Comparing functions for equality is a fundamental concept in mathematics, but it presents unique challenges in the world of programming. This article explores the intricacies of function comparison, drawing parallels between mathematical theory and its practical implementation in Haskell. We’ll delve into why comparing functions in Haskell isn’t always straightforward and discuss potential solutions and their limitations.
Sets and the Foundation of Equality
In mathematics, sets are defined by the uniqueness of their elements. This inherent characteristic implies an underlying equivalence: for any two elements within a set, a comparison for equality must be possible. Haskell mirrors this concept with the Eq typeclass, which encompasses all types whose values can be compared for equality. However, a crucial distinction arises when considering functions.
Functions: A Tale of Two Worlds
Mathematically, a function comprises a codomain, a corange (the set of actual output values), and a one-valued relation. Two functions are considered equal if all three components are equal. This allows for a universal comparison of functions for equality.
In contrast, Haskell functions don’t universally belong to the Eq typeclass. Comparing arbitrary functions for equality is not inherently supported. This discrepancy between the mathematical ideal and the practical implementation in Haskell raises a key question: why does this difference exist, and how can we bridge the gap? For the purpose of this discussion, we will disregard bottom values ($bot$) and side effects like IO functions, focusing primarily on pure functions that have direct counterparts in mathematics. Let’s assume a mapping $phi$ exists that translates Haskell functions to their mathematical equivalents. Our goal is to determine how, given two Haskell functions f and g, we can ascertain whether $phi(f)$ equals $phi(g)$.
Implementing Eq for Functions in Haskell: A Limited Solution
While a general Eq instance for functions is impossible, we can achieve it under specific conditions. For functions with a Boolean corange and a codomain that’s an instance of Eq, we can define an Eq instance like this:
{-# LANGUAGE FlexibleInstances #-}
instance Eq b => Eq (Bool -> b) where
f == g = f True == g True && f False == g FalseThis allows us to compare functions like these:
f :: Bool -> Int
f True = 5
f False = 6
g :: Bool -> Int
g True = 5
g False = 6
h :: Bool -> Int
h True = 42
h False = 0Resulting in:
λ (f == g, f == h)
(True, False)This approach extends to functions with finite, enumerable coranges (Bounded and Enum typeclasses) by comparing outputs for all possible inputs:
{-# LANGUAGE FlexibleInstances #-}
instance (Enum a, Bounded a, Eq b) => Eq (a -> b) where
f == g = and $ zipWith (==) (map f is) (map g is)
where is = enumFromTo minBound maxBoundThis enables comparison for functions like toUpper and toLower when working with Char. However, this method becomes computationally infeasible for larger input types.
Alternative Approaches and Their Shortcomings
Other approaches, such as comparing function references or seeking analytical solutions, prove inadequate. Comparing references fails for identical functions at different memory locations, while analytical solutions are often impractical for complex or non-continuous functions.
The Undecidability Problem
The core issue lies in the undecidability of function equality for infinite functions. Rice’s Theorem states that no general algorithm can determine whether a Turing machine computes a partial function with a non-trivial property. This directly implies the impossibility of universally comparing functions for equality in a computational setting. Attempting to do so would be equivalent to solving the Halting Problem, a known undecidable problem in computer science.
Conclusion: Practical Considerations
When comparing functions for equality in a practical context, it’s crucial to define the specific requirements. Often, comparing outputs for a relevant subset of inputs suffices. A comprehensive comparison for all possible inputs is generally unattainable and unnecessary for most applications. Focus on the specific behavior of the functions within the scope of your program, rather than seeking a universal solution to an inherently undecidable problem.
