metric aggregation function which satisfies that (1, 1) ≤ 1 and that is not homogeneous. Note that 1 = (2, 2) 2 · (1, 1) = 2. Next, consider the mapping F : [0, 1]2 −→ [0, 1]2 defined by F(x) = (x1 /2, x2 /2) for all x ∈ [0, 1]2 . It is clear that F is a projective -contraction. Nevertheless F is not a contraction from ([0, 1]2 , D ) into itself, since D ((F(1, 0), F(0, 1)) = (0, 21 ) = 1, D ((1, 0), (0, 1)) = (0, 1) = 1 and, thus, does not exist c ∈ [0, 1[ such that D (F(1, 0), F(0, 1)) ≤ cD ((1, 0), (0, 1)). Therefore, we have shown that the assumption “ is homogeneous” cannot be deleted in the statement of Theorem 8. Example 10. Let ([0, 1], di )i=1,2 be the family of quasi-metric spaces introduced in Example 9. Define the function : R2+ −→ R+ by (x) = x1 + x2 for all x ∈ R2+ . It is obvious that the function is a homogeneous quasi-metric aggregation function. Moreover, it is clear that does not hold the condition “(1, 1) ≤ 1” because (1, 1) = 2. Next consider the mapping F : [0, 1]2 −→ [0, 1]2 defined by F(x) = ((x1 + x2 )/2, (x1 + x2 )/2) for all x ∈ [0, 1]2 . Then we have
x1 + x2 y1 + y2 1 du (Fi (x), Fi (y) = du , = max{y1 − x1 + y2 − x2 , 0} 2 2 2 ≤
1 1 1 max{y1 − x1 , 0} + max{y2 − x2 , 0} = (du (x1 , y1 ), du (x2 , y2 )) 2 2 2
for all x, y ∈ [0, 1]2 and for i=1,2. So, F is a projective -contraction. However, F is not a contraction from the quasi-metric space ([0, 1]2 , D ) into itself. Indeed, take x, y ∈ [0, 1]2 given by x = (0, 0) and y = (1, 1). Then there does not exist c ∈ [0, 1[ such that D (F(0, 0), F(1, 1)) ≤ cD ((0, 0), (1, 1)), since D (F(0, 0), F(1, 1)) = D ((0, 0), (1, 1)) = 2. Therefore, we have shown that the assumption “(1, . . . , 1) ≤ 1” cannot be deleted in statement of Theorem 8. In the light of Theorem 8, it seems natural to wonder if a contraction from the quasi-metric space (X, D ) into itself is always a projective -contraction whenever the quasi-metric aggregation function is homogeneous and verifies (1, . . . , 1) ≤ 1. However the next example gives a negative answer to the masterminded question. Example 11. Let ([0, 1], di )i=1,2 be again the family of quasi-metric spaces introduced in Example 9. Define the function : R2+ −→ R+ by (x) = (x1 + x2 )/2 for all x ∈ R2+ . It is obvious that the function is a homogeneous quasi-metric aggregation function such that (1, 1) ≤ 1. Consider the mapping F : [0, 1]2 −→ [0, 1]2 defined by F(x) = ((x1 + x2 )/2, 0) for all x ∈ [0, 1]2 . Then we have 1 y1 − x1 y2 − x2 1 y1 − x1 1 y2 − x2 D (F(x), F(y)) = max + , 0 ≤ max , 0 + max ,0 2 2 2 2 2 2 2 =
1 D (x, y) 2
for all x, y ∈ [0, 1]2 . Whence we deduce that F is a contraction from the quasi-metric space ([0, 1]2 , D ) into itself. However, F is not a projective -contraction. Indeed, take x, y ∈ [0, 1]2 given by x = (0, 0) and y = (1, 1). Then du (F1 (x), F1 (y)) = du (0, 1) = 1 and du (x1 , y1 ) = du (x2 , y2 ) = du (0, 1) = 1. Hence we deduce that there
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does not exist c ∈ [0, 1[ such that du (F1 (x), F1 (y)) ≤ c(du (x1 , y1 ), du (x2 , y2 )), since (du (x1 , y1 ), du (x2 , y2 )) = (1, 1) = 1. Therefore F is not a projective -contraction. The below result, which is the main theorem in [25], is a version of Theorem 8 for projective contractions. n n Theorem 12. Let (X i , di )i=1 be a family of arbitrary bicomplete quasi-metric spaces and X = i=1 X i . If is a homogeneous quasi-metric aggregation function and F is a projective contraction from (X, D ) into itself, then the following assertions hold: (1) F is a contraction from the quasi-metric (X, D ) into itself. (2) F has a unique fixed point x 0 ∈ X . Proof. Let x, y ∈ X . Then the monotonicity of and the fact that F is a projective contraction yields D (F(x), F(y)) = (d1 (F1 (x), F1 (y)), . . . , dn (Fn (x), Fn (y))) ≤ (c1 d1 (x1 , y1 ), . . . , cn dn (xn , yn )) ≤ (cd1 (x1 , y1 ), . . . , cdn (xn , yn )), where c = max{c1 , . . . , cn }. From the fact that is homogeneous we deduce that (cd1 (x1 , y1 ), . . . , cdn (xn , yn )) = c(d1 (x1 , y1 ), . . . , dn (xn , yn )) and, hence, D (F(x), F(y)) ≤ cD (x, y) ≤ cD (x, y). Therefore, the mapping F : X −→ X is a contraction from the quasi-metric (X, D ) into itself. Since Lemma 7 yields that the quasi-metric space (X, D ) is bicomplete we obtain from Theorem 3 that F has a unique fixed point x0 ∈ X . This completes the proof. Quasi-metric aggregation function such that (1 . . . , 1) ≤ 1, Y is closed in (X, D ), F is a projective -contraction from (Y, D ) into itself, then the following assertions hold: Notice that Definition 6 can be directly adapted to the case of metric spaces in the obvious manner. In addition, Lemma 7 and Theorem 8 remain valid, in fact the proof follows by the same method, if we replace quasi-metric space, bicomplete and quasi-metric aggregation function by metric space, complete and metric aggregation function in the statement of the aforesaid results, respectively. Thus the following “symmetric” version of Theorem 8 can be obtained: n be a family of arbitrary complete metric spaces and X = n X . If is a homogeneous Theorem 13. Let (X i , di )i=1 i=1 i metric aggregation function such that (1, . . . , 1) ≤ 1 and F is a projective -contraction from (X, D ) into itself, then the following assertions hold:
(1) F is a contraction from the metric space (X, D ) into itself. (2) F has a unique fixed point x 0 ∈ X . Since a metric aggregation function is not monotone in general (see Proposition 1) it could seem strange, in the light of the proof of Theorem 8, that the assumption “monotone” does not stand in statement of Theorem 13. However, by in [10, Theorem 8], a homogeneous metric aggregation function is always monotone. Moreover, observe that a quasi-metric aggregation function is a metric aggregation function and, thus, a slight modification of Examples 9 and 10 shows that the assumptions “ is homogeneous” and “(1, . . . , 1) ≤ 1” cannot be also deleted in the statement of Theorem 13. In particular, in order to prove our last affirmation it is enough to replace the quasi-metric space ([0, 1], du ) by the metric space ([0, 1], d E ) in Examples 9 and 10, where d E is the Euclidean metric. In addition if we consider the metric space ([0, 1], d E ) instead of the quasi-metric space ([0, 1], du ) in Example 11, then we obtain an instance of a contraction from the metric space ([0, 1]2 , D ) into itself that is not a projective -contraction although the
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metric aggregation function is homogeneous and satisfies the inequality (1, . . . , 1) ≤ 1. Furthermore, notice that Theorem 13 allows us to obtain as a particular case Theorem 2 when we consider n=1 and is the identity function. Of course, one can obtain a version of Theorem 13, which can be also found in [25], for projective contractions. n be a family of arbitrary complete metric spaces and X = n X . If is a homogeneous Theorem 14. Let (X i , di )i=1 i=1 i metric aggregation function and F is a projective contraction from (X, D ) into itself, then the following assertions hold:
(1) F is a contraction from the metric (X, D ) into itself. (2) F has a unique fixed point x 0 ∈ X . Metric aggregation function such that (1 . . . , 1) ≤ 1, Y is closed in (X, D ) and F is a projective -contraction from (Y, D ) into itself, then the following assertions hold: In the literature one can find contractive notions related to the projective one. We want to point out that for metric spaces the notion of projective contraction is a special case of the contraction in each variable separately (CIEVS) notion when only two metric spaces are considered to merge. However, contrary to the approach based on CIEVS contractions, our new projective framework (Definition 6 and Theorem 12) presents the advantage that we do not need to require that any metric space has the so-called fixed point property (see, for instance, [19,12] or [11]). 2 with X = X 1 × X 2 , a mapping F : X −→ X is said to be a Let us recall that, given two metric spaces (X i , di )i=1 CIEVS provided that for each x1 ∈ X 1 and x2 ∈ X 2 there exist c1 (x1 ), c2 (x2 ) ∈ [0, 1[ such that d(F1 (x1 , y), F1 (x1 , z)) ≤ c1 (x1 )d((x1 , y), (x1 , z)) for all y, z ∈ X 2 and d(F2 (x, x2 ), F2 (y, x2 )) ≤ c2 (x2 )d((x, x2 ), (y, x2 )) 2 . for all x, y ∈ X 1 , where d : X −→ X is a metric constructed from the family of metrics (di )i=1 Although it seems that there is not apparent relationship between CIEVS and the projective -contractions, we want to emphasize that there exist examples of CIEVS that are not contractions from (X,d) into itself and that, however, they are contractions from (X, D ) into itself when an appropriate metric aggregation function is chosen. A typical example of this situation is given by the mapping F : [0, 1]2 −→ [0, 1]2 defined by F(x) = ((x1 + x2 )/2, (x1 + x2 )/2) for all x ∈ [0, 1]2 , where the set [0,1] is endowed with the Euclidean metric d E . Of course it is not hard to see that F is not a contraction from ([0, 1]2 , d) into itself, where 1
d(x, y) = (d E (x1 , y1 )2 + d E (x2 , y2 )2 ) 2
for all x, y ∈ [0, 1]2 . However, taking : R2+ −→ R+ given by (x) = (x 1 + x2 )/2 for all x ∈ R2+ , we immediately obtain that F is a contraction from ([0, 1]2 , D ) into itself, where D (x, y) = (d E (x1 , y1 ), d E (x2 , y2 )) =
d E (x1 , y1 ) + d E (x2 , y2 ) 2
for all x, y ∈ [0, 1]2 . Observe that D is a metric, since is, by Proposition 1, a metric aggregation function. Furthermore, notice that F is not a projective -contraction which explains that F has more than one fixed point. We end the section noting that apparently the unique connection of the theory of quasi-metric aggregation functions with the aggregation theory is given by the fact that the quasi-metric aggregation functions merge a collection of inputs in such way that their main properties are kept by the output. However, we want to stress that homogeneous quasi-metric aggregation functions are closely related to aggregation functions in the spirit of [16]. For instance, and among other properties, homogeneous quasi-metric aggregation functions present a type of averaging behavior and idempotency. Concretely, we have that every homogeneous quasi-metric aggregation function : Rn+ −→ R+ verifies for all x ∈ Rn+ and t ∈ R+ that (1, . . . , 1) · min{x 1 , . . . , xn } ≤ (x) ≤ (1, . . . , 1) · max{x 1 , . . . , xn } and (t, . . . , t) = t(1, . . . , 1).
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Moreover, as homogeneous quasi-metric aggregation functions we have several important families of aggregation functions as, for instance, the following ones:
n p 1 (1) (x) = ( i=1 wi xi ) p for all x, w ∈ Rn+ , where p ∈ [1, ∞[ and wi > 0 for all i = 1, . . . , n.. n wi xi for all x, w ∈ Rn+ with wi > 0 for all i = 1, . . . , n. (2) (x) = i=1
n n wi = 1 and (3) (x) = i=1 wi x(i) for all x ∈ Rn+ , with w ∈ Rn+ such that wi ∈ [0, 1] for all i = 1, . . . , n, i=1 wi ≥ w j for i < j, where x(i) is the ith largest of the x 1 , . . . , xn (OWA operators, see [43] and Proposition 8.21 in [33]). Observe that the OWAs operators given in the above statement (3) have, in addition to the homogeneity, the property (1, . . . , 1) ≤ 1. Moreover, this property is satisfied by the mappings given in the preceding statement (1) whenever
last n we consider, in addition, that i=1 wi ≤ 1, and by the mappings given in statement (2) whenever we consider further that wi ∈]0, 1] for all i = 1, . . . , n. Besides the above examples, the Choquet integral is another important instance of homogeneous quasi-metric aggregation functions whenever the corresponding fuzzy measure, that is a normalized capacity according to [16]), is a metric inducing fuzzy measure (MIFM for short) in the sense of [4]. Observe that OWAs operators given in statement (3) are retrieved as a particular case of a Choquet integral induced by an MIFM measure. 3. Applications: asymptotic complexity of algorithms and program correctness via aggregation In this subsection we show that the exposed theory in Section 2 is an appropriate mathematical framework for the validation of recursive definitions of programs and for their complexity analysis simultaneously. Concretely we apply our results to discuss, on the one hand, the complexity classes of a collection of recursive programs whose running times of computing hold a coupled system of recurrence equations and, on the other hand, we analyze simultaneously the complexity and the correctness of an easy, but representative, example of an algorithm that computes the factorial function by means of a recursive denotational specification. To this end, let us recall some pertinent concepts about complexity analysis of algorithms, denotational semantics for programming languages and the role of quasi-metric spaces in the aforesaid realms. 3.1. Asymptotic complexity of algorithms, denotational semantics and quasi-metrics In Computer Science the complexity analysis of an algorithm is based on determining mathematically the quantity of resources needed by the algorithm in order to solve the problem for which it has been designed. A typical resource, playing a central role in complexity analysis, is the running time of computing. The aforementioned resource is defined as the time taken by the algorithm to solve a problem, that is, the time elapsed from the moment the algorithm starts to the moment it terminates. Since there are often many algorithms to solve the same problem, one objective of the complexity analysis is to assess which of them is faster when large inputs are considered. To this end, it is required to compare their running time of computing. This is usually done by means of the asymptotic analysis in which the running time of an algorithm is denoted by a function T : N −→ (0, ∞] in such a way that T(n) represents the time taken by the algorithm to solve the problem under consideration when the input of the algorithm is of size n. In general, to determine exactly the function which describes the running time of computing of an algorithm is an arduous task. However, in most situations it is more useful to know the running time of computing of an algorithm in an “approximate” way than in an exact one. For this reason the asymptotic complexity analysis of algorithms is interested in obtaining the “approximate” running time of computing of an algorithm. The O-notation allows one to achieve this. Indeed if f, g : N −→ (0, ∞] denote the running time of computing of algorithms, then the statement g ∈ O( f ) means that there exist n 0 ∈ N and c ∈ R+ such that g(n) ≤ c f (n) for all n ∈ N with n ≥ n 0 (≤ stands for the usual order on R+ ). So the function f gives an asymptotic upper bound of the running time g and, thus, an “approximate” information of the running time of the algorithm. The set O( f ) is called the asymptotic complexity class of f. Hence, from an asymptotic complexity analysis viewpoint, to determine the running time of an algorithm consists of obtaining its asymptotic complexity class. For a deeper treatment of complexity analysis of algorithms we refer the reader to [5]. In 1995, Schellekens [42] introduced a new mathematical framework, known as complexity space, as a part of the development of a topological foundation for the asymptotic complexity analysis of algorithms. This framework is based
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on the notion of quasi-metric space. Let us recall that the complexity space is the pair (C, dC ), where ∞ −n 1 C = f : N −→ (0, ∞] : 0), and h ∈ C such that 0 < h(n) < ∞ for all n ∈ N. Consider the subset Cc of C given by Cc = { f ∈ C : f (1) = c}. Define the mapping G : Cc → Cc by c if n = 0, 1, G( f )(n) = f (n − 1) + h(n) if n ≥ 2
(3)
for all f ∈ Cc . It is clear that a complexity function in Cc is a solution to the recurrence equation (2) if and only if it is a fixed point of the mapping G. Moreover, it is not hard to check that the quasi-metric space (Cc , dC |Cc ) is bicomplete. According to Romaguera et al. [29] the mapping G is a contraction from (Cc , dC |Cc ) into itself. Concretely, we have that dC |Cc (G( f ), G(g)) ≤ 21 dC |Cc ( f, g) for all f, g ∈ Cc .
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Taking into account the two preceding facts we can apply Theorem 3 and, hence, we obtain that the mapping G has a unique fixed point f 0 ∈ C|Cc . So Theorem 3 allows us to guarantee that the recurrence Eq. (2) has a unique solution f 0 which represents the running time of the algorithm under study. With the aim of providing the asymptotic complexity class of f 0 it was proved in [29] that f 0 ∈ O(g) provided that there exists g ∈ C|Cc such that dC |Cc (G(g), g) = 0. Finally, we notice that the original ideas of Schellekens have been also extended, by means of the construction of new specific complexity spaces, to discuss simultaneously the complexity class of several recursive algorithms whose running times of computing hold a coupled system of recurrence equations of the following type (see [9]): T1 (n) = a11 T1 (n − 1) + a12 T2 (n − 1) + · · · + a1k Tk (n − 1) + c1 .. . Tk (n) = ak1 T1 (n − 1) + ak2 T2 (n − 1) + · · · + akk Tk (n − 1) + ck ,
(4)
for all n ∈ N and where ai j , ci ∈ R+ for all i, j = 1, . . . , k and Ti (0) > 0 for all i = 1, . . . , k. Of course the running time of computing of each of the involved recursive algorithms is denoted by the complexity functions Ti with i = 1, . . . , k. The following example of coupled system of recurrence equations whose solution represents the running time of computing of two recursive algorithms can be found in [1] (see also [9]): T1 (n) = T1 (n − 1) + 2T2 (n − 1) + c1 , T2 (n) = 2T1 (n − 1) + 2T2 (n − 1) + c2 ,
(5)
for all n ∈ N and where c1 , c2 ∈ R+ , T1 (0) = c and T2 (0) = d. Often in conjunction with the study of the complexity class of a recursive algorithm is performed the study of its correctness from a denotational semantics viewpoint. Let us recall that in denotational semantics one of the aims consists of giving mathematical models of programming languages in such a way that the meaning of a procedure can be obtained as an element of the constructed model. In particular most programming languages allow to construct procedures by means of recursive definitions in such a way that the meaning of such definition is employed in its own definition. In order to analyze if such a recursive denotational specification of a procedure is meaningful it is usual to make use of a mathematical approach in such a way that the meaning of such recursive denotational specification is obtained as the fixed point of a nonrecursive mapping associated to the denotational specification. In order to clarify these ideas, let us consider the easy, but illustrative, example of a procedure which computes the factorial function. To implement an algorithm that computes the factorial of a nonnegative integer number, the following recursive denotational specification is usually needed: 1 if n = 0 f act(n) = . (6) n f act(n − 1) if n > 0 Of course the above denotational specification has the drawback that the meaning of the symbol fact is expressed in terms of itself. Hence the symbol fact cannot be replaced by its meaning in the denotational specification (6), since the meaning, given by the right-hand side in (6), also contains the symbol. Generally, and following the original ideas of Scott, the meaning of the denotational specification (6) is obtained as the fixed point of a nonrecursive mapping defined from the set of partial functions (or equivalently on the set of words over an alphabet) into itself (see [15]) as follows: 1 if n = 0, H( f ) = (7) n f (n − 1) if n > 0 and n − 1 ∈ dom( f ), where by dom(f) we denote the domain of the partial function f. It is interesting to stress that the Scott model does not incorporate a “metric” tool to measure the degree of approximation of the elements that form the Scott model. In the light of the exposed facts we have that fixed point methods are used in Computer Science in order to discuss the complexity analysis of algorithms and the meaning of recursive denotational specifications for programming languages. Both methods are independent and they can be used without any connexion between them. One of them follows Schellekens’ ideas and the other one follows Scott’s ideas. However, in Section 3.2 we will show that under
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the aggregation point of view we can merge both techniques into one in such a way that we are able to analyze under the same framework, via projective -contractions and fixed point methods, the complexity classes of algorithms whose running times satisfy a coupled system of recurrence equations and, in addition, to analyze simultaneously the complexity class of the running time of computing of an algorithm that performs a computation using a recursive denotational specification and the meaning of such a specification. Specifically, we will apply our theory to obtain, on the one hand, the complexity classes of those recursive algorithms whose running times hold a system of type (5) and, on the other hand, to discuss simultaneously the complexity class and the correctness of a recursive algorithm that performs a computation of the factorial function via the recursive denotational specification of type (6). 3.2. The role of -contractions in asymptotic complexity analysis of algorithms and denotational semantics First we consider the coupled system of recurrence equations (5) and we will show that such a system has a unique solution and, besides, we will provide the asymptotic complexity classes of the running times of computing of both algorithms involved. To this end, consider the complexity spaces (Cc , dC |Cc ) and (Cd , dC |Cd ) where Cc = { f ∈ C : f (0) = c} and Cd = { f ∈ C : f (0) = d}. Define the function : R2+ −→ R+ defined by (x) = max{x1 , x2 } for all x ∈ R2+ . Then it is clear, by Theorem 4, that is a quasi-metric aggregation function. Moreover, it is obvious that is homogeneous and (1, 1) ≤ 1. Since both quasi-metric spaces are bicomplete we deduce, by Lemma 7, that the quasi-metric space (Cc × Cd , (dC |Cc , dC |Cd )) is bicomplete. Next define the mapping F : Cc × Cd −→ Cc × Cd for all ( f, g) ∈ Cc × Cd by F( f, g)(0) = (c, d) and F( f, g)(n) = ( f (n − 1) + 2g(n − 1) + c1 , 2 f (n − 1) + 2g(n − 1) + c2 ) for all n ∈ N with n > 0. A straightforward computation allows to check that F( f, g) ∈ Cc × Cd for all f, g ∈ Cc × Cd . Next we prove that F is a projective -contraction. Indeed, let ( f, g), (h, t) ∈ Cc × Cd . Then we have ∞ 1 1 −n dC |Cc (F1 ( f, g), F1 (h, t)) = − ,0 2 max F1 (h, t)(n) F1 ( f, g)(n) n=0
=
∞
2−n max
n=1
1 1 − ,0 h(n − 1) + 2t(n − 1) + c1 f (n − 1) + 2g(n − 1) + c1
∞ 1 −n 1 1 1 1 1 ≤ max − , 0 + max − ,0 2 2 h(n) f (n) 2 t(n) g(n) n=0
=
∞ n=0
2−n
1 1 1 1 1 1 max − , 0 + max − ,0 2 h(n) f (n) 4 t(n) g(n)
1 1 3 = dC |Cc ( f, h) + dC |Cd (g, t) ≤ (dC |Cc ( f, h), dC |Cd (g, t)). 2 4 4 A similar reasoning yields dC |Cd (F2 ( f, g), F2 (h, t)) ≤ 21 (dC |Cc ( f, h), dC |Cd (g, t)). According to the preceding inequalities we have that F is a projective -contraction and, hence, by Theorem 8, we deduce the existence and uniqueness of a fixed point of F, say ( f 0 , g0 ) ∈ Cc × Cd . Of course, the fixed point ( f 0 , g0 ) is the unique solution to the system (5) which represents the running times of both algorithms involved in the computation,
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that is, f 0 and g0 represent the running times of computing of the algorithms indexed by “1” and “2”, respectively. Observe that F is not a projective contraction. It remains to provide the asymptotic complexity classes of the solution ( f 0 , g0 ) to the coupled system of recurrence equations under study. It is not hard to check that if there exists ( f, g) ∈ Cc × Cd such that D (F( f, g), ( f, g)) = 0, then f 0 ∈ O( f ) and g0 ∈ O(g). Moreover, it is routine to prove that the complexity functions f ∈ Cc and g ∈ Cd given by ⎧ if n = 0, ⎨c f (n) = c + d + c1 + c2 ⎩ if n > 0 e2 − 4 and
⎧ if n = 0, ⎨d g(n) = c + d + c1 + c2 ⎩ if n > 0 e2 − 4
satisfy that D (F( f, g), ( f, g)) = 0 and, thus, that f 0 ∈ O( f ) and g0 ∈ O(g), which is in accordance with the complexity classes that can be found in the literature for those algorithms whose running times of computing hold a coupled system of recurrence equations of type (5) (see [1,9]). Next we apply our results to discuss the running time of computing of an algorithm that performs a computation using a recursive denotational specification and the meaning of such a specification simultaneously. Consider again an algorithm which computes the factorial of a nonnegative integer number through the recursive denotational specification (6). Then it is clear that its running time of computing is the solution to the following recurrence equation c if n = 0, 1, T f act (n) = (8) T f act (n − 1) + c if n > 1, where c ∈ R+ (c > 0) is the time taken by the algorithm to obtain the solution to the problem on the base case. Notice that the recurrence equation (8) is a particular case of the recurrence equation (2) with h(n) = c for all n ∈ N. As we have announced before the Scott model does not incorporate a “metric” tool to measure the degree of approximation of the elements that form the Scott model, for this reason, and with the aim of applying the fixed point framework based on projective contractions and quasi-metric aggregation functions to complexity analysis of algorithms and denotational semantics for programming languages simultaneously, we introduce a quasi-metric on the set of words over an alphabet. Let N∗ be the set of all finite and infinite sequences over the alphabet N. Now, denote by the prefix order on N∗ , i.e. x y ⇐⇒ x is a prefix of y. Next, let (x) denote the length of x. Then we have that (x) ∈ [1, ∞]. Consider the quasi-metric d on N∗ given by d (x, y) = 2−(x,y) − 2−(x) for all x, y ∈ N∗ , where (x, y) denotes the length of the longest common prefix of x and y provided that such a prefix exists, and (x, y) = 0 otherwise. Of course we adopt the convention that 2−∞ = 0. It is well known that the quasi-metric space (N∗ , d ) is bicomplete (see [20]). Moreover, similar to the complexity space case, we have that d (v, w) = 0 ⇐⇒ v w. From now on, given x ∈ N∗ , we will write x = x 0 x1 , . . . whenever (x) = ∞ and x = x0 x1 , . . . , xn−1 whenever (x) = n < ∞. In the remainder we present the promised fixed point method based on projective contractions and quasi-metric aggregation functions in order to prove simultaneously the complexity class of the running time of computing of the recursive algorithm that performs the computation of the factorial function using the denotational specification (6) and the fact that such a denotational specification is meaningful being its meaning the factorial function. Consider again the homogeneous quasi-metric aggregation function : R2+ −→ R+ defined by (x) = max{x1 , x2 } for all x ∈ R2+ .
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Define the mapping F : Cc × N∗ −→ Cc × N∗ by F( f, v) = (G f act ( f ), H (v)) for all f ∈ Cc and v ∈ N∗ , where the expression of G f act concurs with the expression of G given by (3) in Section 3.1 when we take h(n) = c for all n ∈ N, and where the nonrecursive mapping H : N∗ −→ N∗ is defined by 1 if n = 0, H (x) = (9) nxn−1 if n > 0. Hence, we immediately obtain that F is a projective contraction. In fact, from the explanation detailed in Section 3.1 we have that dC |Cc (F1 ( f, v), F1 (g, w)) ≤ 21 dC |Cc ( f, g) for all ( f, v), (g, w) ∈ Cc × N∗ . Moreover, it is a simple matter to check that d (F2 ( f, v), F2 (g, w)) ≤ 21 d (v, w) ( f, v), (g, w) ∈ Cc × N∗ . Indeed, given u, v ∈ N∗ , we have the following possible cases: Case 1. (v, w) = 0. Then, by construction of H, we obtain d (H (v), H (w)) ≤ 21 and that d (v, w) = 1. Case 2. (v, w) = n. Then, by construction of H, we obtain d (H (v), H (w)) = 2−(H (v),H (w)) − 2−(H (v)) = 2−((v,w)+1) − 2−((v)+1) = 21 d (v, w). Furthermore, since the quasi-metric spaces (Cc , dC |Cc ) and (N∗ , d ) are bicomplete we deduce, by Lemma 7, that the quasi-metric space (Cc × N∗ , (dC |Cc , d )) is bicomplete. Then Theorem 12 provides that the mapping F is a contraction from (Cc × N∗ , (dC |Cc , d )) into itself and, thus, it has a unique fixed point ( f f act , w f act ) ∈ Cc × N∗ . By construction of F the unique fixed point ( f f act , w f act ) holds that f f act is the unique fixed point of G f act and w f act is the unique fixed point of H, where f f act is the solution to the recurrence Eq. (8) and w f act ∈ N∗ satisfies that f act f act = 1 and wn = n! for all n ≥ 1. Whence we deduce that w f act represents the meaning of the denotational w0 specification (6), that is the meaning of the factorial function fact, in such a way that f f act (n) is the time taken by the algorithm to compute the factorial of the nonnegative integer number n which is provided by the numerical f act value wn . In order to finish our discussion we need to provide the complexity class of the running time of the algorithm under study. To achieve this goal, suppose that there exists (g, w) ∈ Cc × N∗ such that D (F(g, w), (g, w)) = 0. Then, on the one hand, we have that dC |Cc (G f act (g), g) = 0. It follows that dC |Cc ( f f act , g) = 0 and as a consequence f f act ∈ O(g). On the other hand, we have that d (H (w), w) = 0, which implies that H (w) w. A simple verification shows that H (w) w ⇐⇒ w = w f act . So the fact that D (F(g, w), (g, w)) = 0 fixes the element w ∈ N∗ as w f act , and this last deed provides that the equality D (F(g, w), (g, w)) = 0 gives a tool to determine the complexity class of the running time of computing of our algorithm because of D (F(g, w), (g, w)) = 0 only yields f f act ∈ O(g). Finally if we take gc as the element of C|Cc given by c if n = 0, 1, gc (n) = (10) n + c if n > 1, then we easily can check that D (F(gc , w f act ), (gc , w f act )) = 0. Whence we conclude that f f act ∈ O(gc ), which is in accordance with the complexity class that can be found in the literature (see, for instance, [28]). Therefore the quasi-metric aggregation functions allow to unify in a same framework the fixed point techniques valid for complexity analysis of algorithms and denotational semantics for programming languages in such a way that they provide an “automated” method to discuss simultaneously asymptotic complexity classes and correctness of recursive programs that perform a computation by means of recursive denotational specifications.
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4. Conclusions In this paper we have kept on going deeply in the study of the quasi-metric aggregation problem began in [26]. This time we have introduced a new notion of contraction defined on quasi-metric spaces that have been obtained through quasi-metric aggregation functions. For this new type of mappings, that we have called projective -contractions, we have proved fixed point theorems and we have applied them to model several processes that arise in asymptotic complexity of algorithms and in denotational semantics for programming languages. Acknowledgments The authors acknowledge the support of the Spanish Ministry of Science and Innovation grant MTM2009-10962. The authors are in debt with the anonymous referees whose comments helped them to improve the final version of this paper. References [1] M.D. Atkinson, The complexity of algorithms, in: Computing Tomorrow: Future Research Directions in Computer Science, Cambridge University Press, New York, 1996, pp. 1–20. 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