Martin's Conjecture

Martin’s Conjecture: A Classification of the Naturally Occurring Turing Degrees

1 1 0 1 0 0 1 0 1 0 1 1

Antonio Montalb´an

This paper is about naturally occurring objects in comput- if it can be calculated using a mechanical, step-by-step al- ability theory, the area inside mathematical logic that stud- gorithm. The informal notion of algorithm was already ies the complexity of infinite countable objects. It is about known to the Greeks two millennia ago. But it was not un- the structure that emerges when we look at almost- til the 1930s that Gödel, Turing, and Church, among oth- everywhere behavior with respect to a measure called Mar- ers, proposed the first formal definitions of computable tin’s measure. Posed in the 1970s, Martin’s conjecture was functions. Conceived before the age of computers, those the first indication of this hidden structure. Martin’s con- three equivalent definitions capture what we still call the jecture is currently one of the main open questions in com- class of computable functions: Let us use ℕℕ to denote the putability theory and is of great foundational importance. set of functions from ℕ to ℕ. Today, we say that a function Progress has been slow, and only recently we have started 푓 ∈ ℕℕ is computable if it can be calculated using a com- to appreciate the extent to which Martin’s measure can be puter program in your favorite programming language.1 used to understand the behavior of naturally occurring ob- Which programming language you use is not important. jects in computability theory. You always get the same class of functions as long as the language has some minimum basic functionality. Some Computability Theory languages might be faster or easier to use for certain types Computability theory is the area of logic that studies the of functions, but the class of computable functions from complexity of infinite countable objects. Computer sci- ℕ to ℕ is independent of the language. The restriction to ence and set theory take care of the finite objects and un- functions on the natural numbers is just for simplicity. It countable objects, respectively—well, not exactly, but is not even a real restriction, as any finite object can be more or less. The main notion in computability theory coded by a natural number—that’s what your computer is that of computable function. A function 푓 is computable does when it encodes everything you see on the screen into numbers written in binary. In our examples below, Antonio Montalb´anis a professor of mathematics at the University of California, we will mention finite objects, like strings of characters or Berkeley. His email address is [email protected]. The author was partially supported by NSF grant DMS-1363310 and by a Si- integer polynomials, and assume they are being encoded mons fellowship. The author would also like to thank Pierre Simon for proof- by natural numbers in some standard way without even reading this paper. mentioning it. Communicated by Notices Associate Editor Stephan Ramon Garcia. A subset 퐴 ⊆ ℕ is computable if its characteristic For permission to reprint this article, please contact: 1We impose no restriction on time spent or memory used so long as, on each [email protected]. input, the program gives an answer after finitely many steps, hence using only DOI: https://doi.org/10.1090/noti1940 finitely much memory.

SEPTEMBER 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1209 function 휒퐴 ∶ ℕ → {0, 1} is computable. Here are some is to reduce them to the halting problem. For instance, to examples of noncomputable sets. prove that the word problem is noncomputable, we K: The halting problem is the problem of deciding whether produce an algorithm that decides membership in the halting problem using information about which words in an inputless computer program will eventually halt or 4 run forever without halting, maybe because it enters which finitely presented groups become trivial. an infinite loop or something. We represent this prob- Let us focus on this idea of reducing one problem to another. Consider a function 푔 ∈ ℕℕ. We say that a function lem by the set K of inputless computer programs that ℕ eventually halt. (Here, programs are being viewed as 푓 ∈ ℕ is 푔-computable, or computable in 푔, and write 푓 ≤푇 strings of characters and coded by a single number.) 푔 if the values of 푓 can be calculated using a computer pro- To solve the halting problem, one could try to run the gram that is allowed to consult values of 푔 during its com- given program and see if it halts. The problem is that putation. For example, this program could contain an in- while a program is running and hasn’t halted yet, we struction of the form: “let x:=17; if 푔(x)>27, can’t tell for sure if it is going to stop later or just keep then do this, else do that.” The function 푔 is on going.2 called the oracle of the computation, as we are not specify- WP: The word problem. Consider finitely presented ing how we are getting the values of 푔; they are just given to groups. These are groups that have finitely many gen- us. The relation ≤푇 is transitive and reflexive, and hence a ℕℕ ≤ erators 푎1, 푎2, ..., 푎푘 that satisfy a finite set of relations preordering on . As any preordering does, 푇 naturally 푓 ≡ 푔 푓 ≤ 푔 of the form 푅푖 = 푒, where 푅1, ..., 푅 are words on the induces an equivalence relation, 푇 if 푇 and ℓ ℕ −1 −1 푔 ≤푇 푓, and a partial ordering on the quotient ℕ / ≡푇. letters 푎1, 푎2, ..., 푎푘, 푎1 , ..., 푎푘 . The word problem is to decide, given such a group presentation and an- The equivalence classes are called Turing degrees, and this ℕ other word 푤, whether 푤 is equivalent to the identity partial ordering (ℕ / ≡푇; ≤푇) is denoted by 풟. Each 푒 of the group or not. We define WP to be the set of Turing degree corresponds to a level of complexity, and 풟 tuples ((푎1, 푎2, ..., 푎푘), (푅1, ..., 푅ℓ), 푤) for which the maps out all complexity levels. The properties of this answer is yes. partial ordering have been widely studied since the 1950s. HTP: Hilbert’s tenth problem. Let HTP be the set of poly- Here are some of them. The first four are quite simple, nomials in ℤ[푦0, 푦1, 푦2, ...] that have an integer solu- while the last two require extremely elaborate proofs. tion. Question ten in Hilbert’s famous 1900 list was • 풟 has a least element, 0, that consists of the equiv- whether there was an algorithm to decide member- alence class of the computable functions. ship in HTP. It was answered negatively in 1970 by • Every two degrees have a least upper bound given Matiyasevich, building on work by Martin Davis, Hi- by 푓 ⊕ 푔, where 푓 ⊕ 푔 is the function ℎ such that lary Putnam, and Julia Robinson. ℎ(2푛) = 푓(푛) and ℎ(2푛 + 1) = 푔(푛). TF: Let 푇퐹 be the set of finite presentations • 풟 has the countable predecessor property; i.e., for ((푎1, 푎2, ..., 푎푘), (푅1, ..., 푅ℓ)) of groups that are every 푔, there are only countably many 푓’s that are torsion-free. ≤푇 푔. This is because there are only countably COF: Let 퐶푂퐹 be the set of polynomials in many programs one can write. ℤ[푥, 푦0, 푦1, 푦2, ...] that have integer solutions for all • In particular, every degree is countable, and hence but at most finitely many values of 푥. 풟 has size continuum. TA: Let TA be the set of 1st-order sentences in the lan- • 풟 is fat: There are continuum-size sets of degrees guage of arithmetic {0, 1, +, ×, ≤} that are true about that are pairwise incomparable. the natural numbers. • 풟 is very rich: Every countable partial ordering WF: Let WF be the set of programs 횙 for which there embeds in 풟 as an initial segment. exists no sequence 푎0, 푎1, 푎2, ... of natural numbers There are many more results showing that 풟 is 3 such that 횙(푎푖+1) = 푎푖 for all 푖 ∈ ℕ. extremely complex. An example of a question that has To prove that any of these examples cannot be decided eluded an answer for decades is whether 풟 has a nontrivial by a computer program, one starts with the halting prob- automorphism. lem. For that, one uses a Cantor-style diagonalization argument to define, under the assumption that K is com- The Linearity Phenomenon putable, a computable function that is different from all We wanted to find a ruler that we can use to measure com- computable functions. For the other examples, the idea plexity, a hierarchy of complexity levels that we can use to classify problems. Instead, we’ve got an ordering, ≤푇, 2 The “K” is for Stephen Kleene. 4This isn’t easy at all. One has to encode the nuts and bolts of a Turing machine 3 The notation 횙(푎) = 푏 means that the program 횙 halts on input 푎 and out- into a presentation of a group in such a way that a certain word is equivalent to puts 푏. the identity if and only if the given Turing machine halts.

1210 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 8 that gives us a way of saying that certain problems are more theory, and up through the large-cardinal hypotheses, are complex than others and that some are incomparable. The linearly ordered under consistency strength, giving rise to world turned out messier than we wanted it to be. The ex- what is called the Gödel hierarchy. One can cook up ad hoc amples we gave above are ordered as follows under Turing theories that are incomparable under consistency strength, reduction: but the natural ones are always comparable. Again, we can’t even state this in a precise way, as we don’t know 0 <푇 K ≡푇 WP ≡푇 HTP <푇 TF <푇 COF <푇 TA <푇 WF. what makes a theory natural. Our recent paper with James They do seem to form a hierarchy. There are many more Walsh [10] takes a step toward that problem, proving that examples one can get from elsewhere in mathematics and the transfinite iterates of the consistency operator forma many more from computability theory that are still very spine of theories that is canonical in some sense. This must natural, even if they may look slightly weird to outsiders. be somewhat connected to the linearity phenomenon in 5 All the examples we know are ordered in a line. Further- computability theory, but we are far from understanding more, we know of no natural example strictly in between precisely how they are connected. 0 and K, and all natural examples strictly in between 0 and COF are Turing equivalent to either K or TF. We know Martin’s Measure ℕ many natural examples between COF and TA, though they Let us recall that we’ve got a preordering ≤푇 on ℕ given are nicely ordered in a line like the ordering of the natural by 푔 ≤푇 푓 if 푔 can be computed from 푓. The equivalence numbers. We know many natural examples between TA classes it induces are called Turing degrees, and ≤푇 par- and WF, and they are nicely ordered in a well-ordered line. tially orders the Turing degrees. This partial ordering is an The hierarchy we were looking for seems to exist, but 풟 upper-semilattice, meaning that every two elements have seems too chaotic to help us find it. The contrast between a least upper bound. Also, every countable subset has an the general behavior in 풟 and the behavior of the natu- upper bound, though maybe not a least one. Other than rally occurring objects is so stark that there must be a deep that, this partial ordering is extremely messy and very hard reason behind it. We need to dig deeper. to describe. The following peculiarity is the first indication Martin’s conjecture is a formal statement trying to cap- it is not all chaos. We need two definitions first: A subset ℕ ture the essence of this hierarchy within the Turing degrees. 풜 ⊆ ℕ is ≡푇-invariant if whenever 푓 ≡푇 푔, we have that For now, let us say that Martin’s conjecture gives a for- 푓 ∈ 풜 ⟺ 푔 ∈ 풜. The cone above 푓 ∈ ℕℕ is the set ℕ mal mathematical understanding of the following empiri- {푔 ∈ ℕ ∶ 푔 ≥푇 푓}. cal observation: ℕℕ Theorem (Martin’s Turing determinacy). Every ≡푇-invari- While the infinite sequences in are not linearly ℕ ordered by Turing computability, the naturally oc- ant Borel subset of ℕ either contains a cone or is disjoint from curring sequences are. a cone. What will allow us to formalize this statement is the obser- The topology on ℕℕ we are using to define the Borel vation that, in this setting, there are two properties natu- sets is the product of the discrete topology of ℕ.6 It is im- ral objects have that general objects do not: they relativize portant to point out that the theorem above is not about (to be defined below) and they are constructible. We can Borel sets: it is, essentially, about any set you can define formally define what a constructible relativizable object is, without invoking the black magic of the axiom of choice. and we can then write a formal mathematical statement I stated it using Borel sets because that’s the most we can saying they are nicely well-ordered by complexity. That is prove in ZFC, the standard axiomatization for all of math- Martin’s conjecture. Posed in the seventies, Martin’s con- ematics. Let me just say that if we add a few large-cardinal jecture is considered the most important open question assumptions, assumptions that set theorists believe to be in computability theory and of great foundational impor- true (whatever that means), then the theorem is true for a tance within mathematical logic. We will describe its state- much larger class of sets called 퐿(ℝ), the class of all sets ment in detail after we give more background and context. that are constructible over the reals. For a mathematician Similar Behavior in Proof Theory who isn’t a set theorist, the only way to build a subset of ℕℕ outside 퐿(ℝ) is by using the axiom of choice. The the- Another well-known expression of this linearity phenome- orem thus says that for every ≡푇-invariant constructible non is with axiomatizations of fragments of mathematics: ℕ set 풜, there is an 푓 ∈ ℕ such that either every 푔 ≥푇 푓 is the hundreds of axiomatizations for mathematics that logi- in 풜 or none is. cians have looked at, starting from elementary arithmetic, going up through Peano arithmetic, Zermelo–Fraenkel set 6Thus the Borel subsets of ℕℕ form the smallest class of sets that is closed un- 5A linear ordering is a partial ordering where every two elements are compara- der complements, countable unions, and countable intersections and contains ℕ ble. They are also called total orderings. the basic open sets 풱푛,푚 = {푓 ∈ ℕ ∶ 푓(푛) = 푚}.

SEPTEMBER 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1211 Thinking of cones as large sets, we can define a proba- constant-zero function 푛 ↦ 0∶ ℕ → ℕ. Then the halt- ℕ ′ bility measure on all Borel ≡푇-invariant subsets 풜 of ℕ : ing problem K has degree 0 , and one can show that so 푚(퐴) = 1 if 풜 contains a cone, and 푚(퐴) = 0 if 풜 is do WP and HTP. If we apply the Turing jump again, we disjoint from a cone. It is not hard to see that the intersec- get 0′′. It turns out that TF is Turing equivalent to 0′′. If tion of countably many cones contains a cone.7 Thus, this we take another jump, we get to 0′′′, which happens to measure is 휎-additive. The theorem above says that this be Turing equivalent to COF. We can then keep on iterat- measure is ≡푇-ergodic. A corollary of the theorem is that ing, but we will not get to TA in any finite number of steps. (휔) every Borel ≡푇-to-= invariant function must be constant However, TA is Turing equivalent to 0 , the set of pairs on a set of Martin’s measure 1, where ℱ∶ ℕℕ → ℕℕ is {(푛, 푚) ∶ 푛 ∈ 0(푚)}, where 0(푚) is the 푚th iterate of the ≡푇-to-= invariant if ℱ(푔) = ℱ(푓) whenever 푔 ≡푇 푓. This Turing jump and 휔 denotes the first ordinal number that resembles the ergodicity of Lebesgue measure with respect comes after all the natural numbers. The last example in to Vitali’s equivalence relation: If ℱ∶ [0, 1] → ℝ is a Borel our list, WF, is beyond any iteration of the Turing jump function such that whenever 푥 − 푦 ∈ ℚ, ℱ(푥) = ℱ(푦), along any computable ordinal. then ℱ is constant on a set of Lebesgue measure 1. Before getting deeper into the iterates of the jump, let us The reader might be wondering about examples of ≡푇- take a side step and see an example of how relativization invariant sets where one could apply the theorem. The works in conjunction with Martin’s measure. main tool to produce ≡푇-invariant sets is called relativization. Computable Categoricity To see how relativization works, let’s look at an example Relativization from computable structure theory. The reader may skip Given 푓 ∈ ℕℕ, the class of 푓-computable functions satis- this section if the reader is too eager to learn about Martin’s fies pretty much all the same properties as the class ofcom- conjecture in the next section. Computable structure the- putable functions: the partial 푓-computable functions are ory is the subfield of computability theory where we study closed under composition, recursion, and minimization; the complexity of countable structures. These are mathe- they contain the basic functions, namely the projections, matical structures like groups, fields, linear orderings, or the zero function, and the plus-one function; and there graphs. The objective is to understand the forms complex- is a universal partial 푓-computable function. These im- ity takes in this setting and find connections between com- ply pretty much all the other properties of the computable plexity and algebraic properties (see [11]). functions that are used in almost all the proofs in com- For the sake of simplicity, let us describe our example putability theory. Only if one gets to the inner workings only for the case of groups, even though it works for struc- of a Turing machine can one see a difference between these tures in general. A representation of a countable group 풢 = classes. Except for very few occasions, such as in the proof (퐺; 푒, ∗) is an isomorphic copy of 풢 with domain ℕ. That of undecidability of the word problem, it is extremely rare is, it is a group 풜 = (ℕ; 푒퐴 , ∗퐴 ) ≅ 풢, where 푒퐴 ∈ ℕ and 2 to find a proof that deals with the nuts and bolts of aTuring ∗퐴 ∶ ℕ → ℕ. The point of making the domain of the machine. Since this is so rare, pretty much all the results in group ℕ is that we can now use tools from computability computability theory could be restated using the notion of theory. Such a representation 풜 is said to be computable if 푓-computability instead of plain computability, and these 2 the group operation ∗퐴 ∶ ℕ → ℕ is computable. A group new statements would still be true by pretty much the same 풢 may have many representations, as there are many bi- proofs. This process is called relativization. jections between ℕ and 퐺, some of which might be com- When we relativize the halting problem to 푓, we obtain putable and some not. There may also be many different the set of all programs that are allowed to call 푓 during computable representations with different computational their computation and that eventually halt. We denote properties. Structures whose computable representations ′ this set by 푓 . This is an extremely important operation; all have the same computational properties are said to be it’s called the Turing jump. It is a monotone and order- computably categorical: preserving operation. That is, for all 푓, 푔 ∈ ℕℕ, Definition 1. A group 풢 is said to be computably categor- • 푓 < 푓′ and 푇 ical if any two computable representations of 풢 are com- • 푓 ≤ 푔 ⟹ 푓′ ≤ 푔′. putably isomorphic. That is, if 풜 = (ℕ; 푒퐴 , ∗퐴 ) and ℬ = In particular, the Turing jump is ≡푇-to-≡푇-invariant; (ℕ; 푒 , ∗ ) are isomorphic to 풢, and ∗ and ∗ are com- ′ ′ 퐵 퐵 퐴 퐵 i.e., 푓 ≡푇 푔 implies 푓 ≡푇 푔 . Let us use 0 to denote the putable functions, then there is a computable bijection

푝∶ ℕ → ℕ such that 푝(푒퐴 ) = 푒퐵 and 푝(푛∗퐴 푚) = 푝(푛)∗퐵 7 The intersection of the cones above 푓푛 for 푛 ∈ ℕ contains the cone above 푓 푝(푚). for any upper bound 푓 of all the 푓푛’s, as for instance the function 푓(⟨푛, 푚⟩) = 푛 푚 푓푛(푚) where ⟨푛, 푚⟩ is a number coding the pair (푛, 푚), say 2 3 . This is the case, for instance, of (ℚ; 0, +), because given

1212 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 8 a representation 풜 = (ℕ; 0퐴 , ⊕퐴 ) of (ℚ; 0, +), once we 푤1, ..., 푤ℓ on the letters 푎,̄ ̄푥, ̄푦 and their inverses that de- fix an element 1퐴 that we are going to map to 1ℚ, for any termine the automorphism orbit of 푝̄ in the sense that other 푎 ∈ 풜, all we need to do is search for 푛, 푚 ∈ ℕ a tuple 푞̄ is automorphic to 푝̄ if and only if 푞̄ satisfies ̄ ̄ such that 푣푖[ ̄푥↦ 푞,̄ ̄푦↦ 푏] = 푒 and 푤푗[ ̄푥↦ 푞,̄ ̄푦↦ 푏] ≠ 푒 for all 푖 ≤ 푘 and 푗 ≤ ℓ and some 푏̄ ∈ 퐺<ℕ. 퐴 퐴 퐴 either 푎⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⊕퐴 푎 ⊕퐴 ⋯ ⊕퐴 푎 = 1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⊕퐴 1 ⊕퐴 ⋯ ⊕퐴 1 Theorem ([9]). A group 풢 is computably categorical relative 푚 푛 to Martin’s-almost-all 푓’s if and only if it is ∃-atomic over some or 푎 ⊕ 푎 ⊕ ⋯ ⊕ 푎 ⊕ 1퐴 ⊕ ⋯ ⊕ 1퐴 = 0 , ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟퐴 퐴 퐴 퐴 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟퐴 퐴 퐴 tuple. 푚 푛 The use of Martin’s measure is necessary here. There and then map 푎 to 푛/푚 or −푛/푚. exist groups that are computable categorical and not ∃- This is not the case for (ℚ+; 1, ×): On the standard atomic, and groups that are ∃-atomic but not computable representation of (ℚ+; 1, ×), the power relation {(푎, 푏) ∈ +2 푛 categorical. Such groups need to be constructed in rather ℚ ∶ (∃푛 ∈ ℕ)푎 = 푏} can be easily computed us- ad hoc ways, making sure certain objects aren’t computable ing prime decompositions of the numerators and denom- by diagonalizing against each computable function. How- inators. However, in an arbitrary representation, deciding ever, if 풢 is a naturally defined group, then any proof that if 푏 is a power of 푎 requires going through all powers of shows that it is or isn’t computable categorical must rela- 푎 and checking if any of them equal 푏. One can build a tivize. So it would be either computably categorical rela- 풜 = (ℕ; 푒 , ∗ ) (ℚ+; 1, ×) computable representation 퐴 퐴 of tive to all 푓 or to no 푓. In that case, the usage of “almost all” where the power relation in the theorem above is not really relevant, and we don’t 2 even need to relativize the notion of computable categoric- {(푎, 푏) ∈ ℕ ∶ (∃푛 ∈ ℕ) 푎⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ∗퐴 푎 ∗퐴 ⋯ ∗퐴 푎 = 푏} ity. Thus, for naturally occurring 풢, 풢 is computably cate- 푛 gorical if and only if it is ∃-atomic over some tuple. Since is not computable. In other words, the fact that we can we do not know how to formally speak about naturally oc- compute the group operation doesn’t mean we can com- curring groups, the theorem above is the best we can do. pute the power relation. Since the power relation is computable in the standard representation of (ℚ+; 1, ×), Martin’s Conjecture these two representations cannot be computably isomor- Martin’s conjecture is about ≡푇-to-≡푇-invariant functions. phic. Whenever we have a natural object defined in computabil- Studying the difference between different representa- ity theory, we can relativize it. For example, if FIN is the tions of the same structures and the properties that are set of programs that halt on a finite number of inputs, then independent of representations is central to computable FIN푓 is the set of programs with oracle 푓 that halt on a fi- structure theory, and that’s why the notion of computable nite number of inputs. Whenever we define such an op- categoricity becomes so important. Unfortunately, as erator, we get a ≡푇-to-≡푇-invariant function. That is, if proved by Downey, Kach, Lempp, Lewis, Montalb´an,and 푓 푔 8 푓 ≡푇 푔, then FIN ≡푇 FIN . Thus, one way of analyzing Turetsky [2], there is no structural way to characterize the the naturally occurring objects is to look at the relativiz- groups that are computably categorical. This is unsatis- able ones by studying ≡푇-to-≡푇-invariant functions. If fying. The picture gets brighter, though, if we relativize. we have two such functions, ℱ and 풢, then the set of 푋 푓 ∈ ℕℕ 푓 Given , we say that a group is -computably categori- such that ℱ(푋) ≡푇 풢(푋) is either large or small. We say cal if any two 푓-computable representations are 푓- ▿ that ℱ is equivalent to 풢 on a cone and write ℱ ≡푇 풢 if 풢 computably isomorphic. Now, a given group might be ℱ(푋) ≡푇 풢(푋) for Martin’s-almost-all 푋. Again, for nat- 푓-computably categorical for some 푓’s and not for others. urally defined ℱ and 풢, this is either true for all 푋 or for From Martin’s theorem above, we get that a group 풢 is ei- no 푋, so it is not necessary to consider the cones. But to ther computably categorical relative to Martin’s-almost-all prove general statements, we need to compare on cones, 푓’s or computably categorical relative to Martin’s-almost- as otherwise one can easily build strange functions that 푓 no ’s. The meaning of “Martin’s-almost-all” comes, of satisfy ℱ(푋) ≤푇 풢(푋) for some 푋’s and not others. course, from Martin’s measure. The interesting point here We say that a ≡푇-to-≡푇-invariant function is constant is that we can structurally characterize the groups that are on a cone if, for all 푔 on some cone, ℱ(푔) has the same computably categorical relative to Martin’s-almost-all 푓’s. Turing degree. We include the theorem just to give the reader a taste of the type of structural conditions we deal with. 8One can generalize this example and consider any class 풞 ⊆ ℕ⊆ℕ of partial ℕ ⇀ ℕ 풞푓 <ℕ functions and let be the set of programs that, when run with ora- Definition 2. A group 풢 is ∃-atomic over a tuple 푎̄ ∈ 퐺 cle 푓, produce a partial function in 풞. The operator 푓 ↦ 풞푓 is then uniformly <ℕ if, for every tuple 푝̄ ∈ 퐺 , there exist words 푣1, ..., 푣푘, ≡푇-to-≡푇-invariant.

SEPTEMBER 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1213 Martin’s Conjecture. Every Borel ≡푇-to-≡푇-invariant func- rability on a cone and form a much finer hierarchy than tion is either constant on a cone or ≡푇-equivalent to an iterate the ≡푇-to-≡푇-invariant functions used in Martin’s conjec- (maybe transfinite) of the Turing jump on a cone. ture. By the same argument made throughout this paper, our result gives a full description of the natural many-one It implies, in particular, that the Borel ≡푇-to-≡푇- invariant functions are linearly ordered when compared degrees. Another interesting outcome of our result is that on a cone.9 it provides a concrete link between Martin’s conjecture and The original conjecture is not only for Borel functions the Wadge hierarchy, giving us a better understanding of but for all functions although working not over ZFC but where Martin’s conjecture comes from. The Wadge reducibility compares the complexities of over a system denoted by ZF+AD+DC that doesn’t include ℕ the full axiom of choice, where AD denotes the axiom of sets of sequences in ℕ instead of sets of numbers and, determinacy of infinite games and DC a weaker version in contrast to Turing and many-one reducibilities, has the of the axiom of choice called dependent choice. In that property that all degrees are natural, and hence there is no case, we don’t get that all functions are iterates of the Tur- distinction between the general behavior and that of natu- ing jump, but that they are well-ordered under comparabil- rally occurring objects. It is defined as follows: Given two ℕ ity on a cone and that the successor operator is the jump. sets of sequences, 풜, ℬ ⊆ ℕ , we say that 풜 is Wadge If the reader prefers to assume the axiom of choice but is reducible to ℬ and write 풜 ≤푤 ℬ if there is a continuous ℕ ℕ willing to accept some generally accepted large-cardinal hy- function 퐹∶ ℕ → ℕ such that ℎ ∈ 풜 ⟺ 퐹(ℎ) ∈ ℕ 10 pothesis, the statement would be about functions in 퐿[ℝ] ℬ for all ℎ ∈ ℕ . Wadge [14] showed that the Borel ℕ that are constructible without the use of the axiom of subsets of ℕ are linearly ordered by ≤푤 modulo taking choice. In any case, this conjecture is about much more complements: More precisely, he showed that for any two ̄ than Borel functions. Borel sets 풜 and ℬ, either 풜 ≤푤 ℬ or ℬ ≤푤 풜, where ̄ The conjecture is still open. An important version of it 풜 is the complement of 풜. Martin then showed that the was proved by Ted Slaman and John Steel in [12,13], a ver- Borel Wadge degrees are well-founded (i.e., have no infi- sion that could already be used to claim that the only natu- nite descending chains) and hence well-ordered if one ig- rally occurring Turing degrees are, in a sense, the iterates of nores complements. Thus there is no chaotic behavior on 11 the jump. What Slaman and Steel proved is that the con- the Borel Wadge degrees. jecture holds for uniformly ≡푇-to-≡푇-invariant functions. The beauty of the Wadge hierarchy was present to Mar- These are functions for which, if we know which Turing tin when he postulated his conjecture, though there was programs make 푓 and 푔 equivalent, we can figure out no clear connection between the two. That connection is which programs make ℱ(푓) and ℱ(푔) equivalent. The now made explicit by the Kihara–Montalbánresult, which whole argument we made about how natural objects in- proves that there is an order-preserving one-to-one corre- duce ≡푇-to-≡푇-invariant functions still holds for uniform- spondence between the uniformly ≡푇-to-≡푚-invariant ly ≡푇-to-≡푇-invariant functions. Thus, Slaman and Steel’s functions and the Wadge degrees. result already gets us what we wanted. The simplest nontrivial uniformly ≡푇-to-≡푚-invariant function is the Turing jump operator that maps a sequence Recent Connections with the Wadge Hierarchy 푓 ∈ ℕℕ to its jump 푓′. Under the correspondence in Let us finish this article by mentioning some recent work the Kihara–Montalbánresult, this operator corresponds to related to Martin’s conjecture and Martin’s measure. the Wadge degree of the nonclosed open sets.12 There are Takayuki Kihara and the author have recently discov- many more uniformly ≡푇-to-≡푚-invariant functions that ered a finer version of the uniform Martin’s conjecture [4]. are Turing equivalent to the Turing jump but not many- They looked at the relation of many-one reducibility, which one equivalent to it—there are uncountably many, actu- is a refinement of Turing reducibility that has also been ally, ordered as the ordinal 휔1, the first uncountable ordi- studied extensively over the last seventy years: It reduces nal, if we ignore complements. The connection between one decision problem into another by directly converting the original Martin’s conjecture and the many-one version instances of the former to instances of the latter. Formally, is given by the Turing jump: If 퐹∶ ℕℕ → ℕℕ is uniformly a set 퐴 ⊆ ℕ is 푚-reducible to a set 퐵 ⊂ ℕ, written 퐴 ≤푚 퐵, 10 ℕ if there is a computable function 푓∶ ℕ → ℕ such that Recall that the topology we use on ℕ is the product topology of the dis- ℕ 푛 ∈ 퐴 ⟺ 푓(푛) ∈ 퐵 푛 ∈ ℕ crete topology on , getting a space homeomorphic to that of the irrational real for all . The structure of numbers. the 푚-degrees is as chaotic as that of the Turing degrees. Ki- 11As before, we state these results in terms of Borel sets because that is how much hara and Montalbánproved that the uniformly ≡푇-to-≡푚- we can prove in ZFC, but they are not really about Borel sets. All of this holds for invariant functions are almost well-ordered under compa- all constructible sets in 퐿(ℝ) if one assumes the large-cardinal hypothesis and for all sets if one assumes the axiom of determinacy (AD) and forgets about the 9Recall that a linear ordering is a partial ordering where every two elements are axiom of choice. 12 comparable. All the nonclosed open sets are Wadge equivalent to each other.

1214 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 8 ≡푇-to-≡푇-invariant, then its composition with the Turing [4] Kihara T, Montalbán A. The uniform Martin’s con- ′ jump, 푋 ↦ 퐹(푋) , is uniformly ≡푇-to-≡푚-invariant. jecture for many-one degrees, Trans. Amer. Math. Soc., Hence the hierarchy of uniformly ≡푇-to-≡푇-invariant no. 12 (370):9025–9044, 2018, DOI 10.1090/tran/7519. MR3864404 functions can be studied by analyzing the uniformly ≡푇- to-≡ -invariant functions that are of the form 푋 ↦ 퐹(푋)′. [5] MontalbánA. Counting the back-and-forth types, J. Logic 푚 Comput., no. 4 (22):857–876, 2012, DOI 10.1093/log- Other Recent Work com/exq048. MR2956021 [6] MontalbánA. Rice sequences of relations, Philos. Trans. R. The remaining step to prove the full Martin’s conjecture is Soc. Lond. Ser. A Math. Phys. Eng. Sci., no. 1971 (370):3464– called Steel’s conjecture. It states that all Borel ≡푇-to-≡푇- 3487, 2012, DOI 10.1098/rsta.2011.0333. MR2943216 invariant functions are Turing equivalent on a cone to a [7] MontalbánA. A computability theoretic equivalent to uniformly ≡푇-to-≡푇-invariant function. Recent work by Vaught’s conjecture, Adv. Math. (235):56–73, 2013, DOI Andrew Marks shows how this statement is completely or- 10.1016/j.aim.2012.11.012. MR3010050 thogonal to the uniform Martin’s conjecture, that it is a [8] Montalbán A. A robuster Scott rank, Proc. Amer. whole world on its own, and that it is related to other sub- Math. Soc., no. 12 (143):5427–5436, 2015, DOI jects like Borel combinatorics and ergodic theory. Adam 10.1090/proc/12669. MR3411157 [9] Montalbán A. Effectively existentially-atomic struc- Day and Andrew Marks have given a way to understand tures, Computability and complexity; 2017:221–237. the step from the uniform to the nonuniform case of Mar- MR3629724 tin’s conjecture as a cocycle superrigidity result: There is [10] MontalbánA, Walsh J. On the inevitability of the con- a way to associate to every Turing-invariant function a co- sistency operator, J. Symb. Log., no. 1 (84):205–225, 2019, cycle of the free group on countably many generators so DOI 10.1017/jsl.2018.65. MR3922791 that Martin’s conjecture is true if and only if these cocycles [11] MontalbánA. Computable structure theory: Within the are superrigid on a cone. In particular, they showed that arithmetic. In preparation. Steel’s conjecture would follow from the following state- [12] Slaman TA, Steel JR. Definable functions on degrees, Cabal Seminar 81–85; 1988:37–55, DOI ment: Every Borel cocycle of the shift action of 퐹2, the 10.1007/BFb0084969. MR960895 free group on two generators, on the free part of 2퐹2 is a [13] Steel JR. A classification of jump operators, J. Symbolic conjugate of a homomorphism of 퐹2 on a set of Martin’s Logic, no. 2 (47):347–358, 1982, DOI 10.2307/2273146. 퐹2 measure 1. Here a cocycle is a function 훼∶ 퐹2 × 2 → 퐹2 MR654792 such that 훼(ℎ푔, 푋) = 훼(ℎ, 푔푋)훼(푔, 푋), and a conjugate [14] Wadge WW. Reducibility and determinateness on the Baire of a homomorphism is a cocycle of the form 훼(푔, 푋) = space, ProQuest LLC, Ann Arbor, MI, 1983. Thesis (Ph.D.)– −1 퐹2 푏(푔푋)ℎ(푔)푏 (푋), where 푏∶ 2 → 퐹2 is the conjugat- University of California, Berkeley. MR2633374 ing function and ℎ∶ 퐹2 → 퐹2 is a homomorphism. The extent to which Martin’s measure can be used to study the properties of natural objects in computability other than just Turing or many-one degrees is starting to be appreciated by recent work by the author and Matthew Harrison-Trainor, among others [1, 3, 5–8]. Most of this new work is in computable structure theory, in the vein of the section on computable categoricity above, where by analyzing computational properties of structures by their Martin’s-almost-everywhere behavior, one can obtain structural results that capture the behavior of naturally oc- Antonio Montalb án curring structures. Credits References Opening image by Stephan R. Garcia. [1] Csima BF, Harrison-Trainor M. Degrees of categoricity on Photo of Antonio Montalbánis courtesy of the author. a cone via 휂-systems, J. Symb. Log., no. 1 (82):325–346, 2017, DOI 10.1017/jsl.2016.43. MR3631290 [2] Downey R, Kach A, Lempp S, Lewis-Pye A, Mon- talbán A, Turetsky D. The complexity of computable categoricity, Adv. Math. (268):423–466, 2015, DOI 10.1016/j.aim.2014.09.022. MR3276601 [3] Harrison-Trainor M. Degree spectra of relations on a cone, Mem. Amer. Math. Soc., no. 1208 (253):v+107, 2018, DOI 10.1090/memo/1208. MR3803555

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