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"\n",
"<div class=\"highlight\"><pre><span></span><span class=\"k\">def</span> <span class=\"nf\">min_conflicts</span><span class=\"p\">(</span><span class=\"n\">csp</span><span class=\"p\">,</span> <span class=\"n\">max_steps</span><span class=\"o\">=</span><span class=\"mi\">100000</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Solve a CSP by stochastic hillclimbing on the number of conflicts."""</span>\n",
" <span class=\"c1\"># Generate a complete assignment for all variables (probably with conflicts)</span>\n",
" <span class=\"n\">csp</span><span class=\"o\">.</span><span class=\"n\">current</span> <span class=\"o\">=</span> <span class=\"n\">current</span> <span class=\"o\">=</span> <span class=\"p\">{}</span>\n",
" <span class=\"k\">for</span> <span class=\"n\">var</span> <span class=\"ow\">in</span> <span class=\"n\">csp</span><span class=\"o\">.</span><span class=\"n\">variables</span><span class=\"p\">:</span>\n",
" <span class=\"n\">val</span> <span class=\"o\">=</span> <span class=\"n\">min_conflicts_value</span><span class=\"p\">(</span><span class=\"n\">csp</span><span class=\"p\">,</span> <span class=\"n\">var</span><span class=\"p\">,</span> <span class=\"n\">current</span><span class=\"p\">)</span>\n",
" <span class=\"n\">csp</span><span class=\"o\">.</span><span class=\"n\">assign</span><span class=\"p\">(</span><span class=\"n\">var</span><span class=\"p\">,</span> <span class=\"n\">val</span><span class=\"p\">,</span> <span class=\"n\">current</span><span class=\"p\">)</span>\n",
" <span class=\"c1\"># Now repeatedly choose a random conflicted variable and change it</span>\n",
" <span class=\"k\">for</span> <span class=\"n\">i</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"n\">max_steps</span><span class=\"p\">):</span>\n",
" <span class=\"n\">conflicted</span> <span class=\"o\">=</span> <span class=\"n\">csp</span><span class=\"o\">.</span><span class=\"n\">conflicted_vars</span><span class=\"p\">(</span><span class=\"n\">current</span><span class=\"p\">)</span>\n",
" <span class=\"k\">if</span> <span class=\"ow\">not</span> <span class=\"n\">conflicted</span><span class=\"p\">:</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">current</span>\n",
" <span class=\"n\">var</span> <span class=\"o\">=</span> <span class=\"n\">random</span><span class=\"o\">.</span><span class=\"n\">choice</span><span class=\"p\">(</span><span class=\"n\">conflicted</span><span class=\"p\">)</span>\n",
" <span class=\"n\">val</span> <span class=\"o\">=</span> <span class=\"n\">min_conflicts_value</span><span class=\"p\">(</span><span class=\"n\">csp</span><span class=\"p\">,</span> <span class=\"n\">var</span><span class=\"p\">,</span> <span class=\"n\">current</span><span class=\"p\">)</span>\n",
" <span class=\"n\">csp</span><span class=\"o\">.</span><span class=\"n\">assign</span><span class=\"p\">(</span><span class=\"n\">var</span><span class=\"p\">,</span> <span class=\"n\">val</span><span class=\"p\">,</span> <span class=\"n\">current</span><span class=\"p\">)</span>\n",
" <span class=\"k\">return</span> <span class=\"bp\">None</span>\n",
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]
},
{
"cell_type": "markdown",
"Let's use this algorithm to solve the `eight_queens` CSP."
]
},
{
"cell_type": "code",
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"solution = min_conflicts(eight_queens)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"This is indeed a valid solution. \n",
"<br>\n",
"`notebook.py` has a helper function to visualize the solution space."
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {},
"outputs": [
{
"data": {
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qvMrfn7uovnoCQLshYCM2zZ5Xfv1w8L5XXvOejx4PThO2LwpmjANoZwRstNTC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\nS6Vvboqeb9HQwwaAnLnjS95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1640+/O16pRMnpTFzpePPSKNHRa/Ohq8Mvg49Z35wrXTO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\nYrfnP4SxJKkAAAAASUVORK5CYII=\n",
]
},
{
"cell_type": "markdown",
"Lets' see if we can find a different solution."
]
},
{
"cell_type": "code",
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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7ogjrfS+8pLm8AQBoVlsG7JM7/bc/uq619Sh5eK3/9refaW09AADF1RYBe9L4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\nGzbM/cGrvOfxlwanqc6r/P2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G4P1BPedSkA4K1kHHXbfIe/71Qf/979bztRX+CQAALUHA\nbjNTFwy83rG+MtCGDXN/8CrvefylwWmq8yp/f+7C+uoJAGgtAnYrNTnj+rWQuWovv+o9Hz0enCZs\nXyTMGAeA1BCw28yC2cH7piwI3hdFWO974SXN5Q0ASBYBOyUnd/pvf3Rda+tR8vBa/+1vP9PaegAA\n/BGwW+VU5ayus4Z755DPGj6wLcqlWBsfbqz4h7bXTlNe/sgR3vsRw6oSnTrSWAUAAE1hadKEvfv9\nhpz/PX1GGjqrP71P0K6eUV6dpvx4STryhDRhbH15lKfp2yaNeU9gdSuWK2VZxOzLexvSftlXgDZk\nadKs6BjS3PHDLq583zWvufxCgzUAIBUE7DYTZbGUxasq39f68fnZr8ZTLgAgPbEHbDP7kJm9UPY4\nbmbL4y6nyO7bWl/6DVuSqQcAoHViD9jOuX9zzl3gnLtA0kxJJyU9GHc5WbNiTfS0re7t1lNePZ8D\nABCfpIfEPyHpl865XyVcTttbE/PKnp+/LVq6uO/6FffnAABEk3TAXixpU/VGM1tqZj1mFuc9pXJl\nYY2TCN9+wHvevtt//5anveeg+2qXXLmy8v21l9euGwCg9RK7rMvMhkk6IOnDzrlDIelyPV8/ymVd\nkjT9Cmnfgapj+3/OBA1Z17qjV9j+oLwj3ZaTy7pyJe9tSPtlXwHaMPXLuuZL2h0WrDHgJ3cP3jZ/\nWfgxnSFLjUrSuI+H71++Onw/AKB9JBmwl8hnOLywZoSvEDZ54uBtj9VYFvRYjZt59J0I37+ukdY5\nv7eBgwAAzUokYJvZSEmflPSPSeSfSR0TGjosqRnjV93U4IFDx8daDwBANB1JZOqcOymJ/9nb2Pe3\npV0DAEA9WOmsjUzqTLf8WeelWz4AIBg3/0jYoO+3xmzxRofAP/IBL+DvOyD9cn9jedScIT5zcFMx\nQzX78t6GtF/2FaANI80ST2RIHI0LuxRrwezm7pd92Q3S1meDywUAtC8CdqtNuVPaHz7jq2+bNHau\n9/rQVmli1VD5dbdI9zwSvcjZM6Qd66XH7xrYtu+Ad+23JB2Msjb51G9GLxAAEDuGxBPm+/3WGBaX\nvF52qde7eau0ZFV4+np892vSkssGlxPKZzhcYjguD/LehrRf9hWgDSMNiROwE+b7/Z46Iu3xufC6\nStTz2YvmSNcvkubOlI6dkH66R7p1g/TzvRHqFyVYn98beDkX/1lkX97bkPbLvgK0Ieew29bQroYP\n3bLGC9BBxo2Wpk+Wrp5fuX3HC9Iln2uwUK69BoDU0cNOWOj3G3FofGiH9M6zg7dHrkNVL3roLOn0\nmeaGwt+tB7/uMy/vbUj7ZV8B2pAedtub6SIF7VKwbvSSr/LjzjwvnXouYl41gjUAoHVYOCVt02ov\n6G3dwQH2lqXSsae83nLpcXKnt93PkIsiButp34uQCADQKgyJJyzS9xvQy64OrFfOlR68s/G6LFnl\nzTgvFzgsHrF3zXBc9uW9DWm/7CtAGzJLvB1E/n53j5Tc2xWbrFvqfVIaP6Yy6ag50psno9ehc7T0\nxo8rt319o3TzXT4Be9omqXNx5Lz5zyL78t6GtF/2FaANOYedKRf2R+Cq3nbHEGnaFdIrBxrP+ujx\nyt76rx4Z3NOWxDlrAGhjnMNuN2VB0/VID21vLlj7OXehd912Re+aYA0AbY0h8YQ1/P2eOirtacH1\nz+cfbuq6cIbjsi/vbUj7ZV8B2jDSkDg97HY1tNPr9U5dm0z+U9d5+TcRrAEArUMPO2Gxfr8Rrtmu\nKeahb37dZ1/e25D2y74CtCE97NyZ6QYeM44N2r3SrzN+/uuVxwEAMokedsLS/n6Txq/77Mt7G9J+\n2VeANqSHDQBAXhCwAQDIAAI2AAAZ0A4rnfVK+lULy5vQX2ZLpHR+qaWfMQV5b0PaL0a0X+xa/vkK\n0IbnRkmU+qSzVjOznign97Ms75+Rz5dtfL5sy/vnk9r3MzIkDgBABhCwAQDIgCIG7O+kXYEWyPtn\n5PNlG58v2/L++aQ2/YyFO4cNAEAWFbGHDQBA5hCwAQDIgEIFbDP7lJn9m5m9bGZ/lXZ94mRmf2dm\nh83sZ2nXJQlmNtXMnjKzX5jZS2b2xbTrFDczG2Fmz5vZi/2f8Stp1yluZjbEzP7ZzB5Juy5JMLNX\nzOxfzOwFM+tJuz5xM7OxZvYPZvav/f8W/yjtOsXFzD7U326lx3EzW552vcoV5hy2mQ2R9P9J+qSk\n/ZL+SdIS59zPU61YTMxsjqQ3Jf0359x5adcnbmb2Xknvdc7tNrNRknZJujIv7SdJ5q0OcbZz7k0z\nGypph6QvOueeTblqsTGzFZK6JY12zi1Muz5xM7NXJHU753K5cIqZ3SPpJ865u81smKSRzrm+tOsV\nt/548ZqkWc65Vi7sFapIPeyLJL3snNvrnHtH0mZJn065TrFxzj0t6Wja9UiKc+5159zu/tcnJP1C\n0uR0axUv53mz/+3Q/kduflGb2RRJl0u6O+26oH5mNlrSHEnrJck5904eg3W/T0j6ZTsFa6lYAXuy\npFfL3u9Xzv7DLwoze9P8+2EAAAImSURBVL+kj0p6Lt2axK9/yPgFSYcl/cg5l6fP+A1JX5L0P9Ku\nSIKcpK1mtsvMlqZdmZhNl3RE0ob+0xp3m9nZaVcqIYslbUq7EtWKFLD9FqPNTe+lKMzsPZIekLTc\nOXc87frEzTl3xjl3gaQpki4ys1yc3jCzhZIOO+d2pV2XhM12zl0oab6k/9R/qiovOiRdKOlvnXMf\nlfSWpFzNBZKk/qH+KyR9L+26VCtSwN4vaWrZ+ymSDqRUFzSg/7zuA5Ludc79Y9r1SVL/UOM2SZ9K\nuSpxmS3piv5zvJslXWpmf59uleLnnDvQ/3xY0oPyTsXlxX5J+8tGff5BXgDPm/mSdjvnDqVdkWpF\nCtj/JOmDZjat/xfUYklbUq4TIuqfkLVe0i+cc2vSrk8SzKzLzMb2vz5L0jxJ/5pureLhnLvZOTfF\nOfd+ef/2fuyc+0zK1YqVmZ3dPyFS/UPFfyIpN1dtOOcOSnrVzD7Uv+kTknIz6bPMErXhcLjUHrfX\nbAnn3Gkzu0HS45KGSPo759xLKVcrNma2SdJcSRPMbL+kLzvn1qdbq1jNlnSNpH/pP8crSauccz9I\nsU5xe6+ke/pnqP6epPudc7m8/CmnJkl6sP9WkB2SvuuceyzdKsXuC5Lu7e/07JV0fcr1iZWZjZR3\nJdF/TLsufgpzWRcAAFlWpCFxAAAyi4ANAEAGELABAMgAAjYAABlAwAYAIAMI2AAAZAABGwCADPj/\nAUlr8AXRtSNBAAAAAElFTkSuQmCC\n",
"eight_queens = NQueensCSP(8)\n",
"solution = min_conflicts(eight_queens)\n",
"plot_NQueens(solution)"
},
{
"cell_type": "markdown",
"The solution is a bit different this time. \n",
"Running the above cell several times should give you various valid solutions.\n",
"<br>\n",
"In the `search.ipynb` notebook, we will see how NQueensProblem can be solved using a heuristic search method such as `uniform_cost_search` and `astar_search`."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Helper Functions\n",
"We will now implement a few helper functions that will help us visualize the Coloring Problem. We will make some modifications to the existing Classes and Functions for additional book keeping. To begin we modify the **assign** and **unassign** methods in the **CSP** to add a copy of the assignment to the **assignment_history**. We call this new class **InstruCSP**. This will allow us to see how the assignment evolves over time."
]
},
{
"cell_type": "code",
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
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"import copy\n",
"class InstruCSP(CSP):\n",
" \n",
" def __init__(self, variables, domains, neighbors, constraints):\n",
" super().__init__(variables, domains, neighbors, constraints)\n",
" self.assignment_history = []\n",
" \n",
" def assign(self, var, val, assignment):\n",
" super().assign(var,val, assignment)\n",
" self.assignment_history.append(copy.deepcopy(assignment))\n",
" \n",
" def unassign(self, var, assignment):\n",
" super().unassign(var,assignment)\n",
" self.assignment_history.append(copy.deepcopy(assignment))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Next, we define **make_instru** which takes an instance of **CSP** and returns a **InstruCSP** instance. "
]
},
{
"cell_type": "code",
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"def make_instru(csp):\n",
" return InstruCSP(csp.variables, csp.domains, csp.neighbors, csp.constraints)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We will now use a graph defined as a dictionary for plotting purposes in our Graph Coloring Problem. The keys are the nodes and their corresponding values are the nodes they are connected to."
]
},
{
"cell_type": "code",
"metadata": {
"collapsed": true
},
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"source": [
"neighbors = {\n",
" 0: [6, 11, 15, 18, 4, 11, 6, 15, 18, 4], \n",
" 1: [12, 12, 14, 14], \n",
" 2: [17, 6, 11, 6, 11, 10, 17, 14, 10, 14], \n",
" 3: [20, 8, 19, 12, 20, 19, 8, 12], \n",
" 4: [11, 0, 18, 5, 18, 5, 11, 0], \n",
" 5: [4, 4], \n",
" 6: [8, 15, 0, 11, 2, 14, 8, 11, 15, 2, 0, 14], \n",
" 7: [13, 16, 13, 16], \n",
" 8: [19, 15, 6, 14, 12, 3, 6, 15, 19, 12, 3, 14], \n",
" 9: [20, 15, 19, 16, 15, 19, 20, 16], \n",
" 10: [17, 11, 2, 11, 17, 2], \n",
" 11: [6, 0, 4, 10, 2, 6, 2, 0, 10, 4], \n",
" 12: [8, 3, 8, 14, 1, 3, 1, 14], \n",
" 13: [7, 15, 18, 15, 16, 7, 18, 16], \n",
" 14: [8, 6, 2, 12, 1, 8, 6, 2, 1, 12], \n",
" 15: [8, 6, 16, 13, 18, 0, 6, 8, 19, 9, 0, 19, 13, 18, 9, 16], \n",
" 16: [7, 15, 13, 9, 7, 13, 15, 9], \n",
" 17: [10, 2, 2, 10], \n",
" 18: [15, 0, 13, 4, 0, 15, 13, 4], \n",
" 19: [20, 8, 15, 9, 15, 8, 3, 20, 3, 9], \n",
" 20: [3, 19, 9, 19, 3, 9]\n",
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"Now we are ready to create an InstruCSP instance for our problem. We are doing this for an instance of **MapColoringProblem** class which inherits from the **CSP** Class. This means that our **make_instru** function will work perfectly for it."
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"coloring_problem = MapColoringCSP('RGBY', neighbors)"
]
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"coloring_problem1 = make_instru(coloring_problem)"
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"### CONSTRAINT PROPAGATION\n",
"Algorithms that solve CSPs have a choice between searching and or do a _constraint propagation_, a specific type of inference.\n",
"The constraints can be used to reduce the number of legal values for a another variable, which in turn can reduce the legal values for another variable, and so on.\n",
"<br>\n",
"Constraint propagation tries to enforce _local consistency_.\n",
"Consider each variable as a node in a graph and each binary constraint as an arc.\n",
"Enforcing local consistency causes inconsistent values to be eliminated throughout the graph, \n",
"a lot like the `GraphPlan` algorithm in planning, where mutex links are removed from a planning graph.\n",
"There are different types of local consistency:\n",
"1. Node consistency\n",
"2. Arc consistency\n",
"3. Path consistency\n",
"4. K-consistency\n",
"5. Global constraints\n",
"\n",
"Refer __section 6.2__ in the book for details.\n",
"<br>"
]
},
{
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"## AC-3\n",
"Before we dive into AC-3, we need to know what _arc-consistency_ is.\n",
"<br>\n",
"A variable $X_i$ is __arc-consistent__ with respect to another variable $X_j$ if for every value in the current domain $D_i$ there is some value in the domain $D_j$ that satisfies the binary constraint on the arc $(X_i, X_j)$.\n",
"<br>\n",
"A network is arc-consistent if every variable is arc-consistent with every other variable.\n",
"<br>\n",
"\n",
"AC-3 is an algorithm that enforces arc consistency.\n",
"After applying AC-3, either every arc is arc-consistent, or some variable has an empty domain, indicating that the CSP cannot be solved.\n",
"Let's see how `AC3` is implemented in the module."
]
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{
"cell_type": "code",
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"<div class=\"highlight\"><pre><span></span><span class=\"k\">def</span> <span class=\"nf\">AC3</span><span class=\"p\">(</span><span class=\"n\">csp</span><span class=\"p\">,</span> <span class=\"n\">queue</span><span class=\"o\">=</span><span class=\"bp\">None</span><span class=\"p\">,</span> <span class=\"n\">removals</span><span class=\"o\">=</span><span class=\"bp\">None</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""[Figure 6.3]"""</span>\n",
" <span class=\"k\">if</span> <span class=\"n\">queue</span> <span class=\"ow\">is</span> <span class=\"bp\">None</span><span class=\"p\">:</span>\n",
" <span class=\"n\">queue</span> <span class=\"o\">=</span> <span class=\"p\">[(</span><span class=\"n\">Xi</span><span class=\"p\">,</span> <span class=\"n\">Xk</span><span class=\"p\">)</span> <span class=\"k\">for</span> <span class=\"n\">Xi</span> <span class=\"ow\">in</span> <span class=\"n\">csp</span><span class=\"o\">.</span><span class=\"n\">variables</span> <span class=\"k\">for</span> <span class=\"n\">Xk</span> <span class=\"ow\">in</span> <span class=\"n\">csp</span><span class=\"o\">.</span><span class=\"n\">neighbors</span><span class=\"p\">[</span><span class=\"n\">Xi</span><span class=\"p\">]]</span>\n",
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"`AC3` also employs a helper function `revise`."
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"<body>\n",
"<h2></h2>\n",
"\n",
"<div class=\"highlight\"><pre><span></span><span class=\"k\">def</span> <span class=\"nf\">revise</span><span class=\"p\">(</span><span class=\"n\">csp</span><span class=\"p\">,</span> <span class=\"n\">Xi</span><span class=\"p\">,</span> <span class=\"n\">Xj</span><span class=\"p\">,</span> <span class=\"n\">removals</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Return true if we remove a value."""</span>\n",
" <span class=\"n\">revised</span> <span class=\"o\">=</span> <span class=\"bp\">False</span>\n",
" <span class=\"k\">for</span> <span class=\"n\">x</span> <span class=\"ow\">in</span> <span class=\"n\">csp</span><span class=\"o\">.</span><span class=\"n\">curr_domains</span><span class=\"p\">[</span><span class=\"n\">Xi</span><span class=\"p\">][:]:</span>\n",
" <span class=\"c1\"># If Xi=x conflicts with Xj=y for every possible y, eliminate Xi=x</span>\n",
" <span class=\"k\">if</span> <span class=\"nb\">all</span><span class=\"p\">(</span><span class=\"ow\">not</span> <span class=\"n\">csp</span><span class=\"o\">.</span><span class=\"n\">constraints</span><span class=\"p\">(</span><span class=\"n\">Xi</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">Xj</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">)</span> <span class=\"k\">for</span> <span class=\"n\">y</span> <span class=\"ow\">in</span> <span class=\"n\">csp</span><span class=\"o\">.</span><span class=\"n\">curr_domains</span><span class=\"p\">[</span><span class=\"n\">Xj</span><span class=\"p\">]):</span>\n",
" <span class=\"n\">csp</span><span class=\"o\">.</span><span class=\"n\">prune</span><span class=\"p\">(</span><span class=\"n\">Xi</span><span class=\"p\">,</span> <span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">removals</span><span class=\"p\">)</span>\n",
" <span class=\"n\">revised</span> <span class=\"o\">=</span> <span class=\"bp\">True</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">revised</span>\n",
"</pre></div>\n",
"</body>\n",
"</html>\n"
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{
"cell_type": "markdown",
"metadata": {},
"source": [
"`AC3` maintains a queue of arcs to consider which initially contains all the arcs in the CSP.\n",
"An arbitrary arc $(X_i, X_j)$ is popped from the queue and $X_i$ is made _arc-consistent_ with respect to $X_j$.\n",
"<br>\n",
"If in doing so, $D_i$ is left unchanged, the algorithm just moves to the next arc, \n",
"but if the domain $D_i$ is revised, then we add all the neighboring arcs $(X_k, X_i)$ to the queue.\n",
"<br>\n",
"We repeat this process and if at any point, the domain $D_i$ is reduced to nothing, then we know the whole CSP has no consistent solution and `AC3` can immediately return failure.\n",
"<br>\n",
"Otherwise, we keep removing values from the domains of variables until the queue is empty.\n",
"We finally get the arc-consistent CSP which is faster to search because the variables have smaller domains."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let's see how `AC3` can be used.\n",
"<br>\n",
"We'll first define the required variables."
]
},
{
"cell_type": "code",
"execution_count": 22,
"metadata": {
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"neighbors = parse_neighbors('A: B; B: ')\n",
"domains = {'A': [0, 1, 2, 3, 4], 'B': [0, 1, 2, 3, 4]}\n",
"constraints = lambda X, x, Y, y: x % 2 == 0 and (x + y) == 4 and y % 2 != 0\n",
"removals = []"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We'll now define a `CSP` object."
]
},
{
"cell_type": "code",
"execution_count": 23,
"metadata": {
"collapsed": true
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"outputs": [],
"source": [
"csp = CSP(variables=None, domains=domains, neighbors=neighbors, constraints=constraints)"
]
},
{
"cell_type": "code",
"execution_count": 24,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"False"
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"execution_count": 24,
"metadata": {},
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"source": [
"AC3(csp, removals=removals)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"This configuration is inconsistent."
]
},
{
"cell_type": "code",
"execution_count": 25,
"metadata": {
"collapsed": true
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"outputs": [],
"source": [
"constraints = lambda X, x, Y, y: (x % 2) == 0 and (x + y) == 4\n",
"removals = []\n",
"csp = CSP(variables=None, domains=domains, neighbors=neighbors, constraints=constraints)"
]
},
{
"cell_type": "code",
"execution_count": 26,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"True"
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"execution_count": 26,
"metadata": {},
"output_type": "execute_result"
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],
"source": [
"AC3(csp,removals=removals)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"This configuration is consistent."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## BACKTRACKING SEARCH\n",
"\n",
"For solving a CSP the main issue with Naive search algorithms is that they can continue expanding obviously wrong paths. In backtracking search, we check constraints as we go. Backtracking is just the above idea combined with the fact that we are dealing with one variable at a time. Backtracking Search is implemented in the repository as the function **backtracking_search**. This is the same as **Figure 6.5** in the book. The function takes as input a CSP and few other optional parameters which can be used to further speed it up. The function returns the correct assignment if it satisfies the goal. We will discuss these later. Let us solve our **coloring_problem1** with **backtracking_search**."
]
},
{
"cell_type": "code",
"execution_count": 27,
"metadata": {
"collapsed": true
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"outputs": [],
"source": [
"result = backtracking_search(coloring_problem1)"
]
},
{
"cell_type": "code",
"execution_count": 28,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"{0: 'R',\n",
" 1: 'R',\n",
" 2: 'R',\n",
" 3: 'R',\n",
" 4: 'G',\n",
" 5: 'R',\n",
" 6: 'G',\n",
" 7: 'R',\n",
" 8: 'B',\n",
" 9: 'R',\n",
" 10: 'G',\n",
" 11: 'B',\n",
" 12: 'G',\n",
" 13: 'G',\n",
" 14: 'Y',\n",
" 15: 'Y',\n",
" 16: 'B',\n",
" 17: 'B',\n",
" 18: 'B',\n",
" 19: 'G',\n",
" 20: 'B'}"
]
},
"execution_count": 28,
"metadata": {},
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],
"source": [
"result # A dictonary of assignments."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let us also check the number of assignments made."
]
},
{
"cell_type": "code",
"execution_count": 29,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"21"
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"execution_count": 29,
"metadata": {},
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"source": [
"coloring_problem1.nassigns"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now let us check the total number of assignments and unassignments which is the length of our assignment history."
]
},
{
"cell_type": "code",
"execution_count": 30,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"21"
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},
"execution_count": 30,
"metadata": {},
"output_type": "execute_result"
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"source": [
"len(coloring_problem1.assignment_history)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now let us explore the optional keyword arguments that the **backtracking_search** function takes. These optional arguments help speed up the assignment further. Along with these, we will also point out to methods in the CSP class that help make this work. \n",
"\n",
"The first of these is **select_unassigned_variable**. It takes in a function that helps in deciding the order in which variables will be selected for assignment. We use a heuristic called Most Restricted Variable which is implemented by the function **mrv**. The idea behind **mrv** is to choose the variable with the fewest legal values left in its domain. The intuition behind selecting the **mrv** or the most constrained variable is that it allows us to encounter failure quickly before going too deep into a tree if we have selected a wrong step before. The **mrv** implementation makes use of another function **num_legal_values** to sort out the variables by a number of legal values left in its domain. This function, in turn, calls the **nconflicts** method of the **CSP** to return such values.\n"
]
},
{
"cell_type": "code",
"execution_count": 31,
"metadata": {},
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"\n",
"<div class=\"highlight\"><pre><span></span><span class=\"k\">def</span> <span class=\"nf\">mrv</span><span class=\"p\">(</span><span class=\"n\">assignment</span><span class=\"p\">,</span> <span class=\"n\">csp</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Minimum-remaining-values heuristic."""</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">argmin_random_tie</span><span class=\"p\">(</span>\n",
" <span class=\"p\">[</span><span class=\"n\">v</span> <span class=\"k\">for</span> <span class=\"n\">v</span> <span class=\"ow\">in</span> <span class=\"n\">csp</span><span class=\"o\">.</span><span class=\"n\">variables</span> <span class=\"k\">if</span> <span class=\"n\">v</span> <span class=\"ow\">not</span> <span class=\"ow\">in</span> <span class=\"n\">assignment</span><span class=\"p\">],</span>\n",
" <span class=\"n\">key</span><span class=\"o\">=</span><span class=\"k\">lambda</span> <span class=\"n\">var</span><span class=\"p\">:</span> <span class=\"n\">num_legal_values</span><span class=\"p\">(</span><span class=\"n\">csp</span><span class=\"p\">,</span> <span class=\"n\">var</span><span class=\"p\">,</span> <span class=\"n\">assignment</span><span class=\"p\">))</span>\n",
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