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A growing collection of terms and concepts. These definitions are updated and revised as my understanding evolves.
algorithm
P = NP?
- P: The class of decision problems solvable by a deterministic Turing machine in polynomial time, scaling gracefully ($O(n^k)$) as the input grows.
- NP: The class of decision problems where a proposed solution can be verified in polynomial time.
The question asks whether P = NP: Does the ability to efficiently verify a solution guarantee the ability to efficiently find that solution from scratch?
From first principles, generating novel information (discovery, design, theorem proving) appears fundamentally more computationally expensive than recognizing it. Cryptography is predicated on this asymmetry. It is computationally cheap to verify a digital signature, but prohibitively expensive to forge one.
If P = NP, this asymmetry is an illusion. It implies that structural difficulty does not inherently exist. Any problem where we can efficiently recognize a correct answer would also possess an efficient algorithm to construct that answer, collapsing the distinction between verifying a mathematical proof and discovering it.
computability
Turing completeness
A system is Turing complete if it can simulate a Universal Turing Machine. This requires only a minimal threshold of logical primitives: arbitrary state mutation (memory) and conditional branching (the ability to change behavior based on that memory).
From first principles, computation is entirely substrate-independent. Whether built from silicon logic gates, cellular automata (like Conway's Game of Life), or theoretical hydraulic valves, any system that reaches this threshold can compute anything that is theoretically computable.
The rationalist implication is that absolute computational expressivity is a binary state, not a spectrum. There are no "more powerful" classical computing languages, only more ergonomic ones. Given arbitrary time and memory, a rudimentary Turing-complete system can perfectly simulate the execution of any other, rendering them mathematically equivalent.