Scholarly article on topic 'The supremum asymmetric norm on sequence algebras'

The supremum asymmetric norm on sequence algebras Academic research paper on "Mathematics"

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Abstract of research paper on Mathematics, author of scientific article — L.M. García-Raffi, S. Romaguera, E.A. Sánchez-Pérez

Abstract Recently, E.A. Emerson and C.S. Jutla (SIAM J. Comput., 1999), have successfully applied complexity of tree automata to obtain optimal deterministic exponential time algorithms for some important modal logics of programs. The running time of these algorithms corresponds, of course, to complexity functions which are potential functions and, thus, they do not belong, in general, to any dual p-complexity space. Motivated by these facts we here introduce and study a very general class of complexity spaces, which provides, in the dual context, a suitable framework to carry out a description of the complexity functions that generate exponential time algorithms. In particular, such spaces can be modelled as biBanach semialgebras which are isometrically isomorphic to the positive cone of the asymmetric normed linear space consisting of bounded sequences of real numbers endowed with the supremum asymmetric norm.

Academic research paper on topic "The supremum asymmetric norm on sequence algebras"

Electronic Notes in Theoretical Computer Science 74 (2003)

URL: http://www.elsevier.nl/locate/entcs/volumeT4.html 12 pages

The supremum asymmetric norm on sequence algebras: a general framework to measure complexity distances

L.M. García-Raffi , S. Romaguera , E.A. Sánchez-Pérez

Escuela de Caminos, Departamento de Matemática Aplicada, Universidad

Politécnica de Valencia, 46071 Valencia, Spain. lmgarcia@mat.upv.es, sromague@mat.upv.es, easancpe@mat.upv.es

Abstract

Recently, E.A. Emerson and C.S. Jutla (SIAM J. Comput., 1999), have successfully applied complexity of tree automata to obtain optimal deterministic exponential time algorithms for some important modal logics of programs. The running time of these algorithms corresponds, of course, to complexity functions which are potential functions and, thus, they do not belong, in general, to any dual p-complexity space.

Motivated by these facts we here introduce and study a very general class of complexity spaces, which provides, in the dual context, a suitable framework to carry out a description of the complexity functions that generate exponential time algorithms. In particular, such spaces can be modelled as biBanach semialgebras which are isometrically isomorphic to the positive cone of the asymmetric normed linear space consisting of bounded sequences of real numbers endowed with the supremum asymmetric norm.

Keywords: asymmetric norm, semialgebra, biBanach, exponential time algorithm, supP(n)-complexity space, isometrically isomorphic.

1 Introduction

Throughout this paper the letters R, R+, N and u will denote the set of real numbers, of nonnegative real numbers, of natural numbers and nonnegative integer numbers, respectively.

The complexity (quasi-metric) space was introduced by M. Schellekens [14] in order to develop a topological foundation for the complexity analysis of pro-

1 The authors acknowledge the support of the Spanish Ministry of Science and Technology, grant BFM2000-1111. The first author acknowledge also the support of the Spanish Ministry of Science and Technology, grant TIC2000-1750-C06-01.

©2003 Published by Elsevier Science B. V.

grams and algorithms, based on the notion of a "complexity distance", that is, a generalized metric which intuitively measures relative improvements in the complexity of programs and algorithms. The complexity space accepts, among others, many important kinds of exponential time algorithms. In particular, some applications of this theory to the complexity analysis of Divide & Conquer algorithms were given in [14].

Later on, it was introduced in [12] the so-called dual complexity (quasi-metric) space, to discuss in a more handy context several quasi-metric properties of the complexity space which are interesting from a computational point of view. In fact, while the complexity space cannot be modelled as a quasi-normed cone, the dual space admits a structure of quasi-normed (asymmetric normed, in our terminology) semilinear space [13] and, by other hand, it can be directly used for the complexity analysis of certain algorithms, where the running time of computing is the complexity measure (compare [14] Section 4, and [12] page 313).

Motivated by the fact that, in this dual context, the complexity analysis of algorithms with running time O(2n/nr), 0 < r < 1, cannot be performed via the dual complexity space, the authors have recently introduced [4] the so-called dual ^-complexity space (p > 1), which provides, for p > 1, an appropriate framework to discuss complexity functions that are generated by this kind of algorithms. In particular, it was shown that the dual ^-complexity space is an asymmetric normed semilinear space which is isometrically iso-morphic to the positive cone of (lp, || • ||+p) (see Section 2 for definitions and details).

On the other hand, there is in the last few years a renewed interest in automata of infinite objects due to their intimate relation with temporal and modal logics of programs. Thus, E. A. Emerson and C. S. Jutla [1] have successfully applied complexity of tree automata to obtain optimal deterministic exponential time algorithms in some important modal logics of programs, where by an exponential time algorithm we mean an algorithm with running time O(2P(n)), such that P(n) is a polynomial with P(n) > 0 for all n. This running time corresponds to the function f given by f (n) = 2P(n) for all n, which does not belong to any dual ^-complexity space whenever P(n) > n.

In this paper we show that the supremum asymmetric norms that one can define in a natural way on certain sequence algebras provide an efficient tool to study those complexity functions that generate exponential time algorithms. In this direction, we construct a very general class of asymmetric normed linear spaces whose positive cones constitute a suitable setting for extending Schellekens' idea of complexity distance to the measure of improvements in complexity of exponential time algorithms. Furthermore, these positive cones are biBanach semialgebras which are isometrically isomorphic to the positive cone of the biBanach space (l^, ||-||+00), where ||x||+K) = sup{xn V 0 : n E u} for each x := (xn)ne^ E l^. Schellekens proved in [14] that Divide & Conquer algorithms induce contraction maps on the complexity space. In the last

section, we will show that this fact also follows from our approach.

2 Preliminaries

Our basic references for quasi-uniform and quasi-metric spaces are [3] and [6].

By a quasi-metric on a set X we mean a nonnegative real valued function d on X x X such that for all x,y,z E X : (i) d(x,y) = d(y,x) = 0 ^ x = y; and (ii) d(x, y) < d(x, z) + d(z, y).

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 the function ds defined on X x X by ds(x, y) = d(x, y) V d(y, x), is a metric on X.

A quasi-metric space (X, d) is said to be bicomplete if (X, ds) is a complete metric space.

For each pair x,y E R, let u(x,y) = (y — x) V 0. Then u is a bicomplete quasi-metric on R called the upper quasi-metric on R. Note that us is the usual metric on R.

Now let (E, + , •) be a linear space on R. An asymmetric norm (quasinorm in [2]) on E is a nonnegative real valued function q on E such that for all x,y E E and a E R+ :

(i) q(x) = q(—x) = 0 ^ x = 0; (ii) q(ax) = aq(x), and (iii) q(x + y) <

q(x) + q(y).

The pair (E, q) is then called an asymmetric normed linear space (compare [2], [10]).

Observe that if q is an asymmetric norm on E, then the function q_1 defined on E by q_1(x) = q(—x) is also an asymmetric norm on E.

Furthermore, the function qs defined on E by qs(x) = q(x) V q(—x) for all x E E, is a norm on E.

The asymmetric norm q induces, in a natural way, a quasi-metric dq on E, defined by dq(x, y) = q(y — x), for all x,y E E.

If dq is a bicomplete quasi-metric on E, then (E, q) is called a biBanach space [13].

As usual, for 1 < p < x, we denote by lp the linear space of all infinite sequences x : = (xn)new of real numbers such that ^=0 \ xn \p< x.

It is well known that (lp, ||-||p) is a Banach space, where || • ||p is the norm on lp defined by || x ||p=(E^=0 | xn \p)1/p for all x Elp.

We shall split the norm || • ||p as follows (compare [2], [4]):

For each x E R, let x+ be the nonnegative real number x V 0.

Fix p E [1, x). For each x : = (xn)new E lp define x+ := (x+)new and

||x||+p = ||x+ ||p , Le. ||x||+p = (Er=0(x+)P)1/p. Then ^hp is an asymmetric

norm on lp such that the norm (|H|+p)s is equivalent to ||-||p ([4] Corollary 2).

In order to obtain a general theory of dual complexity it was introduced in [4] the following class of spaces.

For each p e [1, to) set B*p := {f e : Er=0(2"n \f (n)\)p < to}.

If for any f,g e B* and a e R we define f + g and a ■ f in the usual pointwise way, then it easily follows that (B*, + , ■) is a linear space.

Now denote by qp the nonnegative real valued function defined on B* by

qp(f) = (J>-7 (n)+)p)1/p.

For each f eB* let x/ := (2-nf (u))ne^. Thus x/ e lp and we have

qp(f ) = \\x/.

Therefore (B*,qp) is an asymmetric normed linear space.

In Corollary 4 of [4] it is shown that (B*, qp) and (lp, \H\+p) are isometrically isomorphic via the linear mapping 0 : B* ^ lp defined by the rule

(0(f ))(u) = 2~nf (n),

and, hence, (Bp, qp) is a biBanach space. (Let us recall that two (asymmetric) normed linear spaces (X, ||-\\x) and (Y, \\^\y) are isometrically isomorphic if there is a linear mapping 0 from X onto Y such that \\0(x)\\y = \\x\\x for all x e X.)

In our context, a semilinear space (on R+) will be an ordered triple (E, +, ■) such that (E, +) is an Abelian monoid (i.e. an Abelian semigroup with neutral element) and ■ is a function from R+ x E to E such that for all x,y e E and a,b e R+ : a^(b^x) = (ab)x, (a+b)■ x = (a-x) + (b-x), a -(x+y) = (a-x) + (a-y), and 1 ■ x = x.

Observe that every semilinear space is a cone in the sense of Keimel and Roth [5].

An asymmetric normed semilinear space is a pair (F, H^^^) such that F is a (nonempty) subset of an asymmetric normed linear space (E, !■!), where ¡■¡^ denotes the restriction of the asymmetric norm ^H to F, and (F, + , ■ ) is a semilinear space (compare [11], [13]). If the restriction to F of the quasi-metric d||.||, induced by ^H , is bicomplete we say that (F, H^^^) is a biBanach semilinear space.

For each p e [1, to) let C* = {f e Bp : f (n) > 0 for all n e u}. Following [4], the asymmetric normed semilinear space (C*,qp) will be called the dual p-complexity space, where the restriction of qp to C* is also denoted by

For each p e [1, to) denote by l+ the positive cone of lp. It is immediate to see that (l+, \H\+p) is an asymmetric normed semilinear space which is closed in the Banach space (lp, (\H\+p)s), where the restriction of \H\+p to l+ is also denoted by \H\+p.

Furthermore, it is clear that the restriction of the mapping 0 : B* ^ lp defined above to C*, is a linear bijection between the dual p-complexity space

(Cp*,qp) and the positive cone (l+, ||-||+p) which preserves asymmetric norms. Hence, if we define the notion of an isometric isomorphism between asymmetric normed semilinear spaces in the obvious manner, we deduce from the above observations that (C*,qp) and (l+, ||-||+p) are isometrically isomorphic.

Observe that the quasi-metric dqp induced on C* by qp is given by

dqP (f,g)=[Y/2-m((g(n) - f (n)) V 0)pl

In particular (C*, dqi) is exactly the dual complexity space as defined in [12].

We also recall that the so-called complexity space [14] is the quasi-metric space (C,dC), where C = {f E (0, : £~=0 2-n(1/f (n)) < w] and dC is the quasi-metric on C given by dC(f,g) = 2-n((1/g(n) — 1/f (n)) V 0).

Following Schellekens ([14], Section 4), the intuition behind the complexity distance between two functions f,g E C is that dC (f,g) measures relative progress made in lowering the complexity by replacing any program P with complexity function f by any program Q with complexity function g. Hence, if f, g belong to the dual complexity space C*, we deduce that dqi (f, g) measures relative progress made in lowering the complexity by replacing g by f because dqi (f,g) = dC (1/f, 1/g). In particular dqi (f,g) = 0 can be interpreted as g is "more efficient" than f on all inputs.

This computational interpretation of the complexity distance dqi remains valid for each quasi-metric dqp [4]. Thus, the fact that dqp(f,g) = 0, can be interpreted as g is more efficient than f. Furthermore qp(f) = dqp(0,f) measures relative progress made in lowering complexity by replacing f by the "optimal" complexity function 0, assuming that the complexity measure is the running time of computing.

3 The supremum asymmetric norm on sequence algebras

In this section we present the precise context that will be used in order to obtain a robust mathematical model for discussing those complexity functions that generate exponential time algorithms

We start by recalling some pertinent concepts.

Here, by an algebra we mean a linear space E (on R) with a binary (multiplicative) operation that is commutative, has identity element and satisfies for all x,y,z E E and a E R the following conditions: x(yz) = (xy)z, x(y + z) = xy + xz, and a(xy) = (ax)y = (ay)x.

A (n asymmetric) normed algebra is an algebra E with a (n asymmetric) norm ||-|| satisfying HxyH < ||x|| ||y|| for all x,y E E. By a Banach algebra is meant a normed algebra that is also a Banach space, and by a biBanach algebra is meant an asymmetric normed algebra that is also a biBanach space.

As usual we denote by l^ the algebra consisting of all bounded infinite

sequences of real numbers.

It is well known that (lro, ) is a Banach algebra for the usual multiplication operation on lro, where is the supremum norm on lro, i.e. \\x\\ro = sup{\xn\ : n e u} for all x := (xn)ne^ e lro.

As in the lp-case (see Section 1) we may split the norm as follows: For each x : = (xn)ne^ e lro define \\x\\+ro = \\x+\\ro , that is to say, \\x\\+ro = sup{xn V 0 : n e u}.

It is immediate to see that HH^^, is an asymmetric norm on lro. In addition, we have the following facts.

Proposition 1. (\H\+ro)s = WMtt on lro.

Proof. Let x := (xn)„ew e lro. It is clear that \\x\\+ro < \\x\\ro and \\—x\\+ro <

\\x\\ro.

On the other hand, for each e > 0 there is k e u such that

\\xL <e + \xk\ =e + (xkV (-xk)) <e + (\\x\\+ro V \\—^Uro^ We conclude that (Hx\^)s = \\x^ .■

Corollary. (lro, \H\+00) is a biBanach space.

Example 1. Note that (lro, HH^^) is a not an asymmetric normed algebra. Indeed, let x := (xn)new e lro with xn = —1 for all n. Clearly \\xxH^ = 1. However \\xH^ = 0.

For each polynomial P(n), with P(n) > 0 for all n e u, define

Bp(n) ,ro := {f e R- : sup{2-p(n)\f (n)\ : n e u} < to}.

It easily follows that Bp(n) ro is a linear space for the usual pointwise operations.

Observe that, in particular, Bro = f|p{n)>n Bp(n),ro, and C* C B* C Bro for all p > 1 .

Now define a binary operation * on Bp (n) ro as follows: For each f,g e Bp (n) ro let f * g be the element of Bp(n) ro given by the rule

(f*g)(n) = 2-p (n)f (n)g(n).

An easy computation shows that, equipped with the operation *, Bp(n) ro is an algebra with identity element the function e : u ^ R given by e(n) = 2p(n) for all n.

Next denote by qp(n) ro the nonnegative real valued function defined on

Bp(n),ro by

qp(n),ro(f) = sup{2-p(n)f (n)+ : n e u}.

For each f E B*P(n) 0 let x/ := (2 P(n)f (n))new. Then x/ e l0 and we have

qP (n), o(f ) = ||x/

Since |H|+^ is an asymmetric norm on lo it follows that qP(n)o o is an asymmetric norm on BP(n) o and consequently (B*P(n) 0, qP(n)o o) is an asymmetric normed linear space.

We shall show that this space is isometrically isomorphic to (l0, H-H+oo). To this end define a mapping 0 : BP(n) o ^ lo by the rule:

(0(f ))(n) = 2-P (n)f (n),

for all f E B*P(n) o and n E u. Thus 0(f) = x/, where x/ is the element of lo defined above. We then have the following result. (Let us recall that a mapping p from an algebra X to an algebra Y is a homomorphism provided that p is a linear mapping such that p(xy) = p(x)p(y) for all x,y E X).

Proposition 2. 0 is a bijective homomorphism between (B*P(n) 0,qP(n), o) and

(l0 ||-||+J such that qP(n),o(f) = ||0(ffor aU f E BP(n),O

Proof. We first show that 0 is bijective.

Suppose that 0(f) = 0(g). Then 2-P(n) f (n) = 2-P(n)g(n) for all n E u, so f = g. Thus 0 is one-to-one.

Now let x := (xn)ne^ E l0. Then the function f defined by f (n) = 2P(n)xn for all n E u, satisfies 0(f) = x. Hence 0 is onto. We conclude that 0 is bijective.

In order to see that 0 is a homomorphism, let f,g E BP(n) 0 and let a,b E R. Then

0(af + bg)(n) = 2-P(n)(af (n) + bg(n)) = a0(f )(n) + b0(g)(n), for all n E u. Therefore 0 is linear.

Moreover 0(f*g)(n) = 2-P(n)(f*g)(n) = 2-2P(n)f (n)g(n) = 0(f )(n)0(g)(n) for all n E u, and thus 0(f * g) = 0(f )0(g). We have shown that 0 is a homomorphism. Finally, given f E B*P(n) 0 we obtain

||0(f )|+0 = ||x/ |+0 = qP(n) o 0(f), which concludes the proof. ■

Corollary. (B*P(n) 0,qP(n), 0) and (l0, ||-||+0) are isometrically isomorphic. Corollary. (Bp(n) 0,qP(n), 0) is a biBanach space.

4 The supP(n) -complexity space

By a semialgebra we mean a semilinear space E (on R+) with a binary (multiplicative) operation that is commutative, has identity element and satis-

fies for all x,y,z e E and a e R+ the following conditions: x(yz) = (xy)z, x(y + z) = xy + xz, and a(xy) = (ax)y = (ay)x.

By an asymmetric normed semialgebra we mean an asymmetric normed semilinear space (F, H^F) such that F is a semialgebra satisfying \\xy\\F < \\x\\F \\y\\F for all x,y e F. If, in addition, (F, H^F) is a biBanach semilinear space, we say that (F, H^F) is a biBanach semilagebra.

Two asymmetric normed semialgebras (X, H^Hx) and (Y, H^Hy) are called isometrically isomorphic if there is a mapping p from X onto Y such that for all x,y e X and a,b e R+, p(ax + by) = ap(x) + bp(y), p(xy) = p(x)p(y) and \\x\\x = Hp(x)\\y .

Next we obtain a simple but crucial example of an asymmetric normed semialgebra.

Denote by l+ the positive cone of lro, i.e. l+ = {x+ : x e lro}.

It is immediate to see that (l+ , \H\+00) is an asymmetric normed semilinear space which is closed in the Banach space (lro, (\H\+00)s), where the restriction of \M\+ro to lro is also denoted by \\■H+ro .

Clearly lro is a semialgebra and for each x,y e l+ we have \\xyH^ < \\x\\+^ \\y\\+^ (compare Example 1).

Consequently, we obtain the following result.

Proposition 3. (lro, IH ) is a biBanach semialgebra.

For each polynomial P(n), with P(n) > 0 for all n e u, consider the biBanach space (Bp(n) ro, qp(n),ro) constructed in the preceding section and let

Cp(n),ro := {f e Bp: f(n) > 0 for all n e u}.

The restriction of the asymmetric norm qp(n),ro to Cp* (n),ro will be also denoted by qp(n),ro if no confusion arises. Similarly, the restriction of the multiplication operation * to Cp(n) ro is also denoted by *. Therefore Cp(n) ro is a semialgebra for the operation *.

It is clear that the restriction to Cp(n) ro of the mapping 0 : Bp(n) ro ^ lro, defined in Section 3, is a bijective homomorphism between the asymmetric normed semialgebra (C*(n) ro, qp(n),ro) and the positive cone (l+ , \H\+ro) which preserves asymmetric norms.

As a consequence of these observations and Proposition 3 we have the following result.

Proposition 4. (Cp(n) ro,qp(n),ro) and (l+, \H\+ro) are isometrically isomorphic biBanach algebras, and hence Cp(n) ro is a closed subset of the Banach

space (B*P(n),ro, (qp(n),ro)s).

In the following the biBanach semialgebra (Cp(n) ro, qp(n),ro) will be called the supp(n)-complexity space.

Remark 1. Observe that, in particular, Q o 0 = HP(n)>n C*P(n)o 0, and Cp C C, for all p > 1. Furthermore, if P(n) > n for all n E u, the identity element e of the semialgebra CP(n) 0 does not belong to any Cp, p > 1. (Recall that e is defined by e(n) = 2P(n) for all n E u, and we have qP(n)o0(e) = 1.)

Remark 2. If P(n) < Q(n) for all n E u, then C*P{n) 0 C CQ{n) 0 and

qQ(n),o(f) < qP(n),o(f) for all f E CP(n),o

Next we show that the (complexity) quasi-metric induced by the asymmetric norm qP(n)o 0 also provides a suitable interpretation of the functions in supP(n)-complexity space.

Let f be a function from u to R+. As usual, a function g : u ^ R+ is said to be in class O(f (n)) if there is c> 0 such that g(n) < cf (n) for all n E u. Let f E C*P(n) 0 and let g be in class O(f (n)). Then g < cf, for some c> 0.

Obviously g E CP(n) oo.

• If c < 1, we have g < f, and hence

dqp (f,g) = qP (n) , o(g — f) =0.

Thus, as in the case of the dual p-complexity space, condition dqp (n) ^ (f, g) = 0 (with f = g), agrees with the fact that that g is more efficient than f on all inputs. Furthermore qP(n)o 0(f) = dqp(n) ^ (0, f) measures relative progress made in lowering complexity by replacing f by the "optimal" complexity function 0, assuming that the complexity measure is the running time of computing, of course.

• If c > 1, then

qP (n), o(g) — qP (n) , o(f) < qP (n), o(g — f)

= sup{2-P(n)((g(n) — f (n)) V 0)) : n E u] < sup{2-P(n)(c — 1)f (n) : n E u] = (c — 1)qP (n), o(f),

and consequently

qP (n), o(g) < cqP (n) , o(f ) and dqp ln)lX (f,g) < (c — 1) dqp ln),x (0,f).

The theory of Smyth completable quasi-metric spaces provides an efficient setting to give a topological foundation for many kinds of spaces which arise naturally in several fields of Theoretical Computer Science ([8], [12], [13], [14], [16], [17], etc.).

A quasi-metric space (X, d) is Smyth completable if and only if every left K-Cauchy sequence in (X,d) is a Cauchy sequence in (X,ds) ([7], [15]). (Let us recall that a sequence (xn)neN in (X, d) is left K-Cauchy [9] provided that for each e > 0 there is k E N such that d(xn, xm) < e whenever k < n < m.)

A quasi-metric space (X, d) is Smyth complete if and only if every left K-Cauchy sequence in (X,d) has a T(ds)-limit point ([7], [15]), where T(ds)

denotes the topology generated by the metric ds.

It immediately follows that a quasi-metric space is Smyth complete if and only if it is bicomplete and Smyth completable.

We say that an asymmetric normed semilinear space (E, q) is Smyth complete (resp. Smyth completable) if (E, dq) is a Smyth complete (resp. Smyth completable) quasi-metric space.

It was proved in [12] that the dual complexity space is Smyth complete. Generalizing this result the authors proved in [4] that the dual p-complexity space is also Smyth complete for all p > 1.

The following example shows that unfortunately the supp(n)-complexity space is not Smyth completable, and hence not Smyth complete.

Example 2. Let P(n) be a polynomial (with P(n) > 0 for all n e u). Define a sequence (fk)ke^ by fk(n) = 0 for n = 0,1,..., k, and fk(n) = 2p(n for n > k. Clearly fk e C*p(n) ro for all k e u (actually each fk is in class O(2p(n))). Then

dqP(n)^ (fk,fk+i) = sup{2"p(n)((fk+!(n) - fk(n)) V 0)} = 0,

for all k e u. Hence (fk)ke^ is a left K-Cauchy sequence in (Cp(n) ro, dqp(n) ^). However, for each j,k e u with j > k, we have

dqp (B) ^ (fj ,fk) = sup{2-p(n)((fk (n) - fj (n)) V 0)} = 1.

Therefore (Cp(n) ro,qp(n),ro) is not Smyth completable.

5 Contraction mappings

It is known that for applications the complexity space (C ,dC) is typically restricted to functions which range over positive integers which are powers of a given integer b (see Section 6 of [14]).

Let a,b,c e N with a,b > 2, let n range over the set {bk : k e u} and let h eC. A functional $ corresponding to a Divide & Conquer algorithm in the sense of [14], is typically defined by

{c if n =1

af (n/b) + h(n) if n e{bk : k e N}.

We recall that this functional intuitively corresponds to a Divide & Conquer algorithm which recursively splits a given problem into a subproblems of size n/b and which takes h(n) time to recombine the separately solved problems into the solution of the original problem.

It was proved in Theorem 6.1 of [14], that $ is a contraction mapping for dC with contraction constant 1/a. This result was extended in Section 4 of [12] to the dual complexity space (C*, dqi), where the corresponding functional $*

García-Raffi, Romaguera and E.A. Sánchez-Pérez is given, for h EC, by

[ 1/c if n =1

))(n) ={ ' f( /b) k

\frnm if n E{bk : k E N}.

A slight modification of the proof of Theorem 6.1 of [14] shows that such a result also follows in the realm of any dual p-complexity space. We conclude the paper by obtaining an extension of Theorem 6.1 of [14] to the supp(n)-complexity space when P(nk+1) > P(nk) for all n,k E u.

Under the above assumptions, define

Cp(n),m I b,c :

= {f : f is the restriction to arguments n of the form bk, k E u, of f' E Cp(n),M such that f(1) = 1/c}.

Observe that each f E Cp(n) ^ | b,c can be considered as an element of Cp(n)^>, by defining f (n) = 0 whenever n E {bk : k E u}. Thus, if for each f E Cp(n) ^ I b,c, $*(f) is defined as above, we obtain the following.

Proposition 5. Let f,g E Cp^ ,^ | b,c. Then $*(f), $*(g) E Cp^ ,x I b,c, and

dP(„),»(**(f),**(g)) < ldqP(„),»(f,g).

Proof. It is easy to check that $*(f), $*(g) E Cp(n) ^ | b,c. Furthermore

dp M ^ ($*(f), $*(g))

= sup 2-p(n^(_^____) v o)

ne{bk:k€N} V a + g(n/b)h(n) a + f (n/b)h(n) J

< sup 2-p(n) Va(g(n/b) - f (n/b)) v 0

ne{bk :keN} \ a

f(n))\/ o) = id

qp (n)

This completes the proof. ■

< a sup 2-p(n) ((g(n) - f (n)) V 0) = adqp(n), „ (f,g).

a ne{bk:keu} a

Acknowledgement. The authors are grateful to Professor M. Schellekens by suggesting the study of contraction mappings made in Section 5.

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