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<a id="skiptocontent" href="#content">Skip to content</a>
<a href="/fi/">FI</a> · <a href="/en/">EN</a>
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</div>
<div id="content" class="content">
<h1 class="title">A Pythonic FP adventure
<br />
<span class="subtitle">Designing a horrible implementation for LCM</span>
</h1>
<p>
Let's write a naive function to find the largest common multiple of
two integers in Python. As usual in Python, there is only one obvious
solution.
</p>
<p>
We will test it with the following check:
</p>
<div class="org-src-container">
<pre class="src src-python" id="orgd663ae3">return lcm(5, 12) == 60
</pre>
</div>
<p>
Here's the naive function:
</p>
<div class="org-src-container">
<pre class="src src-python">def lcm(a: int, b: int) -> int:
m = 1
n = 1
while abs(m*a) != abs(n*b):
if abs(m*a) < abs(n*b):
m += 1
else:
n += 1
return m*a
return lcm(5, 12) == 60
</pre>
</div>
<pre class="example">
True
</pre>
<p>
Now we should either try to make the function more readable to
increase maintainability, or if it is causing performance issues,
optimize it.
</p>
<p>
Allegedly Python supports the functional programming
paradigm. Functional programs are readable and maintainable so let's
try that out.
</p>
<p>
First we want to turn the loop into a recursive call. Because we do
not want to change the type of the function, we need to introduce an
inner function.
</p>
<div class="org-src-container">
<pre class="src src-python">def lcm(a: int, b: int) -> int:
def inner(m, n):
if abs(m*a) < abs(n*b):
return inner(m + 1, n)
elif abs(n*b) < abs(m*a):
return inner(m, n + 1)
else:
return m*a
return inner(1, 1)
return lcm(5, 12) == 60
</pre>
</div>
<pre class="example">
True
</pre>
<p>
Now our function technically adheres to the functional paradigm. But
we can do better! The return statements seem quite redundant, right?
</p>
<div class="org-src-container">
<pre class="src src-python">def lcm(a: int, b: int) -> int:
def inner(m, n):
return (inner(m + 1, n) if abs(m*a) < abs(n*b) else
inner(m, n + 1) if abs(n*b) < abs(m*a) else
m*a)
return inner(1, 1)
return lcm(5, 12) == 60
</pre>
</div>
<pre class="example">
True
</pre>
<p>
I can only guess why if-expressions in Python look like something out
of Perl.
</p>
<p>
Let's change the inner function to a lambda term as that removes one
return statement.
</p>
<div class="org-src-container">
<pre class="src src-python">def lcm(a: int, b: int) -> int:
inner = lambda m: lambda n: (
inner(m + 1)(n) if abs(m*a) < abs(n*b) else
inner(m)(n + 1) if abs(n*b) < abs(m*a) else
m*a
)
return inner(1)(1)
return lcm(5, 12) == 60
</pre>
</div>
<pre class="example">
True
</pre>
<p>
In the process I also curried the inner function to make it closer to
a true lambda term. Though it is not yet a true lambda term as its
definition is self-referential.
</p>
<p>
Fixing this allows us also to remove the last return statement by
turning the whole function into a lambda expression.
</p>
<p>
The fix seems easy at first – just use fixed point recursion. The
problem is that Haskell Curry's classic Y-combinator, when implemented
directly in Python, gives rise to a stack overflow once any function
is passed to it. Python gets stuck evaluating the Y-combinator as the
argument is evaluated eagerly:
</p>
<blockquote>
<p>
All argument expressions are evaluated before the call is attempted. – <a href="https://docs.python.org/3/reference/expressions.html#calls">https://docs.python.org/3/reference/expressions.html#calls</a>
</p>
</blockquote>
<p>
Here's an example for the factorial function producing stack overflow
even before the number to calculate the factorial for is specified:
</p>
<div class="org-src-container">
<pre class="src src-python">Y = (lambda f: (lambda x: f(x(x)))
(lambda x: f(x(x))))
Y(lambda factorial:
lambda n: 1 if n == 1 else n*factorial(n - 1))
</pre>
</div>
<p>
Luckily Python can be tricked into not evaluating the argument as
eagerly with a different version of the Y-combinator. The main
difference seems to be that inside the combinator, <code>f</code> is not passed
the arguments directly but rather as a lambda form, which allows for
more lazy evaluation. Let's call this new combinator <code>fix</code>. Here's the
definition I found on several internet forums:
</p>
<div class="org-src-container">
<pre class="src src-python" id="org08ab578">fix = lambda f: (
(lambda x: f(lambda v: x(x)(v)))
(lambda x: f(lambda v: x(x)(v)))
)
</pre>
</div>
<p>
I couldn't find motivation for the given combinator, so here's proof
it works as expected:
</p>
<div class="org-src-container">
<pre class="src src-python">fix(g)
= (lambda f: (
(lambda x: f(lambda v: x(x)(v))) # By rewriting
(lambda x: f(lambda v: x(x)(v))) # definition above
))(g)
= (lambda x: g(lambda v: x(x)(v))) # By invoking the function
(lambda x: g(lambda v: x(x)(v))) # application (lambda f: ...)(g)
= g(lambda v:
(lambda x: g(lambda v: x(x)(v))) # By rewriting x in 'x(x)' with
(lambda x: g(lambda v: x(x)(v))) # the argument 'lambda x: ...'.
(v)) # This is a function application.
= g((lambda v: fix(g))(v)) # By rewriting the equality
# fix(g) = (lambda x: ...)(lambda x: ...)
# proven in the first two steps
= g(fix(g)) # By function application
# (lambda v: ...)(v)
</pre>
</div>
<p>
<code>fix(g) = g(fix(g))</code> means that the return value of fix(g) is such a
value that calling g repeatedly on it doesn't change the result,
ie. it is a fixed point of g.
</p>
<p>
Let's try it out with the factorial function we saw failing earlier.
</p>
<div class="org-src-container">
<pre class="src src-python">fix = lambda f: (
(lambda x: f(lambda v: x(x)(v)))
(lambda x: f(lambda v: x(x)(v)))
)
return fix(lambda factorial:
lambda n: 1 if n == 1 else n*factorial(n - 1))(5)
</pre>
</div>
<p>
It works fine. Now we can just use the fixed point operator to define
the recursive inner function.
</p>
<div class="org-src-container">
<pre class="src src-python">fix = lambda f: (
(lambda x: f(lambda v: x(x)(v)))
(lambda x: f(lambda v: x(x)(v)))
)
lcm = lambda a, b: (
fix(lambda inner:
lambda m: lambda n: (
inner(m + 1)(n) if m*a < n*b else
inner(m)(n + 1) if n*b < m*a else
m*a
)
)(1)(1)
)
return lcm(5, 12) == 60
</pre>
</div>
<pre class="example">
True
</pre>
<p>
Expanding fix to make lcm a pure lambda term gives
</p>
<div class="org-src-container">
<pre class="src src-python">lcm = lambda a, b: (
(lambda f: (
(lambda x: f(lambda v: x(x)(v)))
(lambda x: f(lambda v: x(x)(v)))
))
(lambda inner:
lambda m: lambda n: (
inner(m + 1)(n) if m*a < n*b else
inner(m)(n + 1) if n*b < m*a else
m*a
)
)(1)(1)
)
return lcm(5, 12) == 60
</pre>
</div>
<pre class="example">
True
</pre>
</div>
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