250 lines
20 KiB
Plaintext
250 lines
20 KiB
Plaintext
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\">"
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],
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"metadata": {
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"id": "9Qgup23vwdll"
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}
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},
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{
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"cell_type": "markdown",
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"source": [
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"# Boolean logic formulae and exhaustive SAT algorithms\n",
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"\n",
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"The purpose of this Python notebook experiment with an exhuastive algorithm for SAT solving.\n",
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"\n",
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"You can save a local copy of this notebook in your Google account and work through it in Colab (recommended) or you can download the notebook and run it locally using Jupyter notebook or similar. \n",
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"\n",
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"Contact me at iclimbtreesmail@gmail.com if you find any mistakes or have any suggestions."
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],
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"metadata": {
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"id": "uORlKyPv02ge"
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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": 1,
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"metadata": {
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"id": "bbF6SE_F0tU8"
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},
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"outputs": [],
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"source": [
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"# Math library\n",
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"import numpy as np"
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]
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},
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{
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"cell_type": "markdown",
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"source": [
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"# Boolean operators\n",
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"\n",
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"First, let's write some functions for the five Boolean operators discussed in the unit. "
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],
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"metadata": {
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"id": "8z_AvPoU6zck"
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}
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},
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{
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"cell_type": "code",
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"source": [
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"def or_op(x_1, x_2):\n",
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" return x_1 or x_2 ;\n",
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"\n",
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"def and_op(x_1, x_2):\n",
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" return x_1 and x_2\n",
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"\n",
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"def imp_op(x_1, x_2):\n",
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" return not x_1 or x_2\n",
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"\n",
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"def equiv_op(x_1, x_2):\n",
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" return x_1==x_2\n",
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"\n",
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"def not_op(x_1):\n",
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" return not x_1"
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],
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"metadata": {
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"id": "ogfjSL857Dlo"
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},
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"execution_count": 6,
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"outputs": []
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},
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{
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"cell_type": "markdown",
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"source": [
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"# Example Boolean Logic Formula\n",
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"\n",
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"Now let's define and implement a Boolean logic formula. We'll use the one from the text.\n",
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"\n",
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"$$\\phi:= \\bigl(x_{1}\\Rightarrow (\\lnot x_{2}\\land x_{3})\\bigr) \\land \\bigl(x_{2} \\Leftrightarrow (\\lnot x_{3} \\lor x_{1})\\bigr).$$"
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],
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"metadata": {
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"id": "e3a1Ps1D8kUH"
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}
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},
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{
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"cell_type": "code",
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"source": [
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"def phi(x_1, x_2, x_3):\n",
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" phi_val = and_op(imp_op(x_1, and_op(not_op(x_2), x_3)),equiv_op(x_2, or_op(not_op(x_3), x_1)))\n",
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" print(\"phi(\",x_1,\",\",x_2,\",\",x_3,\")=\",phi_val)\n",
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" return phi_val"
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],
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"metadata": {
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"id": "yBPbxHTL8xOl"
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},
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"execution_count": 7,
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"outputs": []
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},
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{
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"cell_type": "markdown",
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"source": [
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"# Exhaustive SAT\n",
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"\n",
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"Now let's implement an exhaustive SAT solver. For each possible combination of $x_1$, $x_2$, and $x_3$, we evaluate the Boolean logic formula $\\phi$. If it evaluates to **true** for any of these combinations then the function is **SAT**. Otherwise, it is **UNSAT**."
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],
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"metadata": {
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"id": "OeRG0RB-9HP4"
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}
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},
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{
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"cell_type": "code",
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"source": [
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"def exhaustive_3(phi):\n",
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" for t in range (np.power(2,3)):\n",
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" x_1 = (t % 2) >= 1\n",
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" x_2 = (t % 4) >= 2\n",
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" x_3 = (t % 8) >= 4\n",
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" phi_val = phi(x_1, x_2, x_3)\n",
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" if phi_val:\n",
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" print(\"SAT\")\n",
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" return\n",
|
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" print(\"UNSAT\")"
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],
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"metadata": {
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"id": "QBj3Ap3t9CVO"
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},
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"execution_count": 14,
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"outputs": []
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},
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{
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"cell_type": "code",
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"source": [
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"exhaustive_3(phi)"
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],
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"metadata": {
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"colab": {
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"base_uri": "https://localhost:8080/"
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},
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"id": "FJGoau6C_fWP",
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"outputId": "a841116e-8e8a-400f-a560-b52c670e6b0a"
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},
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"execution_count": 15,
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"outputs": [
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{
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"output_type": "stream",
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"name": "stdout",
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"text": [
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"phi( False , False , False )= False\n",
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"phi( True , False , False )= False\n",
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"phi( False , True , False )= True\n",
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"SAT\n"
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]
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}
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]
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},
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{
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"cell_type": "markdown",
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"source": [
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"How about if we add another constraint (by logically ANDing another term)?\n",
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"\n",
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"$$\\phi_1:= \\bigl(x_{1}\\Rightarrow (\\lnot x_{2}\\land x_{3})\\bigr) \\land \\bigl(x_{2} \\Leftrightarrow (\\lnot x_{3} \\lor x_{1})\\bigr)\\land\\lnot \\bigl(\\lnot x_3 \\Rightarrow x_2\\bigr).$$"
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],
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"metadata": {
|
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"id": "nnHxv9IHB185"
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}
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},
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{
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"cell_type": "code",
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"source": [
|
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"def phi_1(x_1, x_2, x_3):\n",
|
|
" phi_val = and_op(and_op(imp_op(x_1, and_op(not_op(x_2), x_3)),equiv_op(x_2, or_op(not_op(x_3), x_1))),not_op(imp_op(not_op(x_3), x_2)))\n",
|
|
" print(\"phi(\",x_1,\",\",x_2,\",\",x_3,\")=\",phi_val)\n",
|
|
" return phi_val\n"
|
|
],
|
|
"metadata": {
|
|
"id": "T13sNtofBuLs"
|
|
},
|
|
"execution_count": 24,
|
|
"outputs": []
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"source": [
|
|
"exhaustive_3(phi_1)"
|
|
],
|
|
"metadata": {
|
|
"colab": {
|
|
"base_uri": "https://localhost:8080/"
|
|
},
|
|
"id": "qCm_ZdIgCv_P",
|
|
"outputId": "f1cbb656-9f4d-4b68-b347-876ada5fb926"
|
|
},
|
|
"execution_count": 25,
|
|
"outputs": [
|
|
{
|
|
"output_type": "stream",
|
|
"name": "stdout",
|
|
"text": [
|
|
"phi( False , False , False )= False\n",
|
|
"phi( True , False , False )= False\n",
|
|
"phi( False , True , False )= False\n",
|
|
"phi( True , True , False )= False\n",
|
|
"phi( False , False , True )= False\n",
|
|
"phi( True , False , True )= False\n",
|
|
"phi( False , True , True )= False\n",
|
|
"phi( True , True , True )= False\n",
|
|
"UNSAT\n"
|
|
]
|
|
}
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"source": [
|
|
"You should fined that this is **UNSAT**. We have added too many constraints and the three terms that are **AND**ed together are never simultaneously true."
|
|
],
|
|
"metadata": {
|
|
"id": "nlsUlJQfDB14"
|
|
}
|
|
}
|
|
]
|
|
} |