VDMSL Recursive function minimum value of sequence

Question

I think this should be relatively straightforward, was wondering if anyone knew how to answer this:

Define a recursive function seq-min : N+ -> N which returns the smallest value in a sequence of natural numbers.

I was thinking something along the lines of...

if hd seq < hd tl seq then seq-min([hd seq] concat tl tl seq)
else if hd seq > hd tl seq then seq-min(tl seq)
else if hd seq = hd tl seq then seq-min(tl seq)

Thanks for any help!

You can see the problem with this: since it is testing pairs of values, and recursing on the tail of the list, eventually it will be passed a single value sequence. At that point, the "hd tl seq" will fail. The answer below avoids the need for pairs by passing the minimum "down" with the recursion and just testing the head. — Nick Battle, Jun 03 '14 at 11:49

score 2 · Answer 1 · answered Jun 03 '14 at 11:36

Something like that might work, but it is quite hard to follow - and we are writing a specification, so it helps if it is clear. The following was my first thought. It cheats slightly by using two functions, but I hope it is relatively clear:

    seq_min: seq of nat -> nat
    seq_min(s) ==
            minrec(tl s, hd s)
    pre len s > 0;

    minrec: seq of nat * nat -> nat
    minrec(s, min) ==
            if s = []
            then min
            else if hd s < min
            then minrec(tl s, hd s)
            else minrec(tl s, min);

Note that this does not try to use pairs of values in the comparison, so there is no "tl tl seq" expression etc. The following tests are with VDMJ again:

> p seq_min([])
Error 4055: Precondition failure: pre_seq_min in 'DEFAULT' (z.vdm) at line 5:15
Stopped in 'DEFAULT' (z.vdm) at line 5:15
5:      pre len s > 0;
>
> p seq_min([1])
= 1
Executed in 0.002 secs.
> p seq_min([2,1])
= 1
Executed in 0.014 secs.
> p seq_min([5,2,3,7,6,9,8,3,5,5,2,7,2])
= 2
Executed in 0.004 secs.
>

score 2 · Accepted Answer · answered Jun 03 '14 at 13:10

Here is a slightly different approach, using a single function:

    seq_min: seq of nat -> nat
    seq_min(s) ==
            cases s:
                    [x] -> x,

                    [x] ^ rest ->
                            let min = seq_min(rest) in
                                    if x < min then x else min
            end
    pre len s > 0;

This has the advantage that it is short and intuitive (and a single function). Specifications can be very clear when they are written as a set of "patterns" in a cases expression like this, as each case is explicitly "explained".

VDMSL Recursive function minimum value of sequence

2 Answers2