Rogers Semilattices in the Analytical Hierarchy: The Case of Finite Families

Let $\mathcal{S}$ be a countable set. A numbering of $\mathcal{S}$ is a surjective map $\nu$ from the set of natural numbers $\omega$ onto $\mathcal{S}$. The origins of the theory of numberings can be traced back to the works of Gödel [23] and Kleene [29]. The proof of Gödel’s incompleteness theorems uses an effective numbering of first-order formulae. Kleene (see Theorem XXII in § 65 of Ref. [29]) gave a construction of a universal partial computable function — this result provides a universal computable numbering for the family of all unary partial computable functions. At the end of 1950s, the foundations of the modern theory of numberings were developed by Kolmogorov and Uspenskii [31, 42] and, independently, by Rogers [37].

The algorithmic complexity of different numberings is typically compared via the notion of reducibility between numberings: A numbering $\nu$ is reducible to a numbering $\mu$, denoted by $\nu\leq\mu$, if there is total computable function $f(x)$ such that $\nu(n)=\mu(f(n))$ for all $n\in\omega$. More informally, there is an effective procedure which, given a $\nu$-index of an object from $\mathcal{S}$, computes a $\mu$-index for the same object.

Since the end of 1960s, the research in the theory of numberings has been mainly focused in the area of Rogers semilattices. We give a very brief overview of the classical setting in this area. From now on, we consider only families $\mathcal{S}$ containing subsets of $\omega$, i.e., we always assume that $\mathcal{S}\subset P(\omega)$ and $\mathcal{S}$ is countable.

Let $\mathcal{S}$ be a family of computably enumerable (c.e.) sets. A numbering $\nu$ of the family $\mathcal{S}$ is computable if the set

is c.e. A family $\mathcal{S}$ is computable if it has a computable numbering. In other words, the computability of $\mathcal{S}$ means that one can uniformly enumerate all sets from $\mathcal{S}$. Note that in general, this enumeration allows repetitions.

As simple examples of computable families, one can immediately recall the family of all finite sets and the family of all c.e. sets. A more delicate example can be constructed as follows. If a number $e\not\in\emptyset^{\prime}$, then our family $\mathcal{T}$ contains one-element sets $\{2e\}$ and $\{2e+1\}$. If $e\in\emptyset^{\prime}$, then the set $\{2e,2e+1\}$ belongs to $\mathcal{T}$. It is easy to see that the constructed family $\mathcal{T}$ admits a uniform enumeration and thus, $\mathcal{T}$ is computable. A more interesting feature of $\mathcal{T}$ is the following: Any computable numbering $\nu$ of $\mathcal{T}$ must have repetitions, i.e. there are indices $m\neq n$ with $\nu(m)=\nu(n)$.

In a standard recursion-theoretic way, the notion of reducibility between numberings gives rise to the corresponding upper semilattice: For a computable family $\mathcal{S}$, the Rogers semilattice of $\mathcal{S}$ contains the degrees of all computable numberings of $\mathcal{S}$. As per usual, here two numberings have the same degree if they are reducible to each other. Roughly speaking, the supremum of two numberings is provided by their join, see Section 2.1 for formal details.

To give a flavor of studies of computable families, we mention here two celebrated classical results on Rogers semilattices: Let $\mathcal{S}$ be a computable family, and let $R$ be its Rogers semilattice. Khutoretskii [28] proved that if $R$ contains more than one element, then $R$ is infinite. Selivanov [40] showed that an infinite $R$ cannot be a lattice.

Goncharov and Sorbi [24] started developing the theory of generalized computable numberings. One of their approaches to generalized computations can be summarized as follows. Let $\Gamma$ be a complexity class (e.g., $\Sigma^{0}_{1}$, $d$-$\Sigma^{0}_{1}$, $\Sigma^{0}_{n}$, or $\Pi^{1}_{n}$). A numbering $\nu$ of a family $\mathcal{S}$ is $\Gamma$-computable if the set $G_{\nu}$ from (1) belongs to the class $\Gamma$. We say that a family $\mathcal{S}$ is $\Gamma$-computable if it has a $\Gamma$-computable numbering. Note that the classical notion of a computable numbering becomes a synonym for a $\Sigma^{0}_{1}$-computable numbering.

In a similar way to computable families, one can define the Rogers semilattice $\mathcal{R}_{\Gamma}(\mathcal{S})$ for a $\Gamma$-computable family $\mathcal{S}$, see Section 2.1 for formal details.

We follow the approach of Goncharov and Sorbi, and study the following problem:

The current paper is a continuation of studies developed in Refs. [14, 12]. These papers concentrated on Rogers semilattices of $\Pi^{1}_{n}$-computable families. Dorzhieva [14] showed that for a $\Pi^{1}_{n}$-computable family $\mathcal{S}$, one of the following two conditions holds:

• (a)

  either the Rogers semilattice $\mathcal{R}_{\Pi^{1}_{n}}(\mathcal{S})$ contains only one element,

• (b)

  or the first-order theory $Th(\mathcal{R}_{\Pi^{1}_{n}}(\mathcal{S}))$ is hereditarily undecidable.

The article [12] proves the following: if the semilattice $\mathcal{R}_{\Pi^{1}_{n}}(\mathcal{S})$ contains more than one element, then for any non-zero $m\neq n$ and any $\Pi^{1}_{m}$-computable family $\mathcal{T}$, the structure $\mathcal{R}_{\Pi^{1}_{n}}(\mathcal{S})$ is not isomorphic to $\mathcal{R}_{\Pi^{1}_{m}}(\mathcal{T})$. Further related work is discussed in Section 2.2.

The paper [12] left the following problem open:

While attacking Problem 1.2, we observed the following: In order to apply the known numbering-theoretic techniques in this setting, one needs to employ additional set-theoretic assumptions.

This observation motivated us to organize our paper as follows: We prove a number of results concerning Problem 1.1 under the assumption of Projective Determinacy (PD, see Section 3.2 for the details). Why did we choose PD as an additional axiom? Tanaka [41] already initiated a systematic development of recursion theory for the levels of analytical hierarchy, under the assumption of PD. We found his approach well-suited to our goals.

In order to make our paper accessible to both numbering-theoretic and set-theoretic communities, we tried to make the exposition as self-contained as possible.

The structure of the paper is as follows. Section 2 contains the necessary preliminaries on the theory of numberings. Section 3 discusses the background on the analytical hierarchy. In particular, Section 3.2 introduces consequences of PD, which will be employed in our proofs. It is well-known that under PD, the levels of analytical hierarchy exhibit “flip-flopping” behavior: kindred levels are $\Pi^{1}_{1}$, $\Sigma^{1}_{2}$, $\Pi^{1}_{3}$, $\Sigma^{1}_{4}$, …(see, e.g., Refs. [25, 34]). Thus, following Refs. [2, 41], we use the following conventions:

• •

  For a number $k\in\omega$, $E^{1}_{2k+1}:=\Pi^{1}_{2k+1}$ and $E^{1}_{2k+2}:=\Sigma^{1}_{2k+2}$.

• •

  For a non-zero $n\in\omega$, we consider $E^{1}_{n}$-computable families $\mathcal{S}$. By $\mathcal{R}^{1}_{n}(\mathcal{S})$ we denote the Rogers semilattice $\mathcal{R}_{E^{1}_{n}}(\mathcal{S})$.

Section 4 studies elementary properties of the semilattices $\mathcal{R}^{1}_{n}(\mathcal{S})$ for finite families $\mathcal{S}$. Any such $\mathcal{R}^{1}_{n}(\mathcal{S})$ is a distributive upper semilattice (Proposition 4.1). Note that the proof of this fact does not require PD. While assuming PD, we obtain a criterion for when $\mathcal{R}^{1}_{n}(\mathcal{S})$ has the greatest element (Theorem 4.1). We also show that if $\mathcal{R}^{1}_{n}(\mathcal{S})$ has no greatest element, then it is upwards dense (Theorem 4.2).

Section 5 discusses first-order properties of $\mathcal{R}^{1}_{n}(\mathcal{S})$ for infinite families $\mathcal{S}$. We make use of Dorzhieva’s results from Ref. [14] to establish the following:

1. (1)

   For an infinite $E^{1}_{n}$-computable family $\mathcal{S}$, the upper semilattice $\mathcal{R}^{1}_{n}(\mathcal{S})$ is not weakly distributive (Theorem 5.1). Consequently, $\mathcal{R}^{1}_{n}(\mathcal{S})$ is not distributive.

2. (2)

   For an arbitrary $E^{1}_{n}$-computable family $\mathcal{S}$ which contains at least two elements, the semilattice $\mathcal{R}^{1}_{n}(\mathcal{S})$ is infinite, and it is not a lattice (Corollary 5.1).

Summarizing, our results (together with Lemma 3.1 below) provide a first step to the solution of Problem 1.2 — now we know that under PD, there are at least four different isomorphism types of Rogers $E^{1}_{n}$-semilattices $\mathcal{R}^{1}_{n}(\mathcal{S})$, induced via the following families:

1. (1)

   A one-element family $\mathcal{S}$ — in this case, the structure $\mathcal{R}^{1}_{n}(\mathcal{S})$ is also one-element.

2. (2)

   A finite family $\mathcal{S}$, containing more than one element and possessing the $\subseteq$-least element.

3. (3)

   A finite family $\mathcal{S}$ without the $\subseteq$-least element.

4. (4)

   An infinite $E^{1}_{n}$-computable family $\mathcal{S}$.

The last section discusses further problems. In particular, we consider the following question: What happens to our results, if we replace PD with the Axiom of Constructibility ($V=L$)?

Recall that the Axiom of Dependent Choices (DC) states the following: For any non-empty set $A$ and any set of pairs $P\subseteq A\times A$, we have:

Throughout the paper, we work in set theory ZF$+$DC.

Lower-case letters $x,y,z,\dots$ denote variables that range over $\omega$. Capital letters $X,Y,Z,\dots$ are used for subsets of $\omega$.

By $\leq_{\omega}$ we denote the standard ordering of natural numbers. Recall that $\omega^{\omega}$ is the set of all total functions acting from $\omega$ to $\omega$.

As per usual, $\langle\cdot,\cdot\rangle$ is a standard pairing function over $\omega$. By $\langle\cdot\rangle_{0}$ and $\langle\cdot\rangle_{1}$ we denote computable functions such that for every $n\in\omega$, we have $\langle\langle n\rangle_{0},\langle n\rangle_{1}\rangle=n$.

We treat upper semilattices as structures in the language $L_{usl}=\{\leq,\vee\}$.

Here we discuss known results on the analytical hierarchy, which will be employed in our proofs. Furthermore, the section includes proofs of several useful (but a little bit technical) results on numberings: Lemma 3.1, Proposition 3.1, and Lemma 3.2.

Recall that a predicate $R(x_{1},\dots,x_{m};f_{1},\dots,f_{n})$, where the variables $f_{i}$ denote elements from $\omega^{\omega}$, is recursive if there is an index $e\in\omega$ such that for all $f_{1},\dots,f_{n}$ and $x_{1},\dots,x_{m}$, the following conditions hold:

• (a)

  the value $\varphi_{e}^{f_{1}\oplus\dots\oplus f_{n}}(x_{1},\dots,x_{m})$ is defined;

• (b)

  the predicate $R(x_{1},\dots,x_{m};f_{1},\dots,f_{n})$ is true if and only if

  $$\varphi_{e}^{f_{1}\oplus\dots\oplus f_{n}}(x_{1},\dots,x_{m})=1.$$

If an index $e$ satisfies these conditions, then we say that $e$ witnesses the recursiveness of the predicate $R$.

We follow Ref. [39] and use the following version of a normal form for analytical subsets of $\omega$: Let $n$ be a non-zero natural number. A set $X\subseteq\omega^{m}$ is $\Pi^{1}_{n}$ if and only if there is a recursive predicate $R(x_{1},\dots,x_{m},y;f_{1},\dots,f_{n})$ such that for all $\bar{a}\in\omega^{m}$, we have

where the last quantifiers are as follows:

Mutatis mutandis, a $\Sigma^{1}_{n}$ set $X\subseteq\omega^{m}$ can be represented via the following form:

Let $\Gamma\in\{\Sigma^{1}_{n},\Pi^{1}_{n}\,\colon n\geq 1\}$. By $\breve{\Gamma}$ we denote the dual class:

The next lemma will be useful for transferring various results from a class $\Gamma$ into its dual $\breve{\Gamma}$.

For the sake of convenience, from now on, we will use the following notation:

The section discusses elementary properties of the semilattices $\mathcal{R}^{1}_{n}(\mathcal{S})$, where $\mathcal{S}$ is finite. As a warm-up, we apply standard numbering-theoretic techniques (see, e.g., Ref. [6]) to prove that $\mathcal{R}^{1}_{n}(\mathcal{S})$ is a distributive upper semilattice (Proposition 4.1). We also show that if $\mathcal{S}$ contains more than one element, then $\mathcal{R}^{1}_{n}(\mathcal{S})$ is infinite and cannot be a lattice (Proposition 4.2). Note that these proofs do not require additional set-theoretic assumptions. After that, we obtain a criterion for when $\mathcal{R}^{1}_{n}(\mathcal{S})$ has the greatest element (Theorem 4.1). The proof of this criterion already heavily employs the consequences of PD discussed in Section 3.2. After that, the developed techniques are used to prove the following: If $\mathcal{R}^{1}_{n}(\mathcal{S})$ has no greatest element, then it is upwards dense (Theorem 4.2).

Before proceeding to main results, we observe the following very simple fact:

Recall that an upper semilattice $\mathcal{A}=(A;\leq,\vee)$ is distributive if for all $b,a_{0},a_{1}\in\mathcal{A}$, the following holds:

The next two propositions establish general facts about Rogers $E^{1}_{n}$-semilattices of finite families, note that these facts do not depend on PD.

For an $E^{1}_{n}$ set $A$, consider a one-element family $\mathcal{T}:=\{A\}$. It is obvious that the family $\mathcal{T}$ has only one numbering, and hence, the semilattice $\mathcal{R}^{1}_{n}(\mathcal{T})$ is one-element.

The next proposition shows that if a finite family $\mathcal{S}$ contains more than one element, then for this $\mathcal{S}$, one can recast the results of Khutoretskii [28] and Selivanov [40], mentioned in the introduction.