Computable Operations on Compact Subsets of Metric Spaces with Applications to Fr\'echet Distance and Shape Optimization

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by  Chansu Park

Computable Operations on Compact Subsets
of Metric Spaces with Applications to
Fre´chet Distance and Shape Optimization
Chansu Park∗, Ji-Won Park†, Sewon Park†, Dongseong Seon∗ and Martin Ziegler†
∗Seoul National University, †KAIST School of Computing (Republic of Korea)
7
Abstract—WeextendtheTheoryofComputationonrealnum- while the behaviour on other sequences (~a ) is arbitrary
1 m
0 bers, continuous real functions, and bounded closed Euclidean [Weih00, 4.3]. For example addition R2 R is computed
2 subsets, to compact metric spaces (X,d): thereby generically by convert§ing (~a ) Z2 to b := a /→4+a /4 . A
includingcomputational and optimization problems over higher m ∈ n ⌊ n+2,x n+2,y ⌉
computationaccordingtoEquation(1)runsintimeT(n)ifb
r types,suchasthecompact ‘hyper’spacesof(i)nonemptyclosed n
a subsets of X w.r.t. Hausdorff metric, and of (ii) equicontinuous appears after at most T(n) steps, regardless of ~x dom(f)
M ∈
functions on X. The thus obtained Cartesian closure is shown or (~am). Following[BrWe99, DEFINITION4.8] and [Weih00,
to exhibit the same structural properties as in the Euclidean EXERCISE 5.2.1], call a non-empty compact W Rd
6 case, particularly regarding function pre/image. This allows us computableiffthereexistsafamilyA Zd suchthat⊆W has
2 to assert the computability of (iii) Fre´chet Distances between m ⊆
curves and between loops, as well as of (iv) constrained/Shape Hausdorff distance ≤ 2−m to {~am/2m : ~am ∈ Am , where
O] Optimization. (Am) is required to be uniformly recursive in the se(cid:9)nse that
m A¯ N Zd is decidable. W is co-computable
L I. INTRODUCTION, MOTIVATION, BACKGROUND iQf mth{ere}ex×istsma ⊆unifo×rmly co-r.e. such family. We report:
s. Identifying(andjustifying)the ‘right’conceptsand notions Fact 1: Fix non-empty compact W [0;1]d and com-
[c is crucial for the foundationof a theory.The classical Theory putable total Λ:W →Re. ⊆
ofComputingisbasedonTuringmachineswithdata encoded a) [0;1]d itself is computable. The union of two co-/ com-
2
v inbinaryandruntimetakenintheworst-caseoverallinputsof putable subsets is again co-/computable,and the intersec-
2 lengthn asymptotically—fordiscretedata.TheTheory tion of two co-computable sets is co-computable; cmp.
→∞
0 of Computing with continuousdata also dates back to Turing [Weih00, THEOREM5.1.13].
4 (1937) for single reals and to Grzegorczyk (1957) for real b) A point ~x [0;1]d is computable iff the compact
8 functions;yetthequestfortherightnotionsoverhighertypes singleton ~x∈ [0;1]d is co-computable iff ~x is
.0 is still in progress [Roye97], [Schr09], [KaCo10], [LoNo15] computable{; c}mp⊆. [Weih00, EXAMPLE5.1.12.1]. { }
1 since an input here contains infinite information and cannot c) If W is computable, then it contains some computable
0
7 even be read in full before having to start producing output. point; cmp. [Weih00, EXERCISE 5.1.13].
1 Thepresentworkcontinuesthepursuit[KSZ16]forauniform d) ΛadmitsaruntimeboundT =T(n),i.e.,dependingonly
: treatmentofcomputabilityandcomplexityongeneralcompact ontheoutputprecisionn;andforanysuchboundT,n
v metricspaces(X,d):thusgenericallyincludingoperationson T(n+1)+1isabinarymodulusofcontinuityinthat ~x7→
i
X the ‘higher type’ spaces of (i) nonempty closed subsets of ~x′ 2−T(n+1) implies Λ(~x) Λ(~x′) 2−n+1; c|m−p.
r X w.r.t. the Hausdorff distance, and of (ii) equicontinuous [K|o≤91, THEOREM2.19] a(cid:12)nd [W−eih00, T(cid:12)H≤EOREM7.2.7].
a functionsfromX toanothercompactmetricspaceY w.r.t.the e) If W is co-computable, th(cid:12)en the set (cid:12) ~a ,...,~a :
0 n
supremum norm. We are guided by the structural properties
(cid:8)(cid:0) (cid:1)
exhibited in Computational Logic [KrWe87], [Esca13] and
n N, w~ W j n: ~a Zd ~a /2j w~ 2−j
by the well-established Euclidean case [Ko91], [BrWe99], ∈ ∃ ∈ ∀ ≤ j ∈ ∧| j − |≤
(cid:9)
[Weih00]:
(offiniteinitialsequencesofdyadicsequencesconverging
A. Computing Real Numbers, Functions, Closed Subsets to some w~ W) is co-r.e. and Λ has a recursive runtime
∈
bound; cmp. [Weih00, THEOREMS2.4.7+7.2.5+7.2.7].
Computing a real number r means to produce an integer
f) If W and non-empty compact V [0;1]e are co-
sequence am of numerators of dyadic rationals am/2m ap- computable, then so is Λ−1[V] W⊆; cmp. [Weih00,
proximating r up to absolute error 2−m. And computing a ⊆
≤ EXAMPLE5.1.19.2].
(possibly partial) function f : Rd R means:
⊆ → g) If W is computable, then the image Λ[W] [0;1]e is
⊆
Convert any sequence (~a ) Zd satisfying (1) again computable compact [Weih00, EXAMPLE5.2.11].
m
⊆ h) IfcompactW [0;1]dcoincideswiththeclosureofitsin-
|~x−~am/2m|≤2−m, ~x:=limm~am/2m ∈dom(f), terior,W◦,and⊆bothW and[0;1]d R◦ areco-computable,
to some (b ) Z s.t. y b /2n 2−n for y =f(~x) \
n ⊆ | − n |≤ then they are computable [Zieg02, THEOREM3.1].
j) If non-empty compact W [0;1] is computable, then so The so-called Type-2 Theory of Effectivity considers com-
⊆
are maxW [0;1] and minW [0;1]; cmp. [Weih00, putabilityonsecondcountabletopologicalT spaces[Weih00,
0
∈ ∈
LEMMA 5.2.6]. LEMMA3.2.6]bymeansofpartialencodingsasinfinitebinary
Item d) follows from careful continuity and compactness sequences, that is, over Cantor space. It establishes Cartesian
considerations. It corresponds to, and generalizes the (triv- closure by constructing generic encodings of countable prod-
ial) observation in discrete complexity theory that any total ucts [Weih00, DEFINITION 3.3.3] and of the space of (rela-
computation on 0,1 ∗ admits a worst-case runtime bound tively)continuousfunctions[Weih00, DEFINITION3.3.13]as
depending only {on th}e length n of, but not on the input wellas,forthecaseofacompletemetricspace,ofitsinduced
~x 0,1 n itself. This complexity-theoretic property in turn Hausdorffhyperspace[Weih00, EXERCISE8.1.10].However,
is∈th{e key}to prove Items e) to h) although the latter are only lacking(local/sigma)compactness,properties(d)+(f)+(h)from
concerned with computability. Fact 1 do not hold in general. Indeed several notions of
Item a) is optimal in that there exist computable compact computability, equivalent in the Euclidean case [BrWe99],
V,W [0;1] such that V W is not computable [Weih00, havebeenshowndistinctforseparablemetricspaces[BrPr03].
EXERC⊆ISE 5.2.11]; and, r∩egarding Item f), there exists a In fact, compactness is well-known crucial in Pure as well as
computable Λ : [0;1] [0;1] such that Λ−1[0] = contains in Computable Analysis [Schr04], [Esca13], [Stei16].
→ 6 ∅
no computablepoint[Weih00, EXERCISE6.3.12].Also Ernst
C. Recap:ContinuousFunctionsandCompactMetric Spaces
Specker constructed a recursive and increasing sequence of
integerfractionswhosesupremumisnotcomputable[Weih00, We presume a basic comprehension of mathematical cal-
EXAMPLE1.3.2]. culus and properties of compact metric spaces (X,d). Write
Theformerconditionisknownaslowersemi-computability B(x,r) = x′ X : d(x,x′) < r for the open ball
{ ∈ }
[AlBu10, DEFINITION 2] or left-computability; in fact a real in X with center x X and radius r 0, B(x,r)
∈ ≥
number x is computable iff it is both left and right com- for its closure (unless r = 0). More generally abbreviate
putable [Weih00, LEMMA 4.2.5]. We also record that, with B(S,r) := B(x,r) and similarly for B(S,r), S X.
the notation from Equation (1), strict inequality “f(x) > 0” S◦ denotesSthxe∈iSnterior (=largest open subset), S¯ the cl⊆osure
is equivalent to “ n : b > 1” and thus r.e. (recursively (=leastclosedsuperset)ofS;∂S :=S S◦ istheboundaryof
n
enumerable, aka ∃semi-decidable), but in general undecid- S. Write Bd := ~x Rd :x2+ +x\2 1 for the closed
able [Weih00, EXERCISE 4.2.9]. We use computable for the Euclidean d-dim(cid:8)ensi∈onal uni1t bal·l·,· d−1d:≤=∂(cid:9)Bd for the unit
continuous realm, decidable/co-/recursive/enumerable for the S
sphere.Let (X,Y) denotethe space of continuousfunctions
discrete one. C
f :X Y; (X) in case Y =R.
→ C
B. Overview, Previous and Related Work Definition 2:
The present work generalizes the Theory of Computation a) Amodulusofcontinuityoff :X Y isanon-decreasing
→
from Euclidean unit cubes to compact metric spaces (X,d). right-continuous mapping ω : [0; ) [0; ) such that
∞ → ∞
Being separable, computing here naturally means approxima- ω(t) 0 as t 0 and e f(x),f(x′) ω d(x,x′)
→ → ≤
tion up to error 2−n by a sequence (of indices w.r.t. a fixed holds for all x,x′ X. (cid:0) (cid:1) (cid:0) (cid:1)
∈
partial enumeration ξ : N X) of some countable dense b) A binary modulus of continuity of f : X Y is a non-
subset, thus generalizin⊆g the→dyadic rationals D = a/2n : decreasing mapping µ : N N = 0,1,→... such that
a,n Z canonically employed the real case; cmp. [P{ERi89, e f(x),f(x′) 2−n holds→wheneve{r d(x,x′)} 2−µ(n).
2] o∈r [W}eih00, DEFINITION 8.1.2]. Of course the particular c) A(cid:0)bbreviate (cid:1)≤µ(X,Y) = f : X ≤ Y :
c§hoiceofsaidenumerationξ heavilyaffectsthecomputational f has binary mCodulus of continuity(cid:8)µ ; Lip1 →:= Cid is
proWpeertpiersopitosineduincesD[eBfirnPitri0o3n],4[Swchera0k4].conditions on ξ that d) LtheetsNpaNce=of(nvo0n,-ve1x,p.a.n.)siv:evfnunctiNons,d(cid:9)Lenipo2te:=BaCinre7→nsp+a1c.e,
{ ∈ }
assert the entire Fact 1 to carry over; see Theorem 7. They equipped with the metric β(v¯,w¯)=2−min{n:vn6=wn}.
permit the categorical construction of dense enumerations, e) For ( V,W X consider the distance function
∅ ⊆
again satisfying said conditions, for (i) the compact space of dV : X x inf d(x,v) : v V 0 and
∋ 7→ { ∈ } ≥
non-empty closed subsets of X equipped with the Hausdorff Hausdorff distance dH(V,W) = max sup dV(w) : w
{ ∈
distance, and for (ii) the compact space of equicontinuous W ,sup dW(v) : v V . The sup(cid:8).metric on (X,Y)
} { ∈ } C
functions from X to another compact metric space, identified is denoted e∞(f,g)=max(cid:9) e f(x),g(x) :x∈X .
with their graph: Theorem 10. We demonstrate the relevance f) X : N N denotes K(cid:8)olm(cid:0)ogorov’s (cid:1)metric en(cid:9)tropy,
E →
andapplicabilityoftheseconditionsbyasserting(Theorem14) also known as modulus of uniform boundedness[Kohl08,
the computabilityof the Fre´chetDistance between curvesand DEF18.52]: It is defined such that X can be covered by
between loops; and by asserting computability of the generic 2EX(n) open balls of radius 2−n, but not by 2EX(n)−1.
nonlinear optimization problem max Λ(x) : Φ(x) 0 for If the entropy grows linearly, its asymptotic slope coincides
{ ≤ }
every computable cost function Λ : X R and every com- with the Minkowski-Bouligand or box-counting dimension of
→
putable, feasible, and open constraint Φ : X R including X; otherwise the latter is infinite. Compare also [KSZ16],
→
the contemporary case of Shape Optimization: Theorem 17. [Mayo16]...Thecentralmathematicaltoolofthepresentwork
is compactness, so let us recall some aspects of this concept that formalize and generalize numerical (i.e. Euclidean) grids
relevant in the sequel: to (i) more general compact metric spaces by considering
Fact3:Suppose(X,d)isacompactmetricspace;i.e.,every a relaxation of Hierarchical Space Partitioning whose (ii)
sequenceinX admitsaconvergentsubsequence;equivalently: subsetsareclosedballsthathowever(iii)mayoverlapaslong
everycover B(x ,r ) X byopenballscontainsafinite as (iv) their centers keep distance η from each other:
n n n ⊇
subcover nS≤NB(xn,rn)⊇ X. Fix another compact metric Definition4:Fixacompactmetricspace(X,d)ofdiameter
space (Y,Se). diam(X):=max d(x,x′):x,x′ X 1.
{ ∈ }≤
a) X is complete, that is, every Cauchy sequence converges. a) Form Z,anm-coveringofX isasubsetX X such
m
A non-empty subset of X is closed iff the restriction that X∈ B(x,2−m−1). For η N, X⊆ X
of d turns it into a compact space. Every continuous f : is η-sepa⊇raStexd∈iXfmit holds d(x,x′) 2−η∈for allmdis⊆tinct
X R attains its infimum and supremum. x,x′ X ;η-rectangularif x,x′ ≥X :2η d(x,x′) N.
b) Eve→ry f (X,Y) has compact image := f[X] Y, is b) (X,d∈,ξ,Dm) is a presented (c∀ompac∈t)mmetric s·pace if ξ∈:
uniformly∈cContinuousand thus admits a (binary) m⊆odulus N X is a partial dense enumeration and D : N ⊆N
of continuity. stri→ctly increasing such that, for every m N, the im→age
c) By the Arzela` Ascoli Theorem, a set (X,Y) has ξ [2D(m)] of[2D(m)] dom(ξ)constitutes∈anm-covering
compact closure iff it is a subset of Fµ(X⊆,CY) for some o(cid:2)f X. For(cid:3) strictly inc∩reasing η : N N and injective
binary modulus of continuity µ. C ξ, (X,d,ξ,D) is η-rectangular/η-sepa→rated if, for every
d) By Ko¨nig’sLemma,a non-emptysubsetC NN ofBaire m N,ξ [2D(m)] constitutesanη(m)-rectangular/η(m)-
spacehascompactclosure iff eachnodeha⊆sfinitedegree sep∈arated(cid:2)m-cover(cid:3)ing of X, respectively.
in the tree of finite initial segments c) Presentedmetricspace(X,d,ξ,D)iscomputablycompact
ifdom(ξ)andD :N Narerecursiveandthefollowing
C∗ := ~u=(u0,...un) n∈N, ∃v¯: ~u◦v¯∈C ⊆ N∗ is semi-decidable: → (a,b,n,u,v) u,v dom(ξ),
(cid:8) (cid:12) (cid:9) ∈
e) Theset (X)ofnon-emp(cid:12)tyclosedsubsetsofX equipped (cid:8) (cid:12)
with theKHausdorff distance d from Definition 2f) con- a/2n <d ξ(u),ξ(v) <b/2(cid:12)n N5
H ⊆
stitutes again a compact metric space. We shall call (cid:0) (cid:1) (cid:9)
d) For presented (X,d,ξ,D), a name of a point x X is
K(X),dH the Hausdorff hyper-space over X. a sequence u¯ = (u ), u dom(ξ) [2D(m)∈], with
f) F(cid:0)ix W (cid:1)(X), D : N N a sequence, and ξ : N m m ∈ ∩
∈ K → ⊆ → d ξ(u ),x 2−m. The point x is computable if it
X some (possibly partial) enumeration. Then the subsets m ≤
ad(cid:0)mitsarec(cid:1)ursivename.Itispolynomial-timecomputable
x = x and W := x of Baire
spξa,Dce NNQamre ξc,oDm,mpact, wheξr,De we uSsex∈tWhe aξb,Dbreviations if there exists a Turing machine which prints a name u¯
such that u appears after a number of steps boundedby
x := u [2D(m)] dom(ξ) : d x,ξ(u) 2−m m
ξ,D,m
∈ ∩ ≤ some polynomial in m.
and [M]:=(cid:8) 0,1,...,M 1 . (cid:0) (cid:1) (cid:9)
{ − } e) Fix presented (X,d,ξ,D) and (Y,e,υ,E) and recall that
g) Consider the compactmetric space (X Y,d e), where
× × W∗ dom(ξ)∗ denotes the set of finite initial se-
(d e) (x,y),(x′,y′) := max d(x,x′),e(y,y′) . A ξ,D ⊆
× quences of elements u¯ W . A name of a partial
total fun(cid:0)ction f : X (cid:1)Y is contin(cid:8)uous iff graph(f(cid:9)) := ∈ ξ,D
→ mapping Λ : X Y is a (w.r.t. initial substrings)
x,f(x) : x X is compact, i.e. an element of ⊆ →
∈ monotonic, total mapping Λ∗ : X∗ Y∗ such that,
(cid:8)(cid:0)(X Y(cid:1)). (cid:9) ξ,D → υ,E
K × for every u¯ x with x dom(Λ), the (w.r.t. initial
h) Moreover it holds (d×e)H graph(f),graph(g) ≤ substrings) n∈on-dξe,Dcreasing se∈quence Λ∗(u ,...,u )
(cid:0) (cid:1) 0 m m
e (f,g) (ω+id) (d e) graph(f),graph(g) has unbounded length and supremum(cid:0)v¯ Λ(x)υ,E. W(cid:1)e
≤ ∞ ≤ (cid:16) × H(cid:0) (cid:1)(cid:17) write Λ∗n(u¯) for the n-th element vn of ∈said supremum.
for ω any modulus of continuity of f or g; and Computing Λ means (for a Turing machine with write-
F ⊆
(X,Y) is compact iff graph( ) := graph(f) : f onlyright-movingtape)tocomputesomenameΛ∗.Sucha
C F { ∈
(X Y) is. computationrunsintimet(n)ifittakesatmostthatmany
F}⊆K ×
Itema)justifiestheminimumandmaximuminDefinition2f). steps to output Λ∗n(u¯), independently of u¯∈dom(Λ)ξ,D.
f) Forpresented(X,d,ξ,D), a name ofcompactnon-empty
II. COMPUTING ON A COMPACT METRICSPACE W X is a sequence A¯ = (A ) of finite sets A
m m
⊆ ⊆
Here we extend Fact 1 from Euclidean [0;1]d to arbitrary [2D(m)] dom(ξ) such that, for every m N, the set
∩ ∈
compactmetricspaces.Generalizingtherealcasewithdyadic ξ[A ] X hasHausdorffdistance(Definition2f)atmost
m
⊆
rationals D as canonical countable dense subset, computa- 2−m to W. The empty sequence is a name of . W is
tion on a metric space is commonly defined by operating computableifithasanameA¯=(A )whichisun∅iformly
m
on (sequences of indices wrt.) some fixed countable partial recursiveinthesensethattheset m A N Nis
m{ }× m ⊆ ×
dense enumeration; cmp. [PERi89, 2] or [Weih00, DEFINI- decidable.Ifsaidsetisco-r.e.,WQiscalledco-computable.
TION8.1.2]. Of course, the particula§r choice of said enumer- A standard name A¯ of W satisfies dW ξ(a) <2−m for
ationheavilyaffectswhether,andwhichitemsof,Fact1carry every a Am [2D(m)] dom(ξ), (cid:0)and d(cid:1)W ξ(a) >
∈ ⊆ ∩
over [Schr04]. In Definition 4 below we propose conditions 2−m−1 for every a [2D(m)] dom(ξ) Am. (cid:0) (cid:1)
∈ ∩ \
g) A rounding function for presented (X,d,ξ,D) is a map- N u<D(m+1) D(m), N v <E(m)
∋ − ∋
ping R : dom(ξ) N dom(ξ) such that it holds
× →
R(u,m) [2D(m)] and d ξ R(u,m) ,ξ(u) 2−m−1. D(m+1) E(m)+D(m+1) v+u
[Weih03, DE∈FINITION 6.2.2](cid:16)ca(cid:0)lls an m(cid:1)-coveri(cid:17)ng≤(m+1)- · ξ u ·,υ E(m7→)+v ,
spanning.Variouscommonnotionsofcomputabilityforclosed 7→ (cid:16) (cid:0) (cid:1) (cid:0) (cid:1)(cid:17)
subsets, equivalent over Rd [BrPr03, THEOREM 3.6], are N∋u<D(m+1), N∋v <E(m+1)−E(m)
known to become distinct over more general spaces [BrPr03,
If X and Y are η(m)-separated/rectangular, then so is
THEOREMS 3.9(3)+3.11(4)+3.15(2)]. Definition 4 thus has X Y. Recursive rounding functions for X and Y give
been crafted with great care. For instance, although computa- ×
rise to one for X Y.
tionsaccordingtoe)operateonapproximationsuptoabsolute ×
Items a+e) generalize the case of real vectors. A computable
error 2−m, the requirement in a) of an m-covering to pro-
≤ compact subset need not in turn constitute a computably
videstrictlybetterapproximationsiscrucial.Itemc)strength-
compact metric space: consider for example 1/π [0;1].
ens[Weih00, DEFINITION8.1.2.3]in requiringdom(ξ)to be { }⊆
Remark 6:
recursive, thus asserting a computably compact space X to
a) To every compact (X,d) with partial dense enumeration
be a computable subset of itself in the sense of Item f); see
Theorem 7a) below. It also guarantees the set Xξ∗,D ⊆ N∗ to (ξX:⊆,d,Nξ,→D)Xa, pthreesreenetexdistms esotrmicesDpac:eN. I→f thNe rreenstdreicritnedg
be co-r.e.; see Theorem 7e). The rest of this subsection will
distance
provide further justification by comparison to the Euclidean
Fact 1. Indeed, Definition 4 generalizes the real case: N N dom(ξ) dom(ξ) (u,v) d ξ(u),ξ(v) R
Example 5: × ⊇ × ∋ 7→ ∈
(cid:0) (cid:1)
a) Let Dm := a/2m : a N,0 a < 2m and D := is computable and the natively Π2 set
{ ∈ ≤ }
D denotethesetofdyadicrationalsin[0;1).Define
m m u,n,v ,n ,...v ,n j N, u,v ,...v dom(ξ),
SD(m):=m+1 as well as ̺(0):=0 and inductively ̺: n(cid:0) 1 1 j j(cid:1)(cid:12) ∈ 1 j ∈
[T2hme+n1][\0[;21m],]∋a+,2m̺,7→id+(21a+co1n)/st2imtu+te1s∈aDcmom+1pu\tDabmly. B(cid:0)ξ(u),2−n(cid:1)⊆B(cid:0)ξ(v1(cid:12)),2−n1(cid:1)∪···B(cid:0)ξ(vj),2−nj(cid:1)o
m-rect(cid:0)angular|(f·orm| ally: id-re(cid:1)ctangular) compact space. = u,n,v1,...nj w dom(ξ):d ξ(w),ξ(u) >2−n
It admits a computable rounding function, namely n(cid:0) (cid:1)(cid:12)∀ ∈ (cid:0) (cid:1)
i (cid:12)j : d ξ(w),ξ(v ) <2−ni (2)
i
R(,m) : [2m+n+1] [2m+n] a+2m+n ∨ ∃ ≤ (cid:0) (cid:1) o
· \ ∋ 7→ is actually semi-decidable, then there exists a recursive
2m+ (2a+1)/(2n+1) 1 [2m+1] .
7→ ⌊ − 2⌉ ∈ such D. [BrPr03, DEFINITION 2.6] calls Equation (2),
b) The ‘circle’ [0;1) mod 1 with the metric d(x,y) = with dyadic radii generalized to arbitrary rationals, the
min x y , x y 1 , equippedwith the enumeration effective covering property.
{| − | | − − |}
̺from(a)butnowtakingD(m)=m,isalsocomputably b) Everycompactmetric space (X,d) admits a partialdense
id-rectangular compact. enumeration ξ such that D(m) := X(m + 1) turns
c) Consider Cantor space 0,1 N equipped with the metric (X,d,ξ,D)intoan(m+1)-separatedspEace:Letξ[2D(m)]
β inherited from Baire{spac}e; recall Definition 4d). We enumerate the centers of open balls of radius 2−m−1
turn this into a computably m-rectangular compact space covering X.
0,1 N,β,γ,id+1 as follows: Let γ˜ : N 0,1 ∗ Conversely whenever (X,d,ξ,D) is η-separated, it holds
{ } → { }
(cid:0)enumerateallfiniteb(cid:1)inarystringsinorderoflength,define D(m) X η(m) + 1 : Consider some choice of
≤ E
γ(0) := 0ω, γ(u+1) := γ˜(u) 1 0ω; and truncation 2EX(η(m)+1) ce(cid:0)nters of op(cid:1)en balls of radius 2−η(m)−1
◦ ◦
constitutes a computable rounding function. coveringX;then each one can contain at mostone of the
d) Suppose ψ : X Y is a homeomorphism between points in ξ[2D(m)], since the latter have pairwise distance
computably compa→ct (X,d,ξ,D) and topological space 2−η(m); cmp. [KSZ16, 3].
≥ §
Y. Then Y,d ψ−1,ψ ξ,D constitutes a computable c) Not every convex compact metric space admits a
◦ ◦
compact m(cid:0)etric space (thus jus(cid:1)tifying the notation in b). rectangular enumeration, though: Consider the geodesic
If the former is η(m)-separated/rectangular/has rounding distance on a circle of irrational circumference. Even in
function R, then so does the latter. the Euclidean case, it has been conjectured since Erdo¨s
e) For computably compact metric spaces (X,d,ξ,D) and and Ulam (1946) that no open subset of R2 admits
(Y,e,υ,E), their Cartesian product (X Y,d e,ξ a dense sequence of pairwise rational distances; cmp.
× × ×
υ,D E) becomes again a computably compact metric http://terrytao.wordpress.com/2014/12/20
·
space by defining ξ υ : N X Y inductively on d) Suppose (X,d,ξ,D) is computably compact. Then com-
× ⊆ → ×
D(m) E(m);D(m+1) E(m+1) 1 N: putably compact X,d,ξ,m D(m + 1) admits a
· · − ∩ 7→
(cid:2) (cid:3) recursive rounding(cid:0) function: Given u d(cid:1)om(ξ) and
D(m) E(m)+ D(m+1) D(m) v+u ∈
· − · 7→ m Nthereexists,andDefinition4c)assertsenumeration
(cid:0) (cid:1) ∈
ξ D(m)+u ,υ E(m)+v , of [2D(m+1)] dom(ξ) to find, some v =: R(u,m) with
7→ (cid:16) (cid:0) (cid:1) (cid:0) (cid:1)(cid:17) ∩
d ξ(v),ξ(u) < 2−m−1. (In [2D(m)] dom(ξ), distance even though ξ(a) itself might not even lie in W, the
∩
2−(cid:0)m−1 is fe(cid:1)asible, but not necessarily computably so...) closed ball B ξ(a),2−t(n) does intersect W. Moreover,
e) (Am) is a name of non-empty compact W X iff (i) for every x (cid:0)W B ξ(a(cid:1)),2−t(n) , the finite sequence
⊆ ∈ ∩
every a¯=(am) with am Am satisfying ~a := R(a,0),R(a,1),(cid:0)...,R(a,t(n(cid:1)) 1),a extends to
∈ −
some(cid:0)name a¯ of x, i.e., belongs to W∗ . A(cid:1)bbreviating
n,m:d ξ(a ),ξ(a ) 2−m+2−n (3) ξ,D
∀ m n ≤ y := Λ(x), the machine computing Λ∗ will on input
(cid:0) (cid:1)
ecvoenrsytitxutesWantahmereeoefxsiostmseaxna∈mWea¯asnudchift(hiia)tcEoqnuvaetrisoenly(3to) ea¯ υp(rbond)u,cye Λ∗n2(−a¯n) fo=r:y :b=n Λ∈(x)[.dHomow(υev)e∩r,in[2tEim(ne)]bowuinthd
holds: If∈(A ) constitutes a name of W, then (i) every a¯ t((cid:0)n)itcan(cid:1)no≤tevenreadpast~a;henceΛ∗(a¯)dependsonly
m n
satisfying Equation (3) gives rise to x :lim ξ(a ) X on~a: Λ∗(a¯)=Λ∗(u¯) holds whenever β(a¯,u¯) 2−t(n).
by completeness, and d ξ(a ) 2−m smhowsmx ∈W; Let us dnenote bynΛt(a) the thus well-defined≤compos-
W m n
and (ii) to x ∈ W and e(cid:0)very m(cid:1)∈≤N there exists∈some ite mapping At(n) ∋ a 7→ ~a 7→ a¯ 7→ Λ∗n(a¯) ∈
am Am with d ξ(am),x 2−m, hence satisfying [dom(υ) [2E(n)] satisfying e υ Λt(a) ,y 2−n for
Equa∈tion (3). (cid:0) (cid:1) ≤ all y Λ∩W B ξ(a),2−t(n)(cid:16).(cid:0) n (cid:1) (cid:17) ≤
f) Everystandardnameofsomenon-emptycompactW X ∈ ∩
⊆ Item g) can b(cid:2)e regard(cid:0)ed as a discr(cid:1)e(cid:3)te counterpart to Fact 1j).
is also a name of W: By Definition 4b) there exists,
to every x W, some a dom(ξ) [2D(m)] with Item h) is based on the continuity/adversary argument under-
d ξ(a),x ∈2−m−1; and (A∈m) being a∩standard name lying the sometimes so-called Main Theorem [Weih00, §2.2],
of(cid:0)W req(cid:1)ui≤res a Am whenever dW ξ(a) 2−m−1: Fact 1d+e), and Theorem 7d+e) below.
∈ ≤
thus (Am) is a name of W. (cid:0) (cid:1) A. Computable Operations on a Compact Metric Space
Conversely, to every co-r.e/recursive name (A ) of non-
m For presented metric spaces, the composition of two com-
empty compact W X, there exists a co-r.e./recursive
⊆ putable functions is again computable; computable functions
standard name of W: Indeed every A′ with
m map computable points to computable points; a constant
function is computable iff its value is computable. Moreover,
a′ dom(ξ) [2D(m)] a dom(ξ) [2D(m+3)]:
n ∈ ∩ (cid:12)∃ ∈ ∩ similarly to Fact 1 and [Weih03, DEF4.1], we have:
a A d ξ(a)(cid:12),ξ(a′) 555 2−m−3 Theorem7:Let(X,d,ξ,D)and(Y,e,υ,E)becomputably
m+3
∈ ∧ ≤≤≤ · o ⊆ compact spaces with recursive rounding functions R and
(cid:0) (cid:1)
⊆ A′m ⊆ na′ ∈dom(ξ)∩[2D(m)](cid:12)∃a∈Am+3 ∧ SWaccXordainngd ttootaDleΛfin:iWtion 4gY).aSreupcpoomsepuctoabmlep.act non-empty
d ξ(a),ξ(a′)(cid:12)<<<777 2−m−3 (4) ⊆ →
∧ · o a) X is a computable subset of itself. The union of two co-
(cid:0) (cid:1)
constitutes such a standard name: To a A there /computablesets is again co-/computable;the intersection
m+3
exists some x W with d ξ(a),x ∈2−m−3, hence of two co-computable sets is co-computable.
a′ A′ impl∈ies d ξ(a),ξ((cid:0)a′) <(cid:1)7≤ 2−m−3 and in b) A point x X is computable iff the compact singleton
atu′rn∈ddWomm(cid:0)ξ((xai′))(cid:1) [<2D((cid:0)m(7)]+ A1)′ ·i2(cid:1)m−pmli−es3 d·=ξ(a2)−,mξ(;a′w)hi>le c) I{fxW}⊆isXcoi∈ms cpou-tcaobmle,puittacbolnetaififns{xso}m⊆eXcomispcuotambpleutpaobilnet..
5 2∈−m−3 for e∩very a A\m+m3, and in tur(cid:0)n dW ξ(a′)(cid:1) > d) Λ admits a computable time bound T =T(n) depending
(5· 1) 2−m−3 =2−m∈−1.Nowrecallthatstrict(cid:0)inequa(cid:1)lity only on the output precision n; for any such bound T,
of−dist·ances is r.e., and non-strict is co-r.e. Hence, if n T(n+1)+1 is a binary modulus of continuity of Λ.
7→
(Am) is uniformly co-r.e., then so is the left-hand side e) If W is co-computable,then Wξ∗,D is co-r.e.; and Λ has a
of Equation (4); and if (A ) is even uniformlyrecursive, recursivebinarymodulusofcontinuityandruntimebound.
m
then the right-hand side is uniformly r.e.: now apply the f) If W and non-emptycompact V Y are co-computable,
⊆
next item to its complement. then so is Λ−1[V] W.
⊆
g) Fix pairwise disjoint families X and Z of uniformly g) If non-empty compact W is computable, then the image
m m
co-r.e. subsets of integers. Then there exists a uniformly Λ[W] Y is again (compact and) computable.
⊆
recursive family Y disjoint to Z with X Y : h) If non-empty compact W X coincides with R◦ and
Forgivenx,msearmchinparallelfomrawitnesmsth⊆atxm X both W and X R◦ are c⊆o-computable, then they are
and for one that x Z and report the (negation o6∈f them) computable. \
m
first to succeed. 6∈ j) AtotalΛ:W Y iscomputable(inthesenseofDef.4e)
→
h) Suppose (X,d,ξ,D) and (Y,e,υ,E) are computably iff the compact set graph(Λ) X Y is computable.
⊆ ×
compact with rounding function R : dom(ξ) N Item g) asserts, together with Fact 1j), that maxΛ[W] and
× →
dom(ξ).Let(A ) denoteanameofnon-emptycompact minΛ[W] are computable reals for every computable non-
m m
W X, and Λ∗ a name of Λ : W X Y, empty compact W X and Λ : W [0;1]. And
⊆ ⊆ → ⊆ →
computable in time t(n). Recall that W∗ N∗ denotes Items b+f) imply that, under the same hypothesis and for
ξ,D ⊆
the set of all finite initial segments of sequences in W computable y Y, a unique solution x W to the equation
ξ,D
∈ ∈
and observe that, for every n N and a A , “Λ(x) = y” is computable. Item j) effectivizes Fact 3g), that
t(n)
∈ ∈
is the identification of a function as a transformation with its d ξ(u ),ξ(u ) < 2−m + 2−n and d ξ(u )
m n W m
≤
graphasa‘static’object,andjustifiesourencodingofcompact 3(cid:0) 2−m−2 < 2(cid:1)−m by triangle inequality; (cid:0)hence x(cid:1) :=
·
function spaces in Subsection II-C below; cmp [Bra05, 4]. lim ξ(u ) W.
m m
§ ∈
d) Fix n N and recall from Remark 6h) that a machine
B. Proof of Theorem 7 com∈puting a name Λ∗ of Λ, when presented with any
A
a) By Definition 4b), A :=dom(ξ) [2D(m)] is a name of sequence u¯=(u ) w for some w W, produces
m ∩ m m ∈ ξ,Ξ ∈
X; and uniformly recursive according to Definition 4c). according to Definition 4c) some v = Λ dom(υ)
n n
∈
For A ,B [2D(m)] dom(ξ) with with e Λ(x),υ(v ) 2−n: after a finite number T =
m m n
⊆ ∩ ≤
dH ξ[Am],W ,dH ξ[Bm],V 2−m, Am Bm TA(n,(cid:0)u¯)ofstepsan(cid:1)dinparticular‘knowing’nomorethan
≤ ∪ ⊆
[2D(cid:0)(m)] dom(cid:1)(ξ)(cid:0)has dH ξ[A(cid:1)m Bm],W V 2−m; thefirstT entriesofu¯.ThethusdefinedfunctionT(n, ):
∩ ∪ ∪ ≤ ·
and is uniformly recursiv(cid:0)e/co-r.e. whenever bo(cid:1)th (Am) Wξ,ℓ N(implicitlydependingon )isthereforelocally
→ A
and (B ) are. The cases W = or V = are easily constant,thatis,continuous:T(n,u¯)=T(n,u¯′)whenever
m
treated separately. Finally, for un∅iformly co-r∅.e. An,Bn′ β(u¯,u¯′) 2−T(n,u¯).
subsets of recursive dom(ξ) [2D(m)], Now Fac≤t 3f) asserts W NN to be compact; hence,
ξ,Ξ
∩ ⊆
by Fact 3a), T(n, ) is bounded by some least inte-
Cm := c Am n,n′ N a An b Bn′ : ger T(n) (again de·pending also on ). We show that
∈ ∀ ∈ ∃ ∈ ∃ ∈ A
d ξ(a),ξ(b)(cid:8) 2−n+(cid:12)(cid:12)2−n′ d ξ(c),ξ(a) 2−n+2−m n 7→ T(n + 1) + 1 constitutes a binary modulus of
≤ ∧ ≤ continuity of Λ : Fix x W and consider for each
(cid:0) (cid:1) (cid:0) (cid:1) (cid:9) W ∈
is co-r.e. (since dom(ξ) is) and a name of V W: To m N some u(cid:12) [Ξ(m)] with d x,ξ(u ) 2−m−−−111
twedhv(cid:0)ieeξtrhr(eyadnex(cid:0))xξ,i∈(ξsat(sbnVn)s′,o∩)x(cid:1)m(cid:1)We≤≤cth22e−−rennAe+xamnisd2t−wadnin(cid:0)t′hξ;∈(abdnnA′dξ)n,(,cxsa)i(cid:1),nnxdc≤ebn∩x2′−∈n∈2′,−BVmsno′,, waaevlchlcei∈orcmyrhd′xicn′≤ogi∈ntomciBd;De(eisxmn(cid:12)fi,dwn2e∈ii−tetihdmon−u¯x14′)oaa)nh.damFtshoietrds(cid:0)fithxaris′(cid:0)s,tsξepm(qauurmteeincn)muct(cid:1)relia(cid:1)e≤rus¯≤.u¯′2A∈∈−nmdxxξ′ξff,ooΞ,Ξrr,
∈ ≤
so d ξ(c),ξ(a) 2−n + 2−m: resu(cid:0)lting in(cid:1) c Cm. m T(n) T(n,u¯) bydefinition, ’s outputv¯oninput
≤ ∈ ≥ ≥ A
Conv(cid:0)ersely,toe(cid:1)veryc Cm,therearesequencesan An u¯ coincides up to position n with its output v¯′ on input
∈ ∈
and bn′ Bn′, and thus vn V and wn′ W with u¯′. Triangle inequality thus yields
d ξ(an),∈vn ,d ξ(bn′),wn′ ≤∈ 2−n′; by co∈mpactness d(x,x′) 2−T(n)−1 e Λ(x),Λ(x′) 2−n+1 .
ξ((cid:0)ank)→li(cid:1)mk(cid:0)vnk =:v ∈(cid:1)V and ξ(bn′k)→limkwn′k =: ≤ ⇒ (cid:0) (cid:1)≤
w2−∈nkW+f2o−rns′komimepsulibesseqvue=ncews;=no:wxd(cid:0)ξ(aVnk),Wξ(b;nfi′kn)(cid:1)al≤ly e) RLeetm(aArkm6)em),dWenξ∗o,ℓtecoainccoid-re.es.wniathmethefosretW. According to
∈ ∩
d ξ(c),x ← d ξ(c),ξ(ank) ≤ 2−nk +2−m → 2−m as
k(cid:0) . (cid:1) (cid:0) (cid:1) ~u=(u0,...,un) n N, i,j n:
b) Fo→r c∞omputable x with recursive name u¯ = (u ) n (cid:12) ∈ ∀ ≤
Axξm,D,:=the uumnifocrmonlsytitruetceusrsaivneamseequoefncexo.fCsoinnmgvleemrtsoenl∈ys ujm∈daomm(ξ)A∩m[2:ℓ(dj)](cid:12)ξ(∧amd)(cid:0),ξξ((uuij)),ξ(uj2)−(cid:1)m≤+2−2i−+j 2−j(∧5)
suppose ({Am)}is a co-r.e. name of {x}. {Th}en the sets w∀hich∃is c∈learly co-(cid:0)r.e. with “ m(cid:1)”≤as only unboounded
∀
A′ := R(a′,m) a′ dom(ξ) [2D(m+3)], quantifier and co-r.e. inequality “d ξ(ui),ξ(uj) 2−i+
∀a∈m[2D(m(cid:8)+3)]: a6∈(cid:12)(cid:12)Am+∈3 ∨d ξ(a∩),ξ(a′) <2−m−2 E2−quj”a.tioInnd(e5e)da,nfdofirxeevderjy ~un=,th(eure0(cid:0),e.x.is.t,suans)eqaucce(cid:1)on≤rcdeiangmto
are (i) uniformly semi-decidable,(cid:0)(ii) non-em(cid:1)pty, and (iii(cid:9)) Am suchthat,onetheon≤ehand,dW ξ(am) 2−m ∈0
≤ →
(aini)ytsoeqxuencXe ath′mere∈eAxi′mstscoan′stitduotems(aξ)nam[2eDo(fmx+.3)I]ndweiethd hwehniclee loinmnthξe(aomtnhe)r=h:anxd ∈2−Wj for(cid:0)2s−ommen+(cid:1)s1ub+seq2u−ejnce;
← ≥
d ξ(a′),x∈ 2−m−4, while∈every a ∩ Am+3 satisfies d ξ(amn),ξ(uj) →d x,ξ(uj) . Thisasserts~utoextend
d(cid:0)ξ(a),x(cid:1) ≤2−m−3. Conversely (iii) ∈every a Am+3 to(cid:0)some u¯∈xξ,ℓ(cid:1). (cid:0) (cid:1)
shaa(cid:0)tsisfidesξ(da(cid:1)(cid:0)′)ξ≤,(xa),x<(cid:1)≤3 22−−mm−−3;3haenndcedanξy(RR(a(a′,′,mm∈))),∈xA<′m TpaortfiianldaalgroercituhrmsivAe tcimomepbuotuinngdΛT∗′ ≥froTmtcoom(dp),acetxtWenξd,Ξthtoe
7−m−(cid:0)3. (cid:1) · (cid:0) (cid:1) A′nacceptinginputsu¯fromtheentireset m≥0[2ℓ(m)]by,
c) Let A [2D(m)] dom(ξ) be a uniformly recursive simultaneously to executing (u¯) until iQt prints the n-th
m A
⊆ ∩ output symbol, trying to refute u¯ W just established
name of compact non-empty W X. Then, starting ξ,Ξ
∈
⊆ asco-r.e.Notingthat ′ indeedterminatesonallpossible
wseiathrchanfyora,0an∈d Aac2c,orodniengcaton iRteermataivrkely6ec)oimspguutaatriaonntaeleldy inputsfromcompact Amn≥0[2ℓ(m)]⊆NNandtherefore(d)
to find, some a A with d ξ(a ),ξ(a ) < in some time boundQT′(n) T(n) depending only on n,
m m+2 m n ≥
2−m−1+2−n−1 fora∈lln<m:recallth(cid:0)atstrictinequ(cid:1)ality the following algorithm computes such T′(n):
ofdistancesissemi-decidableaccordingtoDefinition4b). Initialize T′ := 1. Simulate ′ on each ~u
Then u := R(a ,m) dom(ξ) [2D(m)] satisfies T′−1[2ℓ(m)] NT′ until eitherA(ni) it terminates o∈r
m m ∈ ∩ m=0 ⊆
Q
(ii) reads past the finite input. In case (ii), increase h) By Remark 6f) suppose w.l.o.g. that (A ) is a co-r.e.
m
T′ and restart; else output T′ and terminate. standard name of W and (B ) one of X R◦. We show
n
\
f) First observe that, since Λ is continuousaccording to (d), that every family (Cm) with
Λ−1 maps compact/closed V Y to a closed/compact
⊆ A C u dom(ξ) [2D(m)] n>m
snuabmseetooffWW, (aBndk/)okrsXim.iNlaorlwy loente(AofmV)m, adnednoµte:aNc→o-r.Ne. ∃v∈mdo⊆m(ξ)m∩⊆[2D(cid:8)(n)]∈\Bn :d ξ∩(u),ξ(v)(cid:12)(cid:12)<∃2−m (6)
simultaneously a recursive both binary modulus of con- (cid:0) (cid:1) (cid:9)
constitutes a name of W: Since the right-hand side of
tinuity of Λ and runtime bound according to (e). Now
Equation (6) is r.e., the claim then follows with Re-
consider the uniformly recursive mapping Λµ : A
n µ(n) → mark 6g). Indeed, every element u of the right-hand side
dom(υ)∩[2E(n)] according to Remark 5h), and let arises from some v Bn with d ξ(u),x < 2−m for
6∈
x:=ξ(v). (Bn)beingastandardna(cid:0)meofX(cid:1) X◦ implies
\
Cm := a∈dom(ξ)∩[2D(m)] ∀n,n′ ∃a′ ∈Aµ(n′) dX\W◦(x)>2−n and in particular x∈W◦ ⊆W.
∃b∈B(cid:8)n : e υ Λµn′(a′) ,υ(b) (cid:12)(cid:12)≤2−n+2−n′ ∧ nOanmteheimoptlhieers hdanξ(duf)o,rxev<er2y−um∈foAr smo,mbeexing aWst=anWdar◦d.
(cid:0) (cid:0)d ξ(a),ξ(cid:1)(a′) (cid:1) 2−m+2−µ(n′) . Hence there ex(cid:0)ists som(cid:1)e y W◦ with d(x∈,y) < ε :=
∧ ≤ ∈
(cid:0) (cid:1) (cid:9) 2−m d x,ξ(u) ; and in turn some integer n >m such
−
Itisclearlyuniformlyco-r.e.,since(Aµ(n′))and(Bn)are, that 2−n−(cid:0)1 < ε(cid:1) d(x,y) and B(y,3 2−n−1) W◦
− · ⊆
and non-strict inequality is as well, and both dom(ξ) holds; and in turn some v dom(ξ) [2D(n)] with
[2D(µ(n′))] and dom(υ) [2E(n)] are recursive by hy∩- d y,ξ(v) 2−n−1. Then d ξ∈(v),ξ(u) ∩
∩ ≤ ≤
pothesis. We show that (Cm) constitutes a standard (cid:0) d ξ(cid:1)(v),y + d y,x (cid:0) + d x,ξ(cid:1)(u) < 2−m
name of Λ−1[V]: Every a dom(ξ) [2D(m)] with ≤
dΛ−1[V] ξ(a) 2−m−1 belo∈ngs to Cm:∩To x Λ−1[V] (cid:0)≤2−n−1(cid:1) <ε−(cid:0)2−n(cid:1)−1 (cid:0)=2−m−ε(cid:1)
≤ ∈
ewwΛx(eiitxlshlt)daas(cid:0)′(cid:0)xVb,;ξ∈h(∈ae(cid:1)n)A(cid:1)cB]eµ(n≤dn′)wξ2(i−awth)mi,t−ξhe(1(cid:0)ayd′a,)(cid:0)nυxd(,bξe)((cid:1)v2ae−′r)m≤y(cid:1)−n12≤,+−nn′22−∈f−µo(µrNn(n′)y′,t)hae:nar=des tldahanXtadtt\eWrBevi◦|(cid:0)es(cid:0)ξrξy(a(v{vus)zt),a(cid:1)2n·d≥A}2ar−2dni·−ns2a1|a−m(cid:1)nne⊆{−ezl1oeBfm}=(Xeyn2,t3\−on·W|f2ht−◦he.nen{−Tzcr1eihg)ihvs⊆}t-d6∈hWeamBn◦donnismsaitdsprealtithoeeessf
e2(cid:0)−υn(cid:0)′Λ+∈µn′2(−an′).(cid:1),y(cid:1) ≤(cid:0)2−n′ impl(cid:1)ie≤s e(cid:0)υ(cid:0)Λµn′(a′)(cid:1),υ(b)(cid:1) ≤ j) FEiqxuaatiroencu(r6s)iv.∈enamme (A ) of W, a joint recursivetime
m m
CaenoυdnvbΛenrµs∈e(layB′tno)wa,iυ∈th(bCdm)(cid:0)ξ(thae),r2eξ−(eanx′ni+s′)t(cid:1)s2e−≤qnu2′e.−nBmceys+cao2′nm−′µp∈(ancA′t)nµe(ansn′sd), bAacocuonrddianngddtoobminT(ahυrey)omre[2omEdu(7nldu))]s,aoacfncodcrodunintniignfoutroimtyRlyµemr:eaNcruk→r6sihvN)e.ToΛhfµneΛn:
n′ n′ n ≤ µ(n) → ∩
li(cid:0)mk(cid:0)ξ(a′n′k) =(cid:1) x ∈(cid:1) W and limkυ(bnk) = y ∈ V the sets graph(Λ)m :=
for some subsequences. It follows d ξ(a),x 2−m
a2s−nwke+ll2a−sn′ke(cid:0)Λ(0x)a,syk(cid:1) ← e.(cid:0)Tυh(cid:0)eΛrµnef′ko((cid:0)rae′n′kx)(cid:1),υΛ(cid:1)(−bn1≤[ky)](cid:1)an≤d (cid:26)(cid:18)ξ(cid:16)R(cid:0)u,m(cid:1)(cid:17),υ(cid:16)S(cid:0)Λµm+2(u),m(cid:1)(cid:17)(cid:19)(cid:12)(cid:12)(cid:12)
d ξ(a),x →d ξ(a),ξ(→a′ ∞) 2−m+2−∈µ(n′) 2−m u A (cid:12) X Y
as(cid:0)k (cid:1).← (cid:0) n′k (cid:1) ≤ → (cid:12)(cid:12) ∈ µ(m+2)(cid:27) ⊆ ×
g) The →ima∞ge Λ[W] Y is compact by Fact 3b). To see are of the form (ξ (cid:12)(cid:12) υ)[Cm] for uniformly
iratecsccuocrorsdmiivnpegubttaionbai(lreiyt)y,,mafi⊆noxdduaulunrsiefcoourfmrsclioyvnetrienncauumirtsyeivµ(eA:ΛmNµ)m:→AoNf Wof,Λa T(redhce×urcseoi)vnHev(cid:0)egrsrCeampchla(iΛm)⊆,fgorllaop[wDhs((Λ×mfr)o)mm(cid:1)·T≤hEe2(o−mrme)m.] 10adn)d. satisfy
n µ(n) →
dom(υ) [2E(n)] accordingto Remark6h).Nowconsider C. Exponential Objects and Higher-Type Computation
∩
the set
This subsection generalizes Theorem 7 uniformly, that is,
B := Λµ(a) a A with (W,Λ) not fixed but given as input: taken from the
m m ∈ µ(m)
(cid:8) (cid:12) (cid:9) Cartesian product(Example 5e) of the Hausdorff hyper-space
(cid:12)
which is clearly uniformly decidable since µ and A (X) over X for W, and for Λ:X Y from some closed
µ(m) K →
is. We show that (B ) constitutes a name of Λ[W]: To hyper-space of equicontinuous functions to another compact
m
every y = Λ(x) with x W there exists by hypothesis metricspaceY:suchastorenderthisnewinputspaceinturn
some a A with∈d ξ(a),x 2−µ(m); hence compact (Fact 3c). The buzzword ‘hyper’ here stresses our
µ(m)
∈ ≤ climbing up the continuous type hierarchy:
e υ Λµ(a) ,y 2−m by(cid:0) choice(cid:1)of µ and Λµ. Con-
v(cid:16)ers(cid:0)elym, ever(cid:1)y Λ(cid:17)µ≤(a) B arises from some a mA Remark 8: For (X,d) a compact metric space, and bor-
m ∈ m ∈ µ(m) rowing notation to hint at the dual of a topological linear
andinturnsomex W withd ξ(a),x 2−µ(n);hence space, write (X′,d ) for the compact hyper-space X′ :=
∈ ≤ ∞
e υ Λµ(a) ,y 2−m for y :(cid:0)=Λ(x)(cid:1) Λ[W]. Lip (X,[0;1]) of non-expansivereal functions.
(cid:16) (cid:0) m (cid:1) (cid:17)≤ ∈ 1
a) If diam(X) = 1, then X embeds isometrically into X′ e) Partial function evaluation is computable, that is, the
via ı:x d(x, ). mapping
7→ ·
b) Inthissense,X isapropersubsetofX′ sincethereexists
no isometry from X′ to X for reasons of entropy: (X Y) X
K × × ⊇
Consider Z X (non-empty and finite but) of maxi- graph(Λ),x Λ (W,Y), W (X), x W
mz′umdc(azr,dzi′n)a⊆lity1s.ucThhetnhaetvietryhoFlds:∀Zz,z′ ∈0Z,1: izs 1=- (cid:8)(cid:0) (cid:1)(cid:12)(cid:12)∋ ∈gCraph(Λ),x ∈7→KΛ(x) ∈∈Y .(cid:9)
Lip∨schitz; and≥extends to some F˜ X→′ [Ju{ut02}], thus f) The evaluation algori(cid:0)thm from (e)(cid:1)admits a uniformly
∈
having mutual supremum distance 1. This gives rise computable multivalued runtime bound T(Λ,n), i.e., de-
to 2Card(Z) > Card(Z) distinct suc≥h F˜: Mapping them pending only on Λ and the output precision n, that is
isometrically to X would violate maximality of Z X. simultaneously a binary modulus of continuity:
⊆
c) On the other hand every compact space, and in particular
X′,iswell-knownhomeomorphictosomecompactsubset T : (X Y) N graph ( X,Y) N
K × × ⊇ C ⊆ × ∋
of the Hilbert Cube j∈N[0;2−j] =: X. So in this graph(Λ),n Z(cid:0) m N:(cid:1)
topological (rather thaQn metric) sense X′ may actually x,x′ dom(∋Λ)(cid:0): d(x,x′)>2(cid:1)−m⇒e Λ∈(x),Λ(x′) 2−n
admit an embedding into X. ∀ ∈ ∨ ≤
(cid:0) (cid:1)
d) For X = [0;1]d, however, X′ is not homeomorphic to (a g) Function restriction is computable, i.e. the mapping
subset of) X: Fix k N and for (y ,...,y ) [0;1]k let
1 k
∈ ∈
fy~ : [0;1]d [0;1] denote the piecewise linear function (X Y) (X)
with f (j/k→,x ,...,x ) y /k. Then Ψ : [0;1]k K × ×K ⊇
y~ 2 d ≡ j k ∋ graph(Λ),V Λ (W,Y), V,W (X), V W
~y f X′ is well-defined, injective, and continuous: ∈C ∈K ⊆
an7→emby~ed∈ding. An embedding Φ : X′ X would thus (cid:8)(cid:0) ∋ gra(cid:1)p(cid:12)(cid:12)h(Λ),V 7→ graph Λ V = (cid:9)
yield a continuous injective Φ Ψk : (→0;1)k (0;1)d; (cid:0)= graph(Λ)(cid:1) (V Y) (cid:0) (cid:12)(cid:12) ((cid:1)X Y) .
◦ → ∩ × ∈ K ×
contradicting Invariance of Domain for k >d.
h) Type conversion is also computable: partial evaluation
We now turn compact hyper-space (X) into a computably
K
compact metric space, such that any name of W (X) in ( X Y,Z) X (Λ,x) Λ(x, ) ( Y,Z)
∈ K C ⊆ × × ∋ 7→ · ∈ K ⊆
the sense of Definition 4d) is the binary encoding of a name
aswellastheconverse,un-‘Scho¨nfinkeling’[Stra00,p.21].
of W X in the sense of Definition 4f), and vice versa:
⊆
Definition 9: a) For computably compact metric space
j) And so is function image
(X,d,ξ,D), consider (X),d ,ξ ,2D with
H H
K
(cid:0) (cid:1) ( X,Y) (X) (Λ,W) Λ[W] (Y) .
ξ : N b 2j ξ(j):b =1 (X) C ⊆ ×K ∋ 7→ ∈ K
H j j
⊆ ∋ Xj≥0 · 7→ (cid:8) (cid:9) ∈ K k) Suppose Φ : X Y is computable and open in that
→
forb 0,1 incase = j :b =1 dom(ξ), b images Φ[U] Y of open U X are open again. Then
j ∈{ } ∅6 { j }⊆ j≥0 j· the restricted⊆pre-image mappi⊆ng
2j dom ξH otherwise. P
6∈
tbi)alLfeutnCc(t⊆io(cid:0)nXs,(cid:1)ΛY)::=XSW∈YK(Xw)itCh(Wco,mYp)adcetndootmetahien;sestimofilparalry- KR(Y) ∋ V 7→ Φ−1[V] ∈ KR(X)
⊆ →
for ( X,Y). c) Consider the continuous embedding is well-defined and computable.
µ
C ⊆
Here we denote by (X) = W X : W = W◦
( X,Y) Λ graph(Λ) (X Y) , (X) the family ofKsRo-called re(cid:8)gula⊆r subsets of X; re(cid:9)ca⊆ll
C ⊆ ∋ 7→ ∈ K × K
Fact 1h) and Theorem 7h).
justified by Fact 3g+h), by Theorem 7j), and particularly by
Proof of Theorem 10k): Preimage of a continuous open
Item e) of the following uniform result:
mapping commutes with topological closure and interior:
Theorem 10: Let (X,d,ξ,D), (Y,e,υ,E) be computably Φ−1 S◦ = Φ−1[S] ◦andΦ−1 S =Φ−1[S];cmp.[Zieg02,
compact metric spaces with recursive rounding functions. LEM(cid:2)MA(cid:3)4.4a(cid:0)b].Φ−1(cid:1)[V] isthusr(cid:2)eg(cid:3)ular.MoreoverbothW :=
a) Theunionmapping (X) (X) (V,W) V W Φ−1 V and Φ−1 Y V◦ = Y W◦ are co-computable
K ×K ∋ 7→ ∪ ∈ \ \
(X) is computable. acco(cid:2)rdin(cid:3)g to Theor(cid:2)em 7f); h(cid:3)ence W is computable by virtue
K
b) The mappings X x x (X) and (X) of Theorem 7h). This argument is non-uniform, but closer
∋ 7→ { } ∈ K K ⊇
x :x X x x X are computable. inspection shows it to hold uniformly.
{ } ∈ }∋{ }7→ ∈
c) T(cid:8)hereisacomputablemappingconvertinganygivenname
III. APPLICATIONS
ofsomeW (X)intoastandardnameofthesameW.
∈K
d) For computable W (X) and recursive strictly in- We apply the above considerations to two computational
∈ K
creasing µ : N N, graph (W,Y) is a computable problemsovercompactmetricspacesbeyondtheclassicalEu-
µ
→ C
compact subset of (X Y(cid:0)), i.e. a co(cid:1)mputable point in clidean case: a space of homeomorphisms(Subsection III-A),
K ×
(X Y) . and the space of compact subsets (Subsection III-B).
K K ×
(cid:0) (cid:1)
A. Fre´chet Distance
In1906MauriceFre´chetintroducedapseudo-metricforpa-
rameterizedcontinuouscurvesand,in1924,forparameterized
surfaces that in various ways improves over both supremum
and Hausdorff Norm:
Fig.1. a)TwosmoothsimplecurvesA,B:[0;1]→[0;1]2 whoseFre´chet
Definition 11: Let (X,d), (Y,e) be compactmetric spaces.
Distanceisnotattainedbyanyinjectivereparameterizationϕ. b)Twosmooth
a) The Fre´chet Distance of two continuousmappingsA,B : simplecurves A,B:[0;1]→[0;1]2 whoseFre´chet Distance is attained by
acontinuum ofreparameterizations ϕ.
X Y is given by F(A,B)=inf F (A,B), where
ϕ id,ϕ
→
F (A,B):=sup e A α(x) ,B β(x) (7)
α,β x∈X
(cid:16) (cid:0) (cid:1) (cid:0) (cid:1)(cid:17) Remark 13:
with infimum ranging over the set Aut(X) of all homeo- a) The pseudo-metric in Equation 7 is symmetric:
morphisms (i.e. continuous bijections) ϕ:X X. F (A,B) = F (A,B) holds for all bijections
→ id,ϕ α◦β,α◦ϕ◦β
b) ForX =[0;1],theorientedFre´chetDistanceF′(A,B)of α,β :X X, since the set x,ϕ(x) :x X agrees
continuous(notnecessarilysimple)curvesA,B :[0;1] with α→β(y),α ϕ β(y)(cid:8):(cid:0)y X (cid:1). How∈eve(cid:9)rthe inf
Y is defined similarly with the infimum ranging ov→er overϕ(cid:8)(cid:0) A◦ut(X) i◦n D◦efinitio(cid:1)n 11∈is in(cid:9)general‘attained’
Aut′([0;1]): the set of all strictly increasing continuous only by∈non-injective reparametrizations (Figure 1a).
ϕ:[0;1] [0;1] with ϕ(0)=0 and ϕ(1)=1. b) On the other hand, the mapping (α,ϕ) F (A,B) is
c) For X =→1 the unitcircle, the oriented Fre´chet Distance uniformly continuous; namely has modu7→lus oαf,ϕcontinuity
S
F′(A,B) of continuous loops A,B : 1 Y is defined the sum of those of A and B. The sets Aut(X) and
similarly with infimum ranging over ASut′→( 1): the set of Aut′(X) may thus be replaced by their topological clo-
S
all clockwise continuous bijections ϕ: 1 1. sures, Aut(X) and Aut′(X) in (X,X)as propersuper-
d) For X = B2 the Euclidean unit dSisc,→thSe oriented sets,withoutaffectingthevalueoCfF andF′,respectively.
Fre´chet Distance F′(A,B) of continuous 2D surfaces However those closures still lack equicontinuity.
A,B : 1 Y is defined similarly with infimum rang- c) For the smooth simple curves A,B : [0;1] [0;1]2
ing oveSr A→ut′ B2 : the set of all continuous bijections depicted in Figure 1b), their (non-/oriented) Fre´→chet Dis-
ϕ: B2 B2 m(cid:0)app(cid:1)ing some/all clockwise simple curves tance is attained by a continuum of homeomorphisms
in B2 to→clockwise image(s) [Alt09, DEFINITION 2]. ϕ:[0;1]→[0;1], i.e., non-uniquely.
e) More generally fix a d-dimensional orientable compact d) Aut([0;1]) is the disjoint union of the path-connected
subspace of increasing homeomorphisms, i.e. those in
m[Manuinfko8ld4,XC,Oi.Re.O,wLLitAhRdY-th6h5o.4m].olFoogryagnroyuhpoHmde(oXm,oZrp)h∼=ismZ Aut′([0;1]), and the decreasing ones; similarly for
ϕ : X X, the action of composition with ϕ induces Aut( 1). More generally, for any d-dimensional ori-
S
an isom→orphism of the k-th homology group; which for entable compact manifold X, Aut(X) decomposes
k =d can only be multiplication either by 1 or by +1; into the locally arc-connected subspace Aut′(X) of
and the latter ϕ by definition comprise Aut−′(X). orientation-preserving homeomorphisms and that of
orientation-reversingones [Sand60].
The above notions have recently received much attention —
e) To every ϕ Aut′([0;1]) there exist 2-Lipschitz ψ,χ
in Computational Geometry, that is, for polygonalcurves and Aut′([0;1])∈such that ϕ=ψ χ−1; similarly for the non∈-
triangulated surfaces; cf. for instance [AAB*16], [AHK*15], ◦
oriented case.
[BDS14], [Alt09], [Goda91] and both the references and
f) There exists a constant K 2 such that every ϕ
motivating examples therein — as well as for the important Aut′( 1) admits a decompos≥ition ϕ = ψ χ−1 with K∈-
Question12:Withoutrestrictingtopiecewise/combinatorial LipschSitz bijections ψ,χ Aut′( 1); agai◦n, similarly for
inputs,cantheFre´chetDistance(s)becomputedinthesenseof ∈ S
the non-oriented case.
RecursiveAnalysis,thatis,byapproximationuptoguaranteed
g) There exists a constant K 2 such that every Lipschitz-
absoluteerror2−nforeverygivenn∈Nandeverygiven/fixed continuous ϕ Aut′ Bd ≥admits a decomposition ϕ =
pair of continuous/computablefunctions A,B ? ∈
Theorem 14 below gives a positive answer for curves (X = α−1◦β◦γ−1 Bd with(cid:0)K-L(cid:1)ipschitzα,β,γ ∈Aut′ 2Bd ;
[0;1]) and loops (X = 1) but also shows that an optimal similarlyfort(cid:12)henon-orientedcase.Here,usingMin(cid:0)kowsk(cid:1)i
reparametrization ϕ cannSot in general be computable. operations, B(cid:12)d+Bd =2Bd =B(0,2) Rd denotes the
⊆
Recall (Fact 3a) that compactness and continuity guarantee closed Euclidean ball around center 0 with radius 2.
infimum (e.g. in Definition 2f) to exist, be attained, and h) Picking up on b), extend the definition of Fα,β(A,B)
computableaccordingto Fact 1h) and Theorem7g). Our goal accordingtoEquation(7)fromcontinuousfunctionsα,β :
istoarguesimilarlyinEquation(7),onlythatthegroundspace X X to compact relations α,β X X as
→ ⊆ ×
here consists of functions ϕ. A first na¨ıve attempt fails since
sup e A(a),B(b) x X :(x,a) α,(x,b) β .
Aut(X) (X,X) is not compactand the infimum thus not ∃ ∈ ∈ ∈
⊆C (cid:8) (cid:0) (cid:1)(cid:12) (cid:9)
necessarily attained: Then (α,ϕ) F ((cid:12)A,B) still remains continuous w.r.t.
α,ϕ
7→
the Hausdorff metric on (X X) (X X), and it ofallthoseC D D ,withm N,thusconstitutes
m m m
holds F(A,B) = inf FK (A×,B) =×Kinf ×F (A,B) a name of gra⊆ph L×ip ([0;1],[0;1]∈) Aut′([0;1])
ϕ id,ϕ α,β α,β 2 ∩ ⊆
with infimum taken over graph(Aut(X)) (X X); ([0;1]2). (cid:0) (cid:1)
similarly for F′ and Aut′(X): see Figure 2⊆a)K. × b) KBy Remark 13b+e), F′(A,B) coincides with
For motivation, consider a bounded uniformly continuous infχ,ψFχ,ψ(A,B), where the infimum ranges over
0funcbtuiotnsaaltiΦsfy:inPg→‘scRalionngnionvna-crioamncpea’ctΦP(α=,ϕ{)(α=,ϕΦ)(:1,αϕ,/ϕα>). tLhiep (c[0lo;s1e]t,[0;s1u]b)setAuLti′p(2[0(;[01;])1],o[f0;c1o])m∩puAtaubtl′y([0c;o1m])pa×ct
2 ∩
T(cid:9)hen it obviously suffices to consider (α,ϕ) ∈ [0;1]2: a Lip2([0;1],[0;1]) × Lip2([0;1],[0;1]). Moreover said
compact space. Items e+f+g) exhibit a similar property for subset is computable by a); and so is the mapping
(α,ϕ)7→e∞(A◦α,B◦ϕ), but without commutativity. (χ,ψ)7→Fχ,ψ(A,B) on it. Hence Theorem7g) assert its
Theorem 14: Let (Y,e,υ,E) denote a computablycompact imagetobeacomputablesubsetof[0;1],whoseminimum
space of diam(Y) 1. is computable according to Fact 1j). The non-oriented
≤
a) Thecompactsetgraph Lip ([0;1],[0;1]) Aut′([0;1]) case proceeds similary according to Remark 13d).
2 ∩ ⊆ c) By Remark 13b+f) and regarding that
([0;1]2) of graphs(cid:0) of non-decreasing 2-Lipschi(cid:1)tz
K Lip ( 1, 1) Aut′( 1) Lip ( 1, 1) Aut′( 1)
ϕ : [0;1] [0;1] with ϕ(0) = 0 and ϕ(1) = 1 is K S S ∩ S × K S S ∩ S
→ is a computable subset of computably compact
computable.
Lip ( 1, 1) Lip ( 1, 1) as domain of computable
b) Non/orientedFre´chetDistances between continuouspaths K S S × K S S
F,F′ : ([0;1],Y)2 [0;1] are computable. mapping (χ,ψ) 7→ Fχ,ψ(A,B) The non-oriented case
C → proceeds similary.
c) The same holds for non/oriented Fre´chet Distances be-
tween continuous loops F,F′ : ( 1,Y)2 [0;1].
C S →
d) There exist computable smooth A,B : [0;1] [0;1] and
→
strictlyincreasinghomeomorphismϕ:[0;1] [0;1]such
→
that A = B ϕ holds but no computable non-decreasing
◦
surjectionϕsatisfiesA=B ϕ;nordoesanycomputable
◦
non-increasing surjection ϕ.
e) There exist computable smooth simple (=injective) A˜,B˜ :
[0;1] [0;2]2, codomain considered equipped with the
2D ma→ximum norm, such that F′(A˜,B˜) = 1 = F(A˜,B˜)
isattainedbysomestrictlyincreasinghomeomorphismϕ:
[0;1] [0;1] but by no computable non-decreasing/non-
→
increasing surjection ϕ. Fig.2. a)Exampleofconvergence inHausdorff(graph) butnotSupremum
(function) norm. b)Illustrating theproofofTheorem14a).
Regarding higher dimensions, [AlBu10, THEOREM 1] has
asserted at least left/upper semi-computability; recall the
paragraph following Fact 1: Computationally enumerating a Proof of Remark 13:
sequence ϕ dense in separable Aut(X) (X,X), to- e) Let ϕ be non-decreasing. Then the continuous and sur-
n
⊆ C
gether with computabilityand continuityof A,B,e yieldsa jective mapping ϕ˜ : [0;1] t t+ϕ(t) /2 [0;1]
∞ ∋ 7→ ∈
computable sequence Fid,ϕn(A,B) whose infimum coincides satisfies, for t≥t′, (cid:0) (cid:1)
with F(A,B); and for ϕ ranging over a compact space, its
n ϕ˜(t) ϕ˜(t′)=(t t′)/2 + ϕ(t) ϕ(t′) /2 (t t′)/2
covering property asserts that finitely many (balls centered − − − ≥ −
(cid:0) (cid:1)
around)themsufficetoapproximateF(A,B)alsofrombelow. and hence is strictly increasing with 2-Lipschitz inverse
Proof of Theorem 14: χ Aut′([0;1]). It remains to observe that ψ :=ϕ χ is
∈ ◦
a) Recall that a name of continuous ϕ : [0;1] [0;1] is 2-Lipschitz since, again for t t′,
→ ≥
a family of finite sets C D D approximating
graph(ϕ) in Hausdorff mmet⊆ric. Nmo×w itmis easy to enu- ψ t+ϕ(t) ψ t′+ϕ(t′) = ϕ(t) ϕ(t′)
2 − 2 − ≤
merate, uniformly in m N, all those Cm satisfying (cid:0)ϕ(t) (cid:1)ϕ(t′)(cid:0)+t t′ (cid:1)= 2 t+ϕ(t) t′+ϕ(t′) .
the following condition: C∈ is a ‘chain’ of points in the ≤ − − · 2 − 2
m (cid:0) (cid:1)
sense of Go (aka ), starting at the lower left corner f) Applying an isometric rotation we may w.l.o.g. suppose
and proceeding to the upper right such that at at least ϕ(1) = 1. Since 1 is homeomorphic to [0;1) mod 1,
S
every second step ‘up’ is followed by one ‘right’; see this reduces to e).
Figure2b)illustratingtheidea(thatwedeliberatelyrefrain g) Accordingtof),therestrictionofϕto theball’sboundary
fLartiopmm2(o[fs0ot;r1m2]−,a[lm0iz;i1nto]g)sf∩uormAtheuetrs)′u(.[c0Th;h1eC]n) th;haeasngdHraapcuohsndovoferrfesfveledyriysetavϕnecr∈ye aihsdommLi-etLosimapsodcrehpcihotizmsmopusotssψiitd,ioχent:hϕSe(cid:12)(cid:12)1SE1→u=clSiψd1e◦.aχNn−od1tieswkthitBahtdK~x(0-7→,L1ip/~xsL/c)|h~xio|t2zf
m
such C has distance at most 2−m to the graph of some radius 1/L. Thus, applying Alexander’s Trick, χ extends
m
ϕ Lip ([0;1],[0;1]) Aut′([0;1]). The collection C radially to a 2K-Lipschitz homeomorphismof entire B2,
∈ 2 ∩ m