A Philosopher Explains XOR Operations with Extension Methods and TypeScript
Dr. Adrian Mróz
I break stuff to build better solutions | Cloud Engineer | Shaping the future of the media industry
An analytical exploration of array operations through the lens of propositional logic
In this article, I present a philosophical examination of a LeetCode problem that deals with the fundamental logical operation of exclusive disjunction (XOR). Via analytical philosophy and type extension, we'll explore how basic logical principles can elegantly solve complex computational problems.
The Nature of XOR and Its Properties
The basic intuition behind this algorithm comes from the logical properties of exclusive disjunction (XOR) - a truth-functional operator that is true if and only if its operands differ in truth value. In the domain of binary arithmetic, this maps elegantly to bit-wise operations where XOR(a,b) = 1 iff a ≠ b.
The core insight comes from two key logical principles:
The associative property of XOR: (a ⊕ b) ⊕ c ≡ a ⊕ (b ⊕ c)
Order Independence: Consider the logical operation of XORing three propositions. Just as the statement "if p implies q and q implies r, then p implies r" holds true regardless of the order, the sequence of our XOR operations is similarly invariant:
Imagine planning a party
It doesn't matter if we decide on salad first or the main dish first, the end result is the same: we'll end up with exactly one of:
The self-negation property: a ⊕ a ≡ 0
(Self-Cancellation): Imagine having a light switch. If you flip it twice, it returns to its original state. This is logically equivalent to saying "p is exclusively either true or true". Which, by the nature of exclusive disjunction, must be false. Similarly, when we XOR a number with itself:
It is like saying "I'll either take the bus or take the bus" - this isn't really a choice at all! Or "The door is either open or open" - logically meaningless as an exclusive choice
These properties form the philosophical bedrock of our solution: we can reorder our operations freely (associativity) and know that paired propositions will cancel each other out (self-negation). Just as in both formal logic and everyday decision-making, these principles guide our understanding of exclusive choices and their combinations.
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A Three-Part Solution Through Type Extension
The solution is split into three distinct logical operations, implemented through TypeScript's prototype extension mechanism:
exclusiveDisjunction
contributionIfOdd
xorAllNums
The logical validity of this approach rests on the principle that when we apply XOR to a sequence of identical values an even number of times, the result is necessarily 0, while an odd number of times preserves the original value.
Complexity
Time complexity O(n+m)
Where n=∣nums1∣ and m=∣nums2∣
Justification: The algorithm performs a linear traversal of each array exactly once through the reduce operation, and all other operations (length checks, bitwise operations) are O(1).
Space complexity: O(1)
Justification: The algorithm maintains only a constant number of variables regardless of input size, using only accumulator variables for the reduce operation and temporary storage for intermediate XOR results.
Conclusion
The solution demonstrates how foundational logical principles can be extended to array operations while preserving the fundamental properties of XOR operations.
declare global {
interface Array<T> {
/**
* Calculates the exclusive disjunction (XOR) of all elements in the array.
* It combines all elements in the array by comparing each pair of bits:
* if the bits are different, it results in 1; if they are the same, it results in 0.
*
* @returns {number} The result of the XOR operation on all elements of the array.
*/
exclusiveDisjunction(): number;
/**
* Checks if the provided number is odd or even.
* If the number is odd, it returns the result of the `exclusiveDisjunction` operation.
* If the number is even, it returns 0.
*
* @param {number} cardinalityParity - A number used to check if it is odd or even.
* @returns {number} The result of the `exclusiveDisjunction` if the number is odd, or 0 if the number is even.
*/
contributionIfOdd(cardinalityParity: number): number;
}
}
// Extend Array prototype with exclusiveDisjunction
Array.prototype.exclusiveDisjunction = function (): number {
return this.reduce((sum, n) => sum ^ n);
};
// Extend Array prototype with contributionIfOdd
Array.prototype.contributionIfOdd = function (cardinalityParity: number): number {
return (cardinalityParity & 1) ? this.exclusiveDisjunction() : 0;
};
/**
* Computes the XOR of the results from two arrays based on their lengths.
* It checks if the length of each array is odd or even.
* If it's odd, it performs an exclusive disjunction on the array;
* If it's even, it returns 0.
*
* @param {number[]} nums1 - The first array of numbers.
* @param {number[]} nums2 - The second array of numbers.
* @returns {number} The XOR of the results from both arrays based on their lengths.
*/
function xorAllNums(nums1: number[], nums2: number[]): number {
return nums1.contributionIfOdd(nums2.length) ^ nums2.contributionIfOdd(nums1.length);
};
This article is part of a series examining algorithmic problems through the lens of philosophical analysis. For the complete implementation including TypeScript interfaces and detailed documentation, visit the original solution on LeetCode.
#Programming #TypeScript #Algorithms #Philosophy #LogicGates #SoftwareEngineering