Computable counter-examples to the Brouwer fixed-point theorem
aa r X i v : . [ m a t h . G M ] A p r Computable counter-examples to the Brouwerfixed-point theorem
Petrus H. Potgieter
Department of Decision Sciences, University of South Africa (Pretoria), PO Box 392,Unisarand, 0003, Republic of South Africa, [email protected] , [email protected] , Abstract.
This paper is an overview of results that show the Brouwerfixed-point theorem (BFPT) to be essentially non-constructive and non-computable. The main results, the counter-examples of Orevkov andBaigger, imply that there is no procedure for finding the fixed point ingeneral by giving an example of a computable function which does not fixany computable point. Research in reverse mathematics has shown theBFPT to be equivalent to the weak K¨onig lemma in RCA (the systemof recursive comprehension) and this result is illustrated by relating theweak K¨onig lemma directly to the Baigger example. Keywords : Computable analysis, Brouwer fixed-point theorem, weak K¨oniglemma
We consider the Brouwer fixed-point theorem (BFPT) in the following form,where the standard unit interval is denoted by I = [0 , Theorem 1 (Brouwer).
Any continuous function f : I → I has a fixedpoint, i.e. there exists an x ∈ I such that f ( x ) = x . A computable real number is a number for which a Turing machine exists that,on input n , produces a rational approximation with error no more than 2 − n . Acomputable point is a point all the coordinates of which are computable reals.The notation N for the non-negative natural numbers; R c for the set of computable reals; I c for I ∩ R c ; and δX for the boundary of a set X , being X ∩ X c is also used. The two examples discussed use distinct definitions of a computablefunction of real variables. Russian school
In the Russian school of Markov and others, a computablefunction maps computable reals to computable reals by a single algorithmor the function that translates an algorithm approximating the argumentto an algorithm approximating the value of the functions. It need not bepossible to extend a function that is computable in the Russian school toa continuous function on all of the reals. These functions are often called
Markov-computable . Polish school
In the Polish school of Lacombe, Grzegorczyk, Pour-El and Richards,and others, a function is computable on a region if it maps every every com-putable sequence of reals to a computable sequence of reals and it has acomputable uniform modulus of continuity on the region [1].
One can construct a Markov-computable function f through a computable map-ping of descriptions of computable points x ∈ I c to descriptions of f ( x ) ∈ I c ,such that f ( x ) = x ∀ x ∈ I c . That is, no computable point is a fixed point for f . Unfortunately the f whichis constructed in this way, cannot be extended to a continuous function on I .This is the construction of [2], another instance of which can be found in [3]. · · · Fig. 1.
Basic contraction in the Orevkov counter-example
Lemma 1.
Suppose A k is a sequence of rectangles in I with computable ver-tices, disjoint interiors, and such that(i) ∅ 6 = δA j \ S i There exist computable sequences of ra-tional numbers ( a n ) and ( b n ) in the interval I = [0 , such that the intervals J n = [ a n , b n ] have the following properties. Later we shall deduce the fact from the existence of a Kleene tree. i) If n = m then | J n ∩ J m ] ≤ .(ii) If a n = 0 then a n ∈ { b , b , . . . } and if b n = 1 then b n ∈ { a , a , . . . } .(iii) I c ( S n J n , i.e. the J n cover the computable reals in I = [0 , . Now, let ( A n ) n ≥ be any computable enumeration of the J k × J ℓ . This completesthe proof of the lemma, and the example. Let a be any non-computable point in I . Consider the function f which moveseach point half-way to a , f ( x ) = x + 12 ( a − x )and has a single fixed point, namely a itself. The function f is continuous anddefined on all of I and has no computable fixed point. Nevertheless, this is notreally interesting since – the fixed point a has no reasonable description—since it is itself not com-putable; and therefore – the function f has no reasonable description—it is not computable in anysense.One would like to see a function which is computable, defined (and therefore con-tinuous) on all of I and yet avoids fixing any of the computable points I c . Thefollowing example, having appeared in [5] and in [3], modifies the construction ofOrevkov to produce a computable f defined on all of I having no computablefixed point. One uses the intervals J n = [ a n , b n ] of Orevkov’s example and sets C n = [ k,ℓ ≤ n J k × J ℓ after which one defines f progressively, using the sets C n . The points t n = ( v n , v n )where v n = min x ∈ I { x | ( x, x ) C n } are used as “target point” at each stage of the construction, as in Figure 2. Notethat v = lim n →∞ v n is not a computable number and ( v, v ) will be one of the fixed points of f . Definition 1. For any W ⊆ I we define W (cid:4) ε = (cid:8) x ∈ W (cid:12)(cid:12) d (cid:0) x, δW \ δI (cid:1) ≥ ε (cid:9) and W (cid:3) ε = (cid:8) x ∈ W (cid:12)(cid:12) d (cid:0) x, δW \ δI (cid:1) = ε (cid:9) . b t C b t C Fig. 2. The “target points” t n One can define f n such that1. f n moves every point in the interior of C (cid:4) − n n but is the identity outside theset, and is computable;2. f n +1 agrees with f n on C (cid:4) − n · n and therefore3. f = lim n →∞ f n is computable.Every computable point eventually lies in some C (cid:4) − n · n ⊂ (cid:16) C (cid:4) − n n (cid:17) ◦ and is therefore moved by f . Clearly f ( I ) ⊆ I and f will be as required. Infact, f has no fixed point in [ n C n = [ k,ℓ ≥ J k × J ℓ . Also, f has no isolated fixed point—its fixed points all occur on horizontal andvertical lines spanning the height and breadth of the unit square. Further detailsof the construction appear in Appendix A. The construction cannot be applied inthe one-dimensional case because it is impossible to effect a change of directionby continuous rotation. In reverse mathematics it is known that in RCA , the system of recursive com-prehension and Σ -induction, the weak K¨onig lemma, WKL , is equivalent tothe Brouwer FPT [6]. Lemma 3 (WKL , K˝onig). Every infinite binary tree has an infinite branch. The K¨onig lemma does not have a direct computable counterpart. heorem 2 (Kleene [7]). There exists an infinite binary tree, all the com-putable paths of which are finite. The relation of the Kleene tree to the Baigger counterexample is reviewed inthis section. The discussion is informal and attempts only to give the essentialideas that have been revealed by the approach of reverse mathematics. In RCA ,the weak K¨onig lemma WKL has been shown to be equivalent to a number ofother results in elementary analysis, such as the fact that any continuous functionon a compact interval is also uniformly continuous [8]. WKL and RCA can,furthermore, be used to prove G¨odel’s incompleteness theorem for a countablelanguage [9]. f to Kleene tree Let f be a computable function, as in the Baigger example, mapping I toitself—with no computable fixed point. The following auxiliary result will beused to construct the Kleene tree. Lemma 4. Let a computable g : I → [0 , be given. Then there exists a Turing-computable h : N → N such that for any ( n , n , . . . , n , k ) with ≤ n n ≤ n n ≤ and ≤ n n ≤ n n ≤ we have h : ( n , n , . . . , n , k ) ( m , m ) with m ≤ m where m m ≤ min g (cid:18)(cid:20) n n , n n (cid:21) × (cid:20) n n , n n (cid:21)(cid:19) ≤ m m + 1 k . Let g = || f ( x ) − x || and let h be as in the lemma. Note that g ( x ) = 0 if and onlyif x is a fixed point of f . We shall use only the essential consequences that – g ( x ) > x ; and – there exists a (non-computable) x such that g ( x ) = 0.As usual, { , } ∗ denotes the set of finite binary sequences and ab is the con-catenation of a and b . Definition 2. A binary tree is a function t : { , } ∗ → { , } such that t ( ab ) = 0 for all b whenever t ( a ) = 0 . An infinite branch of a tree t is an infinite binary sequence, on all of which finiteinitial segments t takes the values . The tree is computable whenever the function t is Turing-computable and a computable branch is a computable binary sequence which is an infinite branch.Define the Kleene tree as follows. Let t ( i . . . i n ) = n Y m =1 s ( i . . . i m )here s is a function taking values in { , } . This definition of t ensures that t isin fact a tree and if s is computable, t will be a computable tree. The function s will use h to estimate whether g gets close to zero on a specific square and if g has been bounded away from zero on the square, that branch of the tree willterminate.Define s : { , } ∗ → { , } for all sequences i j . . . i n j n of even length by s ( i j . . . i n j n ) = χ { } (cid:18) m m (cid:19) where( m , m ) = h ( i . . . i n , n , i . . . i n + 1 , n , j . . . j n , n , j . . . j n + 1 , n , n )and binary strings have been interpreted as the natural numbers which theyrepresent. Let s take the value 1 on sequences of odd length.The tree t defined in this way is obviously computable. It remains to showthat t is – infinite; and – has no infinite computable branch.Let x be any point where g ( x ) = 0. Then there exist infinite sequences ( i n )and ( j n ) such that x ∈ (cid:20) i . . . i n n , j . . . j n n (cid:21) × (cid:20) i . . . i n + 12 n , j . . . j n + 12 n (cid:21) for all n and therefore, for all n , s ( i j . . . i n j n ) = 1 and so t ( i j . . . i n j n ) = 1 whichproves the existence of an infinite branch, hence that the tree t is infinite.Suppose that t had an infinite computable branch. The branch would corre-spond to a decreasing chain of closed squares, the intersection of which wouldbe non-empty. Let x be a point in the intersection. Since, by construction ofthe tree, g ( x ) ≤ n for all n , g ( x ) = 0 and hence x would be a fixed pointof f . However, by the construction—the branch being computable—the point x would also be computable, contradicting the fact that f has not computablefixed point. Therefore the tree t has no infinite computable branch. f Suppose we are given a computable tree t with no infinite computable branch.This tree can be used to construct a sequence of closed intervals with a com-putable sequence of end-points, covering all the computable real numbers inthe unit interval and for which the corresponding open intervals are pair-wisedisjoint.Using the computable function t , one can enumerate all of the maximal finitebranches of the tree. Say, b ( n ) = b ( n ) . . . b λ ( n ) ( n )nd set J n, = " b ( n ) . . . b λ ( n ) λ ( n ) , b ( n ) . . . b λ ( n ) + λ ( n ) J n,m = (cid:20) b ( n ) . . . b λ ( n ) + 2 − m +1 λ ( n ) , b ( n ) . . . b λ ( n ) + 2 − m λ ( n ) (cid:21) for m ≥ J n,m covers all the computablepoints I c but not all of the unit interval I . It is easy to see that – for every computable x ∈ I c there exists a computable binary sequence ( x n )such that x . . . x n n ≤ x < x . . . x n + 12 n for all n and since t has no infinite computable branch t ( x . . . x ℓ ) = 0 for some least ℓ , in which case x ∈ ∪ m J n,m where b ( n ) = x . . . x ℓ ; – if ( x n ) is an infinite branch of t then, since it is not computable, for all w we have x x . . . = w . . . and thereforelim n x . . . x n + 12 n [ m J ℓ,m for every ℓ .The Baigger example f can now be constructed using the intervals J n,m and bythat construction one obtains a computable f with no computable fixed point,as required. The existence of the Kleene tree can quite easily be derived from the impos-sibility of ensuring the existence of a computable fixed point for a computablefunction (in both Russian and Polish senses), in two dimensions (or higher). Theingenuous constructions of Orevkov and Baigger provide a way of defining a com-putable function with no computable fixed point from the set of intervals derivedfrom the Kleene tree, in a constructive manner. This correspondence is, perhaps,more attractive for the “working mathematician” than the elegant derivation ofthe result in reverse mathematics. In one dimension, any computable f : I → I does have a computable point x ∈ I c such that f ( x ) = x , which can be seen byfairly straight-forward reduction ad absurdum from the assumption that this isnot the case. eferences 1. Pour-El, M.B., Richards, J.I.: Computability in analysis and physics. Perspectivesin Mathematical Logic. Springer-Verlag, Berlin (1989)2. Orevkov, V.P.: A constructive map of the square into itself, which moves everyconstructive point. Dokl. Akad. Nauk SSSR (1963) 55–583. Wong, K.C., Richter, M.K.: Non-computability of competitive equilibrium. Eco-nomic Theory (1) (1999) 1–274. Miller, J.S.: Degrees of unsolvability of continuous functions. J. Symbolic Logic (2) (2004) 555–5845. Baigger, G.: Die Nichtkonstruktivit¨at des Brouwerschen Fixpunktsatzes. Arch.Math. Logik Grundlag. (3-4) (1985) 183–1886. Shioji, N., Tanaka, K.: Fixed point theory in weak second-order arithmetic. Ann.Pure Appl. Logic (2) (1990) 167–1887. Kleene, S.C.: Recursive functions and intuitionistic mathematics, Providence, R. I.,Amer. Math. Soc. (1952) 679–6858. Simpson, S.G.: Which set existence axioms are needed to prove the cauchy/peanotheorem for ordinary differential equations? The Journal of Symbolic Logic (1984) 783–8029. Simpson, S.G.: Subsystems of second order arithmetic. Perspectives in Mathemat-ical Logic. Springer-Verlag, Berlin (1999) Appendix A: details of the construction in Section 3 The constructions should guarantee that at each stage, the function f n movesevery point of D n = (cid:16) C (cid:4) − n n \ C (cid:4) − n · n (cid:17) ◦ in the direction of t n by an amount proportional to its distance to C (cid:3) − n n . Theconstruction of f with this property is trivial. We proceed to construct f n +1 from f n .(i) Extend and modify f n to C (cid:4) − n n +1 so that every point x of (cid:16) C (cid:4) − n n +1 \ C (cid:4) − n · n +1 (cid:17) ◦ is moved in the direction of t n by an amount proportional to d (cid:16) x, C (cid:3) − n n +1 (cid:17) .(ii) Modify the resulting function so that each point in C (cid:4) − n n +1 \ C (cid:4) − n · n +1 is mapped a non-negative amount proportional to its distance to C (cid:3) − ( n +1) n +1 in the direction of t n . n C (cid:3) − n n C (cid:4) − n · n D n − n Fig. 3. Sets used in the construction(iii) By rotation of the direction of the mapping, extend the function to C (cid:4) − ( n +1) n +1 such that every point x of D n +1 = (cid:18) C (cid:4) − ( n +1) n +1 \ C (cid:4) − ( n +1) · n +1 (cid:19) ◦ is mapped in the direction of t n +1 by an amount proportional to d (cid:16) x, C (cid:3) − ( n +1) n +1 (cid:17) .The final step is the only one in which we use the fact that we are working intwo dimensions as this step requires the continuous (computable) rotation of avector in the direction of t n to a vector in the direction of t n +1 .A construction is given explicitly in [5] but it should be clear from the pre-ceding that it can be done in many different ways. The important part of theproof is that the construction is, at each stage, extended at the boundary to“look right” from the outside. This ensures that, eventually every point is in factmoved towards one of a sequence of points that converge to the non-computablefixed point ( v, v ) on the diagonal. The Baigger construction is a somewhat deli-cate construction of a function that is in fact computable but that—somehow—mimics a simple mapping of every point in I in the direction of ( v, vv, v