{ "cells": [ { "cell_type": "markdown", "id": "c9cdfdab", "metadata": {}, "source": [ "# Introduction to AI · What you cannot write down\n", "\n", "A task you perform perfectly, in under a second, without being able to say\n", "how. Then an honest attempt to write the rules for it.\n" ] }, { "cell_type": "code", "execution_count": 1, "id": "1a97a193", "metadata": { "execution": { "iopub.execute_input": "2026-08-17T13:31:14.748986Z", "iopub.status.busy": "2026-08-17T13:31:14.748626Z", "iopub.status.idle": "2026-08-17T13:31:15.937065Z", "shell.execute_reply": "2026-08-17T13:31:15.936827Z" } }, "outputs": [], "source": [ "# hide — plumbing so this file runs both in the repo and in Colab.\n", "try:\n", " from _figkit import save_fig, record\n", "except ImportError: # Colab — no repo, no problem\n", " def save_fig(name, plot, **kw):\n", " import matplotlib.pyplot as plt\n", " fig, ax = plt.subplots(figsize=kw.get('figsize', (7, 4.2))); plot(ax); plt.show()\n", " def record(key, value):\n", " print(f'{key} = {value}'); return value\n", "import numpy as np\n", "from sklearn.datasets import load_digits # bundled — no download\n" ] }, { "cell_type": "code", "execution_count": 2, "id": "5039da55", "metadata": { "execution": { "iopub.execute_input": "2026-08-17T13:31:15.938309Z", "iopub.status.busy": "2026-08-17T13:31:15.938178Z", "iopub.status.idle": "2026-08-17T13:31:15.947423Z", "shell.execute_reply": "2026-08-17T13:31:15.947086Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "n_digits = 1,797\n", "img_size = 8x8\n", "n_pixels = 64\n", "n_classes = 10\n" ] }, { "data": { "text/plain": [ "10" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "#| caption: 1,797 handwritten digits, eight pixels square\n", "D = load_digits()\n", "IMG, Y = D.images, D.target\n", "record('n_digits', f'{len(Y):,}')\n", "record('img_size', f'{IMG.shape[1]}x{IMG.shape[2]}')\n", "record('n_pixels', IMG.shape[1] * IMG.shape[2])\n", "record('n_classes', len(set(Y)))\n" ] }, { "cell_type": "markdown", "id": "46dbc2e1", "metadata": {}, "source": [ "## 1. Read these\n", "\n", "You will get all ten right, and it will take you about a second.\n" ] }, { "cell_type": "code", "execution_count": 3, "id": "bfd8cd38", "metadata": { "execution": { "iopub.execute_input": "2026-08-17T13:31:15.948686Z", "iopub.status.busy": "2026-08-17T13:31:15.948581Z", "iopub.status.idle": "2026-08-17T13:31:15.996151Z", "shell.execute_reply": "2026-08-17T13:31:15.995858Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "figure ten-digits -> what-you-cannot-write-down.ten-digits.{light,dark}.svg\n" ] } ], "source": [ "#| caption: Ten digits, one of each\n", "from matplotlib.patches import Rectangle\n", "from matplotlib.collections import PatchCollection\n", "\n", "picks = [int(np.where(Y == d)[0][1]) for d in range(10)]\n", "\n", "def plot(ax):\n", " # Rectangles, not imshow: imshow embeds a base64 raster and the page's\n", " # SVG guard refuses . Alpha carries the ink so it reads on either theme.\n", " rects, alphas = [], []\n", " for k, idx in enumerate(picks):\n", " ox = k * 9\n", " for r in range(8):\n", " for c in range(8):\n", " v = IMG[idx][r, c] / 16.0\n", " if v > 0.02:\n", " rects.append(Rectangle((ox + c, r), 1, 1)); alphas.append(v)\n", " pc = PatchCollection(rects, facecolor=(0.91, 0.41, 0.25), edgecolor='none')\n", " pc.set_alpha(None); pc.set_facecolors([(0.91, 0.41, 0.25, a) for a in alphas])\n", " ax.add_collection(pc)\n", " ax.set_xlim(0, 90); ax.set_ylim(8.6, -0.6); ax.set_aspect('equal')\n", " ax.set_xticks([]); ax.set_yticks([])\n", " for s in ax.spines.values(): s.set_visible(False)\n", "save_fig('ten-digits', plot, figsize=(7, 1.5))\n" ] }, { "cell_type": "markdown", "id": "119e0b14", "metadata": {}, "source": [ "## 2. Now write the rule\n", "\n", "Here are the features a person actually proposes when asked how they tell\n", "digits apart — ink in each band, symmetry, how wide it is, whether it has a\n", "hole in it. All of them measurable, all of them sensible.\n" ] }, { "cell_type": "code", "execution_count": 4, "id": "8bdd7302", "metadata": { "execution": { "iopub.execute_input": "2026-08-17T13:31:15.997477Z", "iopub.status.busy": "2026-08-17T13:31:15.997390Z", "iopub.status.idle": "2026-08-17T13:31:16.648589Z", "shell.execute_reply": "2026-08-17T13:31:16.648389Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "n_features = 10\n", "feature_list = total ink, ink in the top, ink in the middle, ink at the bottom, …\n" ] }, { "data": { "text/plain": [ "'total ink, ink in the top, ink in the middle, ink at the bottom, …'" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "#| caption: The features a human would name\n", "def holes(img):\n", " \"\"\"Enclosed background regions — the thing that makes 0 and 8 obvious.\"\"\"\n", " bg = img < 4; H, W = bg.shape\n", " seen = np.zeros_like(bg, bool); n = 0\n", " for sr in range(H):\n", " for sc in range(W):\n", " if not bg[sr, sc] or seen[sr, sc]: continue\n", " stack, cells, touches = [(sr, sc)], [], False\n", " seen[sr, sc] = True\n", " while stack:\n", " r, c = stack.pop(); cells.append((r, c))\n", " if r in (0, H-1) or c in (0, W-1): touches = True\n", " for dr, dc in ((1,0),(-1,0),(0,1),(0,-1)):\n", " nr, nc = r+dr, c+dc\n", " if 0 <= nr < H and 0 <= nc < W and bg[nr, nc] and not seen[nr, nc]:\n", " seen[nr, nc] = True; stack.append((nr, nc))\n", " if not touches: n += 1\n", " return n\n", "\n", "def features(img):\n", " ink = img.sum()\n", " return {\n", " 'total ink': ink,\n", " 'ink in the top': img[:3].sum(),\n", " 'ink in the middle':img[3:5].sum(),\n", " 'ink at the bottom':img[5:].sum(),\n", " 'ink on the left': img[:, :4].sum(),\n", " 'ink on the right': img[:, 4:].sum(),\n", " 'left-right symmetry': -abs(img[:, :4].sum() - img[:, 4:][:, ::-1].sum()),\n", " 'how wide it is': np.count_nonzero(img.sum(axis=0) > 2),\n", " 'how tall it is': np.count_nonzero(img.sum(axis=1) > 2),\n", " 'holes in it': holes(img),\n", " }\n", "\n", "NAMES = list(features(IMG[0]))\n", "X = np.array([[features(im)[n] for n in NAMES] for im in IMG], float)\n", "record('n_features', len(NAMES))\n", "record('feature_list', ', '.join(NAMES[:4]) + ', …')\n" ] }, { "cell_type": "markdown", "id": "8b1c52f3", "metadata": {}, "source": [ "## 3. One rule per digit, chosen as well as possible\n", "\n", "For each digit, find the single best `IF feature is between a and b THEN it\n", "is this digit` rule available. Then two conditions. Then three.\n" ] }, { "cell_type": "code", "execution_count": 5, "id": "f39e36d3", "metadata": { "execution": { "iopub.execute_input": "2026-08-17T13:31:16.649661Z", "iopub.status.busy": "2026-08-17T13:31:16.649581Z", "iopub.status.idle": "2026-08-17T13:31:16.750726Z", "shell.execute_reply": "2026-08-17T13:31:16.750490Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "no_rule_fires = 21%\n", "rules_contradict = 43%\n", "exactly_one_rule = 36%\n", "acc_1cond = 2.9%\n", "acc_2cond = 14.4%\n", "acc_3cond = 21.6%\n", "acc_4cond = 23.4%\n", "acc_5cond = 24.2%\n", "best_rule_acc = 24.2%\n", "guessing = 10%\n", "figure rules-vs-you -> what-you-cannot-write-down.rules-vs-you.{light,dark}.svg\n" ] } ], "source": [ "#| caption: Hand-style rules, tuned to the best they can be\n", "def best_rule(target, k):\n", " \"\"\"Best k-condition rule for one digit: pick the k features whose bands\n", " separate it best, and take the tightest band that keeps 80% of them.\"\"\"\n", " is_t = Y == target\n", " scored = []\n", " for j in range(X.shape[1]):\n", " lo, hi = np.percentile(X[is_t, j], [10, 90])\n", " fires = (X[:, j] >= lo) & (X[:, j] <= hi)\n", " prec = (fires & is_t).sum() / max(fires.sum(), 1)\n", " scored.append((prec, j, lo, hi))\n", " scored.sort(reverse=True)\n", " return [(j, lo, hi) for _, j, lo, hi in scored[:k]]\n", "\n", "def accuracy(k, tally=None):\n", " rules = {d: best_rule(d, k) for d in range(10)}\n", " right = 0\n", " for i in range(len(Y)):\n", " votes = [d for d, conds in rules.items()\n", " if all(lo <= X[i, j] <= hi for j, lo, hi in conds)]\n", " if tally is not None:\n", " tally['none' if not votes else 'many' if len(votes) > 1 else 'one'] += 1\n", " if len(votes) == 1 and votes[0] == Y[i]: right += 1\n", " return right / len(Y)\n", "\n", "# The two failures an expert system actually suffers, counted (OUTLINE L7):\n", "# digits NO rule covers, and digits where rules CONTRADICT each other.\n", "tally = {'none': 0, 'one': 0, 'many': 0}\n", "accuracy(3, tally)\n", "record('no_rule_fires', f\"{tally['none']/len(Y):.0%}\")\n", "record('rules_contradict', f\"{tally['many']/len(Y):.0%}\")\n", "record('exactly_one_rule', f\"{tally['one']/len(Y):.0%}\")\n", "\n", "curve = [accuracy(k) for k in range(1, 6)]\n", "for k, a in zip(range(1, 6), curve):\n", " record(f'acc_{k}cond', f'{a:.1%}')\n", "record('best_rule_acc', f'{max(curve):.1%}')\n", "record('guessing', f'{1/10:.0%}')\n", "\n", "def plot(ax):\n", " ax.plot(range(1, 6), [100*c for c in curve], marker='o', lw=2.2)\n", " ax.axhline(10, ls=':', lw=1.2)\n", " ax.text(5, 12, 'guessing', ha='right', fontsize=9)\n", " ax.axhline(97, ls='--', lw=1.2)\n", " ax.text(1.05, 92, 'you, just now, in one second', fontsize=9)\n", " ax.set_xticks(range(1, 6))\n", " ax.set_xlabel('conditions per rule')\n", " ax.set_ylabel('digits identified correctly (%)')\n", " ax.set_ylim(0, 105)\n", "save_fig('rules-vs-you', plot, figsize=(7, 3.8))\n" ] }, { "cell_type": "markdown", "id": "550f2466", "metadata": {}, "source": [ "## 4. Why it cannot work: look at the overlap\n", "\n", "Take the single feature that best separates any two digits, and plot it.\n" ] }, { "cell_type": "code", "execution_count": 6, "id": "e5f0f0bd", "metadata": { "execution": { "iopub.execute_input": "2026-08-17T13:31:16.751894Z", "iopub.status.busy": "2026-08-17T13:31:16.751825Z", "iopub.status.idle": "2026-08-17T13:31:16.840663Z", "shell.execute_reply": "2026-08-17T13:31:16.840402Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "best_feature_1v7 = ink at the bottom\n", "overlap_1v7 = 81%\n", "figure the-overlap -> what-you-cannot-write-down.the-overlap.{light,dark}.svg\n" ] } ], "source": [ "#| caption: The best feature for telling a 1 from a 7\n", "def overlap(a, b):\n", " best = None\n", " for j in range(X.shape[1]):\n", " ma, mb = X[Y == a, j], X[Y == b, j]\n", " lo = max(ma.min(), mb.min()); hi = min(ma.max(), mb.max())\n", " share = (((ma >= lo) & (ma <= hi)).mean() + ((mb >= lo) & (mb <= hi)).mean()) / 2\n", " if best is None or share < best[0]: best = (share, j)\n", " return best\n", "\n", "share, j = overlap(1, 7)\n", "record('best_feature_1v7', NAMES[j])\n", "record('overlap_1v7', f'{share:.0%}')\n", "\n", "def plot(ax):\n", " bins = np.linspace(X[:, j].min(), X[:, j].max(), 26)\n", " ax.hist(X[Y == 1, j], bins=bins, alpha=.75, label='the digit 1')\n", " ax.hist(X[Y == 7, j], bins=bins, alpha=.75, label='the digit 7')\n", " ax.set_xlabel(NAMES[j])\n", " ax.set_ylabel('how many digits')\n", " ax.legend(frameon=False)\n", "save_fig('the-overlap', plot, figsize=(7, 3.4))\n" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.13.5" } }, "nbformat": 4, "nbformat_minor": 5 }