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Mathematical Logic: A Course with Exercises Part II: Recursion Theory, Godel's Theorems, Set Theory, Model Theory

Rene Cori, Daniel Lascar and Donald H. Pelletier

Chapter 5

Recursion theory - all with Video Answers

Educators

Chapter Questions

02:35

Problem 1

Show that every finite subset of $\mathbb{N}$ is primitive recursive.

Angelo Rendina
Angelo Rendina
Numerade Educator
02:54

Problem 2

Show that the function $f$ defined by
$$
\begin{aligned}
f(0) &=f(1)=1, \\
f(n+2) &=f(n)+f(n+1)
\end{aligned}
$$
is primitive recursive.

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
01:07

Problem 3

Set $\mathcal{S}^{*}=\bigcup_{p>0} \mathbb{N}^{p}$ and define a map $\alpha$ from $\mathcal{S}^{*}$ into $\mathbb{N}$ as follows: if $\sigma$ is a sequence of integers of length $p$, then $\alpha(\sigma)=\alpha_{2}\left(p, \alpha_{p}(\sigma)\right)$.
(a) Show that the function $\alpha$ is injective and that its range is a primitive recursive set.
(b) Show that there exists a primitive recursive function $g$ such that if $\sigma=$ $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ and if $b=\sup \left(n, a_{1}, a_{2}, \ldots, a_{n}\right)$, then $\alpha(s) \leq g(b) .$
(c) Show that the function $\phi$ defined by
$$
\phi(p, i, x)= \begin{cases}\beta_{p}^{i}(x) & \text { if } 1 \leq i \leq p \\ 0 & \text { otherwise }\end{cases}
$$
is primitive recursive.
(d) We now define another coding: let $\gamma$ be the function which, with every $\left(a_{0}, a_{1}, a_{2}, \ldots, a_{p}\right) \in \mathcal{S}^{*}$, associates the integer
$$
\gamma\left(\left(a_{0}, a_{1}, \ldots, a_{p}\right)\right)=\pi(0)^{a_{0}+1} \cdot \pi(1)^{a_{1}+1} \cdot \ldots \cdot \pi(p)^{a_{p}+1}
$$
it is understood that the value of $\gamma$ on the empty sequence is 1 . Show that $\gamma$ is an injective map and that its range is a primitive recursive set.
(e) Show that the two codings can be obtained from one another in a primitive recursive way; more precisely, show that there exist two primitive recursive functions $f$ and $h$ of one variable such that
(i) for all $x$ in the range of $\alpha, f(x)=\gamma(\sigma)$, where $\sigma$ is the non-empty sequence satisfying $\alpha(\sigma)=x$;
(ii) for all $x$ in the range of $\gamma, h(x)=\alpha(\sigma)$, where $\sigma$ is the non-empty sequence satisfying $\gamma(\sigma)=x$.

Carson Merrill
Carson Merrill
Numerade Educator
06:10

Problem 4

Show that the function whose value at $n$ is the $n$th digit in the decimal expansion of $e$ (the real number that is the basis for natural logarithms) is primitive recursive.

Chris Trentman
Chris Trentman
Numerade Educator
04:03

Problem 5

(a) Let $p$ be a positive integer. Show that the set
$$
E=\left\{\left(a_{0}, a_{1}, \ldots, a_{p}\right) \in \mathbb{N}^{p+1}:\right.
$$
the polynomial $a_{0}+a_{1} X+\cdots+a_{p} X^{p}$ has a root in $\left.\mathbb{Z}\right\}$
is primitive recursive.
(b) Repeat question (a) replacing $\mathbb{Z}$ by $\mathbb{Q}$.
(c) Show that the set
$$
\begin{aligned}
F=&\left\{\Omega(\sigma): p \text { is an integer, } \sigma=\left(a_{0}, a_{1}, \ldots, a_{p}\right)\right. \text { and the polynomial }\\
&\left.a_{0}+a_{1} X+\cdots+a_{p} X^{p} \text { has a root in } \mathbb{Z}\right\}
\end{aligned}
$$
is primitive recursive. ( $\Omega$ is defined in the section on codings of sequences.)

Nick Johnson
Nick Johnson
Numerade Educator
09:01

Problem 6

Let $L$ be a language whose only symbol $R$ represents a binary predicate and let $F$ be a closed formula of $L$. The spectrum of $F$, which we will denote by $S p(F)$, is defined to be the set
$\{n \in \mathbb{N}: F$ has a model of cardinality $n\} .$
(See Exercise 10 of Chapter 3.)
Show that $S p(F)$ is a primitive recursive set.

Chris Trentman
Chris Trentman
Numerade Educator
02:35

Problem 7

For each of the following functions, construct a Turing machine that computes it: (a) $\lambda x \cdot x^{2}$,
(b) $\lambda x y \cdot x y$,
(c) $\lambda x \cdot x-1$
(d) $\lambda x y \cdot x-y$

Aman Gupta
Aman Gupta
Numerade Educator
13:52

Problem 8

Construct a Turing machine that halts if and only if the integer represented on its first band at the initial instant is even.

Chris Trentman
Chris Trentman
Numerade Educator
23:32

Problem 9

(a) Show that if a partial function $f \in \mathcal{F}_{1}^{*}$ is $T$-computable, then it is computable by a Turing machine that has exactly three bands.
(b) Consider the set $\mathcal{M}_{n}$ of Turing machines that have three bands and $n$ states. Set these machines in operation with an initial configuration in which all bands are clean. If machine $\mathcal{M}$ halts, we let $\sigma(\mathcal{M})$ be the number of strokes written on its second band at the instant it halts; otherwise, we set $\sigma(\mathcal{M})=0$. Show that the set
$$
\left\{\sigma(\mathcal{M}): \mathcal{M} \in \mathcal{M}_{n}\right\}
$$
is bounded. We will denote the upper bound of this set by $\Sigma(n)$.
(c) Let $f$ be a partial function of one variable that is computable by a machine $\mathcal{M}$ in $\mathcal{M}_{n} .$ For every integer $p$, construct a machine $\mathcal{N}_{p}$ with three bands which, when started in an initial configuration in which all bands are clean, begins by writing $p$ strokes on its first band, then returns its head to the beginning of the tape, and continues to behave exactly as $\mathcal{M}$ would.
How many states does $\mathcal{N}_{p}$ have?
(d) Show that the function $\Sigma$ is not $T$-computable.

Chris Trentman
Chris Trentman
Numerade Educator
00:53

Problem 10

Let $f \in \mathcal{F}_{1} .$ Show that $f$ is recursive if and only if its graph
$$
G=\left\{(x, y) \in \mathbb{N}^{2}: y=f(x)\right\}
$$
is recursive.

Adrian Co
Adrian Co
Numerade Educator
44:28

Problem 11

The purpose of this exercise is to provide a direct proof of the fact that Ackerman's function is recursive.
Define the following binary relation $\ll$ on $\mathbb{N}^{3}:(a, b, c) \ll\left(a^{\prime}, b^{\prime}, c^{\prime}\right)$ if and only if
$$
\begin{aligned}
&\sup (a, b, c)<\sup \left(a^{\prime}, b^{\prime}, c^{\prime}\right), \quad \text { or } \\
&\sup (a, b, c)=\sup \left(a^{\prime}, b^{\prime}, c^{\prime}\right) \text { and } a<a^{\prime}, \quad \text { or } \\
&\sup (a, b, c)=\sup \left(a^{\prime}, b^{\prime}, c^{\prime}\right) \text { and } a=a^{\prime} \text { and } b<b^{\prime}, \quad \text { or } \\
&\sup (a, b, c)=\sup \left(a^{\prime}, b^{\prime}, c^{\prime}\right) \text { and } a=a^{\prime} \text { and } b=b^{\prime} \text { and } c \leq c^{\prime} .
\end{aligned}
$$
(a) Show that $\ll$ is a total ordering.
If $\alpha$ and $\beta$ belong to $\mathbb{N}^{3}$, we will say that $\alpha$ is less than or equal (respectively, greater than or equal) to $\beta$ if $\alpha \ll \beta$ (respectively, $\beta \ll \alpha$ ). We will say that $\alpha$ is strictly less (respectively, strictly greater) than $\beta$ if, in addition, $\alpha \neq \beta$.
Show that for all $(a, b, c) \in \mathbb{N}^{3}$, the set
$$
\left\{(x, y, z) \in \mathbb{N}^{3}:(x, y, z) \ll(a, b, c)\right\}
$$
has at most $(\sup (a, b, c)+1)^{3}$ elements. Show that every element $(a, b, c) \in$ $\mathbb{N}^{3}$ has an immediate successor [i.e. there exists an element that is strictly greater than $(a, b, c)$ and is less than or equal to all elements that are strictly greater than $(a, b, c)]$. We will explicitly describe this immediate successor.
(b) Show that there exist three primitive recursive functions $\gamma_{1}, \gamma_{2}$, and $\gamma_{3}$ from $\mathbb{N}$ into $\mathbb{N}$ such that (i) the function $\Gamma$ from $\mathbb{N}$ into $\mathbb{N}^{3}$ defined by
$$
\Gamma(n)=\left(\gamma_{1}(n), \gamma_{2}(n), \gamma_{3}(n)\right)
$$
is a bijection;
(ii) for all integers $n$ and $m, n \leq m$ if and only if $\Gamma(n) \ll \Gamma(m)$.
(c) Let $H$ be the subset of $\mathbb{N}$ defined recursively by the following condition: $n \in H$ if and only if
$$
\begin{array}{lll}
\gamma_{2}(n)=0 & \text { and } & \gamma_{1}(n)=2^{\gamma_{3}(n)} ; & \text { or } \\
\gamma_{3}(n)=0 & \text { and } & \gamma_{1}(n)=1 ; & \text { or }
\end{array}
$$
$$
\gamma_{2}(n) \neq 0 \quad \text { and } \quad \gamma_{3}(n) \neq 0
$$
and there exist integers $p$ and $q$ strictly less than $n$ such that $p \in H, q \in H$, $\gamma_{2}(p)=\gamma_{2}(n), \gamma_{3}(p)=\gamma_{3}(n)-1, \gamma_{2}(q)=\gamma_{2}(n)-1, \gamma_{3}(q)=\gamma_{1}(p)$ and $\gamma_{1}(n)=\gamma_{1}(q)$.
Show that $H$ is primitive recursive.
As in the body of Chapter 5 , let $\zeta$ denote Ackerman's function. Show that, for every integer $n, n \in H$ if and only if $\gamma_{\mathrm{I}}(n)=\zeta\left(\gamma_{2}(n), \gamma_{3}(n)\right)$.
(d) Show that the graph
$$
G=\{(y, x, z): z=\zeta(y, x)\}
$$
of Ackerman's function is primitive recursive. Show that Ackerman's function is recursive.

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
08:42

Problem 12

Show that if $f$ is a function of one variable that is recursive and increasing, then its range is a recursive set. Conversely, show that every infinite recursive set is the range of a strictly increasing recursive function.

Chris Trentman
Chris Trentman
Numerade Educator
01:40

Problem 13

Suppose that $f \in \mathcal{F}_{1}$ is a recursive function and assume that its image is infinite. Show that there exists a recursive function $g \in \mathcal{F}_{1}$ that is recursive and injective and satisfies $\operatorname{lm}(f)=\operatorname{lm}(g)$. Conclude from this that there exists an injective recursive function whose image is not recursive.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:27

Problem 14

Show that every infinite recursively enumerable set includes an infinite recursive set.

Angelo Rendina
Angelo Rendina
Numerade Educator
00:59

Problem 15

Let $\alpha$ be a recursive function that is injective. We set
$$
\begin{aligned}
&A=\operatorname{Ran}(\alpha) \\
&B=\{x: \text { there exists } y>x \text { such that } \alpha(y)<\alpha(x)\}
\end{aligned}
$$
(a) Show that $B$ is recursively enumerable and that its complement is infinite.
(b) Assume that there exists an infinite recursively enumerable subset $C \subset \mathbb{N}$ that is disjoint from $B$. Show that $A$ is recursive.
(c) Show that there exists a recursively enumerable set which has (1) a nonempty intersection with every infinite recursively enumerable set, and (2) an infinite complement.

Srilakshmi E K
Srilakshmi E K
Numerade Educator
01:07

Problem 16

(a) Show that the set of recursive bijections from $\mathbb{N}$ onto $\mathbb{N}$ is a subgroup of the group of permutations of $\mathbb{N}$.
The remainder of this exercise is devoted to showing that this assertion is false if we replace recursive by primitive recursive.
(b) Let $\phi$ be a (total) function of one variable that is recursive but not primitive recursive; let $e$ be an index for a machine $\mathcal{M}$ that computes $\phi$. Consider the function $T$ which associates, with $x$, the time required by $\mathcal{M}$ to compute $\phi(x) ;$ more precisely, $\left.T(x)=\mu t\left[(e, t, x) \in B^{1}\right)\right]$.
Show that if $f$ is a function that satisfies $f(x) \geq T(x)$ for all $x \in \mathbb{N}$, then $f$ is not primitive recursive; also show that the graph $G$ of $T$ is primitive recursive.
(c) We set
$$
g(x)=\sup \{T(y): y \leq x\}+2 x .
$$
Show that $g$ is a strictly increasing recursive function and that it is not primitive recursive. Show that the graph $G_{1}$ and the range $I$ of $g$ are primitive recursive sets.
(d) Show that there is a unique strictly increasing primitive recursive function $g^{\prime}$ whose range is the complement of $I$.
(e) Define the function $h$ by
$$
\begin{aligned}
h(2 x) &=g(x) \\
h(2 x+1) &=g^{\prime}(x)
\end{aligned}
$$
where $g$ and $g^{\prime}$ are the functions defined in (c) and (d) above. Show that $h$ is a bijective recursive function that is not primitive recursive. Show that its inverse, $h^{-1}$, is primitive recursive.

Carson Merrill
Carson Merrill
Numerade Educator
00:43

Problem 17

Exhibit a recursive set $A \subseteq \mathbb{N}^{2}$ such that the set
$$
B=\{x \text { : for all } y \in \mathbb{N},(x, y) \in A\}
$$
is not recursively enumerable.

James Chok
James Chok
Numerade Educator
02:10

Problem 18

Show that there exists a primitive recursive function $\alpha$ of one variable that has the following property: for every integer $x$, if $\phi_{x}^{1}$ is a bijection from $\mathbb{N}$ onto $\mathbb{N}$, then $\alpha(x)$ is an index for the inverse bijection.

Goutam Chand
Goutam Chand
Numerade Educator
01:33

Problem 19

Let $g, \alpha$, and $h$ be partial recursive functions with $g$ and $\alpha$ in $\mathcal{F}_{1}^{*}$ and $h \in \mathcal{F}_{3}^{*}$. Show that there exists one and only one function $f \in \mathcal{F}_{2}^{*}$ such that, for all $x$ and $y$,
$$
\begin{aligned}
f(0, y) &=g(y), \\
f(x+1, y) &=h(f(x, \alpha(y)), y, x),
\end{aligned}
$$
and $f$ is partial recursive.

Harshita Goel
Harshita Goel
Numerade Educator
03:51

Problem 20

Let $A \subseteq \mathbb{N}$ be a recursively enumerable set that is not recursive; let $f$ be a partial recursive function whose domain is $A$ and let $i$ be an index for a Turing machine that computes $f$. Show that the function $\lambda x \cdot T^{1}(i, x)$ cannot be extended to a total recursive function [here, $T^{1}$ is the function defined in the proof of the enumeration theorem whose value is the time required to compute $f(x)$ ].

Chris Trentman
Chris Trentman
Numerade Educator
01:07

Problem 21

The purpose of this exercise is to prove the following fact:
(*) There exists a (total) recursive function $\psi(x, y)$ such that if we set $\psi_{x}=$ $\lambda y . \psi(x, y)$, then the set $\left\{\psi_{x}: x \in \mathbb{N}\right\}$ is precisely the set of all primitive recursive functions of one variable.
(a) Show that if $f \in \mathcal{F}_{p}$, then the following two conditions are equivalent:
(i) $f$ is primitive recursive;
(ii) there exists an index $i$ and a primitive recursive function $g \in \mathcal{F}_{p}$ such that the machine whose index is $i$ computes $f$ and the computation time $T\left(i, x_{1}, x_{2}, \ldots, x_{p}\right)$ is less than or equal to $g\left(x_{1}, x_{2}, \ldots, x_{p}\right)$.
(b) We will make use of Ackerman's function $\zeta$ and of the functions $\zeta_{n}=$ $\lambda x . \zeta(n, x)$. Show that if $f$ is a primitive recursive function of one variable, then there exist two integers $n$ and $A$ such that, for all $x$, we have
$$
f(x) \leq \sup \left(A, \zeta_{n}(x)\right) .
$$
(c) Let $g$ be the function of four variables defined by
$$
\begin{gathered}
g(i, A, n, x)=\mu y \leq \sup (A, \zeta(n, x)) \\
{\left[\exists t \leq \sup (A, \zeta(n, x))(i, t, x, y) \in C^{1}\right] ;}
\end{gathered}
$$
[recall that $(i, t, x, y) \in C^{1}$ means that when the machine whose index is $i$ is set in operation with $x$ on its first band, it will halt at the instant $t$ with output $y$ ]. Show that, for all $i, A$, and $n$, the function $\lambda x \cdot g(i, A, n, x)$ is primitive recursive and that, conversely, if $f$ is any primitive recursive function of one variable, then there exist integers $i, A$, and $n$ such that $f=\lambda x . g(i, A, n, x)$.
(d) Use these results to prove $(*)$.
(e) Show that there exists a recursive set that is not primitive recursive.

Carson Merrill
Carson Merrill
Numerade Educator
08:42

Problem 22

Let $\mathcal{T}$ be a set of partial recursive functions of one variable. We say that $\mathcal{T}$ has a recursive listing if there exists a partial recursive function $F$ of two variables such that, if we set $F_{x}=\lambda y \cdot F(x, y)$, then
$$
\mathcal{T}=\left\{F_{x}: x \in \mathbb{N}\right\} .
$$
Exercise 21 showed that the set of primitive recursive functions has a recursive listing.
(a) Show that the set of total recursive functions does not have a recursive listing.
(b) Show that the set of strictly increasing primitive recursive functions has a recursive listing.
(c) Show that the set of injective primitive recursive functions has a recursive listing.
(d) Let $F \in \mathcal{F}_{2}$ be a recursive function and assume that, for all $x \in \mathbb{N}$, the set
$$
A_{x}=\{F(x, y): y \in \mathbb{N}\}
$$
is infinite. Show that there exists an infinite recursive set, $B$, which is distinct from all the sets $A_{x}$. Conclude from this that the set of strictly increasing recursive functions does not have a recursive listing, nor does the set of injective recursive functions.

Chris Trentman
Chris Trentman
Numerade Educator
14:57

Problem 23

Let $A$ and $B$ be two subsets of $\mathbb{N}$. We say that $A$ is reducible to $B$ and write $A \leq B$ if there exists a (total) recursive function $f$ such that
$x \in A \quad$ if and only if $f(x) \in B$.
(a) Show that the relation $\leq$ is reflexive and transitive.
(b) Assume that $A$ is reducible to $B$. Show that if $B$ is recursively enumerable, then $A$ is recursively enumerable; show also that if $B$ is recursive, then so is $A$.
Set
$$
\begin{aligned}
X &=\left\{x: \phi^{1}(x, x) \text { is defined }\right\} \\
Y &=\left\{\alpha_{2}(x, y): \phi^{1}(x, y) \text { is defined }\right\}
\end{aligned}
$$
(c) Show that a set $A \subseteq \mathbb{N}$ is recursively enumerable if and only if $A \leq Y$.
(d) Let $A$ and $B$ be two subsets of $\mathbb{N}$. Let
$$
C=\{2 n: n \in A\} \cup\{2 n+1: n \in B\} .
$$
Show that $A$ and $B$ are reducible to $C$ and that if $D$ is a subset of $\mathbb{N}$ such that $A$ and $B$ are both reducible to $D$, then $C$ is reducible to $D$.
(e) We will say that $A$ is self-dual if $A \leq \mathbb{N}-A$. Show that for every $B \subseteq \mathbb{N}$, there exists a $C \subseteq \mathbb{N}$ that is self-dual and is such that $B \leq C$.
(f) Let $\mathcal{T}$ be a set of partial recursive functions of one variable that is not empty and is not equal to the set of all partial recursive functions of one variable. Set
$$
A=\left\{x: \phi_{x}^{1} \in \mathcal{T}\right\} .
$$
(i) Show that if the partial.function whose domain is empty belongs to $\mathcal{T}$, then $X \leq \mathbb{N}-A$.
(ii) Show that, in the opposite case, $X \leq A$.
(iii) Show that $A$ is not self-dual.
(g) Show that $Y \leq X$.

Chris Trentman
Chris Trentman
Numerade Educator
03:08

Problem 24

The goal of this exercise is to show that the precautions which we took in defining the unbounded $\mu$-operator (see Definition $5.19$ ) are necessary.
Show that the partial function $\psi(x, y)$ defined by
$$
\psi(x, y)= \begin{cases}\phi^{1}(x, y)-\phi^{1}(x, y) & \text { if } y=0 \\ 0 & \text { otherwise }\end{cases}
$$
is partial recursive.
Define the function $g$ by $g(x)=$ the least integer $y$ such that $\psi(x, y)=0 .$
Show that $g$ is a total function that is not recursive.

Aniket Bajaj
Aniket Bajaj
Numerade Educator
01:07

Problem 25

Consider the following sets:
$$
\begin{aligned}
&A=\left\{x: \phi_{x}^{1}(0) \text { is defined }\right\} \\
&B=\left\{x: \phi_{x}^{1} \text { is a total function }\right\}
\end{aligned}
$$
(a) Show that the complement of $A$ is not recursively enumerable.
(b) Show that there exists a primitive recursive function $f \in \mathcal{F}_{1}$ such that for all $i, i \in A$ if and only if $\alpha(i) \in B$. Show that the complement of $B$ is not recursively enumerable.
(c) Let $F$ be the following partial function:
$$
F(x, y)= \begin{cases}1 & \text { if for all } z<y, \neg B^{1}(e, z, x) \\ \text { undefined } & \text { otherwise }\end{cases}
$$
where $B^{1}$ is the predicate defined in the proof of the enumeration theorem and $e$ is the index of a partial function whose domain is $A$.

Show that the partial function $\lambda y \cdot F(x, y)$ is total if and only if $x \notin A$. Conclude from this that $B$ is not recursively enumerable.
(d) By generalizing the results from (b) and (c), prove the following:
Proposition Let $f$ be a partial recursive function of one variable whose domain is infinite; then neither the set $\left\{x: \phi_{x}^{1}=f\right\}$ nor its complement is recursively enumerable.

Carson Merrill
Carson Merrill
Numerade Educator
12:28

Problem 26

In this exercise, we will give an alternate proof of the fact that there exists a primitive recursive function $\beta$ of one variable such that, for all $i$,
$$
\phi_{i}^{1}=\phi_{\beta(i)}^{1} \quad \text { and } \quad \beta(i)>i .
$$
(See Theorem 5.51.) This proof is based only on the fixed point theorems and no longer involves Turing machines.
(a) Show that there exists a primitive recursive function $\delta$ such that, for all $n$, $\phi_{\delta(n)}^{1}$ is the constant function equal to $n$.
(b) Define the function $\gamma(n, t, z)$ by
$$
\gamma(n, t, z)= \begin{cases}\delta(n) & \text { if } z<t \\ t & \text { otherwise. }\end{cases}
$$
By applying the third version of the fixed point theorem (see Theorem 5.55) to this function, show that there exists a primitive recursive function $h(n, t)$
such that
$$
\phi_{h(n, t)}^{1}= \begin{cases}\phi_{\delta(n)}^{1} & \text { if } h(n, t) \leq t \\ \phi_{t}^{1} & \text { otherwise. }\end{cases}
$$
(c) Show that, for all $t$, the set $A_{t}=\{n: h(n, t) \leq t\}$ has at most $t+1$ elements. Use this to conclude that the desired function $\beta$ exists.

Chris Trentman
Chris Trentman
Numerade Educator
02:02

Problem 27

When we constructed the functions $\phi^{p}$, we used a certain number of codings and, for this purpose, we had to make some completely arbitrary choices. In this exercise, our concern is to know what sort of functions would have been obtained instead of the $\phi^{p}$ if our choices had been different. The only assumption we will make is that these choices are reasonable and sufficient for proving the enumeration theorem and the fixed point theorems.

Let $\Psi=\left\{\psi^{p}: p \geq 1\right\}$ be a family of partial recursive functions such that, for all $p, \psi^{p} \in \mathcal{F}_{p+1}^{*}$. We set
$$
\psi_{x}^{p}=\lambda y_{1} y_{2} \ldots y_{p} . \psi^{p}\left(x, y_{1}, y_{2}, \ldots, y_{p}\right) .
$$
Consider the following conditions on the family $\Psi$ :
-(enu) For every $p>0$, the set $\left\{\psi_{i}^{p}: i \in \mathbb{N}\right\}$ is equal to the set of all partial recursive functions of $p$ variables.
- (smn) For every pair of integers $m$ and $n$, there exists a total recursive function $\sigma_{n}^{m}$ of $n+1$ variables such that for all $i, x_{1}, x_{2}, \ldots, x_{n}, y_{1}, y_{2}, \ldots, y_{m}$, we have
$$
\begin{aligned}
&\psi^{n+m}\left(i, x_{1}, x_{2}, \ldots, x_{n}, y_{1}, y_{2} \ldots, y_{m}\right) \\
&\quad=\psi^{m}\left(\sigma_{n}^{m}\left(i, x_{1}, x_{2}, \ldots, x_{n}\right), y_{1}, y_{2}, \ldots, y_{m}\right)
\end{aligned}
$$
(a) Let $\theta$ be a partial recursive function of two variables. For every integer $x$, we $\operatorname{set} \theta_{x}=\lambda y . \theta(x, y)$. Show that the following two conditions are equivalent:
(i) there exists a family $\Psi=\left\{\psi^{p}: p \geq 1\right\}$ that satisfies conditions (enu) and $(\mathrm{smn})$ and is such that $\psi^{1}=\theta$;
(ii) there exists a recursive function $\beta$ such that, for all $x, \phi_{x}^{1}=\theta_{\beta(x)}$.
(b) Assume once more that the family $\Psi$ satisfies the conditions (enu) and $(\mathrm{smn})$. Show that the fixed point theorems are valid for the family $\Psi$.
(c) Assume that the function $\theta$ satisfies conditions (i) or (ii) from (a). Show that there exist two injective recursive functions $\alpha$ and $\beta$ such that, for all $x$,
$$
\phi_{x}^{1}=\theta_{\beta(x)} \quad \text { and } \quad \theta_{x}=\phi_{\alpha(x)}^{1}
$$
(d) (Difficult!) Under these same hypotheses, show that there exists a recursive function $\varepsilon$ that is total and bijective and is such that, for all $x, \phi_{x}^{1}=\theta_{\varepsilon(x)}$.

Manik Pulyani
Manik Pulyani
Numerade Educator