On aggregation of normed structures

Mathematical and Computer Modelling 54 (2011) 815–827

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On aggregation of normed structures J. Martín, G. Mayor, O. Valero ∗ Department of Math. and Computer Science, University of the Balearic Islands, 07122 Palma de Mallorca, Spain

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Article history: Received 9 October 2010 Received in revised form 22 March 2011 Accepted 22 March 2011 Keywords: Asymmetric distance Aggregation operator Quasi-norm Complexity analysis of algorithms Information fusion Artificial intelligence

abstract In 1981, Borsik and Doboš studied in depth the problem of how to merge, by means of a function, a collection (not necessarily finite) of distance spaces in order to obtain a single one as a result [J. Borsik, J. Doboš, On a product of metric spaces, Math. Slovaca 31 (1981) 193–205]. Later on, Herburt and Moszyńska studied the same problem for the case of normed linear spaces, inspired by the fact that every norm induces in a natural way a distance on a linear space, and analyzed the relationship between the both aforenamed problems [I. Herburt, M. Moszyńska, On metric products, Colloq. Math. 62 (1991) 121–133]. More recently, Romaguera and Schellekens introduced a mathematical approach, based on the notions of asymmetric distance and asymmetric normed linear space, which is suitable for the complexity analysis of programs and algorithms in Computer Science [S. Romaguera, M. Schellekens, Quasi-metric properties of complexity spaces, Topology Appl. 98 (1999) 311–322]. In this paper, motivated by the importance of the information fusion techniques in Artificial Intelligence and by the utility of asymmetric distances and asymmetric norms in Computer Science, we study the Herburt and Moszyńska problem for asymmetric normed linear spaces. In particular we give a general description of how to combine a collection (not necessarily finite) of asymmetric normed linear spaces in order to obtain a single one as output and, in addition, we clear up the relationship between this problem and its analogous of combining asymmetric distance spaces which has been already explored by Mayor and Valero [G. Mayor, O. Valero, Aggregation of asymmetric distances in computer science, Inform. Sci. 180 (2010) 803–812]. Furthermore, it is shown that the asymmetric norms employed, in the spirit of Romaguera and Schellekens, in complexity analysis can be retrieved as a particular case of the developed theory. The last fact opens the possibility of applying a wide range of properties from the general aggregation theory in Artificial Intelligence to the complexity analysis of programs and algorithms in Computer Science. © 2011 Elsevier Ltd. All rights reserved.

1. Introduction In Artificial Intelligence the problem of merging several pieces of input information arises in a natural way, coming from sources of a (possibly) different nature into a simple one in order to take a decision about the plan of action. In a wide range of practical problems, the pieces of information are symbolized by means of some numerical values. So the fusion methods that are based on numerical aggregation functions play a central role in this kind of problems. Moreover, typically, a wide class of the aggregation techniques used impose a constraint in order to select the most suitable aggregation function for the problem to be solved. In general this constraint consists of considering only those functions that merge in such a way that the output data preserves some outstanding and characteristic properties of the inputs. Several applied fields in which are used Artificial Intelligence techniques based on the aforementioned aggregation methods and where one has to confront this type of situations regularly are, among others, Image Processing, Control Theory, Medical Diagnosis or Bioinformatics.

∗

Corresponding author. E-mail addresses: [email protected] (J. Martín), [email protected] (G. Mayor), [email protected] (O. Valero).

0895-7177/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2011.03.030

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J. Martín et al. / Mathematical and Computer Modelling 54 (2011) 815–827

Since the notion of distance plays a distinguishing role in applied sciences, many authors have done research into which functions allow one to merge a collection of distances in order to obtain a single one as a final result. Thus Borsik and Doboš studied in depth the general problem of merging a collection of distances (not necessarily finite) into a single one [1]. After the work of Borsik and Doboš, A. Prada, Trillas and Castiñera have provided a general solution to the aggregation problem of data represented by means of a finite family of pseudodistances and a type of generalized distances [2–4]. Motivated in part by the results presented in [2], Casasnovas and Roselló have introduced and studied several general techniques for merging a finite number of distances into another one with the aim of applying some of their properties to the comparison of biological sequences and to diagnosis problems in medicine [5,6]. Since a norm defined on a linear space induces a distance it seems natural to study, in the spirit of Borsik and Doboš, the problem of merging a collection of norms as in the case of the distance aggregation problem. This question was solved by Herburt and Moszyńska in [7]. Specifically, they proved that both problems are actually equivalent. Recently, it has been shown that asymmetric versions of the notions of distance and norm are appropriate tools to model several processes that arise in a natural way in Computer Science and Bioinformatics. In particular, an efficient framework, based on asymmetric norms and distances, to model the running time of computing in complexity analysis of programs and algorithms have been introduced and developed by García-Raffi, Sánchez-Pérez, Romaguera, Schellekens and Valero in [8–14]. Moreover, asymmetric distances have been employed successfully to describe logic programming processes by Seda in [15,16]. In [17–20], a natural correspondence between similarity measures on biological (nucleotide or protein) sequences and asymmetric distances has been proved, giving practical applications to searches in DNA and protein datasets. Motivated by the work of Rosselló and Casasnovas developed in [5,6], Casasnovas and Valero, and Tirado and Valero have obtained several connections between the asymmetric distance aggregation problem and the theory of computational complexity in [21,22]. Inspired by the fact that the scientific community has shown interest in the use of asymmetric distances in applied research, Mayor and Valero have studied the natural problem of merging a collection (not necessarily finite) of asymmetric distances in the spirit of Borsik and Doboš in [23]. In the same reference, it was shown that the mathematical approach based on asymmetric distances for the computational complexity can be expressed in terms of aggregation functions and, thus, a wide potential class of applications of aggregation theory to computational complexity analysis was open. Since an asymmetric norm induces an asymmetric distance in a similar way to the classical (symmetric) case and the asymmetric norms are suitable for the mathematical foundation of the complexity analysis in Computer Science (see [12,11]), we focus our attention on providing a general description of how to combine a collection (not necessarily finite) of asymmetric norms in order to obtain a single one as output. Concretely, we introduce the notion of asymmetric norm aggregation function which is a generalization of the given one by Herburt and Moszyńska in [7]. Moreover, we extend some results proved in the aforenamed reference to the context of asymmetric norms. In particular, we show that the classes of asymmetric distance aggregation functions and asymmetric norm aggregation functions are exactly the same. As a consequence, we obtain, contrary to the classical case, a characterization of asymmetric norm aggregation functions in terms of monotony and subadditivity. Finally we prove that the developed aggregation theory is a suitable approach for complexity analysis in Computer Science, since the asymmetric norms used in order to quantify the efficiency gained when an algorithm is substituted by another one can be retrieved by means of a family of distinguished asymmetric norm aggregation functions. We organize the document as follows: Section 2 is devoted to introduce the pertinent terminology, concepts and basics of asymmetric distances and asymmetric norms. In the same section we give, on one hand, a detailed exposition of the mathematical preliminaries about the (asymmetric) distances and norms aggregation problems and, on the other hand, about the mathematical foundations, in the sense of Romaguera and Schellekens, of the computational complexity analysis of programs and algorithms. Section 3 is devoted to investigate the problem of asymmetric norms aggregation as well as to establish the aggregation theory as a possible basis for complexity analysis in Computer Science. 2. Preliminaries Throughout this paper we shall use the letters R, R+ , N and Z+ to denote the set of real numbers, the set of nonnegative real numbers, the set of positive integer numbers and the set of nonnegative integer numbers, respectively. In order to fix the terminology let us recall a few concepts. 2.1. Asymmetric distances and asymmetric normed structures In our context by an asymmetric distance (quasi-metric in [24]) 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 distance (metric) on a set X is an asymmetric distance d on X satisfying, in addition, the following condition for all x, y ∈ X :

(iii) d(x, y) = d(y, x).

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Our main reference for asymmetric distances is [24]. An asymmetric distance space (quasi-metric space in [24]) is a pair (X , d) such that X is a (nonempty) set and d is an asymmetric distance on X . Given an asymmetric distance d on X , the nonnegative real-valued function d−1 defined on X × X by d−1 (x, y) = d(y, x) for all x, y ∈ X , is again an asymmetric distance called the conjugate of d. Note that each asymmetric distance d induces, in a natural way, a distance ds on X × X as follows: ds (x, y) = d(x, y) ∨ d−1 (x, y) for all x, y ∈ X , where ∨ stands for the maximum operator. The function u defined on R × R by u(x, y) = (y − x) ∨ 0 for all x, y ∈ R, is an interesting and well-known example of an asymmetric distance (see, for instance, [9]). Moreover, the conjugate of the asymmetric distance u on R is exactly the function u−1 given by u−1 (x, y) = (x − y) ∨ 0 for all x, y ∈ R. The distance induced by u is exactly the Euclidean metric | · | on R, i.e.

|y − x| = u(x, y) ∨ u−1 (y, x) for all x, y ∈ R. As usual we will say that two asymmetric distances d1 and d2 on a set X are equivalent if there exist positive real numbers M , m such that Md1 (x, y) ≤ d2 (x, y) ≤ md1 (x, y) for all x, y ∈ X . When two asymmetric distances d1 , d2 are equivalent we will denote them by d1 ≡ d2 . Let (G, +) be a group with neutral element 0. Following [25], a quasi-norm on G is a nonnegative real-valued function ‖ · ‖ on G such that for all x, y ∈ G:

(i) ‖x‖ = ‖ − x‖ = 0 ⇔ x = 0. (ii) ‖x + y‖ ≤ ‖x‖ + ‖y‖. The pair (G, ‖ · ‖) is called a quasi-normed group. A norm on a group G (see [26,27] and compare page 238 in [28]) is a quasi-norm satisfying, in addition, the following condition for all x ∈ X :

(iii) ‖x‖ = ‖ − x‖. Of course when the quasi-norm is a norm, the pair (G, ‖ · ‖) is called a normed group (see, for instance, page 676 in [27]). If ‖ · ‖ is a quasi-norm on a group G, then the nonnegative real-valued function ‖ · ‖−1 defined on G by

‖ x ‖ −1 = ‖ − x ‖ for all x ∈ G is also a quasi-norm on G. Similar to the case of asymmetric distances the quasi-norm ‖ · ‖−1 is called the conjugate of ‖ · ‖. Observe that a quasi-norm on a group induces, in a natural way, a norm on G which is denoted by ‖ · ‖s and defined by

‖x‖s = ‖x‖ ∨ ‖x‖−1 for all x ∈ G. It is clear that every quasi-norm induces on a group G an asymmetric distance d‖·‖ given by d‖·‖ (x, y) = ‖y − x‖ for all x, y ∈ G. It is evident that the asymmetric distance d‖·‖ is exactly a distance when the quasi-norm ‖ · ‖ on G is in fact a norm. On account of [12], an asymmetric normed linear space is a linear space (V , +, ·) on R endow with a quasi-norm ‖ · ‖ such that (V , ‖ · ‖) is a quasi-normed group and ‖ · ‖ satisfies for all x ∈ V and λ ∈ R+ the extra condition:

(iv) ‖λ · x‖ = λ‖x‖. Usually a quasi-norm satisfying condition (iii) is called an asymmetric norm [12,11]. Of course the notion of normed linear space is retrieved as a particular case of the definition of asymmetric normed linear space when one considers the asymmetric norm as a norm. Let us recall [29] that a normed linear space is an asymmetric linear space (V , ‖ · ‖) such that ‖ · ‖ holds for all x ∈ V and λ ∈ R the additional condition:

(iii′ ) ‖λ · x‖ = |λ| ‖x‖.

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Obviously an asymmetric norm ‖ · ‖ on a linear space V induces an asymmetric distance d‖·‖ by the same method as in the case of quasi-normed groups. A well-known example of asymmetric normed linear space is given by the pair (R, | · |u ), where | · |u is defined by

|x|u = x ∨ 0 for all x ∈ R. Moreover, it is clear that the asymmetric distance u on R is induced by the asymmetric norm | · |u , i.e. u(x, y) = |y − x|u for all x, y ∈ R. In the last years, the theory of asymmetric distances and asymmetric normed linear spaces has turned out to be useful in the theory of complexity analysis of algorithms and programs (see Section 2.3 for a detailed discussion about this topic). 2.2. The (asymmetric) distances and norms aggregation problems According to [1], we will denote by RI and R+ I the set of all real-valued functions and all nonnegative real-valued functions defined on a nonempty set I of indices, respectively. Given x ∈ R+ I we will write xi instead of x(i). From now on we will denote by 0, the element of RI given by 0i = 0 for all i ∈ I. As usual we will consider the set RI ordered by the pointwise order relation ≼, i.e. x ≼ y ⇔ xi ≤ yi for all i ∈ I. Of course + when I = N, we have that RN (R+ N ) matches up with the set of all sequences in R (R ). In the case of the cardinality of I + being finite, say n for some n ∈ N, the set RI (R+ ) will be denoted by R ( R ). n n I Let x, y ∈ RI and λ ∈ R. We denote by x + y and λ · x the elements of RI given by (xi + yi )i∈I and (λxi )i∈I , respectively. + + A function Φ : RI → R is monotone provided that Φ (x) ≤ Φ (y) for all x, y ∈ R+ I with x ≼ y. Moreover, a function + + Φ : R+ → R is said to be subadditive if Φ ( x + y ) ≤ Φ ( x ) + Φ ( y ) for all x , y ∈ R I I . Furthermore, we will say that a + + + function Φ : RI → R is homogeneous provided that Φ (λ · x) = λΦ (x) for all λ ∈ R . + In the sequel, we will denote by OI the set of all functions Φ : R+ I → R such that Φ (x) = 0 ⇔ x = 0. In 1981, Borsik and Doboš studied in depth the problem of how to combine by means of a function a collection of distance spaces in order to obtain a single one as a result [1]. Such functions were called distance (or metric) aggregation functions. + Following [1], a function Φ : R+ I → R is a distance aggregation function if the composite function Φ ◦ δ is a distance on ∏ the set X = i∈I Xi for every indexed family of distance spaces {(Xi , di )}i∈I , where the mapping δ : X × X → R+ I is defined by δ(x, y) = (di (xi , yi ))i∈I . The notion of triangle triplet plays a central role in the aggregation theory of distances. Let us recall that the triplet of nonnegative real numbers (a, b, c ) forms a triangle triplet whenever a ≤ b + c , b ≤ a + c and c ≤ b + a. A characterization, based on the notion of triangle triplet, of those functions which combine a collection (not necessarily finite) of distances into a single one was proved in [1] (see Lemmas 2.3 and 2,4, and Theorem 2.6). In particular the aforementioned result can be enunciated as follows: + Theorem 1. Let Φ : R+ I → R . Then the assertions below are equivalent:

(1) Φ is a distance aggregation function. (2) Φ holds the following properties: (i) Φ ∈ OI . (ii) Let a, b, c ∈ R+ I . If (ai , bi , ci ) is a triangle triplet for all i ∈ I, then so is (Φ (a), Φ (b), Φ (c)). As a consequence of the preceding result an outstanding connection between distance aggregation functions and subadditive functions can be established. More specifically, every distance aggregation function is subadditive. However, it is well known that the converse of the last sentence is not true (see for instance Theorem 8 in [30]). Recently, a version of Theorem 1 in the context of asymmetric distance spaces has been proved in [23]. With this aim, the notion of a distance aggregation function was extended to the asymmetric framework replacing the indexed family of distance spaces by an indexed family of asymmetric distance spaces in the Borsik and Doboš definition. Thus, a function + Φ : R+ I → R ∏ is an asymmetric distance aggregation function if the composite function Φ ◦ δ is an asymmetric distance on the set X = i∈I Xi for every indexed family of asymmetric distance spaces {(Xi , di )}i∈I , where the mapping δ : X × X → R+ I is defined by δ(x, y) = (di (xi , yi ))i∈I . The asymmetric formulation of Theorem 1 can be stated in the following way: + Theorem 2. Let Φ : R+ I → R . Then the assertions below are equivalent:

(1) Φ is an asymmetric distance aggregation function. (2) Φ holds the following properties: (i) Φ ∈ OI . (ii) Let a, b, c ∈ R+ I . If a ≼ b + c, then Φ (a) ≤ Φ (b) + Φ (c). (3) Φ ∈ OI , and Φ is subadditive and monotone. Observe that, contrary to Theorem 1, the preceding result provides a characterization of asymmetric distance functions in terms of monotonicity and subadditivity. Nevertheless, there exist distance aggregation functions which are not monotone (see Example 8 in [23]).

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Since every norm on a linear space induces a distance it seems natural to wonder if one can do research into the aggregation problem for the case of normed linear spaces and, in addition, to analyze the relationship between both problems. This question was solved by Herburt and Moszyńska in [7]. In particular they defined the notion of a norm aggregation function as follows: + A function Φ : R+ I → R is a norm aggregation function if the composite function Φ ◦ δ is a norm on the linear space ∏ V = i∈I Vi for every indexed family of normed linear spaces {(Vi , ‖ · ‖i )}i∈I , where the mapping δ : V → R+ I is defined by δ(x) = (‖xi ‖i )i∈I . After introducing the new aggregation concept, Herburt and Moszyńska gave the next elegant description of those functions which allow us to aggregate a collection (not necessarily finite) of norms into a single one. + Theorem 3. Let Φ : R+ I → R . Then the assertions below are equivalent:

(1) Φ is a norm aggregation function. (2) Φ is an homogeneous distance aggregation function. Furthermore, Herburt and Moszyńska established the following nice connection between distances and norms via aggregation functions. Proposition 4. Let Φ be a norm aggregation function and let {(Vi , ‖ · ‖i )}i∈I be an indexed family∏ of normed linear spaces. Then is defined by δ ( x , y ) = ( d ( x , y )) with V = dΦ ◦δ = Φ ◦ δd‖·‖ , where δd‖·‖ : V × V → R+ d ‖·‖ i ∈ I I i∈Ii Vi . i ‖·‖ Note that in the preceding result the function Φ ◦ δd‖·‖ , on V × V is a distance, since Theorem 3 guarantees that Φ is an homogeneous distance aggregation function. Next we recall some basics of the mathematical approach of complexity analysis of algorithms and programs, as well as the role played by asymmetric distance spaces and asymmetric normed linear spaces in such a framework, in order to motivate our subsequent work (developed in Section 3) and to reveal the interesting fact that this theory can be formulated in terms of aggregation functions (see Section 3.1). 2.3. The mathematical approach for complexity analysis in Computer Science 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. Usually, when one considers a problem there exist many algorithms to solve it. So one objective of the complexity analysis is to evaluate which of them is faster when a data collection is considered. To this end, it is required to compare their running time of computing. This is usually done by means of the asymptotic complexity analysis in which the running time of an algorithm is denoted by a function T : N → R+ in such a way that T (i) (Ti following our notation) represents the time taken by the algorithm to solve the problem under consideration when the input of the algorithm is of size i. 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 → R+ denote the running time of computing of algorithms, then the statement g ∈ O (f ) means that there exists i0 ∈ N and c ∈ R+ such that gi ≤ cfi for all i ∈ N with i ≥ i0 (≤ 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 fuller treatment of complexity analysis of algorithms we refer the reader to [31,32]. In 1999, Romaguera and Schellekens introduced the theory of dual complexity (asymmetric distance) spaces as a part of the development of the mathematical foundation for the complexity analysis in Computer Science [9]. In particular, the dual complexity space is the pair (C ∗ , dC ∗ ), where

 ∗

C =

+

f ∈ RN :

+∞ −

 2 fi < +∞ , −i

i=1

and dC ∗ is the asymmetric distance on C ∗ defined by dC ∗ ( f , g ) =

+∞ −

2−i [(gi − fi ) ∨ 0].

i=1

In the same reference they showed the applicability of the theory to the complexity analysis of algorithms discussing via fixed point arguments the complexity of Divide and Conquer algorithms.

820

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According to [9], it is possible to associate each function of C ∗ with a computational cost in such a way that if f ∈ C ∗ then fi represents the running time of performing some tasks by a program employing an input data of size i. Because of this, the elements of C ∗ are called complexity functions. Moreover, given two functions f , g ∈ C ∗ , the numerical value dC ∗ (f , g ) (the complexity distance from f to g) can be interpreted as a numerical measure of the efficiency gained when the algorithm U, whose running time of computing is represented by g, is substituted by the algorithm V whose running time of computing is represented by f . Hence, if f ̸= g , dC ∗ (f , g ) = 0 provides that g is more ‘‘efficient’’ than f on all inputs (i.e. dC ∗ (f , g ) = 0 ⇔ gi ≤ fi for all i ∈ N) and, thus, the running time of computing does not go down when we replace the program U by the program V . So we can encode the natural order relation ≼ on R+ N , induced by the pointwise order ≤, through the asymmetric distance dC ∗ . In particular the fact that dC ∗ (f , g ) = 0 implies that f ∈ O (g ). Of course the asymmetry of the complexity distance plays a crucial role in this analysis because a (symmetric) distance will provide information about the increase of complexity but it will not be able to indicate which program is more efficient. This provides a sound reason for the use of asymmetric distances, instead of the symmetric ones, in formal methods for complexity analysis in Computer Science. In many situations the running time of an algorithm is symbolized by a function which is obtained by addition of two complexity functions or by a combination of complexity functions multiplied by real numbers. Motivated by this fact, Romaguera and Schellekens considered [10] the set B ∗ given by

 ∗

B =

f ∈ RN :

+∞ −

 2 |fi | < +∞ . −i

i=1

It is clear that B ∗ is a linear space endowed with the usual sum and product by real numbers defined on RN , where the neutral element will be denoted by 0B ∗ . Of course the linear space B ∗ is the natural framework to represent complexity functions that are obtained by linear combinations of another complexity functions. However note that if g ∈ B ∗ is an example of this kind of complexity functions, then g denotes a running time of computing if and only if g ∈ C ∗ . Following the main ideas of Functional Analysis [29], Romaguera and Schellekens introduced in [10] an asymmetric norm ‖ · ‖B ∗ on B ∗ which is defined by

‖f ‖B ∗ =

+∞ −

2−i (fi ∨ 0)

i=1

for all f ∈ B . It is clear that the asymmetric norm ‖ · ‖B ∗ induces an asymmetric distance d‖·‖B ∗ on B ∗ , i.e. d‖·‖B ∗ (f , g ) = ‖g − f ‖B ∗ for all f , g ∈ B ∗ . Furthermore, ∗

dC ∗ (f , g ) = d‖·‖B ∗ (f , g ) for all f , g ∈ C ∗ . The utility of the asymmetric normed linear space (B ∗ , ‖ · ‖B ∗ ) in complexity analysis is provided by the preceding equality. In fact, the numerical value ‖f ‖B ∗ can be interpreted as a kind of ‘‘degree’’ of complexity of an algorithm whenever f ∈ C ∗ , since ‖f ‖B ∗ denotes the complexity distance of f to the ‘‘optimal’’ complexity function 0B ∗ , i.e. dC ∗ (0B ∗ , f ) = ‖f ‖B ∗ for all f ∈ C ∗ . Nevertheless, there are algorithms whose running time of computing cannot be modeled through B ∗ . Indeed, there are algorithms for which the running time is associated with the function f sqrt , given by fi

sqrt

=

i

2 √

i

for all i ∈ N (for a detailed

discussion see [33]). It is clear that f sqrt ̸∈ B ∗ . Consequently the analysis of the relative progress made in lowering the complexity when an algorithm with f sqrt running time is replaced by another one cannot be made in the context of B ∗ . Motivated by this handicap García-Raffi, Romaguera and Sánchez-Pérez extended the linear space B ∗ to a more general one in [12]. They denoted it by Bp∗ , where 1 ≤ p < +∞. The linear space Bp∗ is given by

 ∗

Bp =

f ∈ RN :

 p 2 |fi | < +∞ .

+∞ − 

−i

i=1

Notice that if p = 1 then the above linear space is exactly B ∗ . The new linear space Bp∗ can be endowed with an asymmetric norm ‖ · ‖Bp∗ defined by

 ‖f ‖Bp∗ =

+∞ − 

 1p

2 (fi ∨ 0) −i

p

i=1

for all f ∈ Bp∗ . Again, the asymmetric norm ‖ · ‖Bp∗ induces an asymmetric distance d‖·‖B ∗ on Bp∗ given by p

d‖·‖B ∗ (f , g ) = ‖g − f ‖Bp∗ .

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It is evident that a function f ∈ Bp∗ matches up with the running time of computing of an algorithm if and only if f ∈ Cp∗ , where

 ∗

+

Cp =

f ∈ RN :

+∞ − 

 −i

2 fi

p

< +∞ .

i=1

Moreover, it is clear that f sqrt ∈ Cp∗ ⊂ Bp∗ whenever p > 2. Note that the complexity analysis carried out in B ∗ can be recuperated in Bp∗ and, in particular, it is feasible to measure the improvements in complexity when an algorithms with f sqrt running time is replaced by another one. The pair (Cp∗ , dCp∗ ) is known as the dual p-complexity space, where dCp∗ denotes the restriction of the asymmetric distance d‖·‖B ∗ to Cp∗ . Unfortunately there exist other examples of algorithms whose associated running time of computing is modeled by a function which does not belong to any dual p-complexity space Cp∗ . An example of this kind of algorithms are the so-called exponential time algorithms [34]. The running time of computing of an exponential time algorithm is given by a function f P such that fiP = 2P (i) for all i ∈ N, where P (i) is a polynomial with P (i) ≥ i for each i ∈ N. It is a simple matter to check that fiP ̸∈ Cp∗ ⊂ Bp∗ for any 1 ≤ p < +∞. Hence the analysis of the relative progress made in lowering the complexity when an algorithm with f P running time is replaced by another one cannot be made in the context of any dual p-complexity space. In order to avoid the aforenamed handicap, in [11] it was introduced the most general complexity structure (from among the aforesaid ones) that is a suitable framework to measure the improvements in complexity of f P running time algorithms. On this occasion the complexity structure consists of the asymmetric normed linear space (BP∗,∞ , ‖ · ‖B ∗ ) where P ,∞



 BP∗,∞ =

f ∈ RN :



2−P (i) |fi | < +∞

i∈N

and

‖f ‖Bp∗ =



2−P (i) (fi ∨ 0).

i∈N

Of course the asymmetric norm ‖ · ‖B ∗

P ,∞

d‖·‖B ∗

P ,∞

induces an asymmetric distance d‖·‖B ∗

P ,∞

by means of the equality

(f , g ) = ‖g − f ‖BP∗,∞ .

Clearly a function f ∈ BP∗,∞ matches up with the running time of computing of an algorithm if and only if f ∈ CP∗,∞ , where

 CP∗,∞

 +

= f ∈ RN :



−P (i)

2

fi < +∞ .

i∈N

The pair (CP∗,∞ , dC ∗

P ,∞

) is called the supP -dual complexity space, as usual the asymmetric distance dCP∗,∞ denotes the

restriction of the asymmetric distance d‖·‖B ∗

P ,∞

to CP∗,∞ .

Notice that the complexity analysis carried out in Bp∗ can be recovered in BP∗,∞ , since Bp∗ ⊂ BP∗,∞ for all p ≥ 1 and for

all polynomial P with P (i) ≥ i for all i ∈ N. Furthermore, f P ∈ CP∗,∞ ⊂ BP∗,∞ for all polynomial P with P (i) ≥ i for all i ∈ N. So this new complexity structure is a suitable framework to measure the improvements in complexity of f P running time algorithms. Motivated by Theorem 3, Proposition 4, the aggregation theory developed in [23] for asymmetric distances and, in addition, by the fact that all asymmetric norms that are relevant in the complexity framework introduced above are obtained via aggregation techniques, in the next section we focus our attention on providing a general description of how to combine a collection (not necessarily finite) of asymmetric norms in order to obtain a single one as output and, besides, to clear up the relationship between this problem and its analogous, already explored in [23], of combining asymmetric distances. 3. The aggregation problem for asymmetric normed structures

As announced before in this section we go more deeply into the aggregation problem of asymmetric distance spaces. Particularly we obtain a version of Theorem 3 in the context of asymmetric normed structures and we study the natural question with regard to the relationship between the norms obtained via aggregation and their induced asymmetric distances in the spirit of Proposition 4. For our subsequent dissertation we extend the notion of norm aggregation function to our more general context. Thus, + a function Φ : R+ I → R will ∏ be called a quasi-norm aggregation function whenever the composite function Φ ◦ δ is a quasi-norm on the set G = i∈I Gi for every indexed family of quasi-normed groups {(Gi , ‖ · ‖i )}i∈I , where the mapping δ : G → R+ is defined by δ( x ) = (‖xi ‖i )i∈I for all x ∈ G. I

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+ The preceding definition can be adapted for the case of linear spaces in the obvious manner, i.e. a function Φ : R+ I → R will be ∏called an asymmetric norm aggregation function whenever the composite function Φ ◦ δ is an asymmetric norm on V = i∈I Vi for every indexed family of asymmetric normed linear spaces {(Vi , ‖ · ‖i )}i∈I , where the mapping δ : V → R+ I is defined by δ(x) = (‖xi ‖i )i∈I for all x ∈ V . The following auxiliary result will be useful later on.

Proposition 5. Let a, b, c ∈ R+ such that a ≤ b. Then there exists a quasi-norm (n asymmetric norm) uR2 on R2 in such a way that there exist x, y ∈ R2 with uR2 (x + y) = a, uR2 (x) = b and uR2 (y) = 0. Proof. Define the function uR2 : R2 → R+ by uR 2 ( x ) = x 1 ∨ 0 + x 2 ∨ 0 for all x = (x1 , x2 ) ∈ R2 . It is a simple matter to check that uR2 is a quasi-norm (n asymmetric norm) on R2 . Furthermore, it is easily seen that the elements x, y of R2 given by x = (−a + b, a) and y = (−b, 0) satisfy the required conditions.  + Theorem 6. Let Φ : R+ I → R . Then the assertions below are equivalent:

(1) Φ is a quasi-norm aggregation function. (2) Φ is an asymmetric distance aggregation function. Proof. (1) ⇒ (2). Assume that Φ is a quasi-norm aggregation function. It follows that Φ ∈ OI . Indeed, suppose that there exists x ∈ R+ I such that Φ (x) = 0. Consider the indexed family of quasi-normed groups {(R, ‖ · ‖i )}i∈I with ‖ · ‖i = | · | for all i ∈ I. Then Φ ◦ δ|·| :→ R+ is a quasi-norm on RI , since Φ is a quasi-norm aggregation function. Moreover, if x ∈ R+ I then

Φ ◦ δ|·| (x) = Φ ((|xi |i∈I )) = Φ ((xi )i∈I ) = Φ (x) = 0 and

Φ ◦ δ|·| (−x) = Φ ((| − xi |i∈I )) = Φ ((xi )i∈I ) = Φ (x) = 0. Whence we deduce that x = 0. Next we show that Φ is subadditive. Let a, b ∈ R+ I . Consider again the ∏ indexed family of quasi-normed groups {(R, ‖ · ‖i )}i∈I where ‖ · ‖i = | · | for all i ∈ I. Then Φ ◦ δ|·| u is a quasi-norm on i∈I R = RI , since Φ is a quasi-norm aggregation function. Hence we have that

Φ (a + b) = Φ ((|ai + bi |)i∈I ) = Φ ◦ δ|·| (a + b) ≤ Φ ◦ δ|·| (a) + Φ ◦ δ|·| (b) = Φ ((|ai |)i∈I ) + Φ ((|bi |)i∈I ) = Φ (a) + Φ (b). It remains to prove that Φ is monotone. 2 Let a, b ∈ R+ I such that a ≼ b. Consider the indexed family of quasi-normed groups {(R , ‖ · ‖i )}i∈I where ‖ · ‖i = uR2 for all i ∈ I, and let δu 2 be the mapping associated to the aforesaid family. By Proposition 5 there exist xi , yi ∈ R2 such that R

uR2 (xi + yi ) = a, uR2 (xi ) = b and uR2 (yi ) = 0 for all i ∈ I. Put x = (xi )i∈I ∈ quasi-norm aggregation function and Φ ∈ OI , we obtain that

∏

i∈I

R2 and y = (yi )i∈I ∈

∏

i∈I

R2 . Since Φ is a

Φ (a) = Φ ((uR2 (xi + yi ))i∈I ) = Φ ◦ δuR2 (x + y)

≤ Φ ◦ δuR2 (x)+ Φ ◦ δuR2 (y)

= Φ ((uR2 (xi ))i∈I ) + Φ ((uR2 (yi ))i∈I ) = Φ (b) + Φ (0) = Φ (b). Consequently, by statement (3) in Theorem 2, we deduce that Φ is an asymmetric distance aggregation ∏function. (2) ⇒ (1). Let {(Gi , ‖ · ‖i )}i∈I be an indexed family of quasi-normed groups. Consider x ∈ G = i∈I Gi such that Φ ◦ δ(x) = Φ ◦ δ(−x) = 0. Then ‖xi ‖i = ‖ − xi ‖i = 0 for all i ∈ I, since Φ ∈ OI . Whence we obtain that x = 0. Next we show that Φ ◦ δ(x + y) ≤ Φ ◦ δ(x) + Φ ◦ δ(y) for all x, y ∈ G. Since

‖xi + yi ‖i ≤ ‖xi ‖i + ‖yi ‖i for all i ∈ I we conclude, from the fact that Φ is monotone and subadditive, that

Φ ◦ δ(x + y) = Φ ((‖xi + yi ‖i )i∈I ) ≤ Φ ((‖xi ‖i + ‖yi ‖i )i∈I ) ≤ Φ ((‖xi ‖i )i∈I ) + Φ ((‖yi ‖i )i∈I ) = Φ ◦ δ(x) + Φ ◦ δ(y). Therefore we have shown that Φ ◦ δ is a quasi-norm on G. This completes the proof.



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Similar arguments to those in the proof of the preceding result apply to the next one. + Theorem 7. Let Φ : R+ I → R . Then the assertions below are equivalent:

(1) Φ is an asymmetric norm aggregation function. (2) Φ is an homogeneous asymmetric distance aggregation function. As a consequence of Theorems 2, 6 and 7 we have the following characterizations of quasi-norm and asymmetric norm aggregation functions. + Corollary 8. Let Φ : R+ I → R such that Φ ∈ OI . Then the assertions below are equivalent:

(1) Φ is a quasi-norm aggregation function. (2) Let a, b, c ∈ R+ I . If a ≼ b + c, then Φ (a) ≤ Φ (b) + Φ (c). (3) Φ is subadditive and monotone. + Corollary 9. Let Φ : R+ I → R such that Φ ∈ OI . Then the assertions below are equivalent:

(1) Φ is an asymmetric norm aggregation function. (2) Φ is homogeneous and Φ (a) ≤ Φ (b) + Φ (c) for all a, b, c ∈ R+ I with a ≼ b + c. (3) Φ is homogeneous, subadditive and monotone. It is clear that an asymmetric distance aggregation function is a distance aggregation function (see Corollary 7 in [23]). However there are distance aggregation functions which are not asymmetric aggregation functions (Example 8 in [23]). So it seems natural to wonder if the condition of being a quasi-norm (n asymmetric norm) aggregation function is equivalent to that of being an aggregation norm function. Since every quasi-norm (asymmetric norm) aggregation function is an asymmetric distance aggregation function we have, by Theorem 3, that every quasi-norm (asymmetric norm) aggregation function is also a norm aggregation function. However, the next example shows that there exist norm aggregation functions which are not quasi-norm (asymmetric norm) aggregation functions. + Example 10. Let I = N. Consider the function Φ : R+ N → R given by Φ (0) = 0 and

Φ (x) =



2 1

α(x) ∈]0, 1[ α(x) ≥ 1.

Clearly Φ ∈ ON . Furthermore, it is not hard to see that Φ satisfies the condition (ii) of statement (2) in Theorem 1. So, by 1

1

2 Theorem 3, Φ is a norm aggregation function. However, Φ is not monotone. Indeed, let x 2 , x1 ∈ R+ N such that xi = 1 2



 1

1 2

and

= 1 for all i ∈ N. Clearly x ≺ x , Φ x 2 = 2 and Φ (x ) = 1. Therefore, by statement (3) in Corollary 8, we conclude that Φ is not a quasi-norm (n asymmetric norm) aggregation function. x1i

1

1

Next we discuss what connections between a quasi-norm (n asymmetric norm), the conjugate and its associated norm can be deduced via aggregation functions. Proposition 11. Let Φ be a quasi-norm aggregation function and let {(Gi , ‖·‖i )}i∈I be an indexed family of quasi-normed groups. Then the following statements hold: 1 −1 (1) (Φ ◦ δ)−1 = Φ ◦ δ −1 , where δ −1 : G → R+ (x) = (‖xi ‖− I is defined by δ i )i∈I with ∏ G= + s s s s s (2) (Φ ◦ δ) ≡ Φ ◦ δ , where δ : G → RI is defined by δ (x) = (‖xi ‖i )i∈I with G = i∈I Gi .

∏

i∈I

Gi .

Proof. First of all we note that Φ ◦δ −1 is a quasi-norm on G and that Φ ◦δ s is a norm on G, since Φ is a quasi-norm aggregation function. Let x ∈ G. Then 1 −1 (Φ ◦ δ)−1 (x) = Φ ◦ δ(−x) = Φ ((‖ − x‖i )i∈I ) = Φ ((‖x‖− (x). i )i∈I ) = Φ ◦ δ

Moreover,

(Φ ◦ δ)s (x) = Φ ◦ δ(x) ∨ (Φ ◦ δ)−1 (x). Thus 1 (Φ ◦ δ)s (x) = Φ ◦ δ(x) ∨ Φ ◦ δ −1 (x) = Φ ((‖xi ‖i∈I )) ∨ Φ ((‖xi ‖− i )i∈I ).

Furthermore, 1 Φ ◦ δ s (x) = Φ ((‖xi ‖si )i∈I ) = Φ ((‖xi ‖i ∨ ‖xi ‖− i )i∈I ).

By Corollary 8 Φ is monotone and, thus, 1 −1 Φ ((‖xi ‖i ∨ ‖xi ‖− i )i∈I ) ≥ Φ ((‖xi ‖ii∈I )) ∨ Φ ((‖xi ‖i )i∈I ).

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Hence Φ ◦ δ s (x) ≥ (Φ ◦ δ)s (x). Since Corollary 8 guarantees, in addition, that Φ is subadditive we have that 1 −1 Φ ((‖xi ‖i ∨ ‖xi ‖− i )i∈I ) ≤ Φ ((‖xi ‖i + ‖xi ‖i )i∈I ) 1 ≤ Φ ((‖xi ‖i∈I )) + Φ ((‖xi ‖− i )i∈I )   1 ≤ 2 Φ ((‖xi ‖i∈I )) ∨ Φ ((‖xi ‖− i )i∈I ) 1 1 because ‖xi ‖i ∨ ‖xi ‖− ≤ ‖xi ‖i + ‖xi ‖− for all i ∈ I. Consequently i i

Φ ◦ δ s (x) ≤ 2(Φ ◦ δ)s (x). The proof is complete.



The next result is an asymmetric version of Proposition 4. Proposition 12. Let Φ be a quasi-norm aggregation function and let {(Gi , ‖·‖i )}i∈I be an indexed family of quasi-normed groups. Then the following statements hold: (1) dΦ ◦δ = Φ ◦ δd‖·‖ , where δd‖·‖ : G × G → R+ I is defined by δd‖·‖ (x, y) = (d‖·‖i (x, y))i∈I with G = i∈I Gi . −1 + 1 (2) dΦ ◦δ = dΦ ◦δ −1 . Moreover, dΦ ◦δ −1 = Φ ◦ δd−1 where δd−1 : G × G → RI is defined by δd−1 (x, y) = (d− ‖·‖i (x, y))i∈I with

∏

‖·‖

‖·‖

‖·‖

∏

G = i∈I Gi . s (3) d(Φ ◦δ)s = dsΦ ◦δ = dΦ ◦δ ∨ dΦ ◦δ −1 . Moreover d(Φ ◦δ)s ≡ Φ ◦ δds where δds : G × G → R+ I is defined by δd‖·‖ (x) = ‖·‖ ‖·‖ ∏ s (d‖·‖i (x, y))i∈I with G = i∈I Gi . Proof. By Theorem 6, we have that Φ is an asymmetric distance aggregation function. It follows that Φ ◦ δd‖·‖ and Φ ◦ δd−1 are asymmetric distances on G. Let x, y ∈ G. Then

‖·‖

dΦ ◦δ (x, y) = Φ ◦ δ(y − x) = Φ ((‖y − x‖i )i∈I )

= Φ ((d‖·‖i (x, y))i∈I ) = Φ ◦ δd‖·‖ (x, y) and 1 dΦ ◦δ −1 (x, y) = Φ ◦ δ −1 (y − x) = Φ ((‖y − x‖− i )i∈I )

= Φ ((‖x − y‖i )i∈I ) = Φ ((d‖·‖i (y, x))i∈I ) 1 = Φ ((d− ‖·‖i (x, y))i∈I ) = Φ ◦ δd−1 (x, y). ‖·‖

Moreover, 1 d− 1 (x, y) = dΦ ◦δ −1 (x, y). Φ ◦δ (x, y) = dΦ ◦δ (y, x) = Φ ◦ δd‖·‖ (y, x) = Φ ◦ δd− ‖·‖

This proves assertions (1) and (2). Next we prove assertion (3). The equality dsΦ ◦δ = dΦ ◦δ ∨ dΦ ◦δ −1 follows from (1) and (2). Now we show that d(Φ ◦δ)s = dsΦ ◦δ . To this end, let x, y ∈ G. Then d(Φ ◦δ)s (x, y) = (Φ ◦ δ)s (y − x) = Φ ◦ δ(y − x) ∨ (Φ ◦ δ)−1 (y − x) 1 = Φ ((‖yi − xi ‖i∈I )) ∨ Φ ((‖yi − xi ‖− i )i∈I ) = Φ ◦ δd‖·‖ (x, y) ∨ Φ ◦ δd−1 (x, y) ‖·‖

= dΦ ◦δ (x, y) ∨ dΦ ◦δ−1 (x, y) = dsΦ ◦δ (x, y). Since Φ is an asymmetric distance aggregation function we have that it is also a distance aggregation function and, as a consequence, Φ ◦ δds is a distance on G. ‖·‖

In the sequel we prove that d(Φ ◦δ)s ≡ Φ ◦ δds . ‖·‖

On one hand, we have by the monotonicity of Φ that d(Φ ◦δ)s (x, y) = (Φ ◦ δ)s (y − x) = Φ ◦ δ(y − x) ∨ (Φ ◦ δ)−1 (y − x) 1 = Φ ((‖yi − xi ‖i∈I )) ∨ Φ ((‖yi − xi ‖− i )i∈I ) = Φ ◦ δd‖·‖ (x, y) ∨ Φ ◦ δd−1 (x, y) ‖·‖

= dΦ ◦δ (x, y) ∨ dΦ ◦δ−1 (x, y) 1 = Φ ((d‖·‖i (xi , yi ))i∈I ) ∨ Φ ((d− ‖·‖i (xi , yi ))i∈I )

≤ Φ ((ds‖·‖i (xi , yi ))i∈I ) = Φ ◦ δds‖·‖ (x, y).

J. Martín et al. / Mathematical and Computer Modelling 54 (2011) 815–827

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On the other hand, by the monotonicity and subadditivity of Φ , we obtain 1 Φ ◦ δds‖·‖ (x, y) = Φ ((ds‖·‖i (xi , yi ))) ≤ Φ ((d‖·‖i (xi , yi ))i∈I ) + Φ ((d− ‖·‖i (xi , yi ))i∈I )

= Φ ◦ δd‖·‖ (y − x) + Φ ◦ δd−1 (y − x) ‖·‖

1 s = dΦ ◦δ (x, y) + d− Φ ◦δ (x, y) ≤ 2dΦ ◦δ (x, y) = 2d(Φ ◦δ)s (x, y).

From the preceding inequalities statement (3) follows.



Remark 13. Note that, as a consequence of Corollary 9, Propositions 11 and 12 also hold when we interchange quasi-normed group and quasi-norm aggregation function by asymmetric normed linear space and asymmetric norm aggregation function, respectively. Remark 14. We want to observe that there are several recent works about aggregation via infinitely many argument functions (see, for instance, [35,36] and Appendix A in [37]). Although there are connections between the aforementioned references and our work, the problems studied in the quoted papers are different from the one in this paper. It is necessary to emphasize that the aggregation functions under consideration in [36], i.e. the aggregation functions preserving T transitivity, are closely related to pseudodistance aggregation functions in the sense of [4]. Even though the results obtained as far as here solve the aggregation problem in the spirit of Borsik, Doboš, Herburt and Moszyńska for the case of asymmetric normed structures, we need to present an extended notion of quasi-norm (asymmetric norm) aggregation function in order to develop some connections between complexity analysis in Computer Science and aggregation theory later in Section 3.1. A more general notion than that of asymmetric distance aggregation function was introduced and studied in Section + 4 of [23]. In particular, given a subset Y ⊆ R+ aggregation I , a function Φ : Y → R is called an Y-asymmetric distance∏ function if for every indexed family of asymmetric distance spaces {(Xi , di )}i∈I such that there exists XY ⊆ X = i∈I Xi with δ(XY × XY ) ⊆ Y the function Φ ◦ δ is an asymmetric distance on XY , where the mapping δ : XY × XY → R+ I is defined by δ(x, y) = (di (xi , yi ))i∈I . Obviously the definition of asymmetric distance aggregation function is retrieved as a particular case of the above one whenever the subset Y is exactly R+ I . According to [23], a function Φ : Y → R+ is Y-monotone provided that Φ (x) ≤ Φ (y) for all x, y ∈ Y with x ≼ y. Moreover, a function Φ : Y → R+ is Y-subadditive provided that Φ (x + y) ≤ Φ (x) + Φ (y) for all x, y ∈ Y with x + y ∈ Y. Furthermore, we will say that a function Φ : Y → R+ is Y-homogeneous provided that Φ (λ · x) = λΦ (x) for all λ ∈ R+ and x ∈ Y. In the following, if 0 ∈ Y then we will denote by OI ,Y the set of all functions Φ : Y → R+ such that Φ (x) = 0 ⇔ x = 0. In the light of these new notions the following general version of Theorem 2 was given in [23]. Theorem 15. Let Φ : Y → R+ and Y ⊆ R+ I with 0 ∈ Y. If x + y ∈ Y for all x, y ∈ Y, then the below assertions are equivalent: (1) Φ is a Y-asymmetric distance aggregation function. (2) Φ holds the following properties: (i) Φ ∈ OI ,Y . (ii) Let a, b, c ∈ Y. If a ≼ b + c, then Φ (a) ≤ Φ (b) + Φ (c). (3) Φ ∈ OI ,Y , and Φ is Y-subadditive and Y-monotone. We end the section extending the definition of quasi-norm (asymmetric norm) aggregation function to this general + approach. Thus, given a subset Y ⊆ R+ I , a function Φ : Y → R will be called an Y-quasi-norm aggregation ∏function if for every indexed family of quasi-normed groups {(Gi , ‖ · ‖i )}i∈I such that there exists a subgroup GY ⊆ G = i∈I Gi with δ(GY ) ⊆ Y the function Φ ◦ δ is a quasi-norm on GY , where the mapping δ : GY → Y is defined by δ(x) = (‖xi ‖i )i∈I . In the obvious manner the notion of asymmetric norm aggregation function can be extended to this new context. The next results follow applying the same arguments to those given in Theorems 6 and 7. Theorem 16. Let Φ : Y → R+ and Y ⊆ R+ I with 0 ∈ Y. If x + y ∈ Y for all x, y ∈ Y, then the below assertions are equivalent: (1) Φ is a Y-quasi-norm aggregation function. (2) Φ is a Y-asymmetric distance aggregation function. Theorem 17. Let Φ : Y → R+ and Y ⊆ R+ I with 0 ∈ Y. If x + y ∈ Y for all x, y ∈ Y, then the below assertions are equivalent: (1) Φ is a Y-asymmetric norm aggregation function. (2) Φ is a Y-homogeneous Y-asymmetric distance aggregation function.

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As an immediate consequence of Theorems 15–17 we obtain the following. + Corollary 18. Let Y ⊆ R+ I with 0 ∈ Y and x + y ∈ Y for all x, y ∈ Y, and let Φ : Y → R such that Φ ∈ OI ,Y . Then the assertions below are equivalent:

(1) Φ is a Y-quasi-norm aggregation function. (2) Let a, b, c ∈ Y. If a ≼ b + c, then Φ (a) ≤ Φ (b) + Φ (c). (3) Φ is Y-subadditive and Y-monotone. + Corollary 19. Let Y ⊆ R+ I with 0 ∈ Y and x + y ∈ Y for all x, y ∈ Y, and let Φ : Y → R such that Φ ∈ OI ,Y . Then the below assertions are equivalent:

(1) Φ is a Y-asymmetric norm aggregation function. (2) Φ is Y-homogeneous and Φ (a) ≤ Φ (b) + Φ (c) for all a, b, c ∈ Y with a ≼ b + c. (3) Φ is Y-homogeneous, Y-subadditive and Y-monotone. 3.1. A connection between complexity analysis and aggregation theory In this subsection we show that the asymmetric norms used in the mathematical approach for the complexity analysis of programs and algorithms, exposed in Section 2.3, can be retrieved as a particular case of the aggregation theory developed along Section 3. To this end we present a key result which follows from Corollary 19 and whose easy proof we omit. Corollary 20. The following statements hold: + (1) For each 1 ≤ p < +∞, the function Φp : l+ p → R defined by



∞ − (cj )p Φp (c) =

 1p

j =0

is a l+ p -asymmetric norm aggregation function, where l+ p =

 



∞ − x ∈ R+ : (xj )p N



 1p

j =0

 

< +∞ . 

+ (2) The function Φ∨ : l+ ∞ → R defined by

Φ∨ (c) =



cj

j∈N

is an l+ ∞ -asymmetric norm aggregation function, where

 l+ ∞

 +

= x ∈ RN :



xj < +∞ .

j∈N

In the light of the preceding corollary we are able to show the announced connection between aggregation theory and complexity analysis.   Let 1 ≤ p < +∞. Put lp =

x ∈ RN :

∑∞

j =0

(|xj |)p

 1p

< +∞ . It is clear that lp is a linear space [29]. Now consider

the indexed family of asymmetric normed linear spaces {(R, ‖ · ‖u,i )i∈N } where ‖x‖u,i = 2−i |x|u for all i ∈ N. Then ∏ + + + V = i∈N R = RN and lp ⊂ V . Set Y = lp ⊂ RN . Let VY = lp . Then δ(VY ) ⊆ lp . By Corollary 20 we have that Φp ◦ δ is an asymmetric norm on lp . Moreover, for all f ∈ Bp∗ , the complexity asymmetric norm ‖ · ‖Bp∗ satisfies



+∞ − 

 1p

p 2 (fi ∨ 0)

‖f ‖Bp∗ =

−i

i=1

 =

+∞ − 

 1p −i

2 |fi |u

p

= Φp ◦ δ(xf ),

i =1

where xf = (2−i fi )i∈N ∈ VY = lp . Let P be a polynomial with P (i) ≥ i for all i ∈ N. Consider the indexed family  of asymmetric normed linear spaces {(R+ , ‖ · ‖P ,i )i∈N } where ‖x‖P ,i = 2−P (i) |x|u for all i ∈ N. Put l∞ = {x ∈ RN : j∈N |xj | < +∞}. It is clear that l∞ is a

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827

+ + linear space [29]. In addition, we have that V = i∈N R = RN and that l∞ ⊂ V . Set Y = l∞ ⊂ RN . Let VY = l∞ . Then + δ(VY ) ⊆ l∞ . By Corollary 20 we have that Φ∨ ◦ δ is an asymmetric norm on l∞ . Furthermore, we have that, for all f ∈ BP∗,∞ , the complexity asymmetric norm ‖ · ‖B ∗ satisfies

∏

P ,∞

‖f ‖BP∗,∞ =



=



−P (i)

2

(fi ∨ 0)

i∈N

2−P (i) |fi |u = Φ∨ ◦ δ(xf ),

i∈N

where xf = (2−P (i) fi )i∈N ∈ VY = l∞ . We end the paper emphasizing the special importance of the fact that the asymmetric norms used in complexity analysis can be retrieved as a particular case of the theory of asymmetric norm aggregation functions. Concretely, it opens the possibility of applying a wide range of properties from the general aggregation theory to the complexity analysis of programs and algorithms in Computer Science, which could mean an advantage in solving some problems that arise in a natural way in this field. Acknowledgements The authors acknowledge the support of the Spanish Ministry of Education and Science and FEDER grant MTM200910962. References [1] J. Borsik, J. Doboš, On a product of metric spaces, Math. Slovaca 31 (1981) 193–205. [2] A. Pradera, E. Trillas, E. Castiñeira, On distances aggregation, in: Proceedings of the Information Processing and Management of Uncertainty in Knowledge-Based Systems International Conference, 2000, pp. 693–700. [3] A. 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