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I'm working on practice problems for a test I have, and every example of loop variant decreased with every iteration of the loop. On this one, the values remain the same when a < b. My attempts also got me a loop variant that has a chance of a negative since occasionally a becomes larger than b and vice versa. Any advice on attempting to find and prove the loop variant for this question?

def mystery(a,b):
# Precondition: a >= 0 and b >= 0
while a >= 0 and b >= 0:
    if a < b:
        a, b = b, a
    else:
        a = a - 1
return a

EDIT: For anyone who is interested in this question, my best solution is as follows.

$$f_{1} = a + 2b + 1$$

Andrew Raleigh
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2 Answers2

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Just a hint for now, since this is a practice problem: consider a lexicographic combination of orders.

In some more detail: Suppose you have two maps $f_1:S\to D_1$ and $f_2:S\to D_2$ from your program states $S$ into well-founded ordered domains $(D_1,\le_1)$ and $(D_2,\le_2)$. The lexicographic combination of $\le_1$ and $\le_2$ is the order $\le$ on $D_1\times D_2$ given by $(x_1,y_1)\le(x_2,y_2)$ iff either $x_1\le_1x_2$, or $x_1=x_2,y_1\le_2y_2$. It is also well-founded.

So if $f_1,f_2$ are such that

  • $f_1$ never increases, and
  • whenever $f_1$ does not decrease, $f_2$ does,

then the map $(f_1,f_2):S\to D_1\times D_2$ is a variant proving termination.

Klaus Draeger
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1

Here is an approach involving only one mapping: $$ f = (a+1) \cdot (b+1) + (b - a) $$ Simple case analysis can show that $f$ always decreases as you go through the loop.

Anton Trunov
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