Universal Programs¶

This lesson will teach you about universal programs: these are programs that take $\mathtt{1\#}$ programs as inputs and run them.

To start, enter the following cell:

#@title
!python -m pip install -U setuptools
!python -m pip install -U git+https://github.com/lmoss/onesharp.git@main
from onesharp.interpreter.interpreter import *

u = universal
u

'1#####1###11###11####111111#1#####111###111111###1111111###11111#####1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111###1###1####111#111111111####111##1#####1111###11###11111###111#11111#####1111111111111111111111111111111111111111111111###111111111###111##1#####1111###11###11111###111#11111#####1111111111111111111111111111111111111###111111111###111##1#####1111###11###11111###111#11111#####1111111111111111111111111111111111111111111111###111111111###111##1#####1111###11###11111###111#11111#####111111111111111111111111111111111111111111111111111111###111111111###111##1#####1111###11###11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111###111#11111#####1111111111###1###111#####111111###111###1111##1111####1111#111111####11111111111111111111111111111111111111111111111111111111111111####111#####1###11###1111###11111#1111#111111####11111##1111##111#####1111111###1111###1111##11111##11111####1111#1111111####111111111111111111111111111111111111111111111111111111111111111111111111111###111#####1###11###11111###11111#1111#111111#1111111####1111##111#####111111###111###1111##1111####1111#111111####11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111####111#####1###11###1111###1111#11111#111111####11111#11111#1111##111#####111111###111###1111##1111####1111#111111####11111#####11111111###11###1###111111#####111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111###1###1###11111111####111111#####1111###111#11111#1111####111#####111###111111#111####1#####111111###111###111##1111####111#111111####1111#####111111###111###1##1111####1#111111####111#####111111###111###1##1111####1#111111####111111111111111111111111111111111111111111111111111111111####11#####111###11111###111111111111###11##11##111111####111#11#####1###111###111##11###111#11111111111111####111##11#####11111111111111111111111111111111111111111111111111111111111111111111###1###111##11111#####1###1111111111111111111111####11111#####1111111111111111111111111111###111111111111111111111111111###11111#####111###11###111111111111111111111111111111111111111111111###11#####111111111111111111###111111111###111#11#####1###1###111#111##111##111111111111111111111111111111111111111111111111111111111###111#11#####1###111###111##11###111#111111111111111111####111#111#1111111111111111111111111111111111111111111111###11#####111111111111111111###111111111###111#11#####1###1###111##111##111##11111111111111111111111111111111111###111#11#####1###111###111##11###111#111111111111111111####111#111##111111111111111111111111###11111#####1###1###11111#####1###1###11#####11111111111111111111111###111###111##1111111111111###11#####1###11111###1###11111#111111#111111###11111#11111#111111#111111#1###11#####111111###111###111##1111####111#111111####111#####111111###111###11##1111####11#111111####1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111####11#####1###1###11#####1###1###11#####11111111111111###11###11111111###11#####1###111###1#11111111####1##1111111111####11#####111###11####111####111111#####11###11####1111#####111###11####111####'

Definition 5

A word $u$ is universal if

\[\phifn_u(x) \simeq \phifn_x(\ ) \]

for all words $x$.

This means that if running $x$ on all empty registers eventually gives some output in R1 (and all other registers empty), then that same output would be the result of running $u$ with $p$ in R1 and all other registers empty. And in the other direction, if running $x$ with all empty registers gives some output, say $y$, in R1 (and all registers empty at the end), then we would get the same output $y$ in R1 when we run $u$ with $x$ in R1 and all other registers empty.

Here is a specific example of this. Try running u with 1# 1## in R1. The result is 1#, and this is what happens when we run 1# 1## on all empty registers. So the universal program u takes 1# 1## and simulates a run of that program.

For another example, suppose we start u with self in R1 (and all other registers empty). You can check for yourself that we eventually get self in R1. (Note: the computation might take several minutes.)
This confirms the basic property because running self with all registers empty gives self.

We can even run u on programs that contain u itself. For this, we return to our first example, concerning the word 1# 1##. As we know,

φ_u(1# 1##) = 1#.

Recall also that

φ_write(1# 1##) = 1# 1## 1# 1## 1##

It follows that

φ_{1# 1## 1# 1## 1## + u}( ) = 1#.

And from this we conclude that

φ_u(1# 1## 1# 1## 1## + u) = 1#

This is something you can try for yourself.

The plan for this lesson¶

The goal of this lesson is for you to write a universal program yourself. The next sections guide you, using a series of exercises.

We hasten to make two remarks: first of all, it would be more instructive for you to forget about the rest of this lesson entirely and to just write your own universal program. But this most readers would probably appreciate the hints and will find enough of a challenge in the construction of a long working program. Second, the sections to come do not guide you to write a program thatt is exactly like u above. The u above was designed to be as short as possible. In fact, we leave you with a challenge: write a universal program with the smallest number of instructions. The one above has 300.

Simplification¶

We are going to simplify the requirements in order to make it easier to write a universal program. So we’ll outline how to write a program that is close to a universal program, but not quite a universal program.

We'll write a program $u$ that acts like a universal program, but only for programs which use only R1, R2, and R3. In more detail:

Let $p$ be any program of $\mathtt{1\#}$ which only uses R1, R2, and R3. Let $x$ be any word on our alphabet $\{\mathtt{1},\mathtt{\#}\}$, including possibly the empty word. Then the following are equivalent:

(A’) If $p$ is started with all empty registers, then eventually we halt with $x$ in R1.

(B’) If $u$ is started with $p$ in R1 and with the empty word in all other registers, then eventually we halt with $x$ in R1.

Once again, the work of this section sketches a construction of such a program $u$. You will need to do much of the work yourself in actually getting the program from the sketch. And even when we have $u$, it will not be quite what we want for this lesson overall, because $u$ will only behave right on programs which use the first three registers.
Later on, we’ll come back and drop this simplification to get a program which is truly universal.

At this point, we are ready to sketch a method for you to write a universal program subject to this simplification.

We are going to use registers as follows:

R1: the input program p

R2: an instruction number 1ⁿ

R3: the nth instruction of p

R4: the contents of R1

R5: the contents of R2

R6: the contents of R3

That is, we are going to simulate the computation of a register machine on a program $p$ by using six registers. Our universal program $u$ is what does the simulation, and it works step-by-step. That is, $u$ mimics what a register machine would really do. The only difference is that one step of a real register machine would be done by *many* steps of our universal program.

The registers above hold what is sometimes called a snapshot of a register machine run. This means that they hold all of the relevant information about a register machine computation at one particular point in time: they tell what program is running, which instruction would be run next and what number instruction it is, and also the contents of all of the registers that the program is using. Now a real register machine goes from snapshot to snapshot in one step, but the universal machine that we are building will take many steps to go from one snapshot to the next.

Our universal program is composed of a few sub-programs. We shall illustrate the ideas behind several sub-programs by making a large chart. This chart shows just one example of how a universal program could work, when we take $p$ to be the prorgram

$\mathtt{1\#\#}$ + write

Now to make life easier, we'll write out this program and indicate the instruction numbers above them:

1	2	3	4	5	6	7	8	9
1##	1#####	111111111###	11111###	11#	11##	11##	111111####	11#
10	11	12	13	14	15	16	17	18
11##	111111111####	11#####	111111###	111###	1##	1111####	1#	111111####

What we are going to show is tables of what the various registers show after different parts of a run of a universal program u on this program $\mathtt{1\#\#}$ + write. We hasten to add that the universal program exhibited above does not conform to the tables. (This is because registers work differently in it. Also, the program above is designed to work even without our simplifying assumption.)

Here are the tables; we'll have some words below about them.

R1: the input program p

p

R2: an instruction number 1ⁿ

1

11

11111

1⁶

1⁷

1⁸

R3: the nth instruction of p

1##

1#####

11#

11##

1⁶ ####

R4: the contents of R1

#

R5: the contents of R2

1

1#

1##

R6: the contents of R3

continuing

R1: the input program p

p

R2: an instr number 1ⁿ

11

111

1¹²

1¹⁴

1¹⁷

1¹⁸

R3: the nth instruction of p

1#####

1⁹ ###

11#####

111###

1#

1⁶ ####

R4: the contents of R1

1

R5: the contents of R2

1##

##

R6: the contents of R3

and going further in the move_2,1 part at the end of write

R1: the input program p

p

R2: an instruction number 1ⁿ

1¹²

1¹⁵

1¹⁶

1¹²

1¹⁵

1¹⁶

R3: the nth instruction of p

11#####

1##

1⁴####

11#####

1##

1⁴####

R4: the contents of R1

1

1#

1##

R5: the contents of R2

##

#

R6: the contents of R3

and concluding

R1: the input program p	p	p	p	p	p	1##
R2: an instruction number 1ⁿ	1¹²	1¹²	1¹³	1¹³	1¹⁹
R3: the nth instruction of p		11#####		1^{6 ###}
R4: the contents of R1	1##	1##	1##	1##	1##
R5: the contents of R2
R6: the contents of R3

As promised, here are some words of explanation.

As you can see, there are four different colors of the columns. The colors indicate the sub-programs of u. The columns themselves indicate the register contents when one of the sub-programs begins.

The first color shown is orange. This is the the easiest to explain. The orange sub-program is 11#. So it adds a 1 to R2, indicating that at the very beginning of the execution of the input program, the instruction to look at first is instruction 1.

The beige columns alternate with the green ones. The purpose of the beige program is to take the input program (in R1) and the next instruction number (R2), and to look up the appropriate instruction and put it into R3. This is similar to a program which we saw earlier on. But we need to be sure that the contents of R1 and R2 are not destroyed at the end. (So you will probably need to use registers beyond R6 for all of this.) The way to understand the beige columns is that they tell the contents of the various registers at the beginning of the operation of the "beige sub-program". You will need to write that sub-program. It is a fairly straightforward modification of a program which you already have seen, the one that gives the nthe instruction of an input program. But there are some differences: you need to be sure that the output goes in R3, that the input is preserved, and that if the contents of R2 is one more than the length of the input program in R1, then the program "knows" about this and transfers over to the yellow sub-program shown at the end.

The main action of the steps is done in the green columns; more precisely, the columns shown are the beginnings of the "green" sub-program, and then the results of that sub-program are the beige columns which follow. The exact action depends on what kind of instruction is in R3 in the given green column. For example, if R3 has 11#, then in the next beige column we add a # to R5 (not R2: it is R5 that models what is in R2 of the real machine when we run program). We also must add 1 to R2 in simulating the execution of 11#, since after 11#, we go to the very next instruction. This is how the green columns work when we execute an instruction 11#. The other cases are left for you to think about. Note also that each time the green sub-program runs, R3 is emptied at the end.

The last column color is the yellow one at the very end. Before the yellow one starts up, the beige program has determined (in some sense) that our program $p$ has 18 instructions and we have come to a point where we ask for instruction 19. Thus, the input program has halted. We therefore want to take the contents of R1 as run by the input program (this is the word in R4), and move this to R1. Also, we would like to clear out the contents of whichever registers are not empty. We do this so that the program we are writing will itself come to a good halt on its input.

It might seem silly to have R1 unchanged throughout. But this isn't really what is happening at all. In the course of the beige subprograms, parts of R1 are copied to other registers, and the copied back. So at the end of the beige sub-programs, we have the original program $p$ back in R1.

Similarly, it might seem weird to have R6 completely unused in running a universal program on $p$. This is due to the fact that our program $p$ only uses R1 and R2. If we worked with an example program that used R3, then R6 would not have remained empty in the tables.

Coding unboundedly many registers into one¶

The simplified form of the universal program that we just saw could be modified to work on programs with any fixed finite number of registers. But we want one universal programs that works on all programs, without a pre-established bound. For this, we need a new idea.

We now call upon a technique introduced in a separate notebook in this course, the technique of two-by-two encoding. That notebook shows how we can add an end of register marker to R1, and the work with a simple encoding of symbols by pairs. We took the end of register marker to be ##, and we encoded 1 by 11 and # by 1#.

All of that work was for R1 alone. But we could do the same work for all of the other registers.

Following this, the idea is to simply run all of the encoded registers into one.

For an example of what we mean, suppose that we were dealing with the following snapshot:

R1: 11#

R2:

R3: ##

R4:

R5:

R6: 1

R7: #

R8: ##1

(We understand also that all of the other registers are empty.)

Encoding things as we have explained allows us to code the eight registers, as in the table below:

Contents	R1: 11#	R2:	R3: ##	R4:	R5:	R6: 1	R7: #	R8: ##1
Encoded as	11111###	##	R3:##	##	##	11##	1# ##	1#1#11##

One important point is that the empty registers still get an end-of-register marker ##.

Therefore we would encode the contents of these registers by the single word

11 11 1# ## ## 1# 1# ## ## ## 11 ## 1# ## 1# 1# 11 ##

Without spaces, this is

11111#####1#1#######11##1###1#1#11##

The point is that with some additional work, we can use this single word as an encoded form of all the registers in our program.

Invitation to Computability

Universal Programs

Contents

Universal Programs¶

The plan for this lesson¶

Simplification¶

Coding unboundedly many registers into one¶

Project¶

A remark on the beige and green sub-programs

A remark on the beige and green sub-programs¶

Implementing add# instructions¶

Universal programs of more than one argument¶