The complexity of completions in partial combinatory algebra

Abstract

We discuss the complexity of completions of partial combinatory algebras, in particular, of Kleene’s first model. Various completions of this model exist in the literature, but all of them have high complexity. We show that although there are no computable completions, there exist completions of low Turing degree. We use this construction to relate completions of Kleene’s first model to complete extensions of $\mathrm{PA}$. We also discuss the complexity of pcas defined from nonstandard models of $\mathrm{PA}$.

1. Introduction

Partial combinatory algebra (pca) generalizes the setting of classical combinatory algebra (ca) to structures with a partial application operator. The first entry in the literature is Feferman (1975), which is surprisingly late, some fifty years after the invention of combinatory algebra and the closely related lambda calculus, although the concept of a pca existed before that (see Section 5). Apart from this connection with lambda calculus, pcas have played a notable part in constructive mathematics. At the end of Section 2 below, we list a number of key examples of pcas, and say something about their role in various settings.

Since the application operator in pcas is partial, they can often be naturally represented as c.e. structures in the sense of Selivanov (2003) and Khoussainov (2018). The computable structure theory of pcas as partial c.e. structures was recently studied by Fokina and Terwijn (2024). Since pcas can be seen as abstract models of computation, it is only natural to consider their complexity as algebraic structures from the viewpoint of computability theory. At least for countable pcas, there is a straightforward definition of their complexity in terms of the complexity of a presentation, as in computable model theory. Below we formulate this using numberings (Definition 3.1). Some of the complexity of pcas was studied earlier in Shafer and Terwijn (2021) and Golov and Terwijn (2023). In the current paper, we focus on the complexity of completions, and in particular of completions of what is called Kleene’s first model $\mathcal{K}_1$ , with application defined in terms of partial computable functions on the natural numbers.

A completion of a pca is a total pca (i.e. a combinatory algebra in the classical sense, in which applications are always defined) in which the pca can be embedded. Not every pca has a completion, as was first proved in Klop (1982). On the other hand, Kleene’s $\mathcal{K}_1$ does have a completion. This follows from the sufficient condition for completability given in Bethke et al. (1996). This yields a certain term model $T(\omega )/\!\!\sim$ as a completion of $\mathcal{K}_1$ . In Section 5 below, we will discuss how Scott’s graph model, which is another important example of a pca, can also be seen as a (weak) completion of $\mathcal{K}_1$ . We note, however, that these completions of $\mathcal{K}_1$ have high complexity, which brings up the question of what the optimal complexity of such a completion could be. Although no computable completions of $\mathcal{K}_1$ exist (cf. Theorem 5.2 and the remarks following it), we show that there exist completions of $\mathcal{K}_1$ of low Turing degree (Theorem 5.3). Such completions are close to computable in the sense that the complexity of their halting problem is the same as the standard halting problem.

All this suggests a connection with complete extensions of Peano arithmetic, for which a similar story exists. Note, however, that we are talking here about pcas, that is, the models of a theory, rather than the theory itself. Nevertheless, in Section 6, we show that indeed there is a connection. Finally, in Section 7, we discuss the complexity of pcas resulting from nonstandard models of $\mathrm{PA}$ .

Our notation from computability theory is mostly standard. For unexplained notions, we refer to Odifreddi (1989) or Soare (1987). In particular, $\omega$ denotes the natural numbers, and $\varphi _e$ the $e$ -th partial computable (p.c.) function. For any set $A$ , $A'$ denotes the Turing jump of $A$ , and in particular $\emptyset '$ denotes the halting problem.

2. Partial Combinatory Algebras

To make the paper self-contained, we briefly review the basic definitions from partial combinatory algebra. Our presentation follows van Oosten (2008).

A partial applicative structure (pas) is a set $\mathcal{A}$ together with a partial map $\cdot$ from $\mathcal{A}\times \mathcal{A}$ to $\mathcal{A}$ . We usually write $ab$ instead of $a\cdot b$ , and think of this as “ $a$ applied to $b$ .” If this is defined, we denote this by $ab\!\downarrow$ . By convention, application associates to the left, so we write $abc$ instead of $(ab)c$ . Terms over $\mathcal{A}$ are built from elements of $\mathcal{A}$ , variables, and application. If $t_1$ and $t_2$ are terms, then so is $t_1t_2$ . If $t(x_1,\ldots, x_n)$ is a term with variables $x_i$ , and $a_1,\ldots, a_n \in \mathcal{A}$ , then $t(a_1,\ldots, a_n)$ is the term obtained by substituting the $a_i$ for the $x_i$ . For closed terms (i.e. terms without variables) $t$ and $s$ , we write $t \simeq s$ if either both are undefined, or both are defined and equal. Here application is strict in the sense that for $t_1t_2$ to be defined, it is required that both $t_1$ and $t_2$ are defined.

Definition 2.1. A pas $\mathcal{A}$ is called combinatory complete if for any term $t(x_1,\ldots, x_n,x)$ , $n\geqslant 0$ , with free variables among $x_1,\ldots, x_n,x$ , there exists $b\in \mathcal{A}$ such that for all $a_1,\ldots, a_n,a\in \mathcal{A}$ ,

1. (i) $ba_1\cdots a_n\!\downarrow$ ,

2. (ii) $ba_1\cdots a_n a \simeq t(a_1,\ldots, a_n,a)$ .

A pas $\mathcal{A}$ is a pca if it is combinatory complete. A ca is a pca for which the application operator is total.

Combinatory completeness of pcas can be characterized by the existence of combinators $k$ and $s$ , just as in classical ca and lambda calculus. In van Oosten (2008), it is stated that the following theorem is “essentially due to Feferman (1975).”

Theorem 2.2. (Feferman) A pas $\mathcal{A}$ is a pca if and only if it has elements $k$ and $s$ with the following properties for all $a,b,c\in \mathcal{A}$ :

• • $ka\!\downarrow$ and $kab = a$ ,

• • $sab\!\downarrow$ and $sabc \simeq ac(bc)$ .

In the following, we will always assume that our pcas are nontrivial, that is, have more than one element. This automatically implies that they are infinite and have $k\neq s$ .

The prime example of a pca is Kleene’s first model $\mathcal{K}_1$ that was already mentioned in the introduction. This is a model defined on the natural numbers, with application

\begin{equation*} n \cdot m = \varphi _n(m). \end{equation*}

Thus $\mathcal{K}_1$ models the setting of classical computability theory. We can also relativize this to an arbitrary oracle $X$ , thus obtaining the relativized pca $\mathcal{K}_1^X$ .

Kleene’s second model $\mathcal{K}_2$ , from the book Kleene and Vesley (1965), is a pca defined on Baire space $\omega ^\omega$ . Application $\alpha \cdot \beta$ in this model can be informally described as applying the continuous functional with code $\alpha$ to the real $\beta$ . The original coding of $\mathcal{K}_2$ is a bit cumbersome, but it is essentially equivalent to

\begin{equation*} \alpha \cdot \beta = \Phi _{\alpha (0)}^{\alpha \oplus \beta }, \end{equation*}

where $\Phi _e$ is the $e$ -th Turing functional, and the application is understood to be defined if the RHS is total. This coding, used in Shafer and Terwijn (2021), is easier to work with. See the appendix of Golov and Terwijn (2023) for a proof (and precise statement) of the equivalence with the original coding.

An interesting variant of $\mathcal{K}_2$ , called the van Oosten model, is obtained by extending the domain to include partial functions, cf. van Oosten (1999). $\mathcal{K}_2$ is uncountable, but restricting attention to computable sequences gives a countable pca $\mathcal{K}_2^{\mathrm{eff}}$ . Similarly, restricting to $X$ -computable sequences gives a pca $\mathcal{K}_2^X$ for every $X$ . In Golov and Terwijn (2023), the relations between these and other pcas are studied using embeddings.

Many other examples of pcas can be found in the literature. For example, pcas have been extensively used in constructive mathematics, see Beeson (1985), Troelstra and van Dalen (1988). In particular, they have been used as a basis for models of constructive set theory, as in McCarty (1986), Rathjen (2006), and Frittaion and Rathjen (2021). Pcas are also pivotal in the categorical treatment of realizability, cf. van Oosten (2008, Chapter 2) where they serve as a basis for realizability toposes. In particular, Hyland’s famous effective topos is the realizability topos of $\mathcal{K}_1$ . For a categorical characterization of pcas, see Cockett and Hofstra (2008) (extending early work of Longo and Moggi 1984); for a discussion of pcas in the context of oracles see Kihara (2022).

A pca that has been particularly important in connection with ca and lambda calculus is Scott’s graph model (Scott 1975). This pca is a model of the lambda calculus (see Barendregt 1984), and it is also closely related to the enumeration degrees in computability theory, cf. Odifreddi (1999). We will discuss this model in Section 5, where we also explain how the restriction of this model to the c.e. sets can be seen as a completion of $\mathcal{K}_1$ .

3. Effective Presentations of Pcas

Below, we will call a ca $\mathcal{A}$ $Y$ -computable if $\mathcal{A}$ has a representation such that the application $\cdot$ in $\mathcal{A}$ is $Y$ -computable. We will also require that equality on $\mathcal{A}$ is $Y$ -decidable. The following definition (similar to notions used in Golov and Terwijn 2023) makes this precise, using numberings to represent $\mathcal{A}$ . Recall that a numbering is a surjective function $\gamma :\omega \to \mathcal{A}$ . We think of $n\in \omega$ as a code for $\gamma (n)\in \mathcal{A}$ .

Definition 3.1. Let $\mathcal{A}$ be a pca and $Y\subseteq \omega$ . We call $\mathcal{A}$ partial $Y$ -computable if there exist a numbering $\gamma :\omega \to \mathcal{A}$ and a partial $Y$ -computable function $\psi$ such that for all $n$ and $m$ , $\gamma (n)\cdot \gamma (m)\!\downarrow$ in $\mathcal{A}$ if and only if $\psi (n,m)\!\downarrow$ , and

(1) \begin{equation} \gamma (n)\cdot \gamma (m) = \gamma (\psi (n,m)). \end{equation}

We also require that equality on $\mathcal{A}$ is $Y$ -decidable, meaning that the set $\{(n,m)\mid \gamma (n)=\gamma (m)\}$ is $Y$ -computable. If $\mathcal{A}$ is total, i.e., a ca, then $\psi$ is total, and we simply call $\mathcal{A}$ $Y$ -computable. (This is consistent with Definition 3.2 below.)

Notice that the numbering $\gamma$ in Definition 3.1 is not required to be computable in any way. Also, nontrivial combinatorial algebras are never computable, cf. Barendregt (1984, 5.1.15).

Definition 3.1 focuses on representing application in a pca as a p.c. function. For the record, we also mention another way to define effective representations.

Definition 3.2. We call a pca $\mathcal{A}$ $Y$ -c.e. if there exist a numbering $\gamma :\omega \to \mathcal{A}$ such that the set

(2) \begin{equation} \big \{ (n,m,k)\mid \gamma (n)\cdot \gamma (m)\!\downarrow = \gamma (k) \big \} \end{equation}

is $Y$ -c.e. Again we also require that equality on $\mathcal{A}$ is $Y$ -decidable, meaning that the set $\{(n,m)\mid \gamma (n)=\gamma (m)\}$ is $Y$ -computable. We call $\mathcal{A}$ $Y$ -computable if the set (2) is $Y$ -computable. ¹

As an example, note that $\mathcal{K}_1$ is p.c. in the sense that $a\cdot b$ is a p.c. function on $\omega$ , and that $\mathcal{K}_1$ is c.e. in the sense that $a\cdot b\!\downarrow = c$ is a c.e. relation. We note here that the two definitions are equivalent:

Proposition 3.3. A pca is partial $Y$ -computable if and only if it is $Y$ -c.e.

Proof. We always have c.e. implies p.c.: Given $n,m$ , search for $k$ such that $(n,m,k)$ is in the set (2), and define $\psi (n,m)$ to be the least $k$ found. (Note that $\gamma$ need not be injective, so there may be multiple such $k$ .) Then Equation (1) holds for $\psi (n,m)$ .

The converse direction uses the condition that equality on $\mathcal{A}$ is decidable. Given $\mathcal{A}$ p.c. and $\psi$ satisfying (1), enumerate $(n,m,k)$ if $\psi (n,m)\!\downarrow$ and $\gamma (k) = \gamma (\psi (n,m))$ . This gives an enumeration of Equation (2).

Note that the above definitions are in the spirit of the c.e. structures in Selivanov (2003), where they are called positive structures. These are defined as structures in which the predicates are c.e., and the functions are computable. The latter makes sense for total functions, but in the case of pcas, we are dealing with a partial application operator, in which case it is natural to have this as a c.e. function.

For pcas on $\omega$ (i.e. with $\gamma :\omega \to \mathcal{A}$ the identity), we have that equality on $\mathcal{A}$ is decidable, so the two notions of pca are equivalent. This is the type of pca that was used in Fokina and Terwijn (2024). Without the condition that equality on $\mathcal{A}$ is decidable (or c.e.), it is not clear that the two definitions are equivalent in general, though we do not know of an example of a pca that is p.c. but not c.e.

Finally, in the case of a $Y$ -computable pca or ca $\mathcal{A}$ (which is what we will mostly use below), the requirement that equality on $\mathcal{A}$ is $Y$ -decidable actually follows from the fact that Equation (2) is $Y$ -computable, namely let $n$ be such that $\gamma (n)$ is the identity (which exists in any pca).

4. Embeddings and Isomorphisms

There are at least three notions of embedding for pcas, depending on what structure is required to be preserved. For example, the choice of the combinators $k$ and $s$ from Theorem 2.2 can be regarded as part of the structure or not. For instance, Zoethout (2022, p. 33) does not consider $k$ and $s$ to be part of the structure of a pca. We have the following notions of embedding of pcas:

• • Only preserve applications. This notion was studied in Bethke (1988), Asperti and Ciabattoni (1997), Shafer and Terwijn (2021), and Golov and Terwijn (2023).

• • Besides applications, also preserve $k$ and $s$ , for a particular choice of these combinators. This stronger notion was studied in Bethke et al. (1996, 1999).

• • There is an even weaker notion of embedding, using the notion of applicative morphism, that was introduced in Longley (1994), see also Longley and Normann (2015). Applicative morphisms do not have to preserve applications; instead, there have to be terms in the codomain that simulate applications in the domain. This notion is useful in realizability theory, see van Oosten (2008).

Our primary interest here is the notion of embedding where $k$ and $s$ are not considered part of the signature, but we will also be using the stronger notion of embedding, especially when we talk about completions. To distinguish the two, we will refer to them as weak and strong embeddings. (In Golov and Terwijn 2023, weak embeddings were simply called embeddings.) To distinguish applications in different pcas, we also write $\mathcal{A}\models a\cdot b\!\downarrow$ if this application is defined in $\mathcal{A}$ .

Definition 4.1. For given pcas $\mathcal{A}$ and $\mathcal{B}$ , an injection $f: \mathcal{A} \to \mathcal{B}$ is a weak embedding if for all $a, b \in \mathcal{A}$ ,

(3) \begin{equation} \mathcal{A}\models ab\!\downarrow \; \Longrightarrow \; \mathcal{B}\models f(a)f(b)\!\downarrow \, = f(ab). \end{equation}

If $\mathcal{A}$ embeds into $\mathcal{B}$ , in this way we write $\mathcal{A}\hookrightarrow \mathcal{B}$ . If in addition to (3), for a specific choice of combinators $k$ and $s$ of $\mathcal{A}$ , $f(k)$ and $f(s)$ serve as combinators for $\mathcal{B}$ , we call $f$ a strong embedding.

A (total) ca $\mathcal{B}$ is called a weak completion of $\mathcal{A}$ if there exists a weak embedding $\mathcal{A} \hookrightarrow \mathcal{B}$ . If the embedding is strong, we call $\mathcal{B}$ a strong completion.

Two pcas $\mathcal{A}$ and $\mathcal{B}$ are isomorphic, denoted by $\mathcal{A}\cong \mathcal{B}$ , if there exists a bijection $f:\mathcal{A}\rightarrow \mathcal{B}$ such that for all $a,b\in \mathcal{A}$ , $ab\!\downarrow$ if and only if $f(a)f(b)\!\downarrow$ , and in this case

\begin{equation*} f(a)\cdot f(b) = f(ab). \end{equation*}

Besides the term completion, in the literature also the term extension is used. Bethke et al. (1999, Definition 1.5) call a pca $\mathcal{B}$ an extension of a pca $\mathcal{A}$ if $\mathcal{A} \subseteq \mathcal{B}$ , the application $\cdot _{\mathcal{A}}$ in $\mathcal{A}$ is the restriction of application $\cdot _{\mathcal{B}}$ in $\mathcal{B}$ to the domain of $\cdot _{\mathcal{A}}$ , and $\mathcal{B}$ and $\mathcal{A}$ both have the same combinators $k$ and $s$ as in Theorem 2.2.

Now suppose that $f:\mathcal{A}\hookrightarrow \mathcal{B}$ is a strong embedding. Then $f(\mathcal{A})\subseteq \mathcal{B}$ is an extension in the above sense, where both $f(\mathcal{A})$ and $\mathcal{B}$ have combinators $f(k)$ and $f(s)$ . Note that $\mathcal{A} \cong f(\mathcal{A})$ if we define application in $f(\mathcal{A})$ by $f(\mathcal{A})\models f(a)\cdot f(b) \!\downarrow =f(c)$ if and only if $\mathcal{A}\models a \cdot b\!\downarrow = c$ . So we see that total extensions and completions amount to the same thing, provided that in both cases we have to specify whether to also fix $s$ and $k$ or not.

In Terwijn (2023), it is shown that weak and strong embeddability and completions are different: There exists a pca that is weakly completable, but not strongly completable. ²

5. Complexity of Completions of $\mathcal{K}_1$

It was an important open question in the 1970s whether every pca has a strong completion. The question was raised by Barendregt, Mitschke, and Scott, and discussed at a meeting in Swansea in 1974, cf. Bethke et al. (1999). (Note that this predates Feferman’s paper Feferman 1975.) A negative answer was obtained by Klop (1982), see also Bethke et al. (1999). Other examples of incompletable pcas can be found in Bethke (1987) and Bethke and Klop (1996).

In contrast to these examples, $\mathcal{K}_1$ does have strong completions. This follows from the criterion given in Bethke et al. (1996) about the existence of unique head-normal forms, which is satisfied in $\mathcal{K}_1$ . The completion of $\mathcal{K}_1$ resulting from this is a certain term model $T(\omega )/\!\!\sim$ . On the face of it, the equivalence relation $\sim$ is not computable, since it is essentially equivalence of terms in $\mathcal{K}_1$ . That indeed it cannot be computable follows from Theorem 5.2, and also from the fact that computable combinatorial algebras do not exist.

We now discuss how another famous pca can be seen as a completion of $\mathcal{K}_1$ . Scott’s graph model $\mathcal{G}$ is a pca defined on the power set $\mathcal{P}(\omega )$ , with application defined by

\begin{equation*} X \cdot Y = \big \{ x \mid \exists u (\langle x,u\rangle \in X \wedge D_u\subseteq Y) \big \}. \end{equation*}

Here $D_u$ as always denotes the finite set with canonical code $u$ , and $\langle \cdot, \cdot \rangle$ denotes an effective pairing function. $\mathcal{E}$ is defined as the restriction of $\mathcal{G}$ to the c.e. sets. That $\mathcal{G}$ and $\mathcal{E}$ are (total) cas is implicit in Scott (1975). Note the close connection with enumeration reducibility (cf. Odifreddi 1999, XIV): For all sets $Y$ and $Z$ , $Z\leqslant _e Y$ is equivalent with $X\cdot Y = Z$ for some c.e. set $X$ .

In Golov and Terwijn (2023, Corollary 7.5, it was shown that $\mathcal{K}_1 \hookrightarrow \mathcal{E}$ , so that we can see $\mathcal{E}$ as a weak completion of $\mathcal{K}_1$ . Note that equality on $\mathcal{E}$ is equality of c.e. sets, which is $\Pi ^0_2$ -complete when we represent c.e. sets by their indices. ³ So this is more complicated than equality in the term model $T(\omega )/\!\!\sim$ .

We can see $\mathcal{E}$ as a combination of $\mathcal{K}_1$ and $\mathcal{K}_2$ . Indeed we have

\begin{equation*} \mathcal {K}_1 \hookrightarrow \mathcal {E} \hookrightarrow \mathcal {K}_2 \end{equation*}

(the latter by Golov and Terwijn 2023, Corollary 6.2, so that we can view $\mathcal{E}$ as a kind of middle ground between Kleene’s models. This combination famously gives a model of the $\lambda$ -calculus, as shown in Scott (1975), see Odifreddi (1999, XIV.4).

Below, we use that for an embedding $f:\mathcal{K}_1\hookrightarrow \mathcal{A}$ of $\mathcal{K}_1$ into a pca $\mathcal{A}$ , it suffices to know the value of $f$ on finitely many elements. This observation was also used in Golov and Terwijn (2023, Theorem 4.1), and it can be used to bypass the fact that embeddings such as $f$ do not have to be computable. Below we give a somewhat simpler version of this trick, using the following lemma.

Lemma 5.1. There exist elements $t_n\in \mathcal{K}_1$ , $n\geqslant 1$ , such that for all $n$ and $m$ ,

\begin{align*} t_n \cdot m = \begin{cases} n & \textrm{if}\ m=0 \\ t_{n+1} & \textrm{if}\ m\gt 0. \end{cases} \end{align*}

Proof. By the recursion theorem, let $d\in \omega$ be a code such that

\begin{align*} \varphi _d(n,m) = \begin{cases} n & \text{if $m=0$} \\ S^1_1(d,n+1) & \text{if $m\gt 0$.} \end{cases} \end{align*}

Here $S^1_1$ is the primitive recursive function from the S-m-n-theorem. Define $t_n = S^1_1(d,n)$ . W.l.o.g. we may assume $t_n\gt 0$ for all $n$ . Then $\varphi _{t_n}(m) = \varphi _{S^1_1(d,n)}(m) = \varphi _d(n,m) = t_{n+1}$ for $m\gt 0$ and equal to $n$ otherwise.

In Golov and Terwijn (2023, Corollary 4.2, it was proved that if $\mathcal{K}_1^X\hookrightarrow \mathcal{A}$ is a weak embedding of $\mathcal{K}_1^X$ into a pca $\mathcal{A}$ with $Y$ -c.e. inequality, then $X\leqslant _T Y$ . We obtain a stronger conclusion when we assume that $\mathcal{A}$ is total and $Y$ -computable.

Theorem 5.2. Suppose $\mathcal{A}$ is a $Y$ -computable combinatorial algebra, and that $f:\mathcal{K}_1^X\hookrightarrow \mathcal{A}$ is a weak embedding. Then $X \lt _T Y$ .

Proof. The successor function $S$ in $\mathcal{K}_1$ satisfies $S^n(0)=n$ , but we need an element $t$ such that the $n$ -fold application $t\cdot \ldots \cdot t\cdot 0$ equals $n$ , with the convention that application associates to the left, not to the right. Let $t = t_1$ be as in Lemma 5.1. Then for the $n$ -fold application, we have $t\cdot \ldots \cdot t\cdot 0 = t_n \cdot 0 = n$ for every $n\gt 0$ . Since $f$ is an embedding, we obtain from this

(4) \begin{equation} f(n) = f(t\cdot \ldots \cdot t\cdot 0) = f(t)\cdot \ldots \cdot f(t)\cdot f(0), \end{equation}

with the applications repeated $n$ times. So we see that the image of $f$ is completely determined by $f(t)$ and $f(0)$ .

To show that $Y\not \leqslant _T X$ , let $A$ , $B$ be a $X$ -computably inseparable pair of $X$ -c.e. sets, and let $e$ be a code such that for all $x$ ,

\begin{align*} \varphi _e^X(x) = \begin{cases} 0 & \text{if $x\in A$} \\ 1 & \text{if $x\in B$} \\ \uparrow & \text{otherwise}. \end{cases} \end{align*}

Then we have in particular that

\begin{align*} \mathcal{K}_1^X \models e \cdot x \!\downarrow = 0 &\Longrightarrow \mathcal{A} \models f(e)\cdot f(x) = f(0), \\ \mathcal{K}_1^X \models e \cdot x \!\downarrow = 1 &\Longrightarrow \mathcal{A} \models f(e)\cdot f(x) = f(1). \end{align*}

Since $\mathcal{A}$ is total, for every $x$ the application $f(e)\cdot f(x)$ is always defined in $\mathcal{A}$ , and by Equation (4), it is equal to a term containing only $f(t)$ , $f(0)$ , and application. Because $\mathcal{A}$ is $Y$ -computable, we can compute a code of $f(e)\cdot f(x)$ effectively from $x$ . (All we need is $e$ , and codes of $f(t)$ and $f(0)$ , all of which are fixed.) Furthermore, since the definition of $Y$ -computable pca entails that equality on $\mathcal{A}$ is $Y$ -decidable, we can decide with $Y$ whether $f(e)\cdot f(x)$ is equal to $f(0)$ or $f(1)$ or not. It follows that the set $C = \{x \mid \mathcal{A}\models f(e)\cdot f(x) = f(0)\}$ is $Y$ -computable and separates $A$ and $B$ , and since $A$ and $B$ have no $X$ -computable separation $Y$ is not $X$ -computable.

That $X\leqslant _T Y$ can be shown using a very similar argument. Instead of $\varphi _e^X$ above, use the characteristic function $\varphi _d^X(x)$ which is $0$ if $x\in X$ and $1$ if $x\notin X$ . Then the rest of the argument above, replacing $e$ with $d$ , shows that $X$ is $Y$ -computable. So we have $Y\not \leqslant _T X$ and $X\leqslant _T Y$ , hence $X\lt _T Y$ .

From Theorem 5.2, we see that, in particular, $\mathcal{K}_1$ does not have a computable weak completion, which also follows from the fact that combinatorial algebras are never computable, see Barendregt (1984, 5.1.15). We now show that this is optimal, namely that there exist completions of low Turing degree. (Recall that $Y$ is low if $Y'\leqslant _T \emptyset '$ ).

Theorem 5.3. There exists a strong completion $\mathcal{A}$ of $\mathcal{K}_1$ of low Turing degree, that is, $\mathcal{A}$ is $Y$ -computable such that $Y$ is low.

Proof. The outline of the proof is as follows. We first define a first-order base theory $\mathrm{Cmpl}$ such that each model of $\mathrm{Cmpl}$ gives rise to a strong completion of $\mathcal{K}_1$ . The theory $\mathrm{Cmpl}$ will be consistent because we already know that $\mathcal{K}_1$ has strong completions. We then use standard recursion theory to obtain a complete and consistent extension of $\mathrm{Cmpl}$ of low degree. This does not immediately give a completion of $\mathcal{K}_1$ of low degree, but we use a model-theoretic argument to obtain a completion of the desired complexity.

The language of $\mathrm{Cmpl}$ is two-sorted, ⁴ with a predicate $N(x)$ intended to range over natural numbers, and a predicate $A(x)$ intended to range over a pca $\mathcal{A}$ that is a completion of $\mathcal{K}_1$ . Furthermore, the language has a function symbol $f$ with the intended meaning that $f:\mathcal{K}_1\to \mathcal{A}$ is a strong embedding. The language for the sort $N$ is the same as the language of arithmetic, and for this sort, we take the axioms of $\mathrm{PA}$ . The language of the sort $A$ is that of pcas, with one function symbol $\cdot$ for application in $\mathcal{A}$ . Since $\cdot$ will be required to be total we add it as a function symbol, rather than as a relation symbol, which would have been more appropriate for a partial operation. By arithmetization, we may assume that expressions of the form $\varphi _a(b)\!\downarrow =c$ are directly expressible for the sort $N$ for all standard numbers $a,b,c\in \omega$ , where we represent a number $n\in \omega$ by the term $S^n(0)$ .

So as axioms of $\mathrm{Cmpl}$ , we have the following:

• • The axioms of $\mathrm{PA}$ for the sort $N$ (i.e. all axioms relative to $N$ ).

• • Axioms expressing that $f$ is an embedding from $\mathcal{K}_1$ to $\mathcal{A}$ :

1. − $\forall a \in N (f(a)\in A)$ .

2. − $\forall a,b \in N(f(a)=f(b) \rightarrow a=b)$ .

3. − For all $a,b,c\in \omega$ , we have an axiom

   (5) \begin{equation} \mathcal{K}_1 \models a\cdot b\!\downarrow =c \;\Longrightarrow \; \mathcal{A} \models f(a)\cdot f(b) = f(c). \end{equation}
   Note that the LHS can be expressed for the sort $N$ using the language of arithmetic, using terms $S^n(0)$ to express the natural number $n$ , and the RHS can be expressed for the sort $A$ .

• • To ensure that $f$ is a strong embedding, we fix standard combinators $s$ and $k$ in $\mathcal{K}_1$ satisfying the axioms of Theorem 2.2. Note that these can be expressed for the sort $N$ . Also, note that $s$ and $k$ are just standard numbers, so we do not need to add them to the signature. Next, we add axioms expressing that $f(s)$ and $f(k)$ also satisfy the axioms of Theorem 2.2, but now for the sort $A$ . The existence of these combinators $f(s)$ and $f(k)$ automatically ensures that $\mathcal{A}$ forms a pca. The fact that $\mathcal{A}$ should be total is handled by the fact that application is a function symbol in the language, so no explicit axiom is needed for this.

Taken together, the axioms of $\mathrm{Cmpl}$ express that $f$ is a strong embedding from $\mathcal{K}_1$ to $\mathcal{A}$ . Every model $M$ of $\mathrm{Cmpl}$ gives a strong completion of $\mathcal{K}_1$ as follows. Denote by $M \upharpoonright N$ and $M \upharpoonright A$ the part of $M$ restricted to the sorts $N$ and $A$ . Then $M \upharpoonright N$ is a model of $\mathrm{PA}$ and $\mathcal{A}=M \upharpoonright A$ is a pca. Furthermore, the restriction of $f^M$ to the standard numbers $n\in \omega$ is an injection of $\omega$ into $\mathcal{A}$ , which, by Equation (5), is an embedding of $\mathcal{K}_1$ , which is strong because $f(s)$ and $f(k)$ satisfy the axioms of Theorem 2.2. The values of $f^M$ on possible nonstandard elements of $M \upharpoonright N$ are irrelevant.

Since $\mathrm{Cmpl}$ is a computable axiomatization, by a result of Shoenfield (cf. Cenzer 1999, Theorem 6.1), the set of complete and consistent extensions of it can be represented as a $\Pi ^0_1$ -class, that is, there is a computable tree $T \subseteq 2^{\lt \omega }$ such that the set of infinite paths $[T]$ consists of all complete and consistent extensions of $\mathrm{Cmpl}$ . (We encode sentences by natural numbers, so that paths in $2^\omega$ correspond to sets of sentences.) Note that the theory $\mathrm{Cmpl}$ is consistent because we know by Bethke et al. (1996) that there exists a strong completion of $\mathcal{K}_1$ . In particular, the tree $T$ is infinite, and $[T]$ is nonempty. By the Low Basis Theorem (Jockusch and Soare 1972), $T$ has a path of low Turing degree, which gives us a complete and consistent extension $X$ of $\mathrm{Cmpl}$ of low degree.

Since $X$ is consistent, it has a model $M$ by the completeness theorem, and by the remarks above $M$ defines a strong completion of $\mathcal{K}_1$ , namely $M \upharpoonright A$ . However, there is no guarantee that this completion is $X$ -computable. But we do not need all of $M \upharpoonright A$ ; it suffices to consider the smaller pca $\mathcal{A}$ consisting of all terms built from $f(n)$ for standard numbers $n\in \omega$ (represented as terms $S^n(0))$ , and application. Note that $\mathcal{A}$ is a pca because of the presence of $f(s)$ and $f(k)$ , and $\mathcal{A}$ is total since for $u$ and $v$ of the given form, $u\cdot v$ is again of this form. Not every element of $\mathcal{A}$ is of the form $f(n$ ), for example, it has terms $f(a)\cdot f(b)$ such that $\varphi _a(b)\!\uparrow$ . The pca $\mathcal{A}$ is a sub-pca of $M \upharpoonright A$ in the sense of Shafer and Terwijn (2021). To finish the proof of the theorem, we note that $\mathcal{A}$ is $X$ -computable. Namely, given to terms $u$ and $v$ of the form above, we can simply compute their application as the term $u\cdot v$ . Equality of terms in $\mathcal{A}$ is $X$ -decidable because the theory $X$ is complete and thus contains all equalities $u=v$ and $u\neq v$ of such terms. So the sub-pca $\mathcal{A}$ of $M \upharpoonright A$ is total and $X$ -computable, and hence of low degree since $X$ is low.

6. Complete Extensions of $\mathrm{PA}$

Following modern terminology, we call a Turing degree a PA degree if it is the degree of a complete extension of Peano arithmetic (cf. Downey and Hirschfeldt 2010, p. 84). We will simply call a set PA-complete if it has PA degree.

In this section, we show that every (strong or weak) completion of $\mathcal{K}_1$ computes a PA degree, and vice versa. Since there exist PA-complete sets of low degree (Downey and Hirschfeldt 2010, p. 87), Theorem 5.3 follows from the statement of this equivalence; however, this does not make the proof of Theorem 5.3 superfluous, since the tree $T$ from its proof is used in the proof of the equivalence.

Proposition 6.1. Every PA-complete set computes a strong completion of $\mathcal{K}_1$ .

Proof. By results of Scott and Solovay (cf. Odifreddi 1989, V.5.36), a set $Y$ is PA-complete if and only if it can compute an element of every nonempty $\Pi ^0_1$ -class. In particular, $Y$ can compute an element of $[T]$ for the computable tree $T$ from the proof of Theorem 5.3. By the rest of the proof of Theorem 5.3, this implies that $Y$ computes a strong completion of $\mathcal{K}_1$ , namely the term model defined at the end of the proof.

The following result strengthens Theorem 5.2.

Theorem 6.2. Suppose $\mathcal{A}$ is a $Y$ -computable combinatorial algebra, and that $f:\mathcal{K}_1\hookrightarrow \mathcal{A}$ is a weak embedding. Then $Y$ is PA-complete.

Proof. According to Jockusch and Soare (1972), a set is PA-complete if and only if it can compute a separation of an effectively inseparable pair of c.e. sets. (See also Downey and Hirschfeldt 2010, p. 86.) Now let $A$ , $B$ be a pair of effectively inseparable c.e. sets, for example, we can take the provable and refutable sentences of PA (Odifreddi 1989, p. 513). Then the proof of Theorem 5.2 shows that $Y$ computes a separation of $A$ and $B$ , and hence $Y$ is PA-complete.

Putting Proposition 6.1 and Theorem 6.2 together, we obtain the following characterization:

Corollary 6.3. The following are equivalent for any set $A$ :

1. (i) $A$ computes a weak completion of $\mathcal{K}_1$ ,

2. (ii) $A$ computes a strong completion of $\mathcal{K}_1$ ,

3. (iii) $A$ is PA-complete.

In the case of PA degrees, more is known, namely that they are closed upwards. We do not know whether the degrees of (weak or strong) completions of $\mathcal{K}_1$ are also upwards closed.

7. Nonstandard Models of $\mathrm{PA}$

As mentioned in Beeson (1985, VI.2.5) and van Oosten (2008), every model $M$ of Peano Arithmetic $\mathrm{PA}$ defines a pca on $M$ , with application defined by

(6) \begin{equation} a\cdot b \!\downarrow = c \text{ if } M \models \varphi _a(b)\!\downarrow = c \end{equation}

for all $a,b,c\in M$ . By restricting Equation (6) to standard numbers $a,b,c\in \omega$ , we obtain a pca on $\omega$ , which we will call $\mathcal{K}_1(M)$ . Note that $\mathcal{K}_1(M)$ is just Kleene’s first model $\mathcal{K}_1$ “inside $M$ .” It is a pca because we can pick combinators $k,s \in \mathcal{K}_1$ as in Theorem 2.2 such that $\mathrm{PA}$ proves that they have the required properties.

Note that for $a,b,c\in \omega$ we have that $a\cdot b\!\downarrow = c$ in $\mathcal{K}_1$ if and only if $\exists y(T(a,b,y) \wedge U(y)=c)$ , where $T$ and $U$ are the primitive recursive predicate and function from Kleene’s normal form theorem (cf. Odifreddi 1989). In a nonstandard model $M$ , this $y$ can be nonstandard, so that more computations converge than in reality. In particular, in general, we have

(7) \begin{equation} \mathcal{K}_1 \models a\cdot b \!\downarrow = c \; \Longrightarrow \; \mathcal{K}_1(M) \models a\cdot b \!\downarrow = c, \end{equation}

but not conversely. For example, consider a model $M$ of $\mathrm{PA} + \neg \mathrm{con}(\mathrm{PA})$ , where $\mathrm{con}(\mathrm{PA})$ expresses the consistency of $\mathrm{PA}$ . Such models exist by Gödel’s second incompleteness theorem. If we consider the p.c. function $\varphi _a$ that on input $b$ searches for a proof of an inconsistency in $\mathrm{PA}$ , then the computation $a\cdot b$ will converge in $\mathcal{K}_1(M)$ , but not in $\mathcal{K}_1$ (assuming $\mathrm{PA}$ is consistent).

Note that by Equation (7), we have an embedding $\mathcal{K}_1 \hookrightarrow \mathcal{K}_1(M)$ for every model $M$ of $\mathrm{PA}$ . This is in fact a strong embedding, as the same combinators $s$ and $k$ can be used in $\mathcal{K}_1(M)$ .

Since $\mathrm{PA}$ proves that certain p.c. functions are nontotal, any model of $\mathrm{PA}$ has nontotal p.c. functions. This remains true if we restrict to standard numbers, that is, there are standard numbers $e,n\in \omega$ such that $\mathrm{PA}$ proves that $\varphi _e(n)$ never halts. In particular, we see that $\mathcal{K}_1(M)$ is never a total pca (i.e. a ca). Therefore, we have:

Proposition 7.1. $\mathcal{K}_1(M)$ is never a weak completion of $\mathcal{K}_1$ .

By Tennenbaum’s theorem (cf. Boolos and Jeffrey 1974), there are no computable nonstandard models of $\mathrm{PA}$ . More precisely, there are no nonstandard models in which $+$ is computable. (This is an extension of Tennenbaum’s theorem due to Kreisel.) It follows that there are also no nonstandard models that are c.e., because $+$ is a total operation, and in a c.e. model, it would actually be computable, contradicting Kreisel’s result. So it would seem that the pcas $\mathcal{K}_1(M)$ for nonstandard models $M$ cannot be used for the problems about c.e. pcas discussed in computable structure theory (see Fokina and Terwijn 2024). However, for models $M_0$ and $M_1$ of $\mathrm{PA}$ , we do not have in general that

\begin{equation*} M_0 \not \cong M_1 \Longrightarrow \mathcal {K}_1(M_0) \not \cong \mathcal {K}_1(M_1). \end{equation*}

To see this, let $M_0=\omega$ be the standard model, and let $M_1$ be a nonstandard model that has the same first-order theory as $\omega$ (which exists by the compactness theorem). Then $M_0 \not \cong M_1$ , but $\mathcal{K}_1(M_0) = \mathcal{K}_1(M_1) = \mathcal{K}_1$ .

In particular, we see that it is possible that $\mathcal{K}_1(M)$ is c.e. (in the sense of Definition 3.2) for a nonstandard model $M$ of $\mathrm{PA}$ . This prompts the following question:

Question 7.2. What are the possible c.e. degrees for such $\mathcal{K}_1(M)$ aaaa can $\mathcal{K}_1(M)$ be c.e. but not equal to $\mathcal{K}_1$ aaaa

In the following, we note that though $\mathcal{K}_1(M)$ is always noncomputable, it can have low degree (if we do not require that it is also c.e.).

Proposition 7.3. $\mathcal{K}_1(M)$ is always noncomputable.

Proof. Consider the computably inseparable pair of sets

\begin{align*} A &= \{x\in \omega \mid \varphi _x(x)\!\downarrow =0\} \\ B &= \{x\in \omega \mid \varphi _x(x)\!\downarrow =1\}. \end{align*}

Now consider the set $C = \{x\in \omega \mid \mathcal{K}_1(M) \models x\cdot x\!\downarrow =0\}$ . $C$ is computable from $\mathcal{K}_1(M)$ , and it follows from Equation (7) that it separates $A$ and $B$ , from which it follows immediately that $\mathcal{K}_1(M)$ cannot be computable.

Proposition 7.4. $\mathcal{K}_1(M)$ can have low Turing degree.

Proof. We have to show that there exists a model $M$ and a low set $Y$ such that $\mathcal{K}_1(M)$ is $Y$ -computable (in the sense of Definition 3.2), that is, such that the set

\begin{equation*} Z= \{(a,b,c)\mid \mathcal {K}_1(M)\models a\cdot b\!\downarrow = c\} \end{equation*}

is $Y$ -computable. According to Jockusch and Soare, there exists a complete and consistent extension $X$ of $\mathrm{PA}$ of low Turing degree. Let $M$ be a model with theory $X$ . We can identify the set $Z$ with the set of sentences $a\cdot b\!\downarrow = c$ that hold in $M$ . Since $Z$ is then just a subset of $X$ consisting of sentences of a specific form, there is a computable set $R$ such that $Z = R\cap X$ , and we have $(R\cap X)' \leqslant _T X'\leqslant _T \emptyset '$ so that $Z$ is low.