![](\"http://norvig.com/gesture.png\")
\n",
"\n",
"We'd like to know, for a given keyboard, how confusing is it? How many words have paths on the keyboard that can be confused for other words? We have our work cut out for us:\n",
"\n",
"1. Determine what letters could be confused for each other.\n",
"2. Determine what words/paths can be confused.\n",
"3. Invent some metric for the overall confusingness of a keyboard. \n",
"4. Try to find less-confusing keyboards.\n",
"\n",
"Letter Confusions\n",
"---\n",
"\n",
"So, as a first step, we will make a mapping from each key to the keys that it can be confused with. I'll say that any key within a distance of 1.5 units on the keyboard is a **neighboring** key, and thus a potential confusion:"
]
},
{
"cell_type": "code",
"execution_count": 30,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"def neighboring_keys(kbd, radius=1.5):\n",
" \"Build a dict of {Letter:NeighboringLetters}, e.g. {'Q':'AQW', ...}.\"\n",
" return {A: cat(sorted(B for B in kbd if distance(kbd[A], kbd[B]) <= radius))\n",
" for A in kbd}\n",
"\n",
"cat = ''.join ## Function to join letters (or strings) into one string"
]
},
{
"cell_type": "code",
"execution_count": 31,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"{'A': 'AQSWZ',\n",
" 'B': 'BGHJNV',\n",
" 'C': 'CDFGVX',\n",
" 'D': 'CDEFRSXZ',\n",
" 'E': 'DERSW',\n",
" 'F': 'CDFGRTVX',\n",
" 'G': 'BCFGHTVY',\n",
" 'H': 'BGHJNUVY',\n",
" 'I': 'IJKOU',\n",
" 'J': 'BHIJKMNU',\n",
" 'K': 'IJKLMNO',\n",
" 'L': 'KLMOP',\n",
" 'M': 'JKLMN',\n",
" 'N': 'BHJKMN',\n",
" 'O': 'IKLOP',\n",
" 'P': 'LOP',\n",
" 'Q': 'AQW',\n",
" 'R': 'DEFRT',\n",
" 'S': 'ADESWXZ',\n",
" 'T': 'FGRTY',\n",
" 'U': 'HIJUY',\n",
" 'V': 'BCFGHV',\n",
" 'W': 'AEQSW',\n",
" 'X': 'CDFSXZ',\n",
" 'Y': 'GHTUY',\n",
" 'Z': 'ADSXZ'}"
]
},
"execution_count": 31,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"qwerty_neighbors = neighboring_keys(qwerty) \n",
"qwerty_neighbors"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We see, for example, that **Q**, off in the corner, has only **A**, **W**, and itself as neighbors, while **G**, in the middle of the keyboard, has 8 neighbors. \n",
"\n",
"Word Confusions\n",
"---\n",
"\n",
"Consider each of the letters in the word \"HELLO,\" and all the possible choices for neighbors of each letter:"
]
},
{
"cell_type": "code",
"execution_count": 32,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"['BGHJNUVY', 'DERSW', 'KLMOP', 'KLMOP', 'IKLOP']"
]
},
"execution_count": 32,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"columns = [qwerty_neighbors[L] for L in 'HELLO']\n",
"columns"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"These are five columns of letters, and if we pick one letter from each column, we get a path that is formed by letters that are each confusions of letters in the original word, and so the whole path is a confusion for the original word. So \"JELLO\" is a confusion for \"HELLO\", as would be \"BDKKI\" and \"YWPPP\", except those are not words.\n",
"\n",
"It turns out that there is a library function, `itertools.product`, that will take alternative choices, and generate all possible ways of assembling a sequence consisting of one selection (letter) from each alternative choice."
]
},
{
"cell_type": "code",
"execution_count": 33,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"paths = {cat(picks) for picks in itertools.product(*columns)}"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"How many paths are there?"
]
},
{
"cell_type": "code",
"execution_count": 34,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"5000"
]
},
"execution_count": 34,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"len(paths)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let's look at a few of them:"
]
},
{
"cell_type": "code",
"execution_count": 35,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"['NELKI', 'NWLKL', 'VEMKI', 'VEOOP', 'UWPMK', 'JSLML', 'GRMMO', 'UWKKP']"
]
},
"execution_count": 35,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"random.sample(paths, 8)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"And let's see all the paths that are also words:"
]
},
{
"cell_type": "code",
"execution_count": 36,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"{'BROMO', 'BROOK', 'HELLO', 'JELLO'}"
]
},
"execution_count": 36,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"WORDS & paths"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Only 4 out of 5000; That's pretty good, but it means \"HELLO\" is still a confusing word. We can package this as the function `confusions`:"
]
},
{
"cell_type": "code",
"execution_count": 37,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"def confusions(word, neighbors=neighboring_keys(qwerty)):\n",
" \"All valid words whose paths could be confused with the path for the given word.\"\n",
" columns = [neighbors[L] for L in word]\n",
" return {cat(picks) for picks in itertools.product(*columns)} & WORDS"
]
},
{
"cell_type": "code",
"execution_count": 38,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"{'BROMO', 'BROOK', 'HELLO', 'JELLO'}"
]
},
"execution_count": 38,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"confusions('HELLO')"
]
},
{
"cell_type": "code",
"execution_count": 39,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"{'ALTOS', 'EIDOS', 'SIDLE', 'SKEPS', 'WIELD', 'WORKS', 'WORLD', 'WORMS'}"
]
},
"execution_count": 39,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"confusions('WORLD')"
]
},
{
"cell_type": "code",
"execution_count": 40,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"{'FEARING',\n",
" 'FRAYING',\n",
" 'GEARING',\n",
" 'GRATING',\n",
" 'GRAYING',\n",
" 'GREYING',\n",
" 'REARING',\n",
" 'REEFING',\n",
" 'RESTING',\n",
" 'TEARING',\n",
" 'TESTING'}"
]
},
"execution_count": 40,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"confusions('TESTING')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"So far, so good. But I'm worried about the efficiency of `confusions`.\n",
"\n",
"My first concern is that `WORDS` has 178,691 words, including many unusual words, so there might be too many confusions. I will read in a smaller word set, with only common words:"
]
},
{
"cell_type": "code",
"execution_count": 41,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"! [ -e words.js ] || curl -O https://xkcd.com/simplewriter/words.js\n",
" \n",
"COMMON = set(re.findall('[A-Z]+', open('words.js').read().upper()))"
]
},
{
"cell_type": "code",
"execution_count": 42,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(3614, 'SPECIALS', 'UNDERSTANDINGS')"
]
},
"execution_count": 42,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"len(COMMON), max(COMMON, key=path_length), max(COMMON, key=len)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"More Efficient `confusions`\n",
"---\n",
"\n",
"Another issue is that `confusions` is really slow:"
]
},
{
"cell_type": "code",
"execution_count": 43,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"CPU times: user 3.12 s, sys: 395 ms, total: 3.51 s\n",
"Wall time: 3.52 s\n"
]
},
{
"data": {
"text/plain": [
"{'SOMETHING'}"
]
},
"execution_count": 43,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"%time confusions('SOMETHING')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"It took (on my computer) 3 seconds to compute this. Why so long? Let's count:"
]
},
{
"cell_type": "code",
"execution_count": 44,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[7, 5, 5, 5, 5, 8, 5, 6, 8]"
]
},
"execution_count": 44,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"[len(neighboring_keys(qwerty)[L]) for L in 'SOMETHING']"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"There are 7 × 5 × 5 × 5 × 5 × 8 × 5 × 6 × 8 = 8,400,000 paths for `confusions` to consider. `'PALEOMAGNETISMS'` would have about 10,000 times more paths. Looking at every path is slow, but for most paths, we're wasting our time. For example, one choice for the first two neighboring letters of 'SO' is 'XP', but 'XP' does not start any word in the dictionary. Nevertheless, `itertools.product` will generate 240,000 combinations that start with 'XP', and will then rule them out one at a time. It would be better to stop as soon as we see 'XP', and never consider continuations of this path.\n",
"\n",
"So that gives us the key idea for a more efficient version of `confusions`: *only follow paths that form a prefix of some word.* I'll make a set of all the prefixes of the `COMMON` words:"
]
},
{
"cell_type": "code",
"execution_count": 45,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"def prefixes(words):\n",
" \"Return a set of prefixes (1 to N characters) of this collection of words.\"\n",
" return {word[:i] for word in words for i in range(1, len(word)+1)}\n",
"\n",
"PREFIXES = prefixes(COMMON)"
]
},
{
"cell_type": "code",
"execution_count": 46,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"{'T', 'TH', 'THE', 'THES', 'THESE', 'THEY', 'THO', 'THOS', 'THOSE'}"
]
},
"execution_count": 46,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"prefixes(['THESE', 'THEY', 'THOSE'])"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We can describe the more efficient version of the `confusions` algorithm:\n",
"\n",
"1. Maintain a queue of partial paths, where a partial path is a string representing choices for neighboring letters for a prefix of the word. For example, 'HE' and 'JE' are both partial paths for the word 'HELLO'. Initialize the queue to have just one partial path, the empty string.\n",
"2. Remove a partial path from the queue. Find all possible ways to extend the path by adding a neighboring letter, but only if doing so creates a path that is a prefix of some word in the dictionary. For example, given the word 'HELLO', and the partial path 'JE', consider all the neighbors of 'L' (namely, 'K', 'M', 'L', 'O', or 'P'), but only 'JEM', 'JEL', and 'JEO' are prefixes of words, so add just those to the queue.\n",
"3. When a partial path reaches the same length as the word ('HELLO' in this example), then don't extend it any farther; instead check to see if the path is a word. If it is, yield it as a result.\n",
"\n",
"A word is `confusable` if it has a confusion word (other than itself)."
]
},
{
"cell_type": "code",
"execution_count": 47,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"def confusions(word, neighbors=neighboring_keys(qwerty)):\n",
" \"All valid words whose paths could be confused with the path for this word.\"\n",
" Q = [''] # A queue of partial paths \n",
" while Q:\n",
" path = Q.pop()\n",
" if len(path) < len(word):\n",
" for L in neighbors[word[len(path)]]:\n",
" if path + L in PREFIXES:\n",
" Q.append(path + L)\n",
" elif path in COMMON:\n",
" yield path\n",
"\n",
"def confusable(word, neighbors=neighboring_keys(qwerty)) -> bool: \n",
" \"Is this word confusable with another, given this keyboard's neighboring keys?\"\n",
" return any(c != word for c in confusions(word, neighbors))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let's check the speed (remember the old version took over 3 seconds):"
]
},
{
"cell_type": "code",
"execution_count": 48,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"CPU times: user 119 µs, sys: 0 ns, total: 119 µs\n",
"Wall time: 122 µs\n"
]
},
{
"data": {
"text/plain": [
"{'SOMETHING'}"
]
},
"execution_count": 48,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"%time set(confusions('SOMETHING'))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We went from about 3 seconds to about 100 microseconds: that's 30,000 times faster! We can look at some more confusions:"
]
},
{
"cell_type": "code",
"execution_count": 49,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"{'A': 'S A',\n",
" 'ACCEPT': 'ACCEPT',\n",
" 'AIRS': 'WORE WORD WIFE WIDE SURE SITS SIRS SIDE AIRS',\n",
" 'BEATER': 'NEARER NEARED HEATER HEATED GRAYER GRAYED BESTED BEATER',\n",
" 'BEGAN': 'BEGAN',\n",
" 'BLOWER': 'NOISED BLOWER',\n",
" 'DISCOVERS': 'DISCOVERS',\n",
" 'DRINK': 'DRUNK DRINK',\n",
" 'DROVES': 'DROVES DRIVES',\n",
" 'ENTER': 'ENTER',\n",
" 'EXCEPT': 'EXCEPT',\n",
" 'GATHER': 'GATHER FATHER BAGGER BAGGED',\n",
" 'GUESSING': 'GUESSING',\n",
" 'LEADINGS': 'LEADINGS',\n",
" 'LOOKER': 'POOLED MOONED LOOKER LOOKED KILLER KILLED',\n",
" 'MAPS': 'MAPS',\n",
" 'NEEDED': 'NEEDED BEDDED',\n",
" 'NIGHTS': 'NIGHTS MIGHTS',\n",
" 'RETURNS': 'RETURNS',\n",
" 'ROOMED': 'ROOMED ROLLED FILLED',\n",
" 'SEXING': 'SEXING ADDING',\n",
" 'SHORTEST': 'SHORTEST',\n",
" 'SHOVING': 'SHOVING',\n",
" 'SKINNED': 'SKINNED',\n",
" 'SOUND': 'WOUND SPINS SOUND SKINS',\n",
" 'STOPS': 'STOPS STOOD STOLE DROPS',\n",
" 'THANKS': 'THANKS',\n",
" 'TRIPE': 'TRIPS TRIPE',\n",
" 'TV': 'TV',\n",
" 'VIEWED': 'VIEWED HORSES'}"
]
},
"execution_count": 49,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"{w: ' '.join(confusions(w))\n",
" for w in random.sample(COMMON, 30)}"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Visualizing Paths on a Keyboard\n",
"----\n",
"\n",
"It would be nice to see potentially confusable word paths on a keyboard.\n",
"I'll add a call to `plot_paths` in `show_kbd`:\n",
"\n"
]
},
{
"cell_type": "code",
"execution_count": 50,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"def show_kbd(kbd, name='keyboard', words=()):\n",
" \"Plot the keyboard and show title/stats.\"\n",
" for L in kbd:\n",
" plot_square(kbd[L].x, -kbd[L].y, label=L)\n",
" plot_paths(kbd, words)\n",
" plt.axis('equal'); plt.axis('off')\n",
" plt.title(title(kbd, name));\n",
" plt.show()\n",
" \n",
"def plot_paths(kbd, words):\n",
" \"Plot paths for each word.\"\n",
" for (i, word) in enumerate(words):\n",
" Xs = [kbd[L].x for L in word]\n",
" Ys = [-kbd[L].y for L in word]\n",
" plt.plot(Xs, Ys, '-o')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let's see how it works on three similar paths:"
]
},
{
"cell_type": "code",
"execution_count": 51,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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"**Answered**: \"PALEOMAGNETISMS\" etc.\n", " 2. What words have the highest ratio of path length to word length?
**Answered**: \"PALAPA\" etc.\n", " 3. What is the average segment length, over a typical typing work load?
**Answered**: 3.23 keys, for Qwerty keyboard, on our sample workload.\n", " 4. Is there a better keyboard layout to minimize the average segment length over a work load?
**Answered**: Yes, many layouts at around 1.9 on Qwerty; or 1.7 or 1.8 on more square keyboards.\n", " 5. How often are two words confused because they have similar paths?
**Answered**: On Qwerty, 26% of the words in a small dictionary, and 55% of the words in running text have at least one possible confusion. Other layouts are worse.\n", " 6. Is there a better keyboard layout to minimize confusion?
**Partially Answered**: We found a keyboard with less confusingness than Qwerty. The computation of confusingness takes too long to do very much hillclimbing search. \n", " 7. Is there a better keyboard layout to maximize overall user satisfaction?
**Partially Answered**: We defined a combined metric, and found\n", "a keyboard with a better score. There are no doubt better metrics, and better keyboards to be found.\n", " \n", "\n", "\n", "\n", "Going Beyond\n", "===\n", "\n", "Now it is your turn to answer the open questions, or make up some questions of your own. Good luck! Here are a few ideas to get you started:\n", "\n", "* Hillclimbing just keeps the one best keyboard it has found so far. Other optimization techniques such as\n", "[beam search](http://en.wikipedia.org/wiki/Beam_search) or [genetic algorithms](http://en.wikipedia.org/wiki/Genetic_algorithm) or [ant colony optimization](http://en.wikipedia.org/wiki/Ant_colony_optimization_algorithms) maintain several candidates at a time. Is that a good idea?\n", "\n", "* The code in this notebook emphasizes clarity, not efficiency. Can you modify the code (or perhaps port it to another language) and make it twice as efficient? 10 times? 100 times?\n", "\n", "* What other factors do you think are important to user satisfaction with a keyboard. Can you measure them?\n", "\n", "* Consider the 5 paths below. They all start at 'P', move in a straight line to 'T', and then go to 'S', but they all make different stops along the top row. In other words, the 5 paths all trace exacty the same lines, so they are very confusing, but our definition of `confusions` makes most of them different. Can you think\n", "of a better way to handle confusions for paths like this?" ] }, { "cell_type": "code", "execution_count": 67, "metadata": {}, "outputs": [ { "data": { "image/png": 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mdqeZvW5m/zSzB81s20KnT7gtHzezCVnjzjGzX5YiXjT/tdF++z8z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\n", 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