# Jensen's Device

Jensen's Device is a computer programming technique that exploits call by name. It was devised by Danish computer scientist Jørn Jensen, who worked with Peter Naur at Regnecentralen. They worked on the GIER Algol compiler, one of the earliest correct implementations of ALGOL 60.[1] ALGOL 60 used call by name.

## Description

Jensen's device exploits call by name and side-effects. Call by name is an argument passing convention that delays the evaluation of an argument until it is actually used in the procedure (a consequence of the copy rule for procedures). Algol introduced call by name.

A classic example of Jensen's device is a procedure that computes the sum of a series, ${\displaystyle \Sigma _{k=l}^{u}a_{k}}$:[2][3][4]

``` real procedure Sum(k, l, u, ak)
value l, u;
integer k, l, u;
real ak;
comment k and ak are passed by name;
begin
real s;
s := 0;
for k := l step 1 until u do
s := s + ak;
Sum := s
end;
```

In the procedure, the index variable `k` and summation term `ak` are passed by name. Call by name enables the procedure to change the value of the index variable during execution of the `for` loop. Call by name also causes the `ak` argument to be reevaluated during each iteration of the loop. Typically, `ak` will depend upon the changing (side-effected) `k`.

For example, code to compute the sum of the first 100 terms of a real array `V[]` would be:

``` Sum(i, 1, 100, V[i]).
```

During the execution of `Sum`, the actual argument `i` will increment during each step of the `for` loop, and each of the procedure's evaluations of `ak` will use the current value of `i` to access the successive array elements `V[i]`.

Jensen's device is general. A double summation can be done as:

``` Sum(i, l, m, Sum(j, l, n, A[i,j]))
```

The `Sum` function can be employed for arbitrary functions merely by employing the appropriate expressions. If a sum of integers were desired the expression would be just `Sum(i,1,100,i);`, if a sum of squares of integers, then `Sum(i,1,100,i*i);`, and so on.[5] A slight variation would be suitable for initiating a numerical integration of an expression by a method very similar to that of `Sum`.

The evaluation of `ak` is implemented with a thunk, which is essentially a subroutine with an environment. The thunk is a closure with no arguments. Each time a procedure needs the value of its formal argument, it simply calls the thunk. The thunk evaluates the actual argument in the scope of the calling code (not the scope of the procedure).

In the absence of this pass-by-name facility, it would be necessary to define functions embodying those expressions to be passed according to the protocols of the computer language, or to create a compendium function along with some arrangement to select the desired expression for each usage.

### GPS

Another example is GPS (General Problem Solver), described in D. E. Knuth and J. N. Merner's ALGOL 60 confidential.[6]

```real procedure GPS(I, N, Z, V); real I, N, Z, V;
begin for I := 1 step 1 until N do Z := V; GPS := 1 end;
```

Following is a single statement which finds m-th prime using GPS.

```I := GPS(I, if I=0 then -1.0 else I, P, if I=1 then 1.0 else
if GPS(A, I, Z, if A=1 then 1.0 else
if entier(A)×(entier(I)÷entier(A))=entier(I) ∧ A<I
then 0.0 else Z) = Z then
(if P<m then P+1 else I×GPS(A, 1.0, I, -1.0)) else P)
```

(Note: In the original paper, the expression near the end is `GPS(A, 1.0. I, 0.0)`, due to a corner case in the specification of the semantics of ALGOL 60's for statement.)

## Criticism

Jensen's device relies on call by name, but call by name is subtle and has some problems. Consequently, call by name is not available in most languages. Knuth comments that ALGOL 60 cannot express an `increment(n)` procedure that increases its argument by one; the call `increment(A[i])` does not do the expected action if `i` is a functional that changes with each access.[7] Knuth says, "The use of 'macro' definition facilities to extend language, instead of relying solely on procedures for this purpose, results in a more satisfactory running program."

Others point out that a call by name procedure that swaps its argument can have subtle problems.[8] An obvious swapping procedure is:

```procedure swap(a, b)
integer a, b;
begin
integer temp;
temp := a;
a := b;
b := temp;
end;
```

The procedure does the right thing for many arguments, but the invocation of `swap(i,A[i])` is problematic. Using the Copy Rule leads to the assignments:

``` temp := i;
i := A[i];
A[i] := temp;
```

The problem is the second assignment changes `i`, so the `A[i]` in the third assignment probably will not be the same array element as at the start. If on the other hand the procedure were to be coded the other way around (with b being saved to temp instead of a) then the desired action would result, unless it were invoked as `swap(A[i],i)`

## References

1. ^ Peter Naur's 2005 Turing Award citation Archived 2008-02-12 at the Wayback Machine. mentions his work with Jensen on GIER Algol
2. ^ MacLennan, Bruce J. (1987), Principles of Programming Languages: Design, Evaluation, and Implementation (Second ed.), Holt, Rinehart & Winston, ISBN 0-03-005163-0, pp 141–142
3. ^ Dijkstra, E. W. (November 1961), "Defense of ALGOL 60 (Letter to the Editor)", Communications of the ACM, 4 (11): 502&ndash, 503, doi:10.1145/366813.366844
4. ^ Knuth, D. E. (October 1967), "The Remaining Troublespots in ALGOL 60", Communications of the ACM, 10 (10): 611&ndash, 617, doi:10.1145/363717.363743
5. ^ `Sum` requires a `real` argument for the term, so type conversion is assumed.
6. ^ Donald E. Knuth and Jack N. Merner. 1961. ALGOL 60 confidential. Commun. ACM 4, 6 (June 1961), 268-272. DOI=10.1145/366573.366599 http://doi.acm.org/10.1145/366573.366599
7. ^ Knuth 1967, p. 613. For example, `increment(A[increment(j)])` will increment `j` twice.
8. ^ MacLennan 1987