Computable metric space theory is a
generalization of effective algebra
Alexander Melnikov
UC at Berkeley
CCA 2014
Alexander Melnikov
Computable metric space theory is a generalization of effective alg
I will survey some recent results
that aim to develop computable metric (Banach) space theory
as a natural generalization of effective algebra
Alexander Melnikov
Computable metric space theory is a generalization of effective alg
Introduction
Effective algebra studies algorithmic presentations of countable
algebraic structures such as groups and fields.
Both effective algebra and computable metric/Banach space
theory address essentially the same questions:
- what does it mean for a mathematical structure to be
algorithmically presented?
- which operations and relations on a structure are effective?
- how can we compare algorithmic presentations of a
mathematical structure?
- can we classify algorithmically presented members of a
given abstract class of structures?
and other familiar questions.
Alexander Melnikov
Computable metric space theory is a generalization of effective alg
Introduction
Many methods and ideas of effective algebra can be adjusted
to computable metric/Banach spaces, with nice applications.
There are some related results that pre-date and pre-determine
my research. These include works of:
Pour-El and Richards,
Chris Ash (under Nerode’s supervision),
Hertling,
Kudinov and Korovina.
Alexander Melnikov
Computable metric space theory is a generalization of effective alg
Background on effective algebra
Alexander Melnikov
Computable metric space theory is a generalization of effective alg
Background on effective algebra
For the next 5-7 minutes
forget everything that you know about computable analysis
For now, we are concerned with countably infinite algebraic
structures. Examples include:
finitely and countably generated groups,
fields of the form Zp (αi : i ∈ ω),
sub-ordernings of (Q, <),
and the like.
Alexander Melnikov
Computable metric space theory is a generalization of effective alg
Background on effective algebra
Definition (Rabin 1960, Mal’cev 1961)
A computable presentation of a countably infinite algebraic structure
(A, f1 , . . . , fk ) is an algebraic structure (B, g1 , . . . , gk ) isomorphic to
(A, f1 , . . . , fk ) such that:
the domain of B is ω (the natural numbers),
the operations g1 , . . . , gk of B are Turing computable functions
(or relations).
The notion generalizes several earlier definitions such as
groups with solvable word problem (Dehn) and explicitly
presented fields (van der Waerden).
Alexander Melnikov
Computable metric space theory is a generalization of effective alg
Background on effective algebra
Example
The following algebraic structures have computable presentations:
- the field (Q, +, ×, =),
- the additive group (V∞ , +, =) of the ω-dimensional Q-vector
space,
- (ω, <, =), the well-ordering of the natural numbers,
L
- the ordered abelian group Z<ω
i∈ω Z, +, <lex , =).
lex = (
Alexander Melnikov
Computable metric space theory is a generalization of effective alg
Background on effective algebra
Mal’cev (1961): Computable presentations should be viewed up to
computable isomorphisms.
Example
- all computable presentations of (Q, +, ×, =) are computably
isomorphic (folklore),
- V∞ has ω-many computable presentations up to computable
isomorphism (Mal’cev, Goncharov),
- Z<ω
lex has ω-many computable presentations (essentially Downey
and Kurtz),
- There exists a computable nilpotent group with
exactly two computable presentations, up to computable
isomorphism (Goncharov, Molokov and Romanovsky, 1983).
We also have some general metatheory that involves
non-computable isomorphisms.
Alexander Melnikov
Computable metric space theory is a generalization of effective alg
Background on effective algebra
Another fundamental problem is:
Problem (Nerode and others)
Give conditions for an operation (relation) to be computable
(computably enumerable) in all computable presentations of a
structure.
Example
- V∞ has computable copies with no computable basis (Mal’cev),
- (ω, <, =) has computable presentations where the immediate
successor relation S is not computable (folklore),
- A computable presentation of an orderable abelian group does
not have to be computably orderable (Downey and Kurtz).
We have deep results (Ash-Nerode, Knight, Ventsov and
others) that partially solve the problem. We also know lots
about “standard” algebraic classes (e.g., groups).
Alexander Melnikov
Computable metric space theory is a generalization of effective alg
Background on effective algebra
Problem (goes back to Mal’cev and Rabin)
Can we classify computably presented members in a class?
We have several approaches to this question as well, most of
them are closely related technically. They typically give the
same answer for a standard class:
Example
- Computable vector spaces over a given field are “classifiable”,
- Computable abelian groups are “unclassifiable”.
Most related methods typically involve infinitary computable
formulae. I will not give any formal definitions here.
Alexander Melnikov
Computable metric space theory is a generalization of effective alg
Background on effective algebra
To summarize:
- Effective algebra has accumulated lots of ideas and
machinery.
- Effective algebra and computable analysis are closely
related philosophically (i.e., we often ask similar
questions).
- We can try to extend these ideas and machinery to metric
and Banach space theory.
- We will sometimes succeed, but it will require some neat
new ideas and advanced techniques on top of what is
already known.
Alexander Melnikov
Computable metric space theory is a generalization of effective alg
Computable metric and Banach
spaces
Alexander Melnikov
Computable metric space theory is a generalization of effective alg
Computable metric and Banach spaces
We will have to slightly change our mathematical language:
Instead of the metric function d, we will be using relations d<r and
d>r with the interpretation:
d<r (x, y ) iff d(x, y ) < r ,
d>r (x, y ) iff d(x, y ) > r .
Clearly, the metric function d on a metric space is
completely determined by {d<r , d>r : r ∈ Q}.
Thus, we may speak of a completion of a structure in the
language {d<r , d>r : r ∈ Q}.
Alexander Melnikov
Computable metric space theory is a generalization of effective alg
Computable metric and Banach spaces
The following definition is equivalent to the standard definition
of computability on a metric space that you all know.
Definition
A computable presentation on a Polish metric space M is a
countable algebraic structure (X , d<r , d>r ) such that:
the domain of X is the set of natural numbers,
the relations d<r and d>r are computably enumerable uniformly
in r ,
the completion X of X is isometric to M.
The definition can be naturally extended to other metric
signatures (e.g., to Banach spaces or Banach algebras).
Alexander Melnikov
Computable metric space theory is a generalization of effective alg
Computable metric and Banach spaces
Let’s compare our definitions for countable algebraic and Polish
metric structures:
- A Polish metric space typically has computable
presentations that are not isomorphic in the algebraic
sense.
- For spaces we used computably enumerable relations.
The second “difference“ is not really a difference. Bulgarian and
Russian effective algebraists study algebras with c.e. relations.
But the first difference is essential.
(Pour-El and Richards.) It is natural to view computable presentations
of a Polish metric spaces up to computable isometries between their
completions.
Alexander Melnikov
Computable metric space theory is a generalization of effective alg
Effective algebra is a just a
special case.
Alexander Melnikov
Computable metric space theory is a generalization of effective alg
Effective algebra is a special case!
Theorem (Goncharov)
Every computable algebraic structure can be fully effectively encoded
into a computable connected undirected graph.
This mysterious theorem means:
The map A → GA is uniformly computable and preserves
all effective, algebraic and category-theoretic features of
the input.
Alexander Melnikov
Computable metric space theory is a generalization of effective alg
Effective algebra is a special case!
Fact (Khoussainov and M.)
Every computable algebraic structure can be fully effectively encoded
into a computable perfect Polish metric space.
Proof.
The proof is not hard:
Given a computable algebraic structure A, produce a
computable, connceted graph GA .
Pass to the shortest path metric representation MA of GA .
Smoothen MA by replacing every edge by [0, 1] and
replacing every node by some nice definable computable
compact Polish space.
Alexander Melnikov
Computable metric space theory is a generalization of effective alg
Effective algebra is a special case!
Corollary
There exists a computable perfect Polish space that has exactly two
computable presentations, up to computable isometry.
Corollary
All bizarre counter-examples and monsters from effective algebra can
be carried through (we skip it).
Alexander Melnikov
Computable metric space theory is a generalization of effective alg
Computable categoricity
Alexander Melnikov
Computable metric space theory is a generalization of effective alg
Computable categoricity
Definition (M. 2011)
A computable metric space is computably categorical if completions
of any two computable presentations of the space are computably
isometric.
Theorem (M. 2011)
The following metric spaces are computably categorical:
the Urysohn space,
separable Hilbert spaces with the distance metric (that was
known in some sense),
Cantor space
Also, we have an iff condition for a subspace of Rn to be c.c.
When compared to similar arguments in effective algebra,
proofs for metric spaces tend to be harder.
Alexander Melnikov
Computable metric space theory is a generalization of effective alg
Computable categoricity
Theorem (M. 2011)
The following metric spaces are not computably categorical:
L1 [0, 1] with its usual metric,
C[0, 1] with the sup metric.
The proof for L1 [0, 1] is based on Pour-El and Richards and is a
coding argument.
The proof for C[0, 1] uses a direct diagonalization technique
and a new strategy.
Alexander Melnikov
Computable metric space theory is a generalization of effective alg
Computable categoricity
It is natural to ask how many computable presentations a space
has up to computable isometry.
To state the next result, we need a definition:
Definition
Say that a computable presentation X of a metric space is
rational-valued if:
d(x, y ) is a rational for every x, y ∈ X ,
given (x, y ) ∈ X 2 we can uniformly compute (m, n) ∈ ω 2 such
that d(x, y ) = m
n.
Alexander Melnikov
Computable metric space theory is a generalization of effective alg
Computable categoricity
The following useful abstract result is an extension of a
well-known theorem from effective algebra:
Theorem (M. and Ng, 2012)
Suppose a Polish metric space M has two rational computable
presentations, X and Y, such that:
X is not computably isometric to Y,
there exists a ∆02 - surjective isometry between X and Y that
maps elements of X to elements of Y.
Then M has infinitely many computable presentations that are
pairwise not computably isometric.
The proof is a non-uniform priority argument that uses a new
“preservation” strategy.
Alexander Melnikov
Computable metric space theory is a generalization of effective alg
Computable categoricity
We can apply the result to standard spaces, for instance:
Corollary (M. and Ng, 2012)
The space (C[0, 1], sup) has infinitely many computable
presentations that are pairwise not computably isometric.
Dropping the restrictions on the pair of ∆02 -isometric
presentations (i.e., rational-valuedness) seems to be a rather
difficult problem.
Alexander Melnikov
Computable metric space theory is a generalization of effective alg
Banach spaces
Alexander Melnikov
Computable metric space theory is a generalization of effective alg
Banach spaces
(Pour-El and Richards:) It is natural to restrict ourselves to metric
computable presentations of a Banach space B that
compute all Banach space operations.
Question
Is it really a restriction?
In other words, does every computable structure on the
associated metric space (B, d) necessarily compute
all other standard Banach space operations such as +?
Alexander Melnikov
Computable metric space theory is a generalization of effective alg
Banach spaces
Theorem (M. and Ng, 2012)
There exists a computable presentation of the metric space
(C[0, 1], sup) in which + is not computable.
Recall that C[0, 1] is also a Banach algebra.
Theorem (M. and Ng, 2012)
There exists a computable presentation of the Banach space
(C[0, 1], sup, +) in which × is not computable.
Corollary
(C[0, 1], +, sup) is not computably categorical.
Alexander Melnikov
Computable metric space theory is a generalization of effective alg
Banach spaces
Theorem (M. and Ng, 2012)
There exists a computable presentation of the metric structure
(C[0, 1], sup, +, ×) in which no isometric image of
Id : [0, 1] → [0, 1] is computable.
Corollary
(C[0, 1], +, ×, sup) is not computably categorical.
In contrast, (C[0, 1], +, ×, Id, sup) is computably categorical.
Alexander Melnikov
Computable metric space theory is a generalization of effective alg
Banach spaces
- Our proofs blend effective algebraic and analytic
techniques.
- Our results illustrate that computability of relations is
closely related to computable categoricity.
- The results extend classical earlier works of Pour-El and
Richards in a rather unexpected direction.
Alexander Melnikov
Computable metric space theory is a generalization of effective alg
Isometries, operations, and
syntax
Alexander Melnikov
Computable metric space theory is a generalization of effective alg
Isometries, operations, and syntax
As we have seen, operations do not have to be computable in
all computable presentations of a space.
Problem
Give a necessary and sufficient condition for an operation on a metric
space to be computable in all presentations of the space.
We have already seen, the problem above seems to be related
to:
Problem
Give a necessary and sufficient condition for a metric structure to be
computably categorical.
We can answer these questions for most of the standard
spaces! The answers are abstract but useful.
Alexander Melnikov
Computable metric space theory is a generalization of effective alg
Isometries, operations, and syntax
Example
In a Hilbert space H, the inner product hx, y i can be expressed using
the standard Banach space operations:
hu, v i =
1
(||u + v ||2 − ||u − v ||2 ).
4
Thus, if we can compute the standard operations in a
computable presentation on H, then we can compute the inner
product as well.
One might expect that being expressible is necessary and
sufficient. And this turns to be almost right!
Alexander Melnikov
Computable metric space theory is a generalization of effective alg
Isometries, operations, and syntax
Example (M. 2012)
Let X be a computable presentation on a Hilbert (metric) space
(H, d).
Claim: If the point 0 is computable in X , then + is computable in X .
Proof sketch: Given and x, y ∈ X , find z ∈ X such that:
1
| d(0, z)2 + d(x, y )2 − 2d(x, 0)2 − 2d(y , 0)2 | < δ,
2
| d(y , z) − d(0, x) | < δ,
3
| d(x, z) − d(0, y ) | < δ,
where δ =
2d(0,x)+2d(0,y )+3 .
Then d(x + y , z) < .
For every ∈ Q we can effectively produce a formula Θ such
that X |= Θ (0, x, y , z) guarantees d(x + y , z) < .
Alexander Melnikov
Computable metric space theory is a generalization of effective alg
Isometries, operations, and syntax
All natural examples that we know have the unifom property:
Definition
We say that M is uniformly computably categorical if there exists a
uniform way of producing a computable isometry between any two
computable presentations of M. (We allow finitely many parameters.)
All standardl spaces also satisfy:
Definition
A Polish metric space M is a relatively computably categorical if for
every countable (S, d<r , d>r ) and (X , d<r , d>r ) such that
X ∼
=S∼
= M we have an isometry between the structures that is
S ⊕T X -computable.
Alexander Melnikov
Computable metric space theory is a generalization of effective alg
Isometries, operations, and syntax
All natural examples of computably categorical metric spaces
that we know of are both r.c.c. and u.c.c.:
- Hilbert space
- The Urysohn space
- Cantor space
- The unit simplex in Rn , etc.
Fact (Khoussainov and M.)
There exist computable metric spaces that are computably categorical
but not r.c.c.(or u.c.c.).
We don’t care much since we don’t know any natural example
like that.
Alexander Melnikov
Computable metric space theory is a generalization of effective alg
Isometries, operations, and syntax
Example (Essentially Brattka and Yoshikawa 2005)
A separable Hilbert (metric) space (H, d) is both uniformly and
relatively computably categorical.
Choose any x ∈ X and y ∈ Y.
Declare x a “zero” in X .
Declare y a “zero” in Y.
Reconstruct all Hilbert space operations in X and Y as above
(note it uses some definability).
Run Gram-Schmidt on both X and Y and do back-and-forth.
If we really care about the “standard” 0, we need a parameter
for it.
Alexander Melnikov
Computable metric space theory is a generalization of effective alg
Isometries, operations, and syntax
We are ready to state the following useful:
Theorem (Greenberg, M., and Turetsky 2014)
For a computable Polish space M, TFAE:
1
M is uniformly computably categorical;
2
M is relatively computably categorical;
3
M has a computably enumerable approximate Scott family (to be
discussed).
- An approximate Scott family is a collection of first-order,
existential, and positive formulae with finitely many
parameters that describe orbits of elements up to .
Alexander Melnikov
Computable metric space theory is a generalization of effective alg
Isometries, operations, and syntax
The following result is related to point degree spectra:
Theorem (Greenberg, M., and Turetsky 2014)
For a computable Polish space M and a point z ∈ M, TFAE:
1
2
3
z is uniformly computable in every computable presentation of
M (perhaps, with finitely many parameters);
z is computable with respect to every X such that X ∼
= M;
V
z is definable by a computable conjunction ∈Q Θ of
first-order positive ∃-formulae, such that Θ (x) implies
d(x, z) < .
Alexander Melnikov
Computable metric space theory is a generalization of effective alg
Isometries, operations, and syntax
Similarly, for operations we have:
Theorem (Greenberg, M., and Turetsky 2014)
For a computable Polish space M and an operation F on M, TFAE:
1
2
3
F is uniformly computable in every computable presentation of
M (perhaps, with finitely many parameters);
F is computable with respect to every X such that X ∼
= M.
F has an approximate formal name.
I’ll not define what a formal name is.
Alexander Melnikov
Computable metric space theory is a generalization of effective alg
Isometries, operations, and syntax
Using approximate Scott families, we can significantly
simplify several known proofs (e.g., computable categoricity of
the Urysohn space).
The results can be naturally extended to Banach spaces and
other metric structures.
The main tools for both results are forcing and a careful
analysis of positive Turing operators.
Alexander Melnikov
Computable metric space theory is a generalization of effective alg
Index sets and higher categoricity
Alexander Melnikov
Computable metric space theory is a generalization of effective alg
Index sets and higher categoricity
Suppose K is a class of computable metric structures (e.g.,
computable locally compact metric spaces).
Problem
What does it mean for K to be classifiable or unclassifiable? How can
we measure the difficulty of the classification problem for K?
In effective algebra we use index sets, infinitary computable
formulae, and ∆0n -categoricity.
Alexander Melnikov
Computable metric space theory is a generalization of effective alg
Index sets and higher categoricity
We can effectively list all partial computable presentations of
metric structures:
M0 , M1 , M2 , . . . .
Definition (M. and Nies)
Let K be a class of metric structures closed under isomorphism.
The index set of K is IK = {e : Me ∈ K}.
2 : M ∼ M }.
The isomorphism problem for K is {(i, j) ∈ IK
i =
j
We measure the complexity of these sets using the
(hyper)arithmetical hierarchy.
Alexander Melnikov
Computable metric space theory is a generalization of effective alg
Index sets and higher categoricity
Theorem (M. and Nies, 2013)
(i) The index set of compact computable metric spaces is
Π03 -complete.
(ii) The isomorphism problem for compact computable metric
spaces is Π02 -complete within Π03 .
The proof relies on Gromov’s work. It also uses computable
infinitary formulae.
In fact, within the class of computable Polish spaces, each compact
member is uniquely described up to isometry by a single infinitary
computable Πc3 axiom.
See the book of Ash and Knight for background on infinitary
computable formulae.
Alexander Melnikov
Computable metric space theory is a generalization of effective alg
Index sets and higher categoricity
Definition (M. and Nies)
A computable metric space M is ∆0n -categorical if any two
computable presentations of M are ∆0n -isometric.
So ∆01 -categoricity is just computable categoricity.
If all members of a class K are ∆0n -categorical, then n
reflects the complexity of K.
Alexander Melnikov
Computable metric space theory is a generalization of effective alg
Index sets and higher categoricity
Theorem (M. and Nies, 2013)
(i) Every compact computable metric space is ∆03 -categorical.
(ii) There exists a computable closed subspace of Cantor space
which is not ∆02 -categorical.
We note that all results in this section are closely related
technically.
We can extend (i) to Probability spaces, but we don’t know
whether this is sharp.
Alexander Melnikov
Computable metric space theory is a generalization of effective alg
Selected questions
Alexander Melnikov
Computable metric space theory is a generalization of effective alg
Selected questions
Is ln computably categorical if, and only if, n = 2? (Known
cases n = 1, 2; for both Banach and metric signatures.).
Is there any classical Banach space (e.g., l3 ) that is
computably categorical as a Banach space, but not as a
metric space? The same for Banach algebra vs. Banach
space.
What is the right infinitary computable language for
computable non-compact metric spaces?
Calculate complexities of index sets etc. for standard
classes of metric structures.
I have too many questions and only one slide. Sorry.
Alexander Melnikov
Computable metric space theory is a generalization of effective alg
Thanks!
Alexander Melnikov
Computable metric space theory is a generalization of effective alg