Syntax

LOOP-programs consist of the symbols LOOP, DO, END, :=, + and ; as well as any number of variables and constants. LOOP-programs have the following syntax in modified Backus–Naur form:

${\displaystyle {\begin{array}{lrl}P&::=&x_{i}:=0\\&|&x_{i}:=x_{i}+1\\&|&P;P\\&|&{\mathtt {LOOP}}\,x_{i}\,{\mathtt {DO}}\,P\,{\mathtt {END}}\end{array}}}$

Here, ${\displaystyle Var:=\{x_{0},x_{1},...\}}$ are variable names and ${\displaystyle c\in \mathbb {N} }$ are constants.

Semantics

If P is a LOOP program, P is equivalent to a function ${\displaystyle f:\mathbb {N} ^{k}\rightarrow \mathbb {N} }$. The variables ${\displaystyle x_{1}}$ through ${\displaystyle x_{k}}$ in a LOOP program correspond to the arguments of the function ${\displaystyle f}$, and are initialized before program execution with the appropriate values. All other variables are given the initial value zero. The variable ${\displaystyle x_{0}}$ corresponds to the value that ${\displaystyle f}$ takes when given the argument values from ${\displaystyle x_{1}}$ through ${\displaystyle x_{k}}$.

A statement of the form

xi := 0

means the value of the variable ${\displaystyle x_{i}}$ is set to 0.

A statement of the form

xi := xi + 1

means the value of the variable ${\displaystyle x_{i}}$ is incremented by 1.

A statement of the form

P1; P2

represents the sequential execution of sub-programs ${\displaystyle P_{1}}$ and ${\displaystyle P_{2}}$, in that order.

A statement of the form

LOOP x DO P END

means the repeated execution of the partial program ${\displaystyle P}$ a total of ${\displaystyle x}$ times, where the value that ${\displaystyle x}$ has at the beginning of the execution of the statement is used. Even if ${\displaystyle P}$ changes the value of ${\displaystyle x}$, it won't affect how many times ${\displaystyle P}$ is executed in the loop. If ${\displaystyle x}$ has the value zero, then ${\displaystyle P}$ is not executed inside the LOOP statement. This allows for branches in LOOP programs, where the conditional execution of a partial program depends on whether a variable has value zero or one.

Creating "convenience instructions"

From the base syntax one create "convenience instructions". These will not be subroutines in the conventional sense but rather LOOP programs created from the base syntax and given a mnemonic. In a formal sense, to use these programs one needs to either (i) "expand" them into the code  – they will require the use of temporary or "auxiliary" variables so this must be taken into account, or (ii) design the syntax with the instructions 'built in'.

Example

The k-ary projection function ${\displaystyle U_{i}^{k}(x_{1},...,x_{i},...,x_{k})=x_{i}}$ extracts the i-th coordinate from an ordered k-tuple.

In their seminal paper ^[2] Meyer & Ritchie made the assignment ${\displaystyle x_{j}:=x_{i}}$ a basic statement. As the example shows the assignment can be derived from the list of basic statements.

To create the ${\displaystyle \operatorname {assign} }$ instruction use the block of code below. ${\displaystyle x_{j}:=\operatorname {assign} (x_{i})}$ =_equiv

xj := 0;
LOOP xi DO
  xj := xj + 1
END

Again, all of this is for convenience only; none of this increases the model's intrinsic power.

Addition

Addition is recursively defined as:

${\displaystyle {\begin{aligned}a+0&=a,&{\textrm {(1)}}\\a+S(b)&=S(a+b).&{\textrm {(2)}}\end{aligned}}}$

Here, S should be read as "successor".

In the hyperoperater sequence it is the function ${\displaystyle \operatorname {H_{1}} }$

${\displaystyle \operatorname {H_{1}} (x_{1},x_{2})}$ can be implemented by the LOOP program ADD( x₁, x₂)

LOOP x1 DO
  x0 := x0 + 1
END;
LOOP x2 DO
  x0 := x0 + 1
END

Multiplication

Multiplication is the hyperoperation function ${\displaystyle \operatorname {H_{2}} }$

${\displaystyle \operatorname {H_{2}} (x_{1},x_{2})}$ can be implemented by the LOOP program MULT( x₁, x₂ )

x0 := 0;
LOOP x2 DO
  x0 := ADD( x1, x0)
END

The program uses the ADD() program as a "convenience instruction". Expanded, the MULT program is a LOOP-program with two nested LOOP instructions. ADD counts for one.

More hyperoperators

Given a LOOP program for a hyperoperation function ${\displaystyle \operatorname {H_{i}} }$, one can construct a LOOP program for the next level

${\displaystyle \operatorname {H_{3}} (x_{1},x_{2})}$ for instance (which stands for exponentiation) can be implemented by the LOOP program POWER( x₁, x₂ )

x0 := 1;
LOOP x2 DO
  x0 := MULT( x1, x0 )
END

The exponentiation program, expanded, has three nested LOOP instructions.

Predecessor

The predecessor function is defined as

${\displaystyle \operatorname {pred} (n)={\begin{cases}0&{\mbox{if }}n=0,\\n-1&{\mbox{otherwise}}\end{cases}}}$.

This function can be computed by the following LOOP program, which sets the variable ${\displaystyle x_{0}}$ to ${\displaystyle pred(x_{1})}$.

/* precondition: x2 = 0 */
LOOP x1 DO
  x0 := x2;
  x2 := x2 + 1
END

Expanded, this is the program

/* precondition: x2 = 0 */
LOOP x1 DO
  x0 := 0;
  LOOP x2 DO
    x0 := x0 + 1
  END;
  x2 := x2 + 1
END

This program can be used as a subroutine in other LOOP programs. The LOOP syntax can be extended with the following statement, equivalent to calling the above as a subroutine:

x0 := x1 ∸ 1

Remark: Again one has to mind the side effects. The predecessor program changes the variable x₂, which might be in use elsewhere. To expand the statement x₀ := x₁ ∸ 1, one could initialize the variables x_n, x_n+1 and x_n+2 (for a big enough n) to 0, x₁ and 0 respectively, run the code on these variables and copy the result (x_n) to x₀. A compiler can do this.

Cut-off subtraction

If in the 'addition' program above the second loop decrements x₀ instead of incrementing, the program computes the difference (cut off at 0) of the variables ${\displaystyle x_{1}}$ and ${\displaystyle x_{2}}$.

x0 := x1
LOOP x2 DO
  x0 := x0 ∸ 1
END

Like before we can extend the LOOP syntax with the statement:

x0 := x1 ∸ x2

If then else

An if-then-else statement with if x₁ > x₂ then P1 else P2:

xn1 := x1 ∸ x2;
xn2 := 0;
xn3 := 1;
LOOP xn1 DO
  xn2 := 1;
  xn3 := 0
END;
LOOP xn2 DO
  P1
END;
LOOP xn3 DO
  P2
END;