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Peter Norvig
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Ghost.ipynb
411
Ghost.ipynb
@ -6,9 +6,9 @@
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"source": [
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"# Ghost\n",
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"\n",
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"[*Ghost*](https://en.wikipedia.org/wiki/Ghost_(game)) is a word game in which (quoting Wikipedia):\n",
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"According to [Wikipedia](https://en.wikipedia.org/wiki/Ghost_(game)):\n",
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"\n",
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"> *Ghost is a written or spoken word game in which players take turns adding letters to a growing word fragment, trying not to be the one to complete a valid word. Each fragment must be the beginning of an actual word, and usually some minimum is set on the length of a word that counts, such as three or four letters. The player who completes a word loses.*\n",
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"> **Ghost** *is a written or spoken word game in which players take turns adding letters to a growing word fragment, trying not to be the one to complete a valid word. Each fragment must be the beginning of an actual word, and usually some minimum is set on the length of a word that counts, such as three or four letters. The player who completes a word loses.*\n",
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"\n",
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"I'd like to create a program to allow any two players (human or computer) to play the game, and I'd like to figure out who wins if both players play optimally. The concepts I will need to define, and my implementation choices, are as follows:\n",
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"\n",
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@ -446,24 +446,24 @@
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},
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"outputs": [],
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"source": [
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"def rational(vocab, fragment): \n",
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" \"Select a play that makes opponent lose (if there is one), otherwise any play.\"\n",
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" for play in vocab.legal_plays(fragment):\n",
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" if not win(vocab, play):\n",
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" return play\n",
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" return play # Could return any play here\n",
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"import random\n",
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"\n",
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"def ask(name='Player'):\n",
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"def rational(vocab, fragment): \n",
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" \"Select a play that makes opponent not win (if there is one), otherwise a random play.\"\n",
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" plays = list(vocab.legal_plays(fragment))\n",
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" return next((play for play in plays if not win(vocab, play)),\n",
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" random.choice(plays))\n",
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"\n",
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"def ask(name):\n",
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" \"Return a strategy that asks for the next letter.\"\n",
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" return (lambda _, fragment: \n",
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" fragment + input(\"{}'s letter to add to '{}'? \".format(name, fragment)))"
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" return (lambda _, fragment: input('Player {}, given \"{}\", plays? '.format(name, fragment)))"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Here is a function to play a game:"
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"Here is a function to play a game. You give it a vocabulary, two (or potentially more) strategies, and optionally a `verbose` keyword to say if you want a line printed for each play or not."
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]
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},
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{
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@ -476,11 +476,14 @@
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"source": [
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"from itertools import cycle\n",
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"\n",
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"def play(vocab, *strategies):\n",
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"def play(vocab, *strategies, verbose=True):\n",
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" \"Return (winner, final_fragment) for a game of Ghost between these strategies.\"\n",
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" fragment = ''\n",
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" for strategy in cycle(strategies):\n",
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" play = strategy(vocab, fragment)\n",
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" if verbose:\n",
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" print('Player {}, given \"{}\", plays \"{}\".'\n",
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" .format(strategies.index(strategy), fragment, play))\n",
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" if play not in vocab.legal_plays(fragment):\n",
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" return (winner(fragment + '?'), play) # Player loses for making an illegal play\n",
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" elif play in vocab.words:\n",
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@ -496,10 +499,23 @@
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"collapsed": false
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},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"Player 0, given \"\", plays \"s\".\n",
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"Player 0, given \"s\", plays \"sq\".\n",
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"Player 0, given \"sq\", plays \"squ\".\n",
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"Player 0, given \"squ\", plays \"squo\".\n",
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"Player 0, given \"squo\", plays \"squoo\".\n",
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"Player 0, given \"squoo\", plays \"squoos\".\n",
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"Player 0, given \"squoos\", plays \"squoosh\".\n"
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]
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},
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{
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"data": {
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"text/plain": [
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"(1, 'ply')"
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"(1, 'squoosh')"
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]
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},
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"execution_count": 19,
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@ -518,10 +534,20 @@
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"collapsed": false
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},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"Player 0, given \"\", plays \"h\".\n",
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"Player 0, given \"h\", plays \"ha\".\n",
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"Player 0, given \"ha\", plays \"haa\".\n",
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"Player 0, given \"haa\", plays \"haaf\".\n"
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]
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},
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{
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"data": {
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"text/plain": [
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"(0, 'huddle')"
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"(0, 'haaf')"
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]
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},
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"execution_count": 20,
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@ -544,15 +570,23 @@
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"Player's letter to add to ''? d\n",
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"Player's letter to add to 'dr'? o\n",
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"Player's letter to add to 'droi'? d\n"
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"Player 0, given \"\", plays? d\n",
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"Player 0, given \"\", plays \"d\".\n",
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"Player 1, given \"d\", plays \"dw\".\n",
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"Player 0, given \"dw\", plays? dwa\n",
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"Player 0, given \"dw\", plays \"dwa\".\n",
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"Player 1, given \"dwa\", plays \"dwar\".\n",
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"Player 0, given \"dwar\", plays? dwarv\n",
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"Player 0, given \"dwar\", plays \"dwarv\".\n",
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"Player 1, given \"dwarv\", plays \"dwarve\".\n",
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"Player 0, given \"dwarve\", plays? dwarves\n",
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"Player 0, given \"dwarve\", plays \"dwarves\".\n"
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]
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},
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{
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"data": {
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"text/plain": [
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"(1, 'droid')"
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"(1, 'dwarves')"
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]
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},
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"execution_count": 21,
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@ -561,7 +595,7 @@
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}
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],
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"source": [
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"play(enable1, ask(), rational)"
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"play(enable1, ask(0), rational)"
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]
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},
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{
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@ -655,7 +689,7 @@
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{
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"data": {
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"text/plain": [
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"{'nays', 'nene', 'ngultrum', 'nirvanic', 'nolo', 'null', 'nyctalopia'}"
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"{'nazi', 'nene', 'ngultrum', 'nirvanic', 'nolo', 'null', 'nyctalopia'}"
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]
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},
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"execution_count": 24,
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@ -700,7 +734,7 @@
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" 'dreck',\n",
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" 'drink',\n",
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" 'droit',\n",
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" 'drunk',\n",
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" 'druid',\n",
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" 'dry',\n",
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" 'ewe',\n",
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" 'fjeld',\n",
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@ -714,7 +748,7 @@
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" 'mho',\n",
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" 'nth',\n",
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" 'oquassa',\n",
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" 'praam',\n",
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" 'prase',\n",
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" 'prequel',\n",
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" 'prill',\n",
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" 'pro',\n",
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@ -737,13 +771,13 @@
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" 'vroom',\n",
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" 'wrack',\n",
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" 'wrest',\n",
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" 'wrist',\n",
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" 'wrick',\n",
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" 'wrong',\n",
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" 'wrung',\n",
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" 'wry',\n",
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" 'xanthic',\n",
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" 'xanthin',\n",
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" 'ycleped',\n",
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" 'yperite',\n",
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" 'zucchetto'}"
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]
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},
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@ -807,7 +841,7 @@
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" 'dreck',\n",
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" 'drink',\n",
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" 'droit',\n",
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" 'drunk',\n",
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" 'druid',\n",
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" 'drywall',\n",
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" 'eschatologies',\n",
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" 'eschatology',\n",
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@ -826,7 +860,7 @@
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" 'fjord',\n",
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" 'gjetost',\n",
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" 'hyaenic',\n",
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" 'hybris',\n",
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" 'hybrid',\n",
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" 'hydatid',\n",
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" 'hyena',\n",
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" 'hyenine',\n",
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@ -857,7 +891,7 @@
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" 'plonk',\n",
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" 'pluck',\n",
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" 'plyer',\n",
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" 'quack',\n",
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" 'quaff',\n",
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" 'quell',\n",
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" 'quiff',\n",
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" 'quomodo',\n",
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@ -874,13 +908,13 @@
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" 'vroom',\n",
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" 'wrack',\n",
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" 'wrest',\n",
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" 'wrist',\n",
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" 'wrick',\n",
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" 'wrong',\n",
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" 'wrung',\n",
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" 'wryly',\n",
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" 'xanthic',\n",
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" 'xanthin',\n",
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" 'ycleped',\n",
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" 'yperite',\n",
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" 'zucchetto'}"
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]
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},
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@ -903,7 +937,7 @@
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{
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"data": {
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"text/plain": [
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"(85, ['hyphen', 'ngultrum', 'hyte', 'hybris'])"
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"(85, ['hyphen', 'ngultrum', 'hybrid', 'hyte'])"
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]
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},
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"execution_count": 28,
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@ -928,7 +962,7 @@
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"source": [
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"# SuperGhost\n",
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"\n",
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"In the variant *SuperGhost*, players can add a letter to either the beginning or the end of a fragment, as long as this forms a fragment that is part of some word. I was thinking of SuperGhost when I made the design decision to encapsulate `legal_plays` as a method of `Vocabulary`, rather than as a separate function. Because I did that, I should be able to use all the existing code if I just make a new class, `SuperVocabulary`, that finds *all* fragments (i.e. infixes) rather than just the beginning fragments (i.e. prefixes)."
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"In the variant *SuperGhost*, players can add a letter to either the beginning or the end of a fragment, as long as this forms a fragment that is part of some word. As Wikipedia says, given the fragment `era`, a player might play `bera` or `erad`. I was thinking of SuperGhost when I made the design decision to encapsulate `legal_plays` as a method of `Vocabulary`, rather than as a separate function. Because I did that, I should be able to use all the existing code if I just make a new class, `SuperVocabulary`, that finds *all* fragments (i.e. infixes) rather than just the beginning fragments (i.e. prefixes), and if I change `legal_plays` to add letters to both ends."
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]
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},
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{
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@ -981,27 +1015,14 @@
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},
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{
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"cell_type": "code",
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"execution_count": 35,
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"execution_count": 31,
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"metadata": {
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"collapsed": false
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"[387878, 387844, 1076434, 1076431]"
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]
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},
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"execution_count": 35,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"outputs": [],
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"source": [
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"enable1s = SuperVocabulary(enable1.words)\n",
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"enable1_4s = SuperVocabulary(enable1.words, 4)\n",
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"\n",
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"[len(v.fragments) for v in [enable1, enable1_4, enable1s, enable1_4s]]"
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"enable1_4s = SuperVocabulary(enable1.words, 4)"
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]
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},
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{
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@ -1014,7 +1035,19 @@
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{
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"data": {
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"text/plain": [
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"True"
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"{'atresse',\n",
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" 'ctresse',\n",
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" 'etresse',\n",
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" 'itresse',\n",
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" 'ntresse',\n",
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" 'otresse',\n",
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" 'ptresse',\n",
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" 'rtresse',\n",
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" 'stresse',\n",
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" 'tressed',\n",
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" 'tressel',\n",
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" 'tresses',\n",
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" 'ttresse'}"
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]
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},
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"execution_count": 32,
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@ -1023,9 +1056,8 @@
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}
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],
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"source": [
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"# Can the first player win in SuperGhost with 3-letter words?\n",
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"\n",
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"win(enable1s)"
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"# Example legal plays\n",
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"enable1s.legal_plays('tresse')"
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]
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},
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{
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@ -1046,6 +1078,30 @@
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"output_type": "execute_result"
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}
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],
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"source": [
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"# Can the first player win in SuperGhost with 3-letter words?\n",
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"\n",
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"win(enable1s)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 34,
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"metadata": {
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"collapsed": false
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"True"
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]
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},
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"execution_count": 34,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"# How about with a 4-letter limit?\n",
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"\n",
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@ -1061,7 +1117,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 34,
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"execution_count": 35,
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"metadata": {
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"collapsed": false
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},
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@ -1070,19 +1126,26 @@
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"Peter's fragment, given ''? z\n",
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"Peter's fragment, given 'zq'? zqu\n",
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"Peter's fragment, given 'ezqu'? ezqui\n",
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"Peter's fragment, given 'mezqui'? mezquit\n"
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"Peter's fragment, given ''? q\n",
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"Player 0, given \"\", plays \"q\".\n",
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"Player 1, given \"q\", plays \"mq\".\n",
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"Peter's fragment, given 'mq'? mqu\n",
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"Player 0, given \"mq\", plays \"mqu\".\n",
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"Player 1, given \"mqu\", plays \"umqu\".\n",
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"Peter's fragment, given 'umqu'? umqua\n",
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"Player 0, given \"umqu\", plays \"umqua\".\n",
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"Player 1, given \"umqua\", plays \"kumqua\".\n",
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"Peter's fragment, given 'kumqua'? kumquat\n",
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"Player 0, given \"kumqua\", plays \"kumquat\".\n"
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]
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},
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{
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"data": {
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"text/plain": [
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"(1, 'mezquit')"
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"(1, 'kumquat')"
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]
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},
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"execution_count": 34,
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"execution_count": 35,
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"metadata": {},
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"output_type": "execute_result"
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}
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@ -1095,7 +1158,221 @@
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"I would like to give a concise summary of the strategy for SuperGhost, but my existing `outcomes` function won't do it. That's because it is not enough to know that a particular word results in a win; we have to know in what order the letters of the word are added. I'll leave it as an exercise to find a good way to summarize SuperGhost strategies."
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"I would like to give a concise summary of the strategy for SuperGhost, but my existing `outcomes` function won't do it. That's because it is not enough to know that a particular word results in a win; we have to know in what order the letters of the word are added. I'll leave it as an exercise to find a good way to summarize SuperGhost strategies.\n",
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"\n",
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"# SuperDuperGhost\n",
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"\n",
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"In the variant *SuperDuperGhost*, players have an option to reverse the fragment before adding a letter to the beginning or end. As Wikipedia says, given the fragment `era`, a player might play `bera, erad, nare,` or `aren`.\n",
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"Wikipedia is not clear, but I interpret this as meaning that the fragment played must still be a fragment of a word (not a reversed fragment of a word). Again, all we need is a new subclass:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 36,
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"metadata": {
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"collapsed": true
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},
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"outputs": [],
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"source": [
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"class SuperDuperVocabulary(SuperVocabulary):\n",
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" \"Holds a set of legal words and a set of legal infix fragments of those words.\"\n",
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" \n",
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" def legal_plays(self, fragment):\n",
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" \"All plays that form a valid infix; optionally reverse fragment first.\"\n",
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" return {play for L in alphabet \n",
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" for play in (fragment + L, fragment[::-1] + L,\n",
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" L + fragment, L + fragment[::-1])} & self.fragments"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 37,
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"metadata": {
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"collapsed": true
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},
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"outputs": [],
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"source": [
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"enable1sd = SuperDuperVocabulary(enable1.words)\n",
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"enable1_4sd = SuperDuperVocabulary(enable1.words, 4)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 38,
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"metadata": {
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"collapsed": false
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},
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"outputs": [
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{
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"data": {
|
||||
"text/plain": [
|
||||
"{'atresse',\n",
|
||||
" 'ctresse',\n",
|
||||
" 'dessert',\n",
|
||||
" 'esserts',\n",
|
||||
" 'etresse',\n",
|
||||
" 'itresse',\n",
|
||||
" 'ntresse',\n",
|
||||
" 'otresse',\n",
|
||||
" 'ptresse',\n",
|
||||
" 'rtresse',\n",
|
||||
" 'stresse',\n",
|
||||
" 'tressed',\n",
|
||||
" 'tressel',\n",
|
||||
" 'tresses',\n",
|
||||
" 'ttresse'}"
|
||||
]
|
||||
},
|
||||
"execution_count": 38,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"# Example legal plays\n",
|
||||
"enable1sd.legal_plays('tresse')"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"Now we should check who wins. But when I tried `win(enable1sd)`, I didn't get an answer within the first minute; there are just too many paths to get to the same fragment, so we are repeating a lot of work. The standard response to this problem is a `lru_cache`, which brings the time down to 2 seconds."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 39,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"from functools import lru_cache\n",
|
||||
"\n",
|
||||
"@lru_cache(None)\n",
|
||||
"def win(vocab, fragment=''):\n",
|
||||
" \"Does the player whose turn it is have a forced win?\"\n",
|
||||
" return (fragment in vocab.words or \n",
|
||||
" fragment not in vocab.fragments or\n",
|
||||
" any(not win(vocab, play) \n",
|
||||
" for play in vocab.legal_plays(fragment)))"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 40,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"True"
|
||||
]
|
||||
},
|
||||
"execution_count": 40,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"win(enable1sd)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 41,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"True"
|
||||
]
|
||||
},
|
||||
"execution_count": 41,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"win(enable1_4sd)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"The first player can win with either vocabulary. Here's a sample game."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 42,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"Player 0, given \"\", plays \"b\".\n",
|
||||
"Player 0, given \"b\", plays \"sb\".\n",
|
||||
"Player 0, given \"sb\", plays \"msb\".\n",
|
||||
"Player 0, given \"msb\", plays \"msbu\".\n",
|
||||
"Player 0, given \"msbu\", plays \"emsbu\".\n",
|
||||
"Player 0, given \"emsbu\", plays \"gemsbu\".\n",
|
||||
"Player 0, given \"gemsbu\", plays \"gemsbuc\".\n",
|
||||
"Player 0, given \"gemsbuc\", plays \"gemsbuck\".\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"(0, 'gemsbuck')"
|
||||
]
|
||||
},
|
||||
"execution_count": 42,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"play(enable1sd, rational, rational)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"Let's see how many fragments each vocabulary takes:"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 43,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"[387878, 387844, 1076434, 1076431, 1076434, 1076431]"
|
||||
]
|
||||
},
|
||||
"execution_count": 43,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"[len(v.fragments) for v in [enable1, enable1_4, enable1s, enable1_4s, enable1sd, enable1_4sd]]"
|
||||
]
|
||||
},
|
||||
{
|
||||
@ -1106,12 +1383,14 @@
|
||||
"\n",
|
||||
"Here's a summary of what we have learned. (*Note:* the bold **qursh** means it is a losing word):\n",
|
||||
"\n",
|
||||
"| Game \t| Shortest \t| Winner \t| First Player Outcomes | Second Player Outcomes\n",
|
||||
"|----\t|---\t |---\t |--- |---\n",
|
||||
"| Ghost | 3 \t | Second \t| qaid qiviut qoph **qursh** qurush qwerty | 55 words\n",
|
||||
"| Ghost | 4 \t | First \t| naan nene ngultrum nirvanic nolo null nyctalopia | 85 words\n",
|
||||
"| SuperGhost | 3\t| First \t| ? | ? |\n",
|
||||
"| SuperGhost | 4 \t| First \t| ? | ? |\n",
|
||||
"| Variant \t| Shortest \t| Winner \t| First Player Outcomes | 2nd Outcomes | Fragments\n",
|
||||
"|----\t|---:\t |---\t |--- |--- |---:\n",
|
||||
"| Ghost | 3 \t | Second \t| qaid qiviut qoph **qursh** qurush qwerty | 55 words | 387,878\n",
|
||||
"| Ghost | 4 \t | First \t| naan nene ngultrum nirvanic nolo null nyctalopia | 85 words | 387,844\n",
|
||||
"| Super | 3\t| First \t| ? | ? | 1,076,434\n",
|
||||
"| Super | 4 \t| First \t| ? | ? | 1,076,431\n",
|
||||
"| SuperDuper | 3 | First| ? | ? | 1,076,434\n",
|
||||
"| SuperDuper | 4 | First| ? | ? | 1,076,431\n",
|
||||
"\n",
|
||||
"# Further Work\n",
|
||||
"\n",
|
||||
|
||||
Loading…
Reference in New Issue
Block a user