Scholarly article on topic 'The complexity space of a valued linearly ordered set'

The complexity space of a valued linearly ordered set Academic research paper on "Mathematics"

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Abstract of research paper on Mathematics, author of scientific article — S. Romaguera, E.A. Sánchez-Pérez, O. Valero

Abstract By a valued linearly ordered set (a VLOS for short), we mean a pair (X, ϕ) such that X is a linearly ordered set and ϕ is strictly increasing (= positive monotone) nonnegative real valued function. Clearly, any VLOS is a valuation space. Each VLOS (X, ϕ) generates a linear weightable quasi-metric dϕ on X whose conjugate is order preserving. We show that the Smyth completion of (X, ϕ) also admits the structure of a VLOS . On the other hand, M. Schellekens introduced in 1995, the theory of complexity spaces to develop a topological foundation for the complexity analysis of programs. Here, we introduce the so-called complexity space of a VLOS (X, ϕ) and discuss some of its properties. In particular, we show that it is weightable and preserves Smyth completeness of the (X, ϕ). We apply this complexity approach to the measurement of real numbers and discuss some advantages of our methods with respect to those that use the classical Baire metric.

Academic research paper on topic "The complexity space of a valued linearly ordered set"

Electronic Notes in Theoretical Computer Science 74 (2003)

URL: http://www.elsevier.nl/locate/entcs/volumeT4.html 14 pages

The complexity space of a valued linearly

ordered set

S. Romaguera, E.A. Sánchez-Pérez, O. Valero1

Escuela de Caminos, Departamento de Matemática Aplicada, Universidad

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

Abstract

By a valued linearly ordered set (a VLOS for short), we mean a pair (X, p) such that X is a linearly ordered set and p is a strictly increasing (= positive monotone) nonnegative real valued function. Clearly, any VLOS is a valuation space.

Each VLOS (X, p) generates a linear weightable quasi-metric dv on X whose conjugate is order preserving. We show that the Smyth completion of (X, dv) also admits the structure of a VLOS.

On the other hand, M. Schellekens introduced in 1995, the theory of complexity spaces to develop a topological foundation for the complexity analysis of programs. Here, we introduce the so-called complexity space of a VLOS (X, p) and discuss some of its properties. In particular, we show that it is weightable and preserves Smyth completeness of (X, dv). We apply this complexity approach to the measurement of real numbers and discuss some advantages of our methods with respect to those that use the classical Baire metric.

Mathematics Subject Classification (2000): 54E35, 54F05, 68Q25.

Keywords: linearly ordered set, positive monotone function, valuation, weightable quasi-metric, Smyth completion, complexity space, alphabet, Baire metric.

1 Introduction and preliminaries

Throughout this paper we shall denote by R+, u and N the set of nonnegative real numbers, the set of nonnegative integer numbers and the set of positive integer numbers, respectively.

1 The authors acknowledge the support of the Spanish Ministry of Science and Technology, Plan Nacional I+D+I, and FEDER, under grant BFM2003-02302

©2003 Published by Elsevier Science B. V.

Let us recall that a linear order on a nonempty set X is a (partial) order X on X such that x X y or y X x for all x,y E X. A linearly (ordered) set is a pair (X, X) such that X is a nonempty set and X is a (linear) order on X.

Let (X, X) and (Y, □) be two ordered sets. A mapping f : X ^ Y is said to be monotone if f (x) □ f (y) whenever x X y, and it is called positive monotone if f (x) C f (y) whenever x X y. In case that (Y, □) = (R+, <), with < the usual order on R+, we will say that f is a (positive) monotone function on (X, X).

Note that if (X, X) is a linearly ordered set, then f is positive monotone if and only if it is one-to-one.

Our main references for quasi-pseudo-metric spaces are [3] and [5].

Let us recall that a quasi-pseudo-metric on a set X is a nonnegative real-valued function d on X x X such that for all x,y,z E X : (i) d(x, x) = 0, and (ii) d(x, z) < d(x, y) + d(y, z).

In our context by a quasi-metric we mean a quasi-pseudo-metric d on X such that d(x, y) = d(y, x) = 0 if and only if x = y.

The restriction of a quasi-(pseudo-)metric d on X to any subset of X, will be also denoted by d if no confusion arises.

A quasi-(pseudo-)metric space is a pair (X, d) such that X is a (nonempty) set and d is a quasi-(pseudo-)metric on X.

As usual the associated order <d of a quasi-metric space (X, d) is defined by x <d y & d(x,y) = 0.

Each quasi-pseudo-metric d on a set X induces a topology T(d) on X which has as a base the family of open d-balls {Bd(x,r) : x E X, r > 0}, where Bd(x, r) = {y E X : d(x, y) < r} for all x E X and r > 0.

We say that a quasi-metric d on an ordered set (X, X) is order preserving if x <d y whenever x X y.

If d is a quasi-(pseudo-)metric on X, then the function d-1 defined on X x X by d-1(x,y) = d(y,x), is also a quasi-(pseudo-)metric on X called the conjugate of d, and the function ds defined on X x X by ds(x,y) = d(x,y) V d-1(x,y) is a (pseudo-)metric on X.

A quasi-metric d on a set X is said to be bicomplete if ds is a complete metric on X. In this case we say that (X, d) is a bicomplete quasi-metric space.

A simple but useful example of a (bicomplete) quasi-metric space consists of the pair (R,w), where u is the so-called upper quasi-metric on R, which is defined by u(x,y) = (y — x) V 0, for all x,y E R. Observe that us is the Euclidean metric on R.

2 Generating quasi-metrics from positive monotone functions

In this section we present a general method for generating linear weightable quasi-metrics from monotone functions on linearly ordered sets, which will

be useful later on. In addition, the conjugate quasi-metric is order preserving. The method is essentially an adaptation to our context of the well-known methods to generate metrics and quasi-metrics from valuations on lattices and from semivaluations on semilattices, respectively [1], [15].

Proposition 1. Let p be a monotone function on a linearly ordered set (X, ^ ). Then the real-valued function dv defined on X x X by

d^(x,y) = p(y) - P(x) if x ^ У, dAx,y) = 0 if y ^ x

is a quasi-pseudo-metric on X .such that (d^)-1 is order preserving.

Proof. It is obvious that dv (x, x) = 0 for all x E X. Next we show that for all x,y,z E X, dv(x, z) < dv(x, y) + dv(y, z). Indeed, if z ^ x, then d^(x, z) = 0, so we only consider the case that x < z. In such a case,

if z ^ y, then x ^ y, and hence

Mx,z) = P(z) - P(x) < Py - P(x) = dv(x,y). If y ^ z and x < y we clearly obtain dv(x,z) = d^(x,y) + dv(y,z). It remains to consider the case that y < z and y < x; but then we have that

dv(x, z) = P(z) - P(x) < P(z) - Py = dv(У, z).

Finally, observe that if x,y E X satisfy x ^ y, then, (dv)-l(x,y) = 0. We conclude that (dv)-1 is order preserving on X.B

Proposition 2. Let p be a monotone function on a linearly ordered set (X, Then dv is a quasi-metric on X if and only if p is positive monotone.

Proof. We first suppose that dv is a quasi-metric. Let x,y E X, and assume without loss of generality that x y. If p(x) = p(y), then dv(x,y) = dv(y,x) = 0, so x = y, a contradiction. Therefore p(x) < p(y).

Conversely, if dv(x,y) = dv(y,x) = 0, then p(x) = p(y). Since, by assumption, p is positive monotone, x = y.U

Note that if p is a positive monotone function on a linearly ordered set (X, then ^ is exactly the order <(d^)-i .

Example 1. On (R+, <) consider the identity function id. Clearly, the quasi-metric did is exactly the upper quasi-metric u on R+ defined in Section 1, i.e. did(x,y) = (y - x) V 0.

Example 2. Let a E R+\{0}. Define pa : R+ ^ R+ by pa(x) = ax. It easy to see that pa is a positive monotone function which induces the quasi-metric dVa defined by dVa (x,y) = a(y — x) if x < y and dVa (x,y) = 0 if x > y.

Weightable quasi-metric spaces were introduced by S.G. Matthews [6] as a

part of the study of denotational semantics of dataflow networks. The domain interval, the domain of words and the (dual) complexity space are interesting examples of weightable quasi-metric spaces which appear in several fields of Theoretical Computer Science (see, for instance, [6], [13], [9], [15]).

Let us recall that a quasi-metric space (X, d) is said to be weightable if there exists a function w : X ^ R+ such that for all x,y E X, d(x, y)+w(x) = d(y,x) + w(y). The function w is said to be a weighting function for (X,d) and the quasi-metric d is weightable by the function w.

Observe that (R+,u) is weightable with weighting function the identity function on R+ (compare Example 1 above).

Proposition 3. Let p be a positive monotone function on a linearly ordered set (X, X)- Then dv is a vjeightable quasi-metric on X vnth weighting function p.

Proof. Let x,y E X, and assume that x X y. Then dv(x,y) + p(x) = p(y) — p(x) + p(x). On the other hand dv(y,x) + p(y) = p(y). Thus dv(x,y) + p(x) = d^(y, x) + p(y) for all x,y E XM

In the light of Propositions 2 and 3 we propose the following notion.

Definition 1. A valued linearly ordered set (VLOS for short) is a pair (X, p) such that X is a linearly ordered set and p is a positive monotone function on X.

According to [14], a quasi-metric d on a set X is said to be linear if the associated order <d is linear. The construction given in Proposition 1 immediately shows that if (X, p) is a VLOS, then the quasi-metric dv is linear.

The last result of this section clarifies the relevance of the class VLOS. Furthermore, it will permit us to simplify some proofs in Section 3.

Let (X, d) and (Y, e) be two quasi-metric spaces. A mapping f : (X, d) ^ (Y, e) is an isometry (from (X,d) into (Y, e)) provided that e(f (x),f(y)) = d(x,y) for all x,y E X. It is well known that each isometry from (X,d) into (Y, e) is a one-to-one mapping. If there is an isometry from (X, d) onto (Y, e) we say that (X, d) and (Y, e) are isometric.

Proposition 4. Let (X, p) be a VLOS. Then p is an isometry from the quasi-metric space (X,dv) into the quasi-metric space (R+,u).

Proof. Let x,y E X. If x X y we have p(x) < p(y); otherwise we have p(y) < p(x). So, in any case, we obtain

u(p(x),p(y)) = (p(y) — p(x)) V 0 = Mx y).

Thus p is an isometry from (X,d^) into (R+,u).M

3 The Smyth completion in class VLOS

Smyth completion constitutes an interesting topic from a domain theoretic point of view. In fact, Smyth presented in [16] and [17] a topological framework for denotational semantics based on the theory of Smyth complete (and totally bounded) quasi-uniform and quasi-metric spaces.

Since every weightable quasi-metric space is Smyth completable [4], it follows that for each VLOS (X,p), the quasi-metric space (X,dv) is Smyth completable.

Let us recall that 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) [15], where a sequence (xn)neN in (X, d) is left K-Cauchy [8] 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 limit point in (X,ds) [15].

It immediately follows from the preceding results that a quasi-metric space is Smyth complete if and only if it is bicomplete and Smyth completable. Hence, each weightable bicomplete quasi-metric space is Smyth complete.

In this section we shall prove that the bicompletion of any VLOS is a (Smyth-complete) VLOS.

Let us recall that a quasi-metric space (Y, q) is said to be a bicompletion of the quasi-metric space (X, d) if (Y, q) is a bicomplete quasi-metric space such that (X,d) is isometric to a dense subspace of the metric space (Y,qs). It is well known that each quasi-metric space (X, d) has an (up to isometry) unique bicompletion (X,d) (see [2], [12]).

The bicompletion (X, d) is constructed as follows.

Denote by X the set of all Cauchy sequences in the metric space (X,ds). For each pair {xn}neN, {yn}neN of elements of Y put p({xn}neN, {^n}nen) = limn^^ d(xn, yn). Then p is a bicomplete quasi-pseudo-metric on X. Now let:

R ={({xn}n€N, {yn}n€N)) E X X X : ps({xn}n€N, {yn}n€N) = 0}.

Clearly R is an equivalence relation on X. Denote by X the quotient X/R. For each pair [{in}neN|, [{yn}neN] in X define

d([{xn}n€N], [{yn}n€N]) = p({xn}n€N, {yn}n€N).

In [2] and [12] it was independently proved that (X,d) is a bicomplete quasi-metric space such that (X, d) is isometric to a dense subspace of the metric space (X, (d)s). Therefore (X,d) is the bicompletion of (X,d). Furthermore (d)s = ds on X, and the bicompletion coincides with the standard completion when (X, d) is a metric space.

Definition 2. A VLOS (X, p) is called bicomplete if dv is a bicomplete quasi-metric on X.

By a subspace of a VLOS (X, ф), we mean a VLOS (Y, ) such that Y is a subset of X and is the restriction of ф to Y.

Definition 3. An isometry from a VLOS (X, ф) to a VLOS (Y, ф) is a monotone mapping h : X ^ Y such that ip(h(x)) = ф(х) for all x G X.

Remark. Note that if h : (X, ф) ^ (Y, ф) is an isometry, then h is one-to-one. Thus h and h-1 are positive monotone mappings.

Definition 4. Two VLOS (X, ф) and (Y, ф) are said to be isometric if there is a isometry h from X onto Y.

Proposition 5. Let h be an isometry from a VLOS (X, ф) onto a VLOS (Y, ф). Then the quasi-metric spaces (X,dp) and (Y,dф) are isometric by h.

Proof. Let x,y G X. If h(x) X h(y), then dф(h(x),h(y)) = ф^(у)) — ф(h(x)) = ф(у) — p(x). Since h-1 is monotone, h-1(h(x)) X h-1(h(y)). Hence x X y, so dp(x,y) = ф(у) — p(x). Therefore dф(h(x),h(y)) = dp(x,y). If h(y) X h(x) then y X x and dф(h(x),h(y)) = 0 = dp(x,y).U

Definition 5. Let (X, ф) be a VLOS. We say that a bicomplete VLOS (Y, ф) is a bicompletion of (X, ф) if (X, ф) is isometric to a subspace of (Y, ф) that is dense in the metric space (Y, ^ф)s).

We shall prove that each VLOS has an (up to isometry) unique bicomple-tion.

Let (X, ф) be a VLOS. By Proposition 3, the quasi-metric space (X, dp) is weightable with weighting function ф

Now let (X,dp) be the bicompletion of (X,dp). By Theorem 1 of [7], (XX, dp) is weightable with weighting function фgiven by £>([x]) = limn^œ p(xn) for all x := (xn)neN G X.

Next we define a binary relation Ц on XX as follows:

For each pair x := {xn}neN and y := {xn}neN of elements of XX let

[x] □ [y] ^ ф([x]) < Ф([у]).

Clearly (XX, □) is a linearly ordered set, and we immediately obtain the following result.

Lemma 1. Let (X, ф) be a VLOS. Then (X ,ф) is a VLOS.

By Propositions 2 and 3 and the preceding lemma, dp is a weightable quasi-metric on XX with weighting function ф.

Lemma 2. dp = d$ on X.

Proof. Let x := {xn}neN and y := {xn}neN be two elements of X. Suppose without loss of generality that [x] C [y]. Then p([x]) < p([y]), and hence p(xn) < p(yn) eventually. Therefore

dip([x], [y]) = lim dv(xn,yn) = lim (p(yn) - ^(x„))

n—tt n—tt

= lim p(yn) - lim p(xn) = p([y]) - p([x])

n—n—

= MM, [y]).

We conclude that dp = on XX .■

Corollary. (X, p) is a bicomplete VLOS.

Lemma 3. (X,p) is a bicompletion of (X,p).

Proof. For each x E X denote by X the constant sequence x,x, ...,x,... Since, by Lemma 2, (X,d^) is the (quasi-metric) bicompletion of the quasi-metric space (X, dp), i(X) is dense in (XX, (d^)s) where i denotes the one-to-one mapping from X to XX given by i(x) = [X] for all x E X. Note that [X] consists of all sequences in X which converge to x in the metric space (X, (dp)s). Clearly i is an monotone function, so i(X) is a linearly ordered subset of (X, Q. Finally, since p(i(x)) = p([X]) = p(x) for all x E X, we deduce that (X, p) and (i(X), Xpi(X)) are isometric VLOS. The proof is complete.■

Lemma 4. Let (X, p) be a VLOS. Then any bicompletion of (X, p) is isometric to (XX , p).

Proof. Let (Y,ip) be a bicompletion of (X, p). By Proposition 5, (Y,d^) is a (quasi-metric) bicompletion of (X, dp). Since the bicompletion of a quasi-metric space is unique up to isometry, there is an isometry h from (Y, d^) onto (X, d^). We want to show that h is an isometry from (Y, ip) onto (X, p).

Indeed, let x,y E Y such that x — y. Since ^ is positive monotone, we have

0 < ^(y) - ^(x) = di>(x,y) = dv(h(x),h(y)),

Hence h(x) C h(y). Thus h is (positive) monotone.

On the other hand, it follows from Remark 2 of [7] that p(h) is a weighting function for the quasi-metric d^ on Y. Then, there is a constant function c such that p(h) = ^ + c (see, for instance, Proposition 3.2 of [10]). Since for each x E X, we have p(h(x)) = ^(x), it follows that c = 0, and consequently

p(h(y)) = ^(y) for all y E Y.

We conclude that (Y,ip) and (X,p) are isometric VLOSM

From the above lemmas and the fact, cited above, that each weightable bicomplete quasi-metric space is Smyth complete, we deduce the following.

Theorem 1. Each VLOS (X,p) has a bicompletion (X ,p) which is unique up to isometry. Furthermore (X,dp) is Smyth complete.

4 The complexity space of a VLOS

Let (X, p) be a VLOS. Set

Civ = {f G X- : £ 2—np(f (n)) <

do* iV (f,g) = Y,2—ndv(f (n),g(n))

for all f,g E C*x>v. Note that C*Xp = because constant mappings are in

CX, p.

Note that by Proposition 4, we have

dc*x^(f,g) = J>-n[(p(g(n)) — p(f (n))) V 0].

By f X g we mean that f (n) X g(n) for all n E u. Then (CX, v, X) is clearly an ordered set.

It easy to see that dc*x is a quasi-metric on C*,p whose conjugate quasi-metric is order preserving, and analogous to [11], the quasi-metric space (C*dC*x ) will be called the complexity space of (X, p), and dcx the complexity quasi-metric of (X, p). In particular, if X = R+ and p is the identity function id on R+, then the complexity space of the VLOS (R+,id) is the so-called dual complexity space (see [9]), which consists of the pair (C*,dC*), where

D + \ W . \ ^ i~) — n .

C* = {f G (R+)- : nf(n) <

and dC* is the quasi-metric on C* given by

de* (f,g) = J>-n[(g(n) - f (n)) V 0].

Proposition 6. Let (X, p) be a VLOS. Then the mapping

CX ,v,dc*x iV ) - (C *,de* )

given by the rule

W )(n) = p(f (n))

is a positive monotone isometry.

Proof. First note that ^ is well-defined because for each f E C*Xv one has Er=0 2-np(f (n)) < and thus ) E C*. '

Now let f, g E C*X Then

de* W), tf(g)) = J>"n[(tf(g)(n) - tf(f)(n)) V 0]

= J>-n[(p(g(n)) - p(f(n))) v0] = da*x^(f,g).

Hence ^ is an isometry from (C*XiV,dC*x ) into (C*,dC*). Finally, let f,g E CXp such that f g. Since p is positive monotone it follows that ^(f) < ^(g). We conclude that ^ is positive monotone.■

It is well known [9] that the dual complexity space (C*, dC*) is a weightable quasi-metric space with weighting function W given by W(f) = 2-nf (n).

Combining this result with Proposition 6 we deduce the following.

Proposition 7. The complexity space (C*XiV,dC*x ) is vjeightable with weighting function Wv defined on by Wv(f) = 2-np(f (n)).

It was proved in [9] that the dual complexity space is Smyth complete. In our next theorem we extend this result to any complexity space (C*,v, dC*x ).

Theorem 2. Let (X, p) be a VLOS. Then the following statements are equivalent.

(1) (C*,y,dC*Xv) is bicomplete.

(2) (C*XtV,dC*x ) is Smyth complete.

(3) (X, dp) is Smyth complete

(4) (X, dp) is bicomplete.

Proof. (1) ^ (2). Since the complexity space (CXiV,dC*x ) is a weightable bicomplete quasi-metric space, we deduce that it is Smyth complete.

(2) ^ (3). Let {xk}keN be a left K-Cauchy sequence in (X,dp). Consider the sequence {fk}keN in (C*Xp,dC*x ) where each fk : u ^ X is the constant mapping defined by fk(n) = xk. Next we show that {fk}keN is a left K-Cauchy sequence in (C*dC*x ). Indeed, for each e > 0 there exists k£ E N such that dp(xk,Xj) < e/2 whenever j > k > k£. Thus

dcx, ,(fk, fj) = £ 2-ndp(fk(n), fj(n)) =

Y,2-ndv(xk ,xj) < 2 J] 2-n = e

n=0 n=0

whenever j > k > k£. Since (C*x>v,dc*x ) is Smyth complete, there is f E C*Xp such that limk^^(dcx^)s(f, fk) = 0. Put y = f (0). Since

(dp)s(y, xk) < £ 2-ndp(f (n), fk (n)) + J] 2-ndf (n),f(n))

n=0 n=0

= dCX „ (f,fk ) + dCX (fk,f),

it immediately follows that {xk}keN converges to y in (X, (dp)s). Therefore (X, dp) is Smyth complete.

(3) ^ (4). Obvious.

(4) ^ (1). Since by Proposition 4, (X,dp) is isometric to the subspace (p(X),u) of (R+,u), it follows from our assumption that (p(X),u) is bicom-plete (note that actually it is Smyth complete).

Fix y E p(X). According to the terminology of [11], let

By = {f E (<p(X))" : £ 2-n \y - f (n)| <

and let uBy be the quasi-metric on By given by uBy (f, g) = 'TO=0 2-nu(f (n),g(n)). By Theorem 1 of [11], (By,uBy) is a bicomplete quasi-metric space.

Next we observe that the ^(CX p) = By, where ^ is the isometry defined in Proposition 6.

Indeed, let f E CXp. Then

ro ro ro

E 2-n \y - <fi(f (n))\ < E 2-ny + J] 2->(f (n)) <

n=0 n=0 n=0

so ^(f) E By. Now let f E By. For each n E u there is xn E X such that f (n) = p(xn). Define h E X^ by h(n) = xn for all n E u. Since f E By it immediately follows that J2'TO=0 2-nf (n) < and thus J2'TO=0 2-np(h(n)) < Therefore h E C*Xpp and ^(h) = f. We conclude that CX,p) = By. Then (CX,p), dc^_ ) is isometric to (By, uBy) by Proposition 6, and consequently (CX,P,dc*Xv) is bicomplete.■

It is well known ([9]) that if g E C*, then (C*, (dC*)s) is a compact metric space, where C** = {f E C* : f < g}. From this result and Proposition 6 we deduce the following.

Theorem 3. Let (X,p) be a VLOS and let g E C*Xp, then (C*, (dcx )s) is a compact metric space, where C* = {f E Cx>p : f X g}.

We conclude the paper with an application of the complexity quasi-metrics to the measurement of distances between infinite words over the decimal al-

phabet, and analyze some advantages of our methods with respect to those that use the classical Baire metric.

Let E = {0,1, 2, ...9} and let p : E ^ R+ defined by p(x) = 2-(10-x) for all x E E. Then (E, p) is a VLOS, where E is equipped with the restriction of the usual order on R.

Obviously (E, (dv)s) is a compact metric space, so in particular (E, dv) is Smyth complete.

Denote by Ew the set of all infinite words over E. Each w E Ew will be expressed by w0w1w2..., or by (wn)n€w if no confusion arises.

Clearly, we may assume that the complexity space of (E, p) is the pair (Ew ) where d^ is the weightable quasi-metric on Ew given by

ds- (v,w)

2 ndv(vn,wn),

ds- (v,w)

(---) V 0

It follows from Theorem 2 that (Ew ,d^) is Smyth complete. A typical and well-known metric on Ew is the so-called Baire metric which is given by

D(v, w) = 2-£(v'w) if v = w, and D(w, w) = 0,

for all v,w E Ew, where £(v,w) is defined as the length of the nonempty common prefix of v and w if they exist, and £(v,w) = 0 otherwise.

The following result establishes a useful relation between metrics D and (d№ )s.

Proposition 8. For each v,w E Ew we have

2-11D(v,w) < )s(v,w) < (1 - 2-9)D(v,w).

Proof. Let v,w E Ew. We assume that v = w. Then D(v, w) = 2-i(v'w\ Put £(v, w) = k. Since

(ds-)s(v,w) 2

ds- (v,w) + ds- (w,v)

- 2(ds-)s(v,w),

it follows

(ds-)s(v,w) -> 2

/ —n

n=k 29- 1

2io—wn 210—v

- v2 210 ' Z-^i

2—(k— 1)

29 _ 1 29

D(v, w).

2(d№)s(v,w) > J] 2-n

= 2-(k+i0) = 2—10D(v,w).

This completes the proof. ■

As a consequence we obtain the following well-known result. Corollary. (SQ ,D) is complete.

The following technical result will be useful in the rest of the paper. We omit its easy proof.

Proposition 9. Let u,v,w E XQ be such that un < vn and un < wn for all n E u. If there is n0 E u such that vn < wn for all n > n0, then

(u,w) - (u,v) > 2-n0 [ ---777-— | .

n v ' ' ^ y ' ' — I 2io-wn0 2io-vn0 I

To discuss in our context some advantages of the complexity quasi-metric with respect to the Baire metric, we focus, without loss of generality, in computing distances on the interval [0,1].

For each real number in ]0,1[ admitting a rational decimal expansion, choose exactly this expansion. In addition, identify the real number 0 with 0.000... , and the real number 1 with 0.999...

Consider the unique decimal expansion of all numbers in the interval [0,1] that is obtained in this way, and denote by H the set of such expansions.

Let XQ = {w E XQ : w0 = 0}. Since each x E H can be viewed as an element of XQ, we assume without loss of generality that H is a subset of XQ , in the sequel.

Observe that if {vk}keN and {wk}keN are sequences in H with vk = wk for all k E N, and there is w E H such that D(w,vk) ^ 0, D(w,wk) ^ 0 and £(w,vk) = £(w,wk), then D(w,vk) = D(w,wk) for all k E N. However, Proposition 9 shows that the quasi-metric d^, and thus the metric (d^)s, is able to distinguish between the "distances" from w to vk and from w to wk, respectively, in many interesting cases.

We illustrate this fact with the following example.

Example 3. Denote simply by 0 the word 00000...

Consider the sequence {vk}keN in H given by

v1 := 01000...

v2 := 00100...

k—times

vk :=6q0~0 1000...

> 2"fc(1 - —) - (29 210 )

Clearly we obtain D(0,vk) = 2 k for all k E N. On the other hand

d^ (0,vk) = 2—k b(1) - p(0)] = 2—k [2i0T - ^ ] = 2—k ^

and hence

d№ (0,vk ) = 2—1oD(0,vk).

for all k E N.

Now, we take the sequence {wk}keN in H given by

wT := 011000... w2 := 001100...

k—times

Wk :=600~0 11000...

A straightforward calculation shows that D(0,wk) = 2—k for all k E N. Hence D(0,vk) = D(0,wk) for all k E N, and, obviously, D(w,vk) ^ 0 and D(w,wk) ^ 0.

However, by Proposition 9,

d№ (0,wk) - (0,vk) > 2—(k+T),

which provides a reasonable and desirable relation. Note that the inequality sign is, actually, an equality sign for this case. Therefore, we exactly obtain

dE- (0,wk) = dw (0,vk) + 2—(k+T) — = 2—(k+T) —

and, thus

for all k N.

dw (0, Wk ) = 2H D(0,Wk )

Acknowledgement. The authors are very grateful to Professor Micheal Mac an Airchinnigh for a careful review of the paper.

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