3034e8fcc3
Work around a recent decision in Jupyter Lab and Notebook to render <svg> is inline <img>, breaking tooltips or text selection. (Rerendering all notebooks was painful.) * NEWS: Mention the change. * python/spot/__init__.py: Add a _repr_html_ method to all classes that had a _repr_svg_. It seems Jupyter will use _repr_html_ by default. * python/spot/jupyter.py: SVG replace the _repr_svg_ method by a _repr_html. * tests/python/_altscc.ipynb, tests/python/_autparserr.ipynb, tests/python/_aux.ipynb, tests/python/_mealy.ipynb, tests/python/_partitioned_relabel.ipynb, tests/python/_product_susp.ipynb, tests/python/_product_weak.ipynb, tests/python/_synthesis.ipynb, tests/python/aliases.ipynb, tests/python/alternation.ipynb, tests/python/atva16-fig2a.ipynb, tests/python/atva16-fig2b.ipynb, tests/python/automata-io.ipynb, tests/python/automata.ipynb, tests/python/cav22-figs.ipynb, tests/python/contains.ipynb, tests/python/decompose.ipynb, tests/python/formulas.ipynb, tests/python/games.ipynb, tests/python/gen.ipynb, tests/python/highlighting.ipynb, tests/python/ltsmin-dve.ipynb, tests/python/ltsmin-pml.ipynb, tests/python/parity.ipynb, tests/python/product.ipynb, tests/python/randaut.ipynb, tests/python/satmin.ipynb, tests/python/stutter-inv.ipynb, tests/python/synthesis.ipynb, tests/python/testingaut.ipynb, tests/python/twagraph-internals.ipynb, tests/python/word.ipynb, tests/python/zlktree.ipynb: Update all notebooks.
4304 lines
315 KiB
Text
4304 lines
315 KiB
Text
{
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"cells": [
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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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"outputs": [],
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"source": [
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"import spot\n",
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"from spot.jupyter import display_inline\n",
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"spot.setup(show_default='.bav')"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Support for alternating automata"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"The following automata are what we will use as examples."
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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": 2,
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"metadata": {},
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"outputs": [],
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"source": [
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"aut1, aut2, aut3, aut4, aut5 = spot.automata('''\n",
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"HOA: v1 tool: \"ltl3ba\" \"1.1.3\" name: \"VWAA for FGa && GFb\" States: 6\n",
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"Start: 0 acc-name: co-Buchi Acceptance: 1 Fin(0) AP: 2 \"a\" \"b\" properties:\n",
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"trans-labels explicit-labels state-acc univ-branch very-weak --BODY--\n",
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"State: 0 \"(FG(a) && GF(b))\" [t] 3&1 State: 1 \"GF(b)\" [(1)] 1 [(!1)]\n",
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"2&1 State: 2 \"F(b)\" {0} [(1)] 5 [(!1)] 2 State: 3 \"FG(a)\" {0} [(0)] 4\n",
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"[t] 3 State: 4 \"G(a)\" [(0)] 4 State: 5 \"t\" [t] 5 --END--\n",
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"/* Example from ADL's PSL2TGBA talk. */\n",
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"HOA: v1 States: 3 Start: 0 acc-name: co-Buchi Acceptance: 1 Fin(0) AP:\n",
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"3 \"a\" \"b\" \"p\" --BODY-- State: 0 \"(a;a*;b)*\" {0} [0] 1 [!0] 2 State: 1\n",
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"\"a*;b;(a;a*;b)*\" {0} [0&1&2] 0&1 [!1&2] 1 [!0&!1] 2 [!0&1&2] 0 State:\n",
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"2 [t] 2 --END--\n",
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"HOA: v1 States: 5 Start: 3 acc-name: co-Buchi Acceptance: 1 Fin(0) AP:\n",
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"3 \"a\" \"b\" \"p\" --BODY-- State: 0 \"(a;a*;b)*\" {0} [0] 1 [!0] 2 State: 1\n",
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"\"a*;b;(a;a*;b)*\" {0} [0&1&2] 0&1 [!1&2] 1 [!0&!1] 2 [!0&1&2] 0 State:\n",
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"2 [t] 2 State: 3 [0] 4&0 State: 4 [t] 3 --END--\n",
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"HOA: v1 States: 3 Start: 0 acc-name: co-Buchi Acceptance: 1 Fin(0) AP:\n",
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"3 \"a\" \"b\" \"p\" --BODY-- State: 0 \"(a;a*;b)*\" {0} [0] 1 [!0] 2 State: 1\n",
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"\"a*;b;(a;a*;b)*\" {0} [0&1&2] 0&1 [!1&2] 1 [!0&!1] 2 [!0&1&2] 0 State:\n",
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"2 [t] 2 --END--\n",
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"HOA: v1 tool: \"ltl3dra\" \"0.2.2\" name: \"VWAA for GFa\" States: 3 Start: 0\n",
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"acc-name: co-Buchi Acceptance: 1 Fin(0) AP: 1 \"a\" properties: trans-labels\n",
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"explicit-labels state-acc univ-branch very-weak --BODY-- State: 0 \"GF(a)\"\n",
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"[t] 1&0 State: 1 \"F(a)\" {0} [(0)] 2 [t] 1 State: 2 \"t\" [t] 2 --END--\n",
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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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"metadata": {},
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"source": [
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"## Various display options"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Here is the default output, using the `bav` options as set by default in the first cell. "
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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": 3,
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"metadata": {},
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"outputs": [
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{
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"data": {
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" \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
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"<!-- Generated by graphviz version 2.43.0 (0)\n",
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" -->\n",
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" viewBox=\"0.00 0.00 129.40 360.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
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"<g id=\"graph0\" class=\"graph\" transform=\"scale(0.7246376811594204 0.7246376811594204) rotate(0) translate(4 492.6)\">\n",
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"<title>VWAA for FGa && GFb</title>\n",
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"</g>\n",
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"<g id=\"node3\" class=\"node\">\n",
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"</g>\n",
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"<!-- 0->-1 -->\n",
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"<g id=\"edge2\" class=\"edge\">\n",
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"<title>0->-1</title>\n",
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"<path fill=\"none\" stroke=\"black\" d=\"M71,-378.59C71,-364.81 71,-345.96 71,-335.21\"/>\n",
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"<text text-anchor=\"start\" x=\"71\" y=\"-349.4\" font-family=\"Lato\" font-size=\"14.00\">1</text>\n",
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"</g>\n",
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"<!-- 1 -->\n",
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"<g id=\"node4\" class=\"node\">\n",
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"<title>1</title>\n",
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"<path fill=\"#ffffaa\" stroke=\"black\" d=\"M38,-286C38,-286 12,-286 12,-286 6,-286 0,-280 0,-274 0,-274 0,-262 0,-262 0,-256 6,-250 12,-250 12,-250 38,-250 38,-250 44,-250 50,-256 50,-262 50,-262 50,-274 50,-274 50,-280 44,-286 38,-286\"/>\n",
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"<text text-anchor=\"start\" x=\"8\" y=\"-264.3\" font-family=\"Lato\" font-size=\"14.00\">GF(b)</text>\n",
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"</g>\n",
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"<!-- -1->1 -->\n",
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"<g id=\"edge3\" class=\"edge\">\n",
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"<title>-1->1</title>\n",
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"<path fill=\"none\" stroke=\"black\" d=\"M70.2,-323.83C66.99,-319.93 54.63,-304.95 43.65,-291.62\"/>\n",
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"<polygon fill=\"black\" stroke=\"black\" points=\"39.06,-286.05 45.94,-289.45 41.29,-288.76 43.51,-291.46 43.51,-291.46 43.51,-291.46 41.29,-288.76 41.08,-293.46 39.06,-286.05 39.06,-286.05\"/>\n",
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"</g>\n",
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"<!-- 3 -->\n",
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"<g id=\"node5\" class=\"node\">\n",
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"<title>3</title>\n",
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"<path fill=\"#ffffaa\" stroke=\"black\" d=\"M131.5,-287C131.5,-287 106.5,-287 106.5,-287 100.5,-287 94.5,-281 94.5,-275 94.5,-275 94.5,-261 94.5,-261 94.5,-255 100.5,-249 106.5,-249 106.5,-249 131.5,-249 131.5,-249 137.5,-249 143.5,-255 143.5,-261 143.5,-261 143.5,-275 143.5,-275 143.5,-281 137.5,-287 131.5,-287\"/>\n",
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"<text text-anchor=\"start\" x=\"102.5\" y=\"-271.8\" font-family=\"Lato\" font-size=\"14.00\">FG(a)</text>\n",
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"<text text-anchor=\"start\" x=\"111\" y=\"-256.8\" font-family=\"Lato\" font-size=\"14.00\" fill=\"#1f78b4\">⓿</text>\n",
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"</g>\n",
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"<!-- -1->3 -->\n",
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"<g id=\"edge4\" class=\"edge\">\n",
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"<title>-1->3</title>\n",
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"</g>\n",
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"<!-- 1->1 -->\n",
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"<g id=\"edge5\" class=\"edge\">\n",
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"<title>1->1</title>\n",
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"<path fill=\"none\" stroke=\"black\" d=\"M50.18,-274.72C59.91,-275.02 68,-272.78 68,-268 68,-264.49 63.64,-262.35 57.46,-261.58\"/>\n",
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"<polygon fill=\"black\" stroke=\"black\" points=\"50.18,-261.28 57.3,-258.42 53.67,-261.42 57.17,-261.56 57.17,-261.56 57.17,-261.56 53.67,-261.42 57.04,-264.71 50.18,-261.28 50.18,-261.28\"/>\n",
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"<text text-anchor=\"start\" x=\"68\" y=\"-264.3\" font-family=\"Lato\" font-size=\"14.00\">b</text>\n",
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"</g>\n",
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"<!-- -4 -->\n",
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"<g id=\"node6\" class=\"node\">\n",
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"<title>-4</title>\n",
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"<ellipse fill=\"#ffffaa\" stroke=\"black\" cx=\"26\" cy=\"-180\" rx=\"1.8\" ry=\"1.8\"/>\n",
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"</g>\n",
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"<!-- 1->-4 -->\n",
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"<g id=\"edge6\" class=\"edge\">\n",
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"<title>1->-4</title>\n",
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"<path fill=\"none\" stroke=\"black\" d=\"M17.29,-249.63C13.85,-239.81 10.97,-227.28 13,-216 14.77,-206.18 19.11,-195.51 22.29,-188.56\"/>\n",
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"<polygon fill=\"none\" stroke=\"black\" points=\"24.71,-189.2 25.56,-181.83 20.3,-187.06 24.71,-189.2\"/>\n",
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"<text text-anchor=\"start\" x=\"13\" y=\"-219.8\" font-family=\"Lato\" font-size=\"14.00\">!b</text>\n",
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"</g>\n",
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"<!-- 3->3 -->\n",
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"<g id=\"edge12\" class=\"edge\">\n",
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"<title>3->3</title>\n",
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"<path fill=\"none\" stroke=\"black\" d=\"M143.88,-274.72C153.5,-275.02 161.5,-272.78 161.5,-268 161.5,-264.49 157.19,-262.35 151.09,-261.58\"/>\n",
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"<polygon fill=\"black\" stroke=\"black\" points=\"143.88,-261.28 151.01,-258.42 147.38,-261.42 150.88,-261.57 150.88,-261.57 150.88,-261.57 147.38,-261.42 150.75,-264.71 143.88,-261.28 143.88,-261.28\"/>\n",
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"<text text-anchor=\"start\" x=\"161.5\" y=\"-264.3\" font-family=\"Lato\" font-size=\"14.00\">1</text>\n",
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"<!-- 4 -->\n",
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"<title>4</title>\n",
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"<text text-anchor=\"start\" x=\"106.5\" y=\"-176.3\" font-family=\"Lato\" font-size=\"14.00\">G(a)</text>\n",
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"</g>\n",
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"<title>3->4</title>\n",
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"</g>\n",
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"<title>-4->1</title>\n",
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"</g>\n",
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"<!-- 2 -->\n",
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"<title>2</title>\n",
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"<text text-anchor=\"start\" x=\"14\" y=\"-109.8\" font-family=\"Lato\" font-size=\"14.00\">F(b)</text>\n",
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"<text text-anchor=\"start\" x=\"18\" y=\"-94.8\" font-family=\"Lato\" font-size=\"14.00\" fill=\"#1f78b4\">⓿</text>\n",
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"</g>\n",
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"<!-- -4->2 -->\n",
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"<g id=\"edge8\" class=\"edge\">\n",
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"<title>-4->2</title>\n",
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"</g>\n",
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"<!-- 2->2 -->\n",
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"<g id=\"edge10\" class=\"edge\">\n",
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"<title>2->2</title>\n",
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"<path fill=\"none\" stroke=\"black\" d=\"M46.33,-112.62C55.76,-113.34 64,-111.13 64,-106 64,-102.23 59.56,-100.04 53.46,-99.43\"/>\n",
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"<text text-anchor=\"start\" x=\"64\" y=\"-102.3\" font-family=\"Lato\" font-size=\"14.00\">!b</text>\n",
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"</g>\n",
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"<!-- 5 -->\n",
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"<title>5->5</title>\n",
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"</g>\n",
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]
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},
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"source": [
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"If the state labels take too much space, you can reduce the size of the automaton by forcing states to be numbered with option `1`. The original label is still displayed as a tooltip when the mouse is over the state.\n",
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"\n",
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"Note that passing option `show=...` to `display_inline` is similar to calling `aut.show(...)` on each argument."
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]
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},
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{
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"outputs": [
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"<text text-anchor=\"start\" x=\"134\" y=\"-176.3\" font-family=\"Lato\" font-size=\"14.00\">a</text>\n",
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"</g>\n",
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"</svg>\n",
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"</div><div style='vertical-align:text-top;display:inline-block;'><?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
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"<path fill=\"#ffffaa\" stroke=\"black\" d=\"M96,-214C96,-214 84,-214 84,-214 78,-214 72,-208 72,-202 72,-202 72,-188 72,-188 72,-182 78,-176 84,-176 84,-176 96,-176 96,-176 102,-176 108,-182 108,-188 108,-188 108,-202 108,-202 108,-208 102,-214 96,-214\"/>\n",
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"</g>\n",
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"</g>\n",
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"</g>\n",
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"</a>\n",
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"</g>\n",
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"</g>\n",
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"<polygon fill=\"black\" stroke=\"black\" points=\"83.77,-125.24 84.15,-132.91 82.49,-128.5 81.22,-131.76 81.22,-131.76 81.22,-131.76 82.49,-128.5 78.28,-130.61 83.77,-125.24 83.77,-125.24\"/>\n",
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"<text text-anchor=\"start\" x=\"78\" y=\"-146.8\" font-family=\"Lato\" font-size=\"14.00\">a</text>\n",
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"<g id=\"node4\" class=\"node\">\n",
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"<text text-anchor=\"start\" x=\"47\" y=\"-102.3\" font-family=\"Lato\" font-size=\"14.00\">!a</text>\n",
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"</g>\n",
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"<g id=\"edge9\" class=\"edge\">\n",
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"<title>1->0</title>\n",
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"<path fill=\"none\" stroke=\"black\" d=\"M92.38,-125C91.94,-137.71 91.35,-154.91 90.86,-168.96\"/>\n",
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"<text text-anchor=\"start\" x=\"92\" y=\"-146.8\" font-family=\"Lato\" font-size=\"14.00\">!a & b & p</text>\n",
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"<title>-1</title>\n",
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"</g>\n",
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"<title>1->-1</title>\n",
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"<g id=\"edge10\" class=\"edge\">\n",
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"<title>2->2</title>\n",
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"<path fill=\"none\" stroke=\"black\" d=\"M36.15,-25.6C45.54,-26.77 54,-24.23 54,-18 54,-13.42 49.44,-10.84 43.3,-10.25\"/>\n",
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"</g>\n",
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"<g id=\"edge5\" class=\"edge\">\n",
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"<title>-1->0</title>\n",
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"</g>\n",
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"</g>\n",
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"</g>\n",
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"</svg>\n",
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"<title>3</title>\n",
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"<path fill=\"#ffffaa\" stroke=\"black\" d=\"M131,-353C131,-353 119,-353 119,-353 113,-353 107,-347 107,-341 107,-341 107,-329 107,-329 107,-323 113,-317 119,-317 119,-317 131,-317 131,-317 137,-317 143,-323 143,-329 143,-329 143,-341 143,-341 143,-347 137,-353 131,-353\"/>\n",
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"<text text-anchor=\"middle\" x=\"125\" y=\"-331.3\" font-family=\"Lato\" font-size=\"14.00\">3</text>\n",
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"<g id=\"edge1\" class=\"edge\">\n",
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"<title>I->3</title>\n",
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"</g>\n",
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"<!-- -4 -->\n",
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"<g id=\"node7\" class=\"node\">\n",
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"<title>-4</title>\n",
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"<ellipse fill=\"#ffffaa\" stroke=\"black\" cx=\"106\" cy=\"-258.5\" rx=\"1.8\" ry=\"1.8\"/>\n",
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"</g>\n",
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"<title>3->-4</title>\n",
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"</g>\n",
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"<g id=\"node3\" class=\"node\">\n",
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"<title>0</title>\n",
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"<g id=\"a_node3\"><a xlink:title=\"(a;a*;b)*\">\n",
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"</g>\n",
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"</g>\n",
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"</a>\n",
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"</g>\n",
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"</g>\n",
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"<title>0->1</title>\n",
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"</g>\n",
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"</g>\n",
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"</g>\n",
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"</g>\n",
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"<title>-1->0</title>\n",
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"</g>\n",
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"</g>\n",
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"</g>\n",
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"</svg>\n",
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"<g id=\"a_node2\"><a xlink:title=\"GF(a)\">\n",
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],
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"metadata": {},
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"output_type": "display_data"
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}
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],
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"source": [
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"display_inline(aut1, aut2, aut3, aut4, aut5, show='.bav1')"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"When working with alternating automata, it is quite common to hide \"true states\", and display \"exiting transitions instead\". You can do that with option `u`."
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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": 5,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"</g>\n",
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"</g>\n",
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"</g>\n",
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"</g>\n",
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"</a>\n",
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"</g>\n",
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"</g>\n",
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"<g id=\"edge2\" class=\"edge\">\n",
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"<title>0->1</title>\n",
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"<title>0->T2T0</title>\n",
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"</g>\n",
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"</g>\n",
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"</svg>\n",
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" viewBox=\"0.00 0.00 87.00 299.60\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
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"</g>\n",
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"</g>\n",
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"</g>\n",
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"</g>\n",
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"</g>\n",
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"</svg>\n",
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"</g>\n",
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"</g>\n",
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"</g>\n",
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"</a>\n",
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"</g>\n",
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"</g>\n",
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"<title>1</title>\n",
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"</g>\n",
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"</g>\n",
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"<title>0->1</title>\n",
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"<title>1->0</title>\n",
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"</g>\n",
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"<title>1->1</title>\n",
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"</g>\n",
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}
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"source": [
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"display_inline(aut1, aut2, aut3, aut4, aut5, show='.bav1us')"
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"## Alternation removal"
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]
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"The `remove_alternation()` function works on any alternating automaton that is weak (not necessarily very weak), i.e., in each SCC all transition should belong to the same accepting sets.\n",
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"\n",
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"The second argument of `remove_alternation()`, set to `True` below, simply asks for states to be labeled to help debugging. As the function builds Transition-based Generalized Büchi acceptance, it can be worthwhile to apply `scc_filter()` in an attempt to reduce the number of acceptance sets.\n",
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"\n",
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"The next cell shows this two-step process on our first example automaton."
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]
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"<g id=\"node3\" class=\"node\">\n",
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"<title>1</title>\n",
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"<path fill=\"#ffffaa\" stroke=\"black\" d=\"M55.05,-342C55.05,-342 43.05,-342 43.05,-342 37.05,-342 31.05,-336 31.05,-330 31.05,-330 31.05,-318 31.05,-318 31.05,-312 37.05,-306 43.05,-306 43.05,-306 55.05,-306 55.05,-306 61.05,-306 67.05,-312 67.05,-318 67.05,-318 67.05,-330 67.05,-330 67.05,-336 61.05,-342 55.05,-342\"/>\n",
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"</g>\n",
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"<title>0->1</title>\n",
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"</g>\n",
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"<!-- 1->1 -->\n",
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"<g id=\"edge3\" class=\"edge\">\n",
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"<title>1->1</title>\n",
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"<text text-anchor=\"start\" x=\"89.05\" y=\"-327.8\" font-family=\"Lato\" font-size=\"14.00\">b</text>\n",
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"</g>\n",
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"<!-- 2 -->\n",
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"<g id=\"node4\" class=\"node\">\n",
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"<title>2</title>\n",
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"<path fill=\"#ffffaa\" stroke=\"black\" d=\"M53.55,-240C53.55,-240 30.55,-240 30.55,-240 24.55,-240 18.55,-234 18.55,-228 18.55,-228 18.55,-216 18.55,-216 18.55,-210 24.55,-204 30.55,-204 30.55,-204 53.55,-204 53.55,-204 59.55,-204 65.55,-210 65.55,-216 65.55,-216 65.55,-228 65.55,-228 65.55,-234 59.55,-240 53.55,-240\"/>\n",
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"</g>\n",
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"<g id=\"edge4\" class=\"edge\">\n",
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"<title>1->2</title>\n",
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"<text text-anchor=\"start\" x=\"33.05\" y=\"-269.3\" font-family=\"Lato\" font-size=\"14.00\">!b</text>\n",
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"</g>\n",
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"<g id=\"node5\" class=\"node\">\n",
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"<title>3</title>\n",
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"<path fill=\"#ffffaa\" stroke=\"black\" d=\"M68.55,-36C68.55,-36 45.55,-36 45.55,-36 39.55,-36 33.55,-30 33.55,-24 33.55,-24 33.55,-12 33.55,-12 33.55,-6 39.55,0 45.55,0 45.55,0 68.55,0 68.55,0 74.55,0 80.55,-6 80.55,-12 80.55,-12 80.55,-24 80.55,-24 80.55,-30 74.55,-36 68.55,-36\"/>\n",
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"</g>\n",
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"<!-- 1->3 -->\n",
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"<g id=\"edge5\" class=\"edge\">\n",
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"<title>1->3</title>\n",
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"<path fill=\"none\" stroke=\"black\" d=\"M38.01,-305.87C34.62,-300.28 31,-293.97 28.05,-288 1.69,-234.6 -5.69,-214.57 5.05,-156 12.67,-114.48 32.26,-69.22 45.1,-42.52\"/>\n",
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"<text text-anchor=\"start\" x=\"5.05\" y=\"-174.8\" font-family=\"Lato\" font-size=\"14.00\">a & !b</text>\n",
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"<text text-anchor=\"start\" x=\"15.05\" y=\"-159.8\" font-family=\"Lato\" font-size=\"14.00\" fill=\"#ff4da0\">❶</text>\n",
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"</g>\n",
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"<!-- 4 -->\n",
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"<g id=\"node6\" class=\"node\">\n",
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"<title>4</title>\n",
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"<path fill=\"#ffffaa\" stroke=\"black\" d=\"M129.05,-138C129.05,-138 117.05,-138 117.05,-138 111.05,-138 105.05,-132 105.05,-126 105.05,-126 105.05,-114 105.05,-114 105.05,-108 111.05,-102 117.05,-102 117.05,-102 129.05,-102 129.05,-102 135.05,-102 141.05,-108 141.05,-114 141.05,-114 141.05,-126 141.05,-126 141.05,-132 135.05,-138 129.05,-138\"/>\n",
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"<text text-anchor=\"start\" x=\"113.05\" y=\"-116.3\" font-family=\"Lato\" font-size=\"14.00\">1,4</text>\n",
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"</g>\n",
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"<g id=\"edge6\" class=\"edge\">\n",
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"<title>1->4</title>\n",
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"<path fill=\"none\" stroke=\"black\" d=\"M63.25,-305.95C75.95,-289.88 94.19,-264.66 105.05,-240 115,-217.41 114.21,-210.38 118.05,-186 120.16,-172.67 121.39,-157.65 122.1,-145.41\"/>\n",
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"<polygon fill=\"black\" stroke=\"black\" points=\"122.48,-138.19 125.26,-145.35 122.29,-141.69 122.11,-145.18 122.11,-145.18 122.11,-145.18 122.29,-141.69 118.96,-145.02 122.48,-138.19 122.48,-138.19\"/>\n",
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"<text text-anchor=\"start\" x=\"115.05\" y=\"-225.8\" font-family=\"Lato\" font-size=\"14.00\">a & b</text>\n",
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"<text text-anchor=\"start\" x=\"115.05\" y=\"-211.8\" font-family=\"Lato\" font-size=\"14.00\" fill=\"#1f78b4\">⓿</text>\n",
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"<text text-anchor=\"start\" x=\"131.05\" y=\"-211.8\" font-family=\"Lato\" font-size=\"14.00\" fill=\"#ff4da0\">❶</text>\n",
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"</g>\n",
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"<!-- 2->1 -->\n",
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"<g id=\"edge7\" class=\"edge\">\n",
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"<title>2->1</title>\n",
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"<path fill=\"none\" stroke=\"black\" d=\"M43.6,-240.08C44.09,-245.78 44.62,-252.16 45.05,-258 46.04,-271.45 46.98,-286.47 47.7,-298.68\"/>\n",
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"<polygon fill=\"black\" stroke=\"black\" points=\"48.11,-305.89 44.56,-299.08 47.91,-302.39 47.71,-298.9 47.71,-298.9 47.71,-298.9 47.91,-302.39 50.85,-298.72 48.11,-305.89 48.11,-305.89\"/>\n",
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"<text text-anchor=\"start\" x=\"51.05\" y=\"-276.8\" font-family=\"Lato\" font-size=\"14.00\">b</text>\n",
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"</g>\n",
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"<!-- 2->2 -->\n",
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"<g id=\"edge8\" class=\"edge\">\n",
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"<title>2->2</title>\n",
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"<path fill=\"none\" stroke=\"black\" d=\"M65.93,-231.32C75.51,-231.84 83.55,-228.74 83.55,-222 83.55,-217.05 79.22,-214.06 73.12,-213.03\"/>\n",
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"<text text-anchor=\"start\" x=\"83.55\" y=\"-218.3\" font-family=\"Lato\" font-size=\"14.00\">!b</text>\n",
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"</g>\n",
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"<g id=\"edge9\" class=\"edge\">\n",
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"<title>2->3</title>\n",
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"<path fill=\"none\" stroke=\"black\" d=\"M43.33,-203.78C46,-167.85 52.17,-84.82 55.24,-43.41\"/>\n",
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"<polygon fill=\"black\" stroke=\"black\" points=\"55.78,-36.18 58.4,-43.39 55.52,-39.67 55.26,-43.16 55.26,-43.16 55.26,-43.16 55.52,-39.67 52.12,-42.93 55.78,-36.18 55.78,-36.18\"/>\n",
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"<text text-anchor=\"start\" x=\"51.05\" y=\"-123.8\" font-family=\"Lato\" font-size=\"14.00\">a & !b</text>\n",
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"</g>\n",
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"<!-- 2->4 -->\n",
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"<g id=\"edge10\" class=\"edge\">\n",
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"<title>2->4</title>\n",
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"<path fill=\"none\" stroke=\"black\" d=\"M52.14,-203.99C60.5,-190.43 73.09,-171.27 86.05,-156 90.26,-151.05 95.09,-146.06 99.85,-141.46\"/>\n",
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"<text text-anchor=\"start\" x=\"86.05\" y=\"-174.8\" font-family=\"Lato\" font-size=\"14.00\">a & b</text>\n",
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"</g>\n",
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"<!-- 3->3 -->\n",
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"<g id=\"edge11\" class=\"edge\">\n",
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"<title>3->3</title>\n",
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"<text text-anchor=\"start\" x=\"98.55\" y=\"-21.8\" font-family=\"Lato\" font-size=\"14.00\">a & !b</text>\n",
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"</g>\n",
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"<!-- 3->4 -->\n",
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"<g id=\"edge12\" class=\"edge\">\n",
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"<title>3->4</title>\n",
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"<text text-anchor=\"start\" x=\"121.05\" y=\"-72.8\" font-family=\"Lato\" font-size=\"14.00\">a & b</text>\n",
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"</g>\n",
|
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"<!-- 4->3 -->\n",
|
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"<g id=\"edge13\" class=\"edge\">\n",
|
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"<title>4->3</title>\n",
|
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"<path fill=\"none\" stroke=\"black\" d=\"M104.96,-111.22C93.51,-105.37 79.35,-96.25 71.05,-84 62.99,-72.09 59.46,-56.39 57.96,-43.4\"/>\n",
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"</g>\n",
|
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"<!-- 4->4 -->\n",
|
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"<g id=\"edge14\" class=\"edge\">\n",
|
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"<title>4->4</title>\n",
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"<path fill=\"none\" stroke=\"black\" d=\"M141.21,-129.05C150.59,-130.44 159.05,-127.42 159.05,-120 159.05,-114.55 154.49,-111.48 148.35,-110.78\"/>\n",
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"<text text-anchor=\"start\" x=\"159.05\" y=\"-123.8\" font-family=\"Lato\" font-size=\"14.00\">a & b</text>\n",
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"</g>\n",
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"</g>\n",
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"</svg>\n",
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"</div><div style='vertical-align:text-top;display:inline-block;'><?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
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"<!-- Generated by graphviz version 2.43.0 (0)\n",
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" -->\n",
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"<!-- Pages: 1 -->\n",
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"<svg width=\"148pt\" height=\"360pt\"\n",
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" viewBox=\"0.00 0.00 148.21 360.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
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"<text text-anchor=\"start\" x=\"74.51\" y=\"-458.8\" font-family=\"Lato\" font-size=\"14.00\">Inf(</text>\n",
|
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"<title>0</title>\n",
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"<title>I->0</title>\n",
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"</g>\n",
|
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"<!-- 1 -->\n",
|
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"<g id=\"node3\" class=\"node\">\n",
|
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"<title>1</title>\n",
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"<path fill=\"#ffffaa\" stroke=\"black\" d=\"M54.03,-312C54.03,-312 42.03,-312 42.03,-312 36.03,-312 30.03,-306 30.03,-300 30.03,-300 30.03,-288 30.03,-288 30.03,-282 36.03,-276 42.03,-276 42.03,-276 54.03,-276 54.03,-276 60.03,-276 66.03,-282 66.03,-288 66.03,-288 66.03,-300 66.03,-300 66.03,-306 60.03,-312 54.03,-312\"/>\n",
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"</g>\n",
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"<!-- 0->1 -->\n",
|
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"<g id=\"edge2\" class=\"edge\">\n",
|
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"<title>0->1</title>\n",
|
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"<path fill=\"none\" stroke=\"black\" d=\"M48.03,-362.8C48.03,-350.36 48.03,-333.36 48.03,-319.5\"/>\n",
|
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"<polygon fill=\"black\" stroke=\"black\" points=\"48.03,-312.18 51.18,-319.18 48.03,-315.68 48.03,-319.18 48.03,-319.18 48.03,-319.18 48.03,-315.68 44.88,-319.18 48.03,-312.18 48.03,-312.18\"/>\n",
|
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"<text text-anchor=\"middle\" x=\"52.53\" y=\"-333.8\" font-family=\"Lato\" font-size=\"14.00\">1</text>\n",
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"</g>\n",
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"<!-- 1->1 -->\n",
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"<g id=\"edge3\" class=\"edge\">\n",
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"<title>1->1</title>\n",
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"<path fill=\"none\" stroke=\"black\" d=\"M66.18,-301.6C75.57,-302.77 84.03,-300.23 84.03,-294 84.03,-289.42 79.46,-286.84 73.32,-286.25\"/>\n",
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"<text text-anchor=\"start\" x=\"84.03\" y=\"-290.3\" font-family=\"Lato\" font-size=\"14.00\">b</text>\n",
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"</g>\n",
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"<!-- 2 -->\n",
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"<g id=\"node4\" class=\"node\">\n",
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"<title>2</title>\n",
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"text/plain": [
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]
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"metadata": {},
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"output_type": "display_data"
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}
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],
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"source": [
|
|
"nba1t = spot.remove_alternation(aut1, True)\n",
|
|
"nba1 = spot.scc_filter(nba1t, True)\n",
|
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"display_inline(aut1.show('.bav1u'), nba1t, nba1)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
|
|
"Let's apply this process to the other 4 automata (which are not very-weak, unlike `aut1`). The states marked with `~` are part of a break-point construction."
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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": 8,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"</g>\n",
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"</g>\n",
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"</svg>\n",
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"<g id=\"edge1\" class=\"edge\">\n",
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"<title>I->0</title>\n",
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"</g>\n",
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"<text text-anchor=\"start\" x=\"237.55\" y=\"-537.8\" font-family=\"Lato\" font-size=\"14.00\">a</text>\n",
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"<!-- 1->0 -->\n",
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"<g id=\"edge3\" class=\"edge\">\n",
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"<text text-anchor=\"start\" x=\"268.55\" y=\"-537.8\" font-family=\"Lato\" font-size=\"14.00\">!a</text>\n",
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"<title>1->2</title>\n",
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"<title>3</title>\n",
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"<g id=\"edge5\" class=\"edge\">\n",
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"<title>2->3</title>\n",
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"<path fill=\"none\" stroke=\"black\" d=\"M164.16,-370.36C176.7,-364.23 192.2,-355.48 204.05,-345 206.03,-343.25 222.36,-320.96 235.43,-302.97\"/>\n",
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"<text text-anchor=\"start\" x=\"227.05\" y=\"-326.3\" font-family=\"Lato\" font-size=\"14.00\">a & !b & p</text>\n",
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"<!-- 4 -->\n",
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"<g id=\"node6\" class=\"node\">\n",
|
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"<title>4</title>\n",
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"<path fill=\"#ffffaa\" stroke=\"black\" d=\"M181.05,-123C181.05,-123 141.05,-123 141.05,-123 135.05,-123 129.05,-117 129.05,-111 129.05,-111 129.05,-99 129.05,-99 129.05,-93 135.05,-87 141.05,-87 141.05,-87 181.05,-87 181.05,-87 187.05,-87 193.05,-93 193.05,-99 193.05,-99 193.05,-111 193.05,-111 193.05,-117 187.05,-123 181.05,-123\"/>\n",
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"</g>\n",
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"<!-- 2->4 -->\n",
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"<g id=\"edge6\" class=\"edge\">\n",
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"<title>2->4</title>\n",
|
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"<path fill=\"none\" stroke=\"black\" d=\"M120.04,-372.93C104.74,-367.2 84.57,-357.95 70.05,-345 25.64,-305.42 -13.71,-284.45 5.05,-228 20.27,-182.2 31.57,-170.14 70.05,-141 85.28,-129.47 104.77,-121.3 121.8,-115.77\"/>\n",
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"<text text-anchor=\"start\" x=\"5.05\" y=\"-231.8\" font-family=\"Lato\" font-size=\"14.00\">a & b & p</text>\n",
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"</g>\n",
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"<!-- 3->0 -->\n",
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"<g id=\"edge7\" class=\"edge\">\n",
|
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"<title>3->0</title>\n",
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"<path fill=\"none\" stroke=\"black\" d=\"M275.1,-297.03C280.34,-302.17 285.2,-308.26 288.05,-315 323.49,-398.89 303.78,-511.05 292.37,-559.74\"/>\n",
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"<text text-anchor=\"start\" x=\"308.05\" y=\"-435.8\" font-family=\"Lato\" font-size=\"14.00\">!a & !b</text>\n",
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"</g>\n",
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"<!-- 3->2 -->\n",
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"<g id=\"edge8\" class=\"edge\">\n",
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"<title>3->2</title>\n",
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"<path fill=\"none\" stroke=\"black\" d=\"M224.02,-279.95C197.32,-281.58 158.42,-288.71 139.05,-315 130.56,-326.51 131.2,-342.52 134.02,-355.78\"/>\n",
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"<text text-anchor=\"start\" x=\"139.05\" y=\"-326.3\" font-family=\"Lato\" font-size=\"14.00\">a & !b & p</text>\n",
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"<!-- 5 -->\n",
|
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"<g id=\"node7\" class=\"node\">\n",
|
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"<title>5</title>\n",
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"<path fill=\"#ffffaa\" stroke=\"black\" d=\"M327.05,-36C327.05,-36 307.05,-36 307.05,-36 301.05,-36 295.05,-30 295.05,-24 295.05,-24 295.05,-12 295.05,-12 295.05,-6 301.05,0 307.05,0 307.05,0 327.05,0 327.05,0 333.05,0 339.05,-6 339.05,-12 339.05,-12 339.05,-24 339.05,-24 339.05,-30 333.05,-36 327.05,-36\"/>\n",
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|
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"<g id=\"edge9\" class=\"edge\">\n",
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"<title>3->5</title>\n",
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"<path fill=\"none\" stroke=\"black\" d=\"M280.42,-274.66C300.21,-270.68 325.27,-261.99 338.05,-243 379.74,-181.01 347.41,-86.61 328.39,-42.83\"/>\n",
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"<text text-anchor=\"start\" x=\"357.05\" y=\"-144.8\" font-family=\"Lato\" font-size=\"14.00\">!a & b & p</text>\n",
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"<!-- 6 -->\n",
|
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"<g id=\"node8\" class=\"node\">\n",
|
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"<title>6</title>\n",
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"<path fill=\"#ffffaa\" stroke=\"black\" d=\"M211.05,-210C211.05,-210 171.05,-210 171.05,-210 165.05,-210 159.05,-204 159.05,-198 159.05,-198 159.05,-186 159.05,-186 159.05,-180 165.05,-174 171.05,-174 171.05,-174 211.05,-174 211.05,-174 217.05,-174 223.05,-180 223.05,-186 223.05,-186 223.05,-198 223.05,-198 223.05,-204 217.05,-210 211.05,-210\"/>\n",
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"</g>\n",
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"<!-- 3->6 -->\n",
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"<g id=\"edge10\" class=\"edge\">\n",
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"<title>3->6</title>\n",
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"<path fill=\"none\" stroke=\"black\" d=\"M246.56,-260.84C242.84,-250.72 237.28,-237.98 230.05,-228 226.74,-223.44 222.76,-219.04 218.62,-214.99\"/>\n",
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"<g id=\"edge11\" class=\"edge\">\n",
|
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"<title>4->0</title>\n",
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"<path fill=\"none\" stroke=\"black\" d=\"M193.05,-106.13C254.14,-107.05 384.24,-112.41 417.05,-141 434.44,-156.16 431.05,-167.92 431.05,-191 431.05,-484 431.05,-484 431.05,-484 431.05,-541.55 352.68,-568.53 311.17,-578.78\"/>\n",
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"</g>\n",
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"<title>4->2</title>\n",
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"<path fill=\"none\" stroke=\"black\" d=\"M146.97,-123.25C142.84,-128.74 138.51,-134.96 135.05,-141 114.15,-177.45 106.44,-186.83 98.05,-228 88.61,-274.3 111.71,-327 127.89,-356.59\"/>\n",
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"<title>2->3</title>\n",
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"</g>\n",
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"</svg>\n",
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"source": [
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"nba2, nba3, nba4, nba5 = [spot.scc_filter(spot.remove_alternation(a, True), True) for a in (aut2, aut3, aut4, aut5)]\n",
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"display_inline(nba2, nba3, nba4, nba5)"
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"The following demonstrates that very weak (non-alternating) Büchi automata can be complemented via alternation removal."
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]
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},
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"</g>\n",
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"</g>\n",
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"</g>\n",
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"</g>\n",
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"</g>\n",
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"</svg>\n",
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"<title>0->1</title>\n",
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"<title>0->2</title>\n",
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"<title>1->4</title>\n",
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"</g>\n",
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"<title>1->6</title>\n",
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"<path fill=\"none\" stroke=\"black\" d=\"M310.66,-314.38C320.62,-316.02 330,-313.56 330,-307 330,-302.08 324.73,-299.46 317.89,-299.16\"/>\n",
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"<text text-anchor=\"start\" x=\"340\" y=\"-310.8\" font-family=\"Lato\" font-size=\"14.00\">!b</text>\n",
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"<text text-anchor=\"start\" x=\"346\" y=\"-296.8\" font-family=\"Lato\" font-size=\"14.00\" fill=\"#ff4da0\">❶</text>\n",
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"</g>\n",
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"<!-- 3->3 -->\n",
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"<g id=\"edge9\" class=\"edge\">\n",
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"<title>3->3</title>\n",
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"<path fill=\"none\" stroke=\"black\" d=\"M172.66,-227.38C182.62,-229.02 192,-226.56 192,-220 192,-215.08 186.73,-212.46 179.89,-212.16\"/>\n",
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"<text text-anchor=\"start\" x=\"192\" y=\"-216.3\" font-family=\"Lato\" font-size=\"14.00\">!a & !b</text>\n",
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"</g>\n",
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"<!-- 3->4 -->\n",
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"<g id=\"edge10\" class=\"edge\">\n",
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"<title>3->4</title>\n",
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"<path fill=\"none\" stroke=\"black\" d=\"M141.75,-208.67C117.17,-190.86 66.7,-154.29 38.4,-133.78\"/>\n",
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"<polygon fill=\"black\" stroke=\"black\" points=\"32.38,-129.42 39.89,-130.98 35.21,-131.47 38.05,-133.53 38.05,-133.53 38.05,-133.53 35.21,-131.47 36.2,-136.08 32.38,-129.42 32.38,-129.42\"/>\n",
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"<text text-anchor=\"start\" x=\"108\" y=\"-172.8\" font-family=\"Lato\" font-size=\"14.00\">!a & b</text>\n",
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"<text text-anchor=\"start\" x=\"110\" y=\"-158.8\" font-family=\"Lato\" font-size=\"14.00\" fill=\"#1f78b4\">⓿</text>\n",
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"</g>\n",
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"<!-- 3->5 -->\n",
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"<g id=\"edge11\" class=\"edge\">\n",
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"<title>3->5</title>\n",
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"<path fill=\"none\" stroke=\"black\" d=\"M169.57,-207.91C177.34,-201.3 187.09,-192.55 195,-184 206.41,-171.67 207.02,-166.71 218,-154 223.07,-148.13 228.86,-141.99 234.2,-136.52\"/>\n",
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"<text text-anchor=\"start\" x=\"218\" y=\"-172.8\" font-family=\"Lato\" font-size=\"14.00\">a & !b</text>\n",
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"</g>\n",
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"<!-- 3->6 -->\n",
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"<g id=\"edge12\" class=\"edge\">\n",
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"<title>3->6</title>\n",
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"<path fill=\"none\" stroke=\"black\" d=\"M161.07,-202.63C172.07,-167.17 198.12,-83.12 210.81,-42.2\"/>\n",
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"<polygon fill=\"black\" stroke=\"black\" points=\"212.94,-35.31 213.88,-42.93 211.91,-38.66 210.87,-42 210.87,-42 210.87,-42 211.91,-38.66 207.86,-41.07 212.94,-35.31 212.94,-35.31\"/>\n",
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"<text text-anchor=\"start\" x=\"193\" y=\"-121.8\" font-family=\"Lato\" font-size=\"14.00\">a & b</text>\n",
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"<text text-anchor=\"start\" x=\"193\" y=\"-107.8\" font-family=\"Lato\" font-size=\"14.00\" fill=\"#1f78b4\">⓿</text>\n",
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"<text text-anchor=\"start\" x=\"209\" y=\"-107.8\" font-family=\"Lato\" font-size=\"14.00\" fill=\"#ff4da0\">❶</text>\n",
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"</g>\n",
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"<!-- 4->4 -->\n",
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"<g id=\"edge13\" class=\"edge\">\n",
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"<title>4->4</title>\n",
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"<path fill=\"none\" stroke=\"black\" d=\"M35.78,-122.49C45.31,-123.26 54,-121.76 54,-118 54,-115.24 49.32,-113.7 43.05,-113.38\"/>\n",
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"<text text-anchor=\"start\" x=\"54\" y=\"-121.8\" font-family=\"Lato\" font-size=\"14.00\">!a & !b</text>\n",
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"<text text-anchor=\"start\" x=\"66\" y=\"-106.8\" font-family=\"Lato\" font-size=\"14.00\" fill=\"#1f78b4\">⓿</text>\n",
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"</g>\n",
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"<!-- 4->4 -->\n",
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"<g id=\"edge14\" class=\"edge\">\n",
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"<title>4->4</title>\n",
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"<path fill=\"none\" stroke=\"black\" d=\"M34.89,-124.49C58.95,-129.88 94,-127.72 94,-118 94,-109.19 65.22,-106.59 41.91,-110.19\"/>\n",
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"<polygon fill=\"black\" stroke=\"black\" points=\"34.89,-111.51 41.19,-107.12 38.33,-110.86 41.77,-110.22 41.77,-110.22 41.77,-110.22 38.33,-110.86 42.35,-113.31 34.89,-111.51 34.89,-111.51\"/>\n",
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"<text text-anchor=\"start\" x=\"94\" y=\"-121.8\" font-family=\"Lato\" font-size=\"14.00\">!a & b</text>\n",
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"<text text-anchor=\"start\" x=\"112\" y=\"-107.8\" font-family=\"Lato\" font-size=\"14.00\" fill=\"#ff4da0\">❶</text>\n",
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"</g>\n",
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"<!-- 4->6 -->\n",
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"<g id=\"edge15\" class=\"edge\">\n",
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"<title>4->6</title>\n",
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"<path fill=\"none\" stroke=\"black\" d=\"M33.36,-108.49C56.01,-96.04 99.82,-72.34 138,-54 157.01,-44.87 178.99,-35.32 195.09,-28.51\"/>\n",
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"<polygon fill=\"black\" stroke=\"black\" points=\"201.6,-25.77 196.37,-31.39 198.38,-27.13 195.15,-28.49 195.15,-28.49 195.15,-28.49 198.38,-27.13 193.93,-25.58 201.6,-25.77 201.6,-25.77\"/>\n",
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"<text text-anchor=\"start\" x=\"150.5\" y=\"-71.8\" font-family=\"Lato\" font-size=\"14.00\">a</text>\n",
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"</g>\n",
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"<!-- 5->5 -->\n",
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"<g id=\"edge16\" class=\"edge\">\n",
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"<title>5->5</title>\n",
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"</g>\n",
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"<!-- 5->6 -->\n",
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"<g id=\"edge17\" class=\"edge\">\n",
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"<title>5->6</title>\n",
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"</g>\n",
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"<!-- 6->6 -->\n",
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"<g id=\"edge18\" class=\"edge\">\n",
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"<title>6->6</title>\n",
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"<text text-anchor=\"start\" x=\"265.5\" y=\"-21.8\" font-family=\"Lato\" font-size=\"14.00\">1</text>\n",
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"</g>\n",
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"</g>\n",
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"</svg>\n",
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"</div>"
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],
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"text/plain": [
|
|
"<IPython.core.display.HTML object>"
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|
]
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|
},
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"metadata": {},
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"output_type": "display_data"
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|
}
|
|
],
|
|
"source": [
|
|
"pos = spot.automaton(\"\"\"HOA: v1 name: \"(a & (Fa R XFb)) | (!a & (G!a U\n",
|
|
"XG!b))\" States: 6 Start: 0 AP: 2 \"a\" \"b\" acc-name: Buchi Acceptance: 1\n",
|
|
"Inf(0) properties: trans-labels explicit-labels state-acc\n",
|
|
"semi-deterministic --BODY-- State: 0 [0] 1 [!0] 2 [!0] 3 State: 1 [!1]\n",
|
|
"1 [1] 4 State: 2 {0} [!1] 2 State: 3 [!0] 3 [!0] 5 State: 4 {0} [t] 4\n",
|
|
"State: 5 {0} [!0&!1] 5 --END--\"\"\")\n",
|
|
"altneg = spot.dualize(pos)\n",
|
|
"neg = spot.remove_alternation(altneg)\n",
|
|
"display_inline(pos, altneg.show('.bvu'), neg)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 10,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"# Issue #382.\n",
|
|
"w = spot.parse_word('cycle{!a&b}').as_automaton()\n",
|
|
"assert pos.intersects(w) != neg.intersects(w)"
|
|
]
|
|
}
|
|
],
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|
"metadata": {
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|
"kernelspec": {
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"display_name": "Python 3 (ipykernel)",
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"language": "python",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.11.7"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 2
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}
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