Agentic

unlike its siblings on this branch of the wiki — linear, quadratic, integer — this “programming” really is about writing programs. functional programming is the discipline of building software out of expressions that are evaluated rather than statements that are executed: no assignment, no mutation, no time. what remains is algebra, and algebra is something you can reason about. 𐃏

a gan trains a generator by making it play a game against a learned critic: the generator \(G\) maps noise to samples, the discriminator \(D\) tries to tell those samples from real data, and each improves by exploiting the other’s current weakness — density estimation recast as a two-player minimax game (goodfellow et al. 2014, generative adversarial networks). 𐃏 the framework is treated in ch. 20 of (Goodfellow, Ian, 2016).

a single gaussian is a committed statement: one bump, symmetric, thin tails. real data is usually several stories overlaid — different regimes, different subpopulations — and a gaussian mixture says so explicitly: each point was generated by one of \(k\) gaussians, we just don’t get told which. 𐃏 fitting one is the canonical latent-variable problem, and the algorithm that fits it — expectation-maximisation — is one of the great workhorses of statistics.

most optimisation asks for the best; goal programming asks for good enough, several times over. 𐃏 you attach a numeric target to each of several objectives, measure how far the plan misses each target, and minimise the misses you dislike. the philosophy is herbert simon’s satisficing¹ — real decision makers do not maximise a grand utility function, they set aspiration levels and stop when they are met — and the machinery is pure linear programming: goal programming was invented by charnes and cooper as an LP device² and remains the most-used technique in practical multi-criteria decision making precisely because it never leaves LP territory.

In 1931 a twenty-five-year-old in Vienna published a paper proving that mathematics cannot fully formalise itself: any consistent formal system rich enough to do arithmetic contains true statements it cannot prove, and — worse — cannot even prove its own consistency (Gödel, Kurt, 1931). 𐃏 This page walks the actual argument: what the theorem says precisely, the two technical inventions that power it (Gödel numbering and diagonalisation), and — just as important — the long list of things it does not say.

graph theory began in 1736 when euler noticed that a walking-tour puzzle about seven bridges in königsberg was really a question about dots and lines — and that the geometry (lengths, positions, the river) was irrelevant. 𐃏 that act of abstraction — keep the incidence structure, discard everything else — is the whole subject. a graph is the minimal mathematical model of “things, some pairs of which are related”, which is why it keeps turning up: networks, molecules, dependencies, state machines, social graphs (Epp, Susanna S., 2019).

hashiwokakero (“build bridges”, nikoli) hands you a grid of numbered islands and asks you to join them with bridges until every number is spent. it is the friendliest possible introduction to constraint satisfaction: the constraints are few and visual, propagation alone solves most human-published puzzles, and when it doesn’t, you get to write a backtracking search. this page documents my solver at code/private/hashi/ — a go rewrite of a uni assignment originally in c — including the debugging session that writing this page forced on it. 𐃏

The Large Movie Review Dataset (aclImdb v1.0): 50,000 polar movie reviews from IMDB, split evenly for training and testing. For over a decade the default benchmark for binary sentiment classification — simple enough to load with one function call, large enough that word order and negation actually matter.

Provenance#

Collected at Stanford and released alongside Maas, Daly, Pham, Huang, Ng and Potts, Learning Word Vectors for Sentiment Analysis, ACL 2011. The dataset page (and download) lives at ai.stanford.edu/~amaas/data/sentiment. The original motivation was learning sentiment-aware word representations; the review corpus it shipped with promptly outlived the method.

take a linear program and add one word — integer — and the complexity class jumps from polynomial to NP-hard. that word buys expressive power nothing continuous can match: yes/no decisions, either/or logic, fixed costs, sequencing, assignment. integer programming is the lingua franca of operations research precisely because “decide” is not a convex verb. 𐃏