on the uniqueness of fixed points of endofunctors in a category of complete metric spaces

Information Processing Letters 29 (1988) 277-281

North-Holland

8 December 1988

ON THE UNIQUENESS OF FIXED POINTS OF ENDOFUNCTORS IN A CATEGORY OF COMPLETE METRIC SPACES

Mila E. MAJSTER-CEDERBAUM

Fakultiir ftir Mathematik und Informatrk, Unioersrtiit Mannheim, Semmorgehiiude AS, D-6X00 Mmnheim, Fed. Rep. Germuny

Communicated by R. Wilhelm Received 11 April 1988 Revised 11 July 1988

In 1982, De Bakker and Zucker proposed to use complete metric spaces for the semantic definition of programming

languages that allow for concurrency and synchronisation. The use of the tools of metric topology has been advocated by

Nivat and his colleagues already in the seventies and metric topology was successfully applied to various problems (Nivat,

1979/1980). Recently, the question under which circumstances fixed point equations involving complete metric spaces can be

(uniquely) solved has attracted attention (see, e.g., America and Rutten. 1988. and the present author, 1987). America and

Rutten (1988) provide a criterion for the existence of a solution, namely the contractiveness of the respective functor.

Contractiveness together with an additional criterion, the horn-contractiveness was shown by America and Rutten (1988) to

guarantee uniqueness. The problem of uniqueness is the topic of our contribution.

Keywords: Endofunctor, metric space, synchronization, concurrency

1. Mathematical preliminaries f is called an (isometric) embedding if, Vx, y E M,,

A metric spuce is a pair (M, d) with M a set and d a mapping, d: M X M + [0, 11 which satis-

fies: ’

d,(f(x), f(y)) =d,(x, Y>.

f is called an isometty if f is onto and an (isometric) embedding.

(a) Vx,y E M, (d(x, y) = 0 ax =y),

(b) V’x,y E M, d(x, y) = d(y, x), (c) VX,.Y,ZE M, d(x, y)<d(x, z)+

d(z, Y).

It is well known [4] that every metric space (M,d) can be embedded into a ‘unique’ ‘minimal’ complete metric space, called the completion of

(M, d). A sequence (x,) in a metric space (M, d) is a

Cauchy sequence, whenever VE > 0, 3 N E N, V’n, m > N, d(x,, x,) < E. The metric space (M, d) is called an complete if every Cauchy sequence converges to an element of M.

Let (M,, d,), (M2, d,) be metric spaces. A function f : M, + M2 is called non-distance-in-

creasing if, Vx, y E M,,

d,(f(x), f(y)) G d,(x, Y).

Let C denote the category that has complete metric spaces as objects. The arrows in C are the non-distance-increasing functions. Let M, , M2 be complete metric spaces, and let i : M, + MI be an embedding and j : M, + M, be a non-distance-increasing function. j is called a cut for i if i 0 j = id,,. 2 For embedding i: M, -+ M2 with cut j we put L = (i, j) and write M, -‘M2 and define

B(L) = sup { dnr,(X. i(j(x>))}.

’ 0 < d(x, y) 6 1 can always be obtained for an arbitrary

metric d: M x M - R by substituting d-(x, y) by

4% .Y)/(d(& y)+I).

’ Throughout the paper, the composition f 0 g of functions

stands for Xx.g(f(x)).

0020-0190/88/$3.50 0 1988, Elsevier Science Publishers B.V. (North-Holland) 277

Volume 29. Number 6 INFORMATION PROCESSING LE-M-ERS 8 December 1988

We say that a functor 9 : C + C preserves emhed- dings iff .Fe is an embedding for every embedding e. A functor F that preserves embeddings is said to preserve cuts if .Fc is a cut for Fe, whenever c is a cut for e. A functor .F: C + C that preserves cuts is called contracting if there exists an E, 0 < E < 1, such that, for all D --s’E E C,

1 = (i, j),

6(.%-l) <E.??(L)

where FL = (.Fi, Tj). Note that we have slightly modified the defini-

tion of [l]. just in order to be able to include the empty space as an object.

2. The problem setting

In [3], De Bakker and Zucker proposed a prom- ising approach to the definition of semantics for programming languages based on complete metric spaces as follows: the meaning function Me maps a program X (in a language 2) to its meaning Me[X] which is an element of a certain complete metric space constructed for the language _F as a solution of a certain fixed point equation .FX = X.

This technique has successfully been applied [3,17], for example to different variants of Hoare’s communicating sequential process [S]. Ever since, the question of existence and uniqueness of solutions of such fixed point equations as well as the connection to other models for semantics, for example the denotational approach based on CPOs (complete partial orders) [5,13,14,15,16], has attracted interest. The latter question has been partly investigated in [6,9,17]. Recently, in [l,lO], general criteria have been developed to ensure existence and uniqueness of solutions of equations FX = X, where .F is an endofunctor in a category of complete metric spaces. In particular in [lo], two different criteria, each of which guarantees the existence and uniqueness of fixed points, are pre- sented. In addition, in [lo] an interesting example of a fixed point equation is discussed that cannot be solved by standard techniques. In [l] it is shown that a functor that is contracting (as defined above) has a fixed point. It is then shown in [l] that contractiveness together with hom-con-

27X

tractiveness guarantees that the fixed point is unique (up to isometry). Here we discuss the problem of uniqueness.

3. Fixpoints of contracting functors

In this section we deal with the problem how the fixed points of a contracting functor are re- lated. In particular we show that every contracting functor has a unique ‘minimal’ fixed point and that every contracting function .F with 1 .F,0 1 = 1 has a fixed point that is unique up to isometry.

3.1. Remark. Let (M, d) be a complete metric space. Let

/z,( M) = { U c M : lJ is nonempty and closed}

and let d:, denote the Hausdorff metric on bc( M),

i.e., for x. ye M, X, Y ~fi~(M), let

d&Y, Y)

=max{ sup {A( .\ lz x

x, Y)}, sup {a(.~. X)}j. Vt Y

It is well known that ( %z~( M), Jb,) is a complete metric space (see, e.g., [4]).

Let now D +&E E C, L = (i. j). In particular, D is isometrically embedded into E via i and we can view D as a closed subset of E, i.e.. an element of #,( E). Let d; be the Hausdorff metric on fic( E); then, we can talk about d-,( D, E) or d”( D, E) for ease of notation, considering D and E as elements of jc( E). ’

3.2. Remark. Let D +L E E C, I = (i, ,j). If Vx E E

d,(x, i(.j(x))) G P.

then

8(D, E)</L.

’ Strictly speaking, the ‘distance of D and E as elements of #c(E)’ also depends on the choice of the embedding i. But,

as this choice will always be clear from the context, we omit its indication.

Volume 29. Number 6 INFORMATION PROCESSING LETTERS 8 December 1988

This can easily be seen by the definition of the Hausdorff metric.

3.3. Remark. Let D +‘E E C. Then,

J(D, E) <a(~)

by the above remark and the definition of S(l).

3.4. Theorem. Let F : C + C be a contracting functor and let ,kli denote the empty space.

(a) There is a fixed point A4, of 9J that can be

embedded into any other fixed point A4 of 9”.

(b) If 1 F@ ( = 1, then there is a unique fixed

point of .F up to isometry.

Proof. (a) If .F,tJ =fl, the statement is trivial. Let now 9,0 + 8. Clearly, ,t? is initial in the categories M and C. In particular, there is a unique embedding

x, :fzI -.Ffl.

By iteration we get X, =.F’(h,) : F’fl +F’+‘$, i 2 1, is an embedding. Let U.F’,5 denote the direct limit of this sequence in M. Then the com-

pletion of UF’@, which we denote by U.F’@, is the direct limit of this sequence in C. Let I, : 9’0 --) UF’@ be the canonical embeddings, i > 0, i.e.,

1,=x, 0 I,+,. (1)

Let us now choose an arbitrary one-element space {x0 }. Clearly, there is an embedding

e,: {x0} --+.F@

as 9-B + ,Ef. Moreover, there is a unique embedding

i,:O+ {x0}

and we have

h, = i, 0 e, (2)

as $ is initial. Let now a, : S’{ x0} +F’+ ‘{x0}, i > 0, be defined as

u,=e, 0 .Fi,,

0, =*u,_,, i>, 1. (3)

Clearly, u, is an embedding from .F’{ x0} to .F’+t{ x0 }. Let U9’{ x0 } denote the direct limit in M with respect to { a, }.

We are going to embed F’{ x,,} into Ug’,0 in a way that is compatible with u,. Let in particular

e, 0 1,:(x,} +US’,@,

9 ‘e II o l,+r : ~-‘~%I 1 -UP’,0 fori>l.

These embeddings of F’{xO} into U.F’$ are compatible with the embedding u, : F’{ x0 } -+ F’+‘{ x0}, as

e, 0 I, = a, 0 (.Fe, 0 I,)

by equations (1) (2) and (3). And, in general,

F ‘e 0 o I,+,

= Ac’e 0 o h,+, a I,+, by (1)

=Ple C-l+1 00-+ ho o I,+2 by definition

zz /fl”‘e o-+1, 0°/

r,+1 10 0 A e. 0 l,+2 by (2)

=9u, 0 F'+'e, 0 I,,, by (3)

=(T 0 gl+’ I * eo 0 [r+Z by definition. (4)

As U9’{ x0 } is the direct limit with respect to the {a, } we conclude that there is a unique embedding

e : U9’{ x0} + U.F’,0

such that

h, 0 e=yF’e 0 o !,+I’ (5)

where h, is the canonical embedding of _F’{ x0} intoUF’{x,}, i>O.

Let now N be an arbitrary fixed point of 9 and let h : FN + N be an isometry. Clearly, there is a unique embedding

E”:@+N,

hence

E,: 3’0 + N, i>, 1,

E, = F& ,-I o h (6)

are embeddings of 9’fl into N. We claim that

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Volume 29, Number 6 INFORMATION PROCESSING LE7TERS 8 December 1988

they are compatible with the X,. First we observe where h : .FN + N is an isometry. Clearly, g, is a that cut for E, and, consequently,

E,,=& 0 9~~ 0 h

zz x0 o El by the initiality of fl

and assuming E( = A, 0 E, + , we get

& ,+1 =9-e, 0 h

=9-(X, 0 E,+,) 0 h

=p”x, 0 .FE,+, 0 h

= h ,+, 0 E,+~ by definition. (7)

Again, as UY’fl is the direct limit with respect to {X,} we conclude that there is a unique embedding

E, 0 g, = (FE,_, 0 h) 0 (h-’ 0 9-g,_,)

=s(E,_, o g,-,).

Clearly, a((~,, g,)) < 1 and

a&,+,> %+I))

f: uF’fl+ N

such that

E, = 1, O f,

hence

(8)

= 8( .FE, 0 h, hK’ 0 ,Fg,)

= ,“E;d,(x. h(.~.E,(~g,(h~‘(x)))))

= ,~~~dh.(hb)~ h(~+%,()i))))

= sup d(y. FE,.%-g,(y)) y t.YN

= 6 ( .FE, , Fg, > < E . 6 ( E, , g, ) .

Hence, a((_~,. g,)) G E’-’ with E < 1 and, by Re- mark 3.3, d,,,, (.F’,@, N ) converges to zero. Hence, N is the limes of the 9’fi in j,(N), i 2 1. As N was chosen arbitrarily, as the embedding sequence X, : 3’,0 +,‘+y3 was constructed independently of N, and as the embedding E, : .F’,@ + N are compatible with the X,, we conclude that all fixed points coincide up to isometry. •I

e 0 f:UF{xo}-N

is an embedding. We now construct cuts c, for u, by

L’() = A=, 1 co: ~{xol + (x0)

c, =.Fc,p,, i>, 1.

If 9 is contracting, it has been shown in [l,lO]

that US’{ x0 } is a fixed point of 9, hence we

put M, = US’{ x0 } to get the desired result. (b) Let now in addition 1 .F,d 1 = 1. Let .F,@ =

{ y, }. We again consider the sequence 9’0, i > 0,

with embeddings X,, i > 0, as defined in (a). We define a cut for X,, i >, 1, by

d, = xx.yo, d, =_Fd,_,, i > 2.

Let now, as in (a), N with the embeddings

be another fixed point of 9 E, : F’fl - N defined in (6)

that are compatible with the X, (see (8)). Let

g, = XX.Y”, g, : N ---j { Y,, > 3

g,=h-’ o .gg,_,, g,: N -g’fl,

3.5. Remark. Let 9, : C ---) C be a contracting functor. and let .F1 : C + C be a functor such that

i.e., weakly contracting. Then, 9, 0 YZ and .& 0 9, are trivially contracting. Hence, we may start with one contracting functor 9, : C + C, as for example the functor

S,(X)={p,}u(AxX), Aaset,

in [3], where the metric d on A X X is given by

&u, x), (u’, x’)) = i

1 if u # u’,

$d(x, x’) otherwise,

and d is the metric of X. Then we can construct a variety of functors with unique fixed points using various functors in the construction process that are just weakly contracting.

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Volume 29. Number 6 INFORMATION PROCESSING LETTERS 8 December 1988

4. Conclusion

We have shown that for a contracting functor 9: C -+ C there is a ‘minimal’ fixed point. This result fits in between the existence proof of [l] and their uniqueness result. The uniqueness result of [I] is based on a change of category (instead of C the category of complete metric spaces with a base point is considered) together with the property of horn-contractivity-see [l]-expressing a condition of .F with respect to morphisms. In contrast to this, our approach to guarantee uniqueness is by satisfying that the functor .F applied to the empty set yields a one-element space. When we look at the original paper of [3] and review the construction of spaces, i.e.,

PO = { PO I 7

P r+, = {PO) u/f +c@ x P,) =Ft(P,)*

we see that we can always interpret P,, as 90, i.e., the condition for uniqueness is satisfied. In other words, we may interpret the ‘nil’ process as the result of applying 9 to the empty space. The relation between horn-contractivity and the condition 1 F$I = 1 will be dealt with separately in a forthcoming paper.

References

[I] P. America and J.J.M.M. Rutten, Solving Reflexive Do-

main Equations in a Category of Comblete Metric Spaces,

Rept. CS-R8709, Centre for Mathematics and Computer

Science Amsterdam, 1987: also in: 3rd Workshop on

Mathematical Foundatron of Programming Languages, Lec-

ture Notes in Computer Science (Springer, Berlin, 1988).

[2] M. Arbib and E. Manes, Structures and Functors

(Academic Press, New York, 1975).

[3] J.W. de Bakker and J. Zucker. Processes and the denotational semantics of concurrency, InformatIon and Control

54 (1982) 70-120.

[4] R. Engelking, Genernf Topofog,: (Polish Scientific Pub- lishers, Warsaw, 1977).

[5] N. Francez, C.A.R. Hoarse, D. Lehman and W.P. De Roever, Semantics of nondeterminism, concurrency and

communication, J. Comput. System Ser. 19 (1979) 290-308.

[6] W.G. Golson and W.C. Rounds, Connections between

two theories of concurrency, Information and Control 57

(1983) 102-124.

[7] H. Hahn, Reelle Funktionen (Chelsea, New York, 1948).

[S] C.A.R. Hoare, Communicating sequential processes, Comm. ACM 21 (1978) 666-677.

[9] M.E. Majster-Cederbaum and F. Zetzsche, The relation between Hoare-like and De Bakker-style semantics for

CSP, In preparation.

[lo] M.E. Majster-Cederbaum and F. Zetzsche, Foundatron for

Semantics In Complete Metric Spaces, Tech. Rept. l/1987,

Reihe Informatik. FakultPt fir Mathematik und Infor-

matik. Univ. Mannheim, 1987: also: submitted for pub-

lication.

[ll] M. Nivat, Infinite words, infinite trees, infinite computa- tions, Foundations of Computer Science III..?. Mathemati-

cal Centre Tracts, No. 109 (CWI, Amsterdam, 1979).

(121 M. Nivat. Synchronisation of concurrent processes, In: R.V. Book. ed., Formal Language Theov (Academic Press,

New York, 1980) 429-454.

[13] D.S. Scott, Data types as lattice, SIAM J. Comput. 5

(1976) 522-587.

[14] D.S. Scott, Continuous lattices, toposes, algebraic geome- try and logic, In: Proc. 1971 Dalhousie ConjY, Lecture

Notes in Mathematics, Vol. 274 (Springer, Berlin, 1972)

97-136.

1151 M.B. Smyth, Power domains, J. Comput. System Ser. 16

(1978) 23-26.

[16] M.B. Smyth and G.D. Plotkin, The category-theoretic solution of recursive domain equations. SfAM J. Comput.

11 (1982) 761-781.

[17] F. Zetzsche, Untersuchung Zweier Denotationaler Seman-

trkdefinitionen uon CSP, Ph.D. Thesis, Univ. Mannheim, 1987.

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on the uniqueness of fixed points of endofunctors in a category of complete metric spaces

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