Why Complexity Is Measured Asymptotically, Not by Counting Steps

A common objection to P versus NP is that you could 'cheat' — precompute every answer into a giant table, then 'solve' any instance in one lookup. Complexity theory is built specifically to be immune to this. (See P vs NP Problem: Fundamentals Explained for the core question.) ## Complexity is a function of input size The quantity that matters is not the raw step count for one input but **how the step count grows as the input grows**, expressed in Big O notation. A brute-force Sudoku search scales like 9^n (exponential in the number of cells n); a hoped-for polynomial method scales like n^3. A lookup table does not beat this because *building* the table required doing all the work, and the table itself is astronomically large. ## The abstract step model To stop people hiding a billion operations inside 'one step', complexity uses an abstract machine (Turing machine) where each primitive operation — read a memory cell, compare two numbers, write a value — counts as exactly one step. You cannot define a single step that secretly performs unbounded work. This makes the framework immune to reframing tricks and independent of hardware constants. ## Why 'the answer is hidden in pi' fails Suppose the digits of pi contained every Sudoku solution. You would still need a fast function mapping a specific puzzle to the *position* in pi where its solution sits — and that mapping is essentially as hard as solving the puzzle directly. Every 'the answer already exists somewhere' argument smuggles the original difficulty into the step of *finding the right part*. If you ever found a fast puzzle-to-position rule, *that rule would itself be the polynomial algorithm* — pi would be irrelevant. The magic is never in the container; it is in the mapping, which is exactly what an algorithm is. ## Precomputation versus intrinsic difficulty Measuring cost 'across all of civilization' rather than per run is a legitimate engineering idea — it is simply called a **database**, and engineers use it constantly (dynamic programming is the principled version, valid whenever the total stored work stays polynomial, as in computing Fibonacci numbers). But complexity theory asks a different question: the *intrinsic* difficulty of a problem, independent of work already done. ## A hardness proof is timeless Because it concerns the logical structure of computation, a valid proof of P ≠ NP would be true regardless of which computers exist or what has been precomputed — as enduring as Euclid's proofs. This timelessness is itself the deep reason precomputation cannot resolve P versus NP: a mathematical proof cannot rest on contingent facts about our particular moment in history.