Taking the path computably travelled
Johanna N. Y. Franklin, Dan Turetsky

TL;DR
This paper explores the concept of lowness for paths in Baire and Cantor spaces, establishing their equivalence and linking them to lowness for isomorphism, thereby advancing understanding of computability properties in these spaces.
Contribution
It introduces the notion of lowness for paths in Baire and Cantor spaces and proves their equivalence along with their connection to lowness for isomorphism.
Findings
Lowness for paths in Baire space and Cantor space are equivalent.
Lowness for these spaces is also equivalent to lowness for isomorphism.
The paper establishes foundational relationships between these notions of lowness.
Abstract
We define a real to be low for paths in Baire space (or Cantor space) if every class with an -computable element has a computable element. We prove that lowness for paths in Baire space and lowness for paths in Cantor space are equivalent and, furthermore, that these notions are also equivalent to lowness for isomorphism.
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Taxonomy
TopicsComputability, Logic, AI Algorithms · Benford’s Law and Fraud Detection · Cellular Automata and Applications
Taking the path computably travelled
Johanna N.Y. Franklin
Department of Mathematics
Room 306, Roosevelt Hall
Hofstra University
Hempstead, NY 11549-0114
USA
and
Dan Turetsky
Department of Mathematics
Victoria University of Wellington
Wellington, New Zealand
Abstract.
We define a real to be low for paths in Baire space (or Cantor space) if every class with an -computable element has a computable element. We prove that lowness for paths in Baire space and lowness for paths in Cantor space are equivalent and, furthermore, that these notions are also equivalent to lowness for isomorphism.
The first author was supported in part by Simons Foundation Collaboration Grant #420806.
1. Introduction
Lowness notions are common objects of study in computability theory. Examples include lowness and superlowness in degree theory, lowness for randomness, lowness for genericity, array computability, jump-traceability, and lowness for isomorphism and lowness for categoricity in computable structure theory. Each of these notions characterizes a class of reals which are in some way no more useful as an oracle than the empty set.
One way to understand such notions is via tasks and instances: a real satisfies the lowness notion associated with a task if every instance of the task which has an -computable solution also has a computable solution. For example, we can consider the original lowness notion: a real is low if , but by the Schoenfeld Limit Lemma, this can be understood in our framework by saying that an instance is an and the task is to compute a limit approximation to . Thus, is low if and only if every which is limit-computable from is limit-computable from .
Another well-known example is lowness for Martin-Löf randomness (see the text by Downey and Hirschfeldt [DH10] for background on algorithmic randomness). Here, an instance is again an , and the task is to derandomize , i.e., to capture with a Martin-Löf test. A real is low for randomness if every which can derandomize is already derandomized by .
In this vein, Franklin and Solomon initiated the study of lowness for isomorphism, where an instance is a pair of computable structures and the task is to compute an isomorphism between the structures [FS14].
Definition 1**.**
A real is low for isomorphism if every pair of computable structures with an -computable isomorphism between them have a computable isomorphism between them.
We refer the reader to the text by Ash and Knight [AK00] for background on computable structures. Franklin and Solomon showed that nontrivial examples of such reals exist. For example, they showed that if is 2-generic, then is low for isomorphism [FS14]; an extension of this result was given by Franklin and Turetsky in [FT18].
Note that being low as described by some task does not mean that every -computable solution to an instance of the task is itself a computable solution. An instance may have multiple solutions, some of which are -computable but not computable. However, in such a case, the instance will also have computable solutions. For example, consider two computable copies of . It is a simple exercise to show that these two copies have an isomorphism in every Turing degree. Thus, even if is low for isomorphism, there will be an -computable isomorphism between these copies which is not a computable isomorphism. However, there is also a computable isomorphism between these two copies.
We observe that we can understand the collection of isomorphisms between two computable structures as a -class in Baire space.
Definition 2**.**
For computable structures and , define
[TABLE]
In general, the statement that a function is surjective is , but we avoid that difficulty by including the function’s inverse: we can write down a -formula for by stating that is an embedding of into and that . The existence of the inverse ensures that is surjective.
As any isomorphism from to computes its inverse, we can understand lowness for isomorphism as the lowness notion in which the instances are -classes of the form and the task is to compute an element of the class. It is then natural to consider the related lowness notion in which an instance is any -class and the task is to compute an element of the class. This gives a priori two notions, depending on whether one considers -classes in Baire space or in Cantor space.
Definition 3**.**
A real is low for paths for Baire space (or low for paths for Cantor space) if every class (respectively, ) with an -computable element has a computable element.
We prove the following:
Theorem 4**.**
For , the following are equivalent:
- (1)
* is low for paths for Baire space;* 2. (2)
* is low for paths for Cantor space;* 3. (3)
* is low for isomorphism.*
is obvious, as every -class in Cantor space is itself a -class in Baire space. follows from our discussion of . We will show and then .
The proof of relies on the following result of Simpson [Sim07]:
Lemma 5**.**
If and are nonempty -classes, then there is a -class with .
Here, is Muchnik equivalence: every element of computes an element of , and every element of computes an element of .
Proof of Theorem 4, .
Suppose is low for paths for Cantor space, and let be a -class with an -computable element . We must show that has a computable element. Fix a nonempty -class with no computable elements, e.g., the completions of Peano arithmetic, and let be as in Lemma 5.
As and , there is with . As is low for paths for Cantor space, there must be a computable . Using again , there must be with , and so is computable. Since contains no computable elements, is a computable element as desired. ∎
As being low for paths for Baire space is equivalent to being low for paths for Cantor space, we shall refer to them both as simply low for paths.
Now we turn to the proof of . The following lemma is the heart of this result.
Lemma 6**.**
For every class , there are computable structures and such that .
Proof.
Fix a computable tree such that and so that if is not a leaf, then both and are in . We can ensure has this property by replacing it with . This is a computable tree with this property, and it does not change .
The language of our structures will consist of unary relation symbols for , a unary relation symbol , a ternary relation symbol , and a constant symbol . The universe for both and will be (we caution the reader not to confuse with ). and will be identical in all ways except for their interpretations of .
In both and ,
- •
will hold if and only if ,
- •
will hold if and only if is a leaf of and , and
- •
will hold if and only if the following conditions are satisfied:
- –
;
- –
; and
- –
is even.
Finally, in , refers to the element , but in , refers to the element .
Now, suppose is an isomorphism. Because of , it must be that
[TABLE]
for every . We will say that * swaps * if (and, thus, if ).
Claim 6.1**.**
Suppose has the property that for every ,
[TABLE]
Then respects if and only if, for every not a leaf of , swaps either 0 or 2 of .
Proof.
Suppose for . Then respects precisely if, for every not a leaf, is even. This will be even if and only if 0 or 2 of the are 1, which holds if and only if swaps 0 or 2 of . ∎
We may make the following observations about an isomorphism :
- •
Because of and , must swap .
- •
Because of , must not swap any leaf .
- •
By 6.1, if swaps , it must swap exactly one of or .
So from any isomorphism , we can recursively compute a by “following the swaps.” More precisely:
- (1)
We define . 2. (2)
If , then inductively we know that swaps , and thus is not a leaf of . Let be the unique element of such that swaps . Define .
This shows that .
Conversely, suppose . We wish to compute an isomorphism from . We will define by swapping along and fixing its values elsewhere. More precisely:
- (1)
If , define for . 2. (2)
If , define for .
Clearly, respects each of the . Since , we have that . For any leaf , we know that , so and respects . For any , does not swap any of ; for any , swaps and precisely one of , . Thus, by 6.1, respects . Therefore, is an isomorphism, and this shows that . ∎
Now we can complete the proof of our result.
Proof of Theorem 4, .
Suppose is low for isomorphism, and let be a -class with an -computable element . We must show that has a computable element. By Lemma 6, we can fix computable structures and with . Then there is with . As is low for isomorphism, there is a computable . Then there is with , and is the desired computable element. ∎
This result has a very pleasing corollary. Franklin and McNicholl introduced the notion of lowness for isometry, where an instance is a pair of computable metric spaces and the task is to compute an isometry between the structures [FM].
Definition 7**.**
A real is low for isometry if every pair of computable structures with an -computable isometry between them have a computable isometry between them.
We refer the reader to Pour-El and Richards [PER89] for background on computable metric spaces. McNicholl and Stull have further studied this in the particular case where the metric spaces are Banach spaces [MS]. Franklin and McNicholl showed that a real is low for isomorphism if and only if it is low for isometry [FM].
We observe that, given two computable metric spaces, one can construct the -class of all isometries between these two spaces in a way similar to that for the class for computable structures. Thus we derive as a corollary one direction of Franklin and McNicholl’s result: if is low for isomorphism, then it is low for isometry.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[AK 00] C.J. Ash and J. Knight. Computable Structures and the Hyperarithmetical Hierarchy . Number 144 in Studies in Logic and the Foundations of Mathematics. North-Holland, 2000.
- 2[DH 10] Rodney G. Downey and Denis R. Hirschfeldt. Algorithmic Randomness and Complexity . Springer, 2010.
- 3[FM] Johanna N.Y. Franklin and Timothy H. Mc Nicholl. Degrees of and lowness for isometric isometry. In progress.
- 4[FS 14] Johanna N.Y. Franklin and Reed Solomon. Degrees that are low for isomorphism. Computability , 3(2):73–89, 2014.
- 5[FT 18] Johanna N.Y. Franklin and Dan Turetsky. Lowness for isomorphism and degrees of genericity. Computability , 7(1):1–6, 2018.
- 6[MS] Timothy H. Mc Nicholl and D.M̃. Stull. The isometry degree of a computable copy of ℓ p superscript ℓ 𝑝 \ell^{p} . Submitted.
- 7[PER 89] Marian B. Pour-El and J. Ian Richards. Computability in analysis and physics . Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1989.
- 8[Sim 07] Stephen G. Simpson. An extension of the recursively enumerable Turing degrees. J. Lond. Math. Soc. (2) , 75(2):287–297, 2007.
