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"\n",
"<div class=\"highlight\"><pre><span></span><span class=\"k\">def</span> <span class=\"nf\">dpll</span><span class=\"p\">(</span><span class=\"n\">clauses</span><span class=\"p\">,</span> <span class=\"n\">symbols</span><span class=\"p\">,</span> <span class=\"n\">model</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""See if the clauses are true in a partial model."""</span>\n",
" <span class=\"n\">unknown_clauses</span> <span class=\"o\">=</span> <span class=\"p\">[]</span> <span class=\"c1\"># clauses with an unknown truth value</span>\n",
" <span class=\"k\">for</span> <span class=\"n\">c</span> <span class=\"ow\">in</span> <span class=\"n\">clauses</span><span class=\"p\">:</span>\n",
" <span class=\"n\">val</span> <span class=\"o\">=</span> <span class=\"n\">pl_true</span><span class=\"p\">(</span><span class=\"n\">c</span><span class=\"p\">,</span> <span class=\"n\">model</span><span class=\"p\">)</span>\n",
" <span class=\"k\">if</span> <span class=\"n\">val</span> <span class=\"ow\">is</span> <span class=\"bp\">False</span><span class=\"p\">:</span>\n",
" <span class=\"k\">return</span> <span class=\"bp\">False</span>\n",
" <span class=\"k\">if</span> <span class=\"n\">val</span> <span class=\"ow\">is</span> <span class=\"ow\">not</span> <span class=\"bp\">True</span><span class=\"p\">:</span>\n",
" <span class=\"n\">unknown_clauses</span><span class=\"o\">.</span><span class=\"n\">append</span><span class=\"p\">(</span><span class=\"n\">c</span><span class=\"p\">)</span>\n",
" <span class=\"k\">if</span> <span class=\"ow\">not</span> <span class=\"n\">unknown_clauses</span><span class=\"p\">:</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">model</span>\n",
" <span class=\"n\">P</span><span class=\"p\">,</span> <span class=\"n\">value</span> <span class=\"o\">=</span> <span class=\"n\">find_pure_symbol</span><span class=\"p\">(</span><span class=\"n\">symbols</span><span class=\"p\">,</span> <span class=\"n\">unknown_clauses</span><span class=\"p\">)</span>\n",
" <span class=\"k\">if</span> <span class=\"n\">P</span><span class=\"p\">:</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">dpll</span><span class=\"p\">(</span><span class=\"n\">clauses</span><span class=\"p\">,</span> <span class=\"n\">removeall</span><span class=\"p\">(</span><span class=\"n\">P</span><span class=\"p\">,</span> <span class=\"n\">symbols</span><span class=\"p\">),</span> <span class=\"n\">extend</span><span class=\"p\">(</span><span class=\"n\">model</span><span class=\"p\">,</span> <span class=\"n\">P</span><span class=\"p\">,</span> <span class=\"n\">value</span><span class=\"p\">))</span>\n",
" <span class=\"n\">P</span><span class=\"p\">,</span> <span class=\"n\">value</span> <span class=\"o\">=</span> <span class=\"n\">find_unit_clause</span><span class=\"p\">(</span><span class=\"n\">clauses</span><span class=\"p\">,</span> <span class=\"n\">model</span><span class=\"p\">)</span>\n",
" <span class=\"k\">if</span> <span class=\"n\">P</span><span class=\"p\">:</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">dpll</span><span class=\"p\">(</span><span class=\"n\">clauses</span><span class=\"p\">,</span> <span class=\"n\">removeall</span><span class=\"p\">(</span><span class=\"n\">P</span><span class=\"p\">,</span> <span class=\"n\">symbols</span><span class=\"p\">),</span> <span class=\"n\">extend</span><span class=\"p\">(</span><span class=\"n\">model</span><span class=\"p\">,</span> <span class=\"n\">P</span><span class=\"p\">,</span> <span class=\"n\">value</span><span class=\"p\">))</span>\n",
" <span class=\"k\">if</span> <span class=\"ow\">not</span> <span class=\"n\">symbols</span><span class=\"p\">:</span>\n",
" <span class=\"k\">raise</span> <span class=\"ne\">TypeError</span><span class=\"p\">(</span><span class=\"s2\">"Argument should be of the type Expr."</span><span class=\"p\">)</span>\n",
" <span class=\"n\">P</span><span class=\"p\">,</span> <span class=\"n\">symbols</span> <span class=\"o\">=</span> <span class=\"n\">symbols</span><span class=\"p\">[</span><span class=\"mi\">0</span><span class=\"p\">],</span> <span class=\"n\">symbols</span><span class=\"p\">[</span><span class=\"mi\">1</span><span class=\"p\">:]</span>\n",
" <span class=\"k\">return</span> <span class=\"p\">(</span><span class=\"n\">dpll</span><span class=\"p\">(</span><span class=\"n\">clauses</span><span class=\"p\">,</span> <span class=\"n\">symbols</span><span class=\"p\">,</span> <span class=\"n\">extend</span><span class=\"p\">(</span><span class=\"n\">model</span><span class=\"p\">,</span> <span class=\"n\">P</span><span class=\"p\">,</span> <span class=\"bp\">True</span><span class=\"p\">))</span> <span class=\"ow\">or</span>\n",
" <span class=\"n\">dpll</span><span class=\"p\">(</span><span class=\"n\">clauses</span><span class=\"p\">,</span> <span class=\"n\">symbols</span><span class=\"p\">,</span> <span class=\"n\">extend</span><span class=\"p\">(</span><span class=\"n\">model</span><span class=\"p\">,</span> <span class=\"n\">P</span><span class=\"p\">,</span> <span class=\"bp\">False</span><span class=\"p\">)))</span>\n",
"</pre></div>\n",
"</body>\n",
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]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"psource(dpll)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The algorithm uses the ideas described above to check satisfiability of a sentence in propositional logic.\n",
"It recursively calls itself, simplifying the problem at each step. It also uses helper functions `find_pure_symbol` and `find_unit_clause` to carry out steps 2 and 3 above.\n",
"<br>\n",
"The `dpll_satisfiable` helper function converts the input clauses to _conjunctive normal form_ and calls the `dpll` function with the correct parameters."
]
},
{
"cell_type": "code",
"execution_count": 49,
"metadata": {},
"outputs": [
{
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"<head>\n",
" <title></title>\n",
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"td.linenos { background-color: #f0f0f0; padding-right: 10px; }\n",
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"\n",
"<div class=\"highlight\"><pre><span></span><span class=\"k\">def</span> <span class=\"nf\">dpll_satisfiable</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Check satisfiability of a propositional sentence.</span>\n",
"<span class=\"sd\"> This differs from the book code in two ways: (1) it returns a model</span>\n",
"<span class=\"sd\"> rather than True when it succeeds; this is more useful. (2) The</span>\n",
"<span class=\"sd\"> function find_pure_symbol is passed a list of unknown clauses, rather</span>\n",
"<span class=\"sd\"> than a list of all clauses and the model; this is more efficient."""</span>\n",
" <span class=\"n\">clauses</span> <span class=\"o\">=</span> <span class=\"n\">conjuncts</span><span class=\"p\">(</span><span class=\"n\">to_cnf</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">))</span>\n",
" <span class=\"n\">symbols</span> <span class=\"o\">=</span> <span class=\"nb\">list</span><span class=\"p\">(</span><span class=\"n\">prop_symbols</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">))</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">dpll</span><span class=\"p\">(</span><span class=\"n\">clauses</span><span class=\"p\">,</span> <span class=\"n\">symbols</span><span class=\"p\">,</span> <span class=\"p\">{})</span>\n",
"</pre></div>\n",
"</body>\n",
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"psource(dpll_satisfiable)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let's see a few examples of usage."
]
},
{
"cell_type": "code",
"execution_count": 50,
"metadata": {},
"outputs": [],
"source": [
"A, B, C, D = expr('A, B, C, D')"
]
},
{
"cell_type": "code",
"execution_count": 51,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"{A: True, B: True, C: False, D: True}"
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},
"execution_count": 51,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"dpll_satisfiable(A & B & ~C & D)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"This is a simple case to highlight that the algorithm actually works."
]
},
{
"cell_type": "code",
"execution_count": 52,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
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},
"execution_count": 52,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"dpll_satisfiable((A & B) | (C & ~A) | (B & ~D))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"If a particular symbol isn't present in the solution, \n",
"it means that the solution is independent of the value of that symbol.\n",
"In this case, the solution is independent of A."
]
},
{
"cell_type": "code",
"execution_count": 53,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"{A: True, B: True}"
]
},
"execution_count": 53,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"dpll_satisfiable(A |'<=>'| B)"
]
},
{
"cell_type": "code",
"execution_count": 54,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
]
},
"execution_count": 54,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"dpll_satisfiable((A |'<=>'| B) |'==>'| (C & ~A))"
]
},
{
"cell_type": "code",
"execution_count": 55,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
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},
"execution_count": 55,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"dpll_satisfiable((A | (B & C)) |'<=>'| ((A | B) & (A | C)))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### 2. WalkSAT algorithm\n",
"This algorithm is very similar to Hill climbing.\n",
"On every iteration, the algorithm picks an unsatisfied clause and flips a symbol in the clause.\n",
"This is similar to finding a neighboring state in the `hill_climbing` algorithm.\n",
"<br>\n",
"The symbol to be flipped is decided by an evaluation function that counts the number of unsatisfied clauses.\n",
"Sometimes, symbols are also flipped randomly, to avoid local optima. A subtle balance between greediness and randomness is required. Alternatively, some versions of the algorithm restart with a completely new random assignment if no solution has been found for too long, as a way of getting out of local minima of numbers of unsatisfied clauses.\n",
"<br>\n",
"<br>\n",
"Let's have a look at the algorithm."
]
},
{
"cell_type": "code",
"execution_count": 56,
"metadata": {},
"outputs": [
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"\n",
"<div class=\"highlight\"><pre><span></span><span class=\"k\">def</span> <span class=\"nf\">WalkSAT</span><span class=\"p\">(</span><span class=\"n\">clauses</span><span class=\"p\">,</span> <span class=\"n\">p</span><span class=\"o\">=</span><span class=\"mf\">0.5</span><span class=\"p\">,</span> <span class=\"n\">max_flips</span><span class=\"o\">=</span><span class=\"mi\">10000</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Checks for satisfiability of all clauses by randomly flipping values of variables</span>\n",
"<span class=\"sd\"> """</span>\n",
" <span class=\"c1\"># Set of all symbols in all clauses</span>\n",
" <span class=\"n\">symbols</span> <span class=\"o\">=</span> <span class=\"p\">{</span><span class=\"n\">sym</span> <span class=\"k\">for</span> <span class=\"n\">clause</span> <span class=\"ow\">in</span> <span class=\"n\">clauses</span> <span class=\"k\">for</span> <span class=\"n\">sym</span> <span class=\"ow\">in</span> <span class=\"n\">prop_symbols</span><span class=\"p\">(</span><span class=\"n\">clause</span><span class=\"p\">)}</span>\n",
" <span class=\"c1\"># model is a random assignment of true/false to the symbols in clauses</span>\n",
" <span class=\"n\">model</span> <span class=\"o\">=</span> <span class=\"p\">{</span><span class=\"n\">s</span><span class=\"p\">:</span> <span class=\"n\">random</span><span class=\"o\">.</span><span class=\"n\">choice</span><span class=\"p\">([</span><span class=\"bp\">True</span><span class=\"p\">,</span> <span class=\"bp\">False</span><span class=\"p\">])</span> <span class=\"k\">for</span> <span class=\"n\">s</span> <span class=\"ow\">in</span> <span class=\"n\">symbols</span><span class=\"p\">}</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_flips</span><span class=\"p\">):</span>\n",
" <span class=\"n\">satisfied</span><span class=\"p\">,</span> <span class=\"n\">unsatisfied</span> <span class=\"o\">=</span> <span class=\"p\">[],</span> <span class=\"p\">[]</span>\n",
" <span class=\"k\">for</span> <span class=\"n\">clause</span> <span class=\"ow\">in</span> <span class=\"n\">clauses</span><span class=\"p\">:</span>\n",
" <span class=\"p\">(</span><span class=\"n\">satisfied</span> <span class=\"k\">if</span> <span class=\"n\">pl_true</span><span class=\"p\">(</span><span class=\"n\">clause</span><span class=\"p\">,</span> <span class=\"n\">model</span><span class=\"p\">)</span> <span class=\"k\">else</span> <span class=\"n\">unsatisfied</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">append</span><span class=\"p\">(</span><span class=\"n\">clause</span><span class=\"p\">)</span>\n",
" <span class=\"k\">if</span> <span class=\"ow\">not</span> <span class=\"n\">unsatisfied</span><span class=\"p\">:</span> <span class=\"c1\"># if model satisfies all the clauses</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">model</span>\n",
" <span class=\"n\">clause</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\">unsatisfied</span><span class=\"p\">)</span>\n",
" <span class=\"k\">if</span> <span class=\"n\">probability</span><span class=\"p\">(</span><span class=\"n\">p</span><span class=\"p\">):</span>\n",
" <span class=\"n\">sym</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=\"nb\">list</span><span class=\"p\">(</span><span class=\"n\">prop_symbols</span><span class=\"p\">(</span><span class=\"n\">clause</span><span class=\"p\">)))</span>\n",
" <span class=\"k\">else</span><span class=\"p\">:</span>\n",
" <span class=\"c1\"># Flip the symbol in clause that maximizes number of sat. clauses</span>\n",
" <span class=\"k\">def</span> <span class=\"nf\">sat_count</span><span class=\"p\">(</span><span class=\"n\">sym</span><span class=\"p\">):</span>\n",
" <span class=\"c1\"># Return the the number of clauses satisfied after flipping the symbol.</span>\n",
" <span class=\"n\">model</span><span class=\"p\">[</span><span class=\"n\">sym</span><span class=\"p\">]</span> <span class=\"o\">=</span> <span class=\"ow\">not</span> <span class=\"n\">model</span><span class=\"p\">[</span><span class=\"n\">sym</span><span class=\"p\">]</span>\n",
" <span class=\"n\">count</span> <span class=\"o\">=</span> <span class=\"nb\">len</span><span class=\"p\">([</span><span class=\"n\">clause</span> <span class=\"k\">for</span> <span class=\"n\">clause</span> <span class=\"ow\">in</span> <span class=\"n\">clauses</span> <span class=\"k\">if</span> <span class=\"n\">pl_true</span><span class=\"p\">(</span><span class=\"n\">clause</span><span class=\"p\">,</span> <span class=\"n\">model</span><span class=\"p\">)])</span>\n",
" <span class=\"n\">model</span><span class=\"p\">[</span><span class=\"n\">sym</span><span class=\"p\">]</span> <span class=\"o\">=</span> <span class=\"ow\">not</span> <span class=\"n\">model</span><span class=\"p\">[</span><span class=\"n\">sym</span><span class=\"p\">]</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">count</span>\n",
" <span class=\"n\">sym</span> <span class=\"o\">=</span> <span class=\"n\">argmax</span><span class=\"p\">(</span><span class=\"n\">prop_symbols</span><span class=\"p\">(</span><span class=\"n\">clause</span><span class=\"p\">),</span> <span class=\"n\">key</span><span class=\"o\">=</span><span class=\"n\">sat_count</span><span class=\"p\">)</span>\n",
" <span class=\"n\">model</span><span class=\"p\">[</span><span class=\"n\">sym</span><span class=\"p\">]</span> <span class=\"o\">=</span> <span class=\"ow\">not</span> <span class=\"n\">model</span><span class=\"p\">[</span><span class=\"n\">sym</span><span class=\"p\">]</span>\n",
" <span class=\"c1\"># If no solution is found within the flip limit, we return failure</span>\n",
" <span class=\"k\">return</span> <span class=\"bp\">None</span>\n",
"</pre></div>\n",
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"source": [
"psource(WalkSAT)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The function takes three arguments:\n",
"<br>\n",
"1. The `clauses` we want to satisfy.\n",
"<br>\n",
"2. The probability `p` of randomly changing a symbol.\n",
"<br>\n",
"3. The maximum number of flips (`max_flips`) the algorithm will run for. If the clauses are still unsatisfied, the algorithm returns `None` to denote failure.\n",
"<br>\n",
"The algorithm is identical in concept to Hill climbing and the code isn't difficult to understand.\n",
"<br>\n",
"<br>\n",
"Let's see a few examples of usage."
]
},
{
"cell_type": "code",
"execution_count": 57,
"outputs": [],
"source": [
"A, B, C, D = expr('A, B, C, D')"
]
},
{
"cell_type": "code",
"execution_count": 58,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"{A: True, B: True, C: False, D: True}"
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},
"execution_count": 58,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"WalkSAT([A, B, ~C, D], 0.5, 100)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"This is a simple case to show that the algorithm converges."
]
},
{
"cell_type": "code",
"execution_count": 59,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
]
},
"execution_count": 59,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"WalkSAT([A & B, A & C], 0.5, 100)"
]
},
{
"cell_type": "code",
"execution_count": 60,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"{A: True, B: True, C: True, D: True}"
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},
"execution_count": 60,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"WalkSAT([A & B, C & D, C & B], 0.5, 100)"
]
},
{
"cell_type": "code",
"execution_count": 61,
"metadata": {},
"outputs": [],
"source": [
"WalkSAT([A & B, C | D, ~(D | B)], 0.5, 1000)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"This one doesn't give any output because WalkSAT did not find any model where these clauses hold. We can solve these clauses to see that they together form a contradiction and hence, it isn't supposed to have a solution."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"One point of difference between this algorithm and the `dpll_satisfiable` algorithms is that both these algorithms take inputs differently. \n",
"For WalkSAT to take complete sentences as input, \n",
"we can write a helper function that converts the input sentence into conjunctive normal form and then calls WalkSAT with the list of conjuncts of the CNF form of the sentence."
]
},
{
"cell_type": "code",
"execution_count": 62,
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"outputs": [],
"source": [
"def WalkSAT_CNF(sentence, p=0.5, max_flips=10000):\n",
" return WalkSAT(conjuncts(to_cnf(sentence)), 0, max_flips)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now we can call `WalkSAT_CNF` and `DPLL_Satisfiable` with the same arguments."
]
},
{
"cell_type": "code",
"execution_count": 63,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"{A: False, B: False, C: True, D: False}"
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},
"execution_count": 63,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"WalkSAT_CNF((A & B) | (C & ~A) | (B & ~D), 0.5, 1000)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"It works!\n",
"<br>\n",
"Notice that the solution generated by WalkSAT doesn't omit variables that the sentence doesn't depend upon. \n",
"If the sentence is independent of a particular variable, the solution contains a random value for that variable because of the stochastic nature of the algorithm.\n",
"<br>\n",
"<br>\n",
"Let's compare the runtime of WalkSAT and DPLL for a few cases. We will use the `%%timeit` magic to do this."
]
},
{
"cell_type": "code",
"execution_count": 64,
"outputs": [],
"source": [
"sentence_1 = A |'<=>'| B\n",
"sentence_2 = (A & B) | (C & ~A) | (B & ~D)\n",
"sentence_3 = (A | (B & C)) |'<=>'| ((A | B) & (A | C))"
]
},
{
"cell_type": "code",
"execution_count": 65,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"6.78 ms ± 238 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)\n"
]
}
],
"source": [
"%%timeit\n",
"dpll_satisfiable(sentence_1)\n",
"dpll_satisfiable(sentence_2)\n",
"dpll_satisfiable(sentence_3)"
]
},
{
"cell_type": "code",
"execution_count": 66,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"4.64 ms ± 65.3 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)\n"
]
}
],
"source": [
"%%timeit\n",
"WalkSAT_CNF(sentence_1)\n",
"WalkSAT_CNF(sentence_2)\n",
"WalkSAT_CNF(sentence_3)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"On an average, for solvable cases, `WalkSAT` is quite faster than `dpll` because, for a small number of variables, \n",
"`WalkSAT` can reduce the search space significantly. \n",
"Results can be different for sentences with more symbols though.\n",
"Feel free to play around with this to understand the trade-offs of these algorithms better."
]
},
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{
"cell_type": "markdown",
"metadata": {},
"source": [
"### SATPlan"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"In this section we show how to make plans by logical inference. The basic idea is very simple. It includes the following three steps:\n",
"1. Constuct a sentence that includes:\n",
" 1. A colection of assertions about the initial state.\n",
" 2. The successor-state axioms for all the possible actions at each time up to some maximum time t.\n",
" 3. The assertion that the goal is achieved at time t.\n",
"2. Present the whole sentence to a SAT solver.\n",
"3. Assuming a model is found, extract from the model those variables that represent actions and are assigned true. Together they represent a plan to achieve the goals.\n",
"\n",
"\n",
"Lets have a look at the algorithm"
]
},
{
"cell_type": "code",
"execution_count": 67,
"metadata": {},
"outputs": [
{
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"\n",
"<div class=\"highlight\"><pre><span></span><span class=\"k\">def</span> <span class=\"nf\">SAT_plan</span><span class=\"p\">(</span><span class=\"n\">init</span><span class=\"p\">,</span> <span class=\"n\">transition</span><span class=\"p\">,</span> <span class=\"n\">goal</span><span class=\"p\">,</span> <span class=\"n\">t_max</span><span class=\"p\">,</span> <span class=\"n\">SAT_solver</span><span class=\"o\">=</span><span class=\"n\">dpll_satisfiable</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Converts a planning problem to Satisfaction problem by translating it to a cnf sentence.</span>\n",
"<span class=\"sd\"> [Figure 7.22]"""</span>\n",
"\n",
" <span class=\"c1\"># Functions used by SAT_plan</span>\n",
" <span class=\"k\">def</span> <span class=\"nf\">translate_to_SAT</span><span class=\"p\">(</span><span class=\"n\">init</span><span class=\"p\">,</span> <span class=\"n\">transition</span><span class=\"p\">,</span> <span class=\"n\">goal</span><span class=\"p\">,</span> <span class=\"n\">time</span><span class=\"p\">):</span>\n",
" <span class=\"n\">clauses</span> <span class=\"o\">=</span> <span class=\"p\">[]</span>\n",
" <span class=\"n\">states</span> <span class=\"o\">=</span> <span class=\"p\">[</span><span class=\"n\">state</span> <span class=\"k\">for</span> <span class=\"n\">state</span> <span class=\"ow\">in</span> <span class=\"n\">transition</span><span class=\"p\">]</span>\n",
"\n",
" <span class=\"c1\"># Symbol claiming state s at time t</span>\n",
" <span class=\"n\">state_counter</span> <span class=\"o\">=</span> <span class=\"n\">itertools</span><span class=\"o\">.</span><span class=\"n\">count</span><span class=\"p\">()</span>\n",
" <span class=\"k\">for</span> <span class=\"n\">s</span> <span class=\"ow\">in</span> <span class=\"n\">states</span><span class=\"p\">:</span>\n",
" <span class=\"k\">for</span> <span class=\"n\">t</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"n\">time</span><span class=\"o\">+</span><span class=\"mi\">1</span><span class=\"p\">):</span>\n",
" <span class=\"n\">state_sym</span><span class=\"p\">[</span><span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"n\">t</span><span class=\"p\">]</span> <span class=\"o\">=</span> <span class=\"n\">Expr</span><span class=\"p\">(</span><span class=\"s2\">"State_{}"</span><span class=\"o\">.</span><span class=\"n\">format</span><span class=\"p\">(</span><span class=\"nb\">next</span><span class=\"p\">(</span><span class=\"n\">state_counter</span><span class=\"p\">)))</span>\n",
"\n",
" <span class=\"c1\"># Add initial state axiom</span>\n",
" <span class=\"n\">clauses</span><span class=\"o\">.</span><span class=\"n\">append</span><span class=\"p\">(</span><span class=\"n\">state_sym</span><span class=\"p\">[</span><span class=\"n\">init</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">])</span>\n",
"\n",
" <span class=\"c1\"># Add goal state axiom</span>\n",
" <span class=\"n\">clauses</span><span class=\"o\">.</span><span class=\"n\">append</span><span class=\"p\">(</span><span class=\"n\">state_sym</span><span class=\"p\">[</span><span class=\"n\">goal</span><span class=\"p\">,</span> <span class=\"n\">time</span><span class=\"p\">])</span>\n",
"\n",
" <span class=\"c1\"># All possible transitions</span>\n",
" <span class=\"n\">transition_counter</span> <span class=\"o\">=</span> <span class=\"n\">itertools</span><span class=\"o\">.</span><span class=\"n\">count</span><span class=\"p\">()</span>\n",
" <span class=\"k\">for</span> <span class=\"n\">s</span> <span class=\"ow\">in</span> <span class=\"n\">states</span><span class=\"p\">:</span>\n",
" <span class=\"k\">for</span> <span class=\"n\">action</span> <span class=\"ow\">in</span> <span class=\"n\">transition</span><span class=\"p\">[</span><span class=\"n\">s</span><span class=\"p\">]:</span>\n",
" <span class=\"n\">s_</span> <span class=\"o\">=</span> <span class=\"n\">transition</span><span class=\"p\">[</span><span class=\"n\">s</span><span class=\"p\">][</span><span class=\"n\">action</span><span class=\"p\">]</span>\n",
" <span class=\"k\">for</span> <span class=\"n\">t</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"n\">time</span><span class=\"p\">):</span>\n",
" <span class=\"c1\"># Action 'action' taken from state 's' at time 't' to reach 's_'</span>\n",
" <span class=\"n\">action_sym</span><span class=\"p\">[</span><span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"n\">action</span><span class=\"p\">,</span> <span class=\"n\">t</span><span class=\"p\">]</span> <span class=\"o\">=</span> <span class=\"n\">Expr</span><span class=\"p\">(</span>\n",
" <span class=\"s2\">"Transition_{}"</span><span class=\"o\">.</span><span class=\"n\">format</span><span class=\"p\">(</span><span class=\"nb\">next</span><span class=\"p\">(</span><span class=\"n\">transition_counter</span><span class=\"p\">)))</span>\n",
"\n",
" <span class=\"c1\"># Change the state from s to s_</span>\n",
" <span class=\"n\">clauses</span><span class=\"o\">.</span><span class=\"n\">append</span><span class=\"p\">(</span><span class=\"n\">action_sym</span><span class=\"p\">[</span><span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"n\">action</span><span class=\"p\">,</span> <span class=\"n\">t</span><span class=\"p\">]</span> <span class=\"o\">|</span><span class=\"s1\">'==>'</span><span class=\"o\">|</span> <span class=\"n\">state_sym</span><span class=\"p\">[</span><span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"n\">t</span><span class=\"p\">])</span>\n",
" <span class=\"n\">clauses</span><span class=\"o\">.</span><span class=\"n\">append</span><span class=\"p\">(</span><span class=\"n\">action_sym</span><span class=\"p\">[</span><span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"n\">action</span><span class=\"p\">,</span> <span class=\"n\">t</span><span class=\"p\">]</span> <span class=\"o\">|</span><span class=\"s1\">'==>'</span><span class=\"o\">|</span> <span class=\"n\">state_sym</span><span class=\"p\">[</span><span class=\"n\">s_</span><span class=\"p\">,</span> <span class=\"n\">t</span> <span class=\"o\">+</span> <span class=\"mi\">1</span><span class=\"p\">])</span>\n",
"\n",
" <span class=\"c1\"># Allow only one state at any time</span>\n",
" <span class=\"k\">for</span> <span class=\"n\">t</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"n\">time</span><span class=\"o\">+</span><span class=\"mi\">1</span><span class=\"p\">):</span>\n",
" <span class=\"c1\"># must be a state at any time</span>\n",
" <span class=\"n\">clauses</span><span class=\"o\">.</span><span class=\"n\">append</span><span class=\"p\">(</span><span class=\"n\">associate</span><span class=\"p\">(</span><span class=\"s1\">'|'</span><span class=\"p\">,</span> <span class=\"p\">[</span><span class=\"n\">state_sym</span><span class=\"p\">[</span><span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"n\">t</span><span class=\"p\">]</span> <span class=\"k\">for</span> <span class=\"n\">s</span> <span class=\"ow\">in</span> <span class=\"n\">states</span><span class=\"p\">]))</span>\n",
"\n",
" <span class=\"k\">for</span> <span class=\"n\">s</span> <span class=\"ow\">in</span> <span class=\"n\">states</span><span class=\"p\">:</span>\n",
" <span class=\"k\">for</span> <span class=\"n\">s_</span> <span class=\"ow\">in</span> <span class=\"n\">states</span><span class=\"p\">[</span><span class=\"n\">states</span><span class=\"o\">.</span><span class=\"n\">index</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">)</span> <span class=\"o\">+</span> <span class=\"mi\">1</span><span class=\"p\">:]:</span>\n",
" <span class=\"c1\"># for each pair of states s, s_ only one is possible at time t</span>\n",
" <span class=\"n\">clauses</span><span class=\"o\">.</span><span class=\"n\">append</span><span class=\"p\">((</span><span class=\"o\">~</span><span class=\"n\">state_sym</span><span class=\"p\">[</span><span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"n\">t</span><span class=\"p\">])</span> <span class=\"o\">|</span> <span class=\"p\">(</span><span class=\"o\">~</span><span class=\"n\">state_sym</span><span class=\"p\">[</span><span class=\"n\">s_</span><span class=\"p\">,</span> <span class=\"n\">t</span><span class=\"p\">]))</span>\n",
"\n",
" <span class=\"c1\"># Restrict to one transition per timestep</span>\n",
" <span class=\"k\">for</span> <span class=\"n\">t</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"n\">time</span><span class=\"p\">):</span>\n",
" <span class=\"c1\"># list of possible transitions at time t</span>\n",
" <span class=\"n\">transitions_t</span> <span class=\"o\">=</span> <span class=\"p\">[</span><span class=\"n\">tr</span> <span class=\"k\">for</span> <span class=\"n\">tr</span> <span class=\"ow\">in</span> <span class=\"n\">action_sym</span> <span class=\"k\">if</span> <span class=\"n\">tr</span><span class=\"p\">[</span><span class=\"mi\">2</span><span class=\"p\">]</span> <span class=\"o\">==</span> <span class=\"n\">t</span><span class=\"p\">]</span>\n",
"\n",
" <span class=\"c1\"># make sure at least one of the transitions happens</span>\n",
" <span class=\"n\">clauses</span><span class=\"o\">.</span><span class=\"n\">append</span><span class=\"p\">(</span><span class=\"n\">associate</span><span class=\"p\">(</span><span class=\"s1\">'|'</span><span class=\"p\">,</span> <span class=\"p\">[</span><span class=\"n\">action_sym</span><span class=\"p\">[</span><span class=\"n\">tr</span><span class=\"p\">]</span> <span class=\"k\">for</span> <span class=\"n\">tr</span> <span class=\"ow\">in</span> <span class=\"n\">transitions_t</span><span class=\"p\">]))</span>\n",
"\n",
" <span class=\"k\">for</span> <span class=\"n\">tr</span> <span class=\"ow\">in</span> <span class=\"n\">transitions_t</span><span class=\"p\">:</span>\n",
" <span class=\"k\">for</span> <span class=\"n\">tr_</span> <span class=\"ow\">in</span> <span class=\"n\">transitions_t</span><span class=\"p\">[</span><span class=\"n\">transitions_t</span><span class=\"o\">.</span><span class=\"n\">index</span><span class=\"p\">(</span><span class=\"n\">tr</span><span class=\"p\">)</span> <span class=\"o\">+</span> <span class=\"mi\">1</span><span class=\"p\">:]:</span>\n",
" <span class=\"c1\"># there cannot be two transitions tr and tr_ at time t</span>\n",
" <span class=\"n\">clauses</span><span class=\"o\">.</span><span class=\"n\">append</span><span class=\"p\">(</span><span class=\"o\">~</span><span class=\"n\">action_sym</span><span class=\"p\">[</span><span class=\"n\">tr</span><span class=\"p\">]</span> <span class=\"o\">|</span> <span class=\"o\">~</span><span class=\"n\">action_sym</span><span class=\"p\">[</span><span class=\"n\">tr_</span><span class=\"p\">])</span>\n",
"\n",
" <span class=\"c1\"># Combine the clauses to form the cnf</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">associate</span><span class=\"p\">(</span><span class=\"s1\">'&'</span><span class=\"p\">,</span> <span class=\"n\">clauses</span><span class=\"p\">)</span>\n",
"\n",
" <span class=\"k\">def</span> <span class=\"nf\">extract_solution</span><span class=\"p\">(</span><span class=\"n\">model</span><span class=\"p\">):</span>\n",
" <span class=\"n\">true_transitions</span> <span class=\"o\">=</span> <span class=\"p\">[</span><span class=\"n\">t</span> <span class=\"k\">for</span> <span class=\"n\">t</span> <span class=\"ow\">in</span> <span class=\"n\">action_sym</span> <span class=\"k\">if</span> <span class=\"n\">model</span><span class=\"p\">[</span><span class=\"n\">action_sym</span><span class=\"p\">[</span><span class=\"n\">t</span><span class=\"p\">]]]</span>\n",
" <span class=\"c1\"># Sort transitions based on time, which is the 3rd element of the tuple</span>\n",
" <span class=\"n\">true_transitions</span><span class=\"o\">.</span><span class=\"n\">sort</span><span class=\"p\">(</span><span class=\"n\">key</span><span class=\"o\">=</span><span class=\"k\">lambda</span> <span class=\"n\">x</span><span class=\"p\">:</span> <span class=\"n\">x</span><span class=\"p\">[</span><span class=\"mi\">2</span><span class=\"p\">])</span>\n",
" <span class=\"k\">return</span> <span class=\"p\">[</span><span class=\"n\">action</span> <span class=\"k\">for</span> <span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"n\">action</span><span class=\"p\">,</span> <span class=\"n\">time</span> <span class=\"ow\">in</span> <span class=\"n\">true_transitions</span><span class=\"p\">]</span>\n",
"\n",
" <span class=\"c1\"># Body of SAT_plan algorithm</span>\n",
" <span class=\"k\">for</span> <span class=\"n\">t</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"n\">t_max</span><span class=\"p\">):</span>\n",
" <span class=\"c1\"># dictionaries to help extract the solution from model</span>\n",
" <span class=\"n\">state_sym</span> <span class=\"o\">=</span> <span class=\"p\">{}</span>\n",
" <span class=\"n\">action_sym</span> <span class=\"o\">=</span> <span class=\"p\">{}</span>\n",
"\n",
" <span class=\"n\">cnf</span> <span class=\"o\">=</span> <span class=\"n\">translate_to_SAT</span><span class=\"p\">(</span><span class=\"n\">init</span><span class=\"p\">,</span> <span class=\"n\">transition</span><span class=\"p\">,</span> <span class=\"n\">goal</span><span class=\"p\">,</span> <span class=\"n\">t</span><span class=\"p\">)</span>\n",
" <span class=\"n\">model</span> <span class=\"o\">=</span> <span class=\"n\">SAT_solver</span><span class=\"p\">(</span><span class=\"n\">cnf</span><span class=\"p\">)</span>\n",
" <span class=\"k\">if</span> <span class=\"n\">model</span> <span class=\"ow\">is</span> <span class=\"ow\">not</span> <span class=\"bp\">False</span><span class=\"p\">:</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">extract_solution</span><span class=\"p\">(</span><span class=\"n\">model</span><span class=\"p\">)</span>\n",
" <span class=\"k\">return</span> <span class=\"bp\">None</span>\n",
"</pre></div>\n",
"</body>\n",
"</html>\n"
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"<IPython.core.display.HTML object>"
]
},
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],
"source": [
"psource(SAT_plan)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let's see few examples of its usage. First we define a transition and then call `SAT_plan`."
]
},
{
"cell_type": "code",
"execution_count": 68,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [