e778965730
This is to workaround differences in minidom's pretty-printing that occurred between Python 3.7 and 3.8. * python/spot/jupyter.py (SVG): New class. * python/spot/__init__.py: Use it. * tests/python/_altscc.ipynb, tests/python/alternation.ipynb, tests/python/automata.ipynb, tests/python/formulas.ipynb, tests/python/gen.ipynb, tests/python/highlighting.ipynb, tests/python/ltsmin-dve.ipynb, tests/python/ltsmin-pml.ipynb, tests/python/product.ipynb, tests/python/randaut.ipynb, tests/python/testingaut.ipynb, tests/python/twagraph-internals.ipynb, tests/python/word.ipynb: Adjust.
4304 lines
299 KiB
Text
4304 lines
299 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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"<text text-anchor=\"start\" x=\"15.5\" y=\"-287.91\" font-family=\"Lato\" font-size=\"14.00\">GF(b)</text>\n",
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"<!-- 3 -->\n",
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"<text text-anchor=\"start\" x=\"128.5\" y=\"-295.41\" font-family=\"Lato\" font-size=\"14.00\">FG(a)</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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"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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"Let's make sure that option `u` and `s` (to display SCCs) work well together:"
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"</g>\n",
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"<!-- 0 -->\n",
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"</g>\n",
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"</g>\n",
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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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"</g>\n",
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"<!-- 2 -->\n",
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"<g id=\"node7\" class=\"node\">\n",
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"<title>2</title>\n",
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"<g id=\"a_node7\"><a xlink:title=\"F(b)\">\n",
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"</a>\n",
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"</g>\n",
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"</g>\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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"<text text-anchor=\"start\" x=\"71.74\" y=\"-75.17\" font-family=\"Lato\" font-size=\"14.00\">!b</text>\n",
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"</g>\n",
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"<!-- T5T2 -->\n",
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"<!-- 2->T5T2 -->\n",
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"<g id=\"edge9\" class=\"edge\">\n",
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"<title>2->T5T2</title>\n",
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"<text text-anchor=\"start\" x=\"26.87\" y=\"-22.8\" font-family=\"Lato\" font-size=\"14.00\">b</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=\"M133.91,-167.12C143.73,-168.42 152.87,-166.29 152.87,-160.74 152.87,-156.57 147.73,-154.34 141.01,-154.03\"/>\n",
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"<text text-anchor=\"start\" x=\"152.87\" y=\"-157.04\" font-family=\"Lato\" font-size=\"14.00\">a</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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" \"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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"<!-- Pages: 1 -->\n",
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"<svg width=\"160pt\" height=\"360pt\"\n",
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" viewBox=\"0.00 0.00 160.15 360.00\" 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=\"66\" y=\"-503.8\" font-family=\"Lato\" font-size=\"14.00\">Inf(</text>\n",
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"<text text-anchor=\"start\" x=\"88\" y=\"-503.8\" font-family=\"Lato\" font-size=\"14.00\" fill=\"#1f78b4\">⓿</text>\n",
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"<text text-anchor=\"start\" x=\"104\" y=\"-503.8\" font-family=\"Lato\" font-size=\"14.00\">)&Inf(</text>\n",
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"<text text-anchor=\"start\" x=\"140\" y=\"-503.8\" font-family=\"Lato\" font-size=\"14.00\" fill=\"#ff4da0\">❶</text>\n",
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"<text text-anchor=\"start\" x=\"156\" y=\"-503.8\" font-family=\"Lato\" font-size=\"14.00\">)</text>\n",
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"<text text-anchor=\"start\" x=\"69\" y=\"-489.8\" font-family=\"Lato\" font-size=\"14.00\">[gen. Büchi 2]</text>\n",
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"<!-- I -->\n",
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"<!-- 0 -->\n",
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"<g id=\"node2\" class=\"node\">\n",
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"<title>0</title>\n",
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"<text text-anchor=\"middle\" x=\"115\" y=\"-422.3\" font-family=\"Lato\" font-size=\"14.00\">0</text>\n",
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"</g>\n",
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"<!-- I->0 -->\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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"<!-- 1 -->\n",
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"<g id=\"node3\" class=\"node\">\n",
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"<title>1</title>\n",
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"<text text-anchor=\"start\" x=\"105\" y=\"-320.3\" font-family=\"Lato\" font-size=\"14.00\">1,3</text>\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=\"M115,-407.58C115,-391.52 115,-367.52 115,-349.52\"/>\n",
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"<polygon fill=\"black\" stroke=\"black\" points=\"115,-342.22 118.15,-349.22 115,-345.72 115,-349.22 115,-349.22 115,-349.22 115,-345.72 111.85,-349.22 115,-342.22 115,-342.22\"/>\n",
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"<text text-anchor=\"start\" x=\"118.5\" y=\"-378.8\" 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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"<polygon fill=\"black\" stroke=\"black\" points=\"138.61,-314.85 145.55,-311.57 142.11,-314.78 145.61,-314.72 145.61,-314.72 145.61,-314.72 142.11,-314.78 145.68,-317.87 138.61,-314.85 138.61,-314.85\"/>\n",
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"<text text-anchor=\"start\" x=\"163.5\" 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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"<ellipse fill=\"#ffffaa\" stroke=\"black\" cx=\"27\" cy=\"-120\" rx=\"27\" ry=\"18\"/>\n",
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"<text text-anchor=\"start\" x=\"17\" y=\"-116.3\" font-family=\"Lato\" font-size=\"14.00\">1,4</text>\n",
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"</g>\n",
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"<!-- 1->2 -->\n",
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"<g id=\"edge4\" class=\"edge\">\n",
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"<title>1->2</title>\n",
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|
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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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"</g>\n",
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|
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"<title>1->3</title>\n",
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"<path fill=\"none\" stroke=\"black\" d=\"M105.26,-306.83C102.32,-301.09 99.48,-294.45 98,-288 95.01,-275.01 95.01,-270.99 98,-258 98.95,-253.87 100.46,-249.66 102.19,-245.65\"/>\n",
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|
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"<text text-anchor=\"start\" x=\"98\" y=\"-269.3\" font-family=\"Lato\" font-size=\"14.00\">!b</text>\n",
|
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"</g>\n",
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|
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"<g id=\"node6\" class=\"node\">\n",
|
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"<title>4</title>\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=\"M131.74,-309.85C149.12,-295.03 175.22,-269.17 186,-240 196.49,-211.61 186.04,-136.28 174,-102 165.7,-78.37 148.67,-55.33 135.2,-39.54\"/>\n",
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"<polygon fill=\"black\" stroke=\"black\" points=\"130.42,-34.07 137.39,-37.27 132.72,-36.7 135.02,-39.34 135.02,-39.34 135.02,-39.34 132.72,-36.7 132.65,-41.41 130.42,-34.07 130.42,-34.07\"/>\n",
|
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|
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|
|
"</g>\n",
|
|
"<!-- 2->2 -->\n",
|
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"<g id=\"edge7\" class=\"edge\">\n",
|
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"<title>2->2</title>\n",
|
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"<path fill=\"none\" stroke=\"black\" d=\"M50.61,-129.15C61.97,-130.28 72,-127.23 72,-120 72,-114.35 65.88,-111.26 57.79,-110.71\"/>\n",
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"<polygon fill=\"black\" stroke=\"black\" points=\"50.61,-110.85 57.55,-107.57 54.11,-110.78 57.61,-110.72 57.61,-110.72 57.61,-110.72 54.11,-110.78 57.68,-113.87 50.61,-110.85 50.61,-110.85\"/>\n",
|
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"<text text-anchor=\"start\" x=\"72\" y=\"-123.8\" font-family=\"Lato\" font-size=\"14.00\">a & b</text>\n",
|
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|
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"<text text-anchor=\"start\" x=\"89\" y=\"-109.8\" font-family=\"Lato\" font-size=\"14.00\" fill=\"#ff4da0\">❶</text>\n",
|
|
"</g>\n",
|
|
"<!-- 2->4 -->\n",
|
|
"<g id=\"edge8\" class=\"edge\">\n",
|
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"<title>2->4</title>\n",
|
|
"<path fill=\"none\" stroke=\"black\" d=\"M24.07,-101.85C22.62,-87.68 22.96,-67.72 33,-54 43.99,-38.98 62.7,-30.36 79.57,-25.44\"/>\n",
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"<polygon fill=\"black\" stroke=\"black\" points=\"86.67,-23.55 80.72,-28.4 83.28,-24.45 79.9,-25.35 79.9,-25.35 79.9,-25.35 83.28,-24.45 79.09,-22.31 86.67,-23.55 86.67,-23.55\"/>\n",
|
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"<text text-anchor=\"start\" x=\"33\" y=\"-72.8\" font-family=\"Lato\" font-size=\"14.00\">a & !b</text>\n",
|
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"<text text-anchor=\"start\" x=\"43.5\" y=\"-57.8\" font-family=\"Lato\" font-size=\"14.00\" fill=\"#ff4da0\">❶</text>\n",
|
|
"</g>\n",
|
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"<!-- 3->1 -->\n",
|
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"<g id=\"edge9\" class=\"edge\">\n",
|
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"<title>3->1</title>\n",
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"<path fill=\"none\" stroke=\"black\" d=\"M115,-240.22C115,-256.21 115,-280.21 115,-298.25\"/>\n",
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"<polygon fill=\"black\" stroke=\"black\" points=\"115,-305.58 111.85,-298.58 115,-302.08 115,-298.58 115,-298.58 115,-298.58 115,-302.08 118.15,-298.58 115,-305.58 115,-305.58\"/>\n",
|
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"<text text-anchor=\"start\" x=\"118.5\" y=\"-276.8\" font-family=\"Lato\" font-size=\"14.00\">b</text>\n",
|
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"<text text-anchor=\"start\" x=\"115\" y=\"-261.8\" font-family=\"Lato\" font-size=\"14.00\" fill=\"#1f78b4\">⓿</text>\n",
|
|
"</g>\n",
|
|
"<!-- 3->2 -->\n",
|
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"<g id=\"edge10\" class=\"edge\">\n",
|
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"<title>3->2</title>\n",
|
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"<path fill=\"none\" stroke=\"black\" d=\"M101.63,-205.81C86.56,-188.68 61.96,-160.72 45.15,-141.62\"/>\n",
|
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"<polygon fill=\"black\" stroke=\"black\" points=\"40.2,-136 47.19,-139.18 42.51,-138.63 44.83,-141.26 44.83,-141.26 44.83,-141.26 42.51,-138.63 42.46,-143.34 40.2,-136 40.2,-136\"/>\n",
|
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"<text text-anchor=\"start\" x=\"85\" y=\"-174.8\" font-family=\"Lato\" font-size=\"14.00\">a & b</text>\n",
|
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"<text text-anchor=\"start\" x=\"86\" y=\"-160.8\" font-family=\"Lato\" font-size=\"14.00\" fill=\"#1f78b4\">⓿</text>\n",
|
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"<text text-anchor=\"start\" x=\"102\" y=\"-160.8\" 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=\"edge11\" class=\"edge\">\n",
|
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"<title>3->3</title>\n",
|
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"<path fill=\"none\" stroke=\"black\" d=\"M141.47,-231.23C153.31,-232.11 163.55,-229.03 163.55,-222 163.55,-216.45 157.17,-213.37 148.66,-212.74\"/>\n",
|
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"<polygon fill=\"black\" stroke=\"black\" points=\"141.47,-212.77 148.45,-209.59 144.97,-212.76 148.47,-212.74 148.47,-212.74 148.47,-212.74 144.97,-212.76 148.48,-215.89 141.47,-212.77 141.47,-212.77\"/>\n",
|
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"<text text-anchor=\"start\" x=\"163.55\" y=\"-218.3\" font-family=\"Lato\" font-size=\"14.00\">!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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"<path fill=\"none\" stroke=\"black\" d=\"M119.51,-204.02C120.85,-198.33 122.18,-191.93 123,-186 131.01,-127.84 137.65,-111.91 128,-54 127.37,-50.2 126.37,-46.26 125.22,-42.45\"/>\n",
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"<text text-anchor=\"start\" x=\"133\" y=\"-123.8\" font-family=\"Lato\" font-size=\"14.00\">a & !b</text>\n",
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"</g>\n",
|
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"<!-- 4->2 -->\n",
|
|
"<g id=\"edge13\" class=\"edge\">\n",
|
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"<title>4->2</title>\n",
|
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"<path fill=\"none\" stroke=\"black\" d=\"M106.33,-35.14C97.7,-48.88 84.26,-68.75 70,-84 64.04,-90.37 56.91,-96.62 50.19,-102.03\"/>\n",
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"<polygon fill=\"black\" stroke=\"black\" points=\"44.67,-106.36 48.23,-99.56 47.43,-104.2 50.18,-102.04 50.18,-102.04 50.18,-102.04 47.43,-104.2 52.12,-104.52 44.67,-106.36 44.67,-106.36\"/>\n",
|
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"<text text-anchor=\"start\" x=\"94\" 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->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=\"M142.47,-27.23C154.31,-28.11 164.55,-25.03 164.55,-18 164.55,-12.45 158.17,-9.37 149.66,-8.74\"/>\n",
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"source": [
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"nba1t = spot.remove_alternation(aut1, True)\n",
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"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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"metadata": {},
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"source": [
|
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"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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"<title>1->0</title>\n",
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"</g>\n",
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"<!-- 1->3 -->\n",
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"<g id=\"edge7\" class=\"edge\">\n",
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"</g>\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=\"M124.95,-25.75C135.92,-26.49 145.41,-23.91 145.41,-18 145.41,-13.48 139.85,-10.9 132.31,-10.28\"/>\n",
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"<text text-anchor=\"start\" x=\"148.91\" y=\"-21.8\" font-family=\"Lato\" font-size=\"14.00\">1</text>\n",
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"</g>\n",
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"<!-- 3->0 -->\n",
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"<g id=\"edge9\" class=\"edge\">\n",
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"<title>3->0</title>\n",
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"<path fill=\"none\" stroke=\"black\" d=\"M124.75,-117.88C133.94,-123.6 143.57,-131.34 149.41,-141 189.18,-206.78 171.09,-216.14 161.41,-243 159.42,-248.53 156.13,-253.81 152.46,-258.51\"/>\n",
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"<text text-anchor=\"start\" x=\"173.41\" y=\"-188.3\" font-family=\"Lato\" font-size=\"14.00\">!a & b & p</text>\n",
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"</g>\n",
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"<!-- 3->1 -->\n",
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"<g id=\"edge10\" class=\"edge\">\n",
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"<title>3->1</title>\n",
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"<text text-anchor=\"start\" x=\"10.41\" y=\"-144.8\" font-family=\"Lato\" font-size=\"14.00\">a & !b & p</text>\n",
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"</g>\n",
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"<!-- 3->2 -->\n",
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"<g id=\"edge11\" class=\"edge\">\n",
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"<title>3->2</title>\n",
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"<path fill=\"none\" stroke=\"black\" d=\"M100.41,-86.8C100.41,-74.36 100.41,-57.36 100.41,-43.5\"/>\n",
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"<text text-anchor=\"start\" x=\"100.41\" y=\"-57.8\" 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=\"edge12\" class=\"edge\">\n",
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"<title>3->3</title>\n",
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"<text text-anchor=\"start\" x=\"152.86\" y=\"-101.3\" font-family=\"Lato\" font-size=\"14.00\">a & b & p</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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" viewBox=\"0.00 0.00 237.28 360.00\" 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=\"222.74\" y=\"-662.8\" font-family=\"Lato\" font-size=\"14.00\" fill=\"#1f78b4\">⓿</text>\n",
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"</g>\n",
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"<!-- I->0 -->\n",
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"<g id=\"edge1\" class=\"edge\">\n",
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"<title>I->0</title>\n",
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"<path fill=\"none\" stroke=\"black\" d=\"M197.49,-639.85C197.49,-638.21 197.49,-623.85 197.49,-610.37\"/>\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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"<text text-anchor=\"start\" x=\"184.49\" y=\"-479.3\" font-family=\"Lato\" font-size=\"14.00\">~0,4</text>\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=\"M187.37,-568.25C184.19,-562.43 181.09,-555.64 179.49,-549 176.35,-536.04 176.2,-531.92 179.49,-519 180.58,-514.69 182.32,-510.34 184.32,-506.23\"/>\n",
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"<text text-anchor=\"start\" x=\"183.99\" y=\"-537.8\" font-family=\"Lato\" font-size=\"14.00\">a</text>\n",
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"</g>\n",
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"<!-- 1->0 -->\n",
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"<g id=\"edge3\" class=\"edge\">\n",
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"<title>1->0</title>\n",
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"<path fill=\"none\" stroke=\"black\" d=\"M198.31,-501.22C198.15,-517.21 197.91,-541.21 197.73,-559.25\"/>\n",
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"<text text-anchor=\"start\" x=\"199.99\" y=\"-537.8\" font-family=\"Lato\" font-size=\"14.00\">!a</text>\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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"<ellipse fill=\"#ffffaa\" stroke=\"black\" cx=\"198.49\" cy=\"-381\" rx=\"28.7\" ry=\"18\"/>\n",
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"<text text-anchor=\"start\" x=\"184.49\" y=\"-377.3\" font-family=\"Lato\" font-size=\"14.00\">3,~1</text>\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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"<path fill=\"none\" stroke=\"black\" d=\"M198.49,-464.58C198.49,-448.52 198.49,-424.52 198.49,-406.52\"/>\n",
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"<text text-anchor=\"start\" x=\"198.49\" y=\"-428.3\" font-family=\"Lato\" font-size=\"14.00\">a</text>\n",
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"</g>\n",
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"<!-- 2->3 -->\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=\"M182.93,-365.83C176.4,-359.64 168.86,-352.17 162.49,-345 151.32,-332.44 152.8,-325.24 139.49,-315 126.67,-305.14 110.68,-297.57 96.1,-292.04\"/>\n",
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"<polygon fill=\"black\" stroke=\"black\" points=\"89.52,-289.64 97.17,-289.08 92.81,-290.84 96.09,-292.04 96.09,-292.04 96.09,-292.04 92.81,-290.84 95.01,-294.99 89.52,-289.64 89.52,-289.64\"/>\n",
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"<text text-anchor=\"start\" x=\"162.49\" y=\"-326.3\" font-family=\"Lato\" font-size=\"14.00\">a & b & p</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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"</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=\"M211.93,-364.69C216.17,-358.92 220.31,-352.04 222.49,-345 226.43,-332.26 222.56,-328.33 222.49,-315 222.04,-237.66 243.81,-214.74 220.49,-141 219.05,-136.47 216.82,-132 214.28,-127.85\"/>\n",
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"<polygon fill=\"black\" stroke=\"black\" points=\"210.39,-122.02 216.9,-126.1 212.34,-124.93 214.28,-127.85 214.28,-127.85 214.28,-127.85 212.34,-124.93 211.66,-129.59 210.39,-122.02 210.39,-122.02\"/>\n",
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"<text text-anchor=\"start\" x=\"229.49\" 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=\"M53.11,-297.15C51.83,-310.44 51.27,-329.19 55.49,-345 79.41,-434.79 147.67,-524.71 180.04,-563.77\"/>\n",
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"<polygon fill=\"black\" stroke=\"black\" points=\"184.55,-569.16 177.64,-565.81 182.3,-566.47 180.06,-563.79 180.06,-563.79 180.06,-563.79 182.3,-566.47 182.47,-561.77 184.55,-569.16 184.55,-569.16\"/>\n",
|
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"<text text-anchor=\"start\" x=\"99.49\" y=\"-435.8\" font-family=\"Lato\" font-size=\"14.00\">!a & !b</text>\n",
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|
"</g>\n",
|
|
"<!-- 3->2 -->\n",
|
|
"<g id=\"edge8\" class=\"edge\">\n",
|
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"<title>3->2</title>\n",
|
|
"<path fill=\"none\" stroke=\"black\" d=\"M55.16,-297.2C55.84,-311.8 59.17,-332.34 71.49,-345 95.23,-369.41 134.31,-377.26 162.86,-379.56\"/>\n",
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"<polygon fill=\"black\" stroke=\"black\" points=\"169.87,-380.04 162.68,-382.71 166.38,-379.8 162.89,-379.57 162.89,-379.57 162.89,-379.57 166.38,-379.8 163.1,-376.42 169.87,-380.04 169.87,-380.04\"/>\n",
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"<text text-anchor=\"start\" x=\"71.49\" y=\"-326.3\" font-family=\"Lato\" font-size=\"14.00\">a & !b & p</text>\n",
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"</g>\n",
|
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"<!-- 5 -->\n",
|
|
"<g id=\"node7\" class=\"node\">\n",
|
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"<title>5</title>\n",
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"<ellipse fill=\"#ffffaa\" stroke=\"black\" cx=\"109.49\" cy=\"-18\" rx=\"28.7\" ry=\"18\"/>\n",
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"<text text-anchor=\"start\" x=\"95.49\" y=\"-14.3\" font-family=\"Lato\" font-size=\"14.00\">3,~0</text>\n",
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"</g>\n",
|
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"<!-- 3->5 -->\n",
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"<g id=\"edge9\" class=\"edge\">\n",
|
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"<title>3->5</title>\n",
|
|
"<path fill=\"none\" stroke=\"black\" d=\"M41.8,-261.97C21.69,-236.4 -11.76,-184.78 4.49,-141 20.76,-97.16 60.62,-58.5 86.28,-36.99\"/>\n",
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"<polygon fill=\"black\" stroke=\"black\" points=\"91.85,-32.4 88.45,-39.29 89.14,-34.63 86.44,-36.86 86.44,-36.86 86.44,-36.86 89.14,-34.63 84.44,-34.42 91.85,-32.4 91.85,-32.4\"/>\n",
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"<text text-anchor=\"start\" x=\"4.49\" 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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"<ellipse fill=\"#ffffaa\" stroke=\"black\" cx=\"97.49\" cy=\"-192\" rx=\"41.69\" ry=\"18\"/>\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=\"M80.33,-264.31C87.91,-258.76 95.33,-251.62 99.49,-243 103.32,-235.05 103.8,-225.54 102.98,-216.96\"/>\n",
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"<!-- 4->0 -->\n",
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"<g id=\"edge11\" class=\"edge\">\n",
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"<title>4->0</title>\n",
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"</g>\n",
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"<title>1->3</title>\n",
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"<title>2->2</title>\n",
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"<title>3->0</title>\n",
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"</g>\n",
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"</svg>\n",
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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 demonstrate that very weak (non-alternating) Büchi automata can be complemented via alternation removal."
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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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"</g>\n",
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"</g>\n",
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"<!-- T5T4 -->\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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" viewBox=\"0.00 0.00 280.08 360.00\" 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 id=\"edge2\" class=\"edge\">\n",
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"<title>0->1</title>\n",
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"<path fill=\"none\" stroke=\"black\" d=\"M61.03,-392.38C52.5,-375.64 38.89,-348.96 29.26,-330.08\"/>\n",
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"<title>0->2</title>\n",
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"</g>\n",
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"<!-- 1->1 -->\n",
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"<title>1->1</title>\n",
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|
"</g>\n",
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|
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|
|
"<g id=\"node5\" class=\"node\">\n",
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"<title>3</title>\n",
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"<text text-anchor=\"middle\" x=\"114\" y=\"-14.3\" font-family=\"Lato\" font-size=\"14.00\">3</text>\n",
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|
"</g>\n",
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"<g id=\"edge5\" class=\"edge\">\n",
|
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"<title>2->3</title>\n",
|
|
"<path fill=\"none\" stroke=\"black\" d=\"M111.58,-292.26C107.23,-286.02 102.4,-278.4 99,-271 82.21,-234.44 79.61,-223.84 74,-184 65.82,-125.9 77.48,-109.33 97,-54 98.53,-49.65 100.46,-45.13 102.46,-40.85\"/>\n",
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"<text text-anchor=\"start\" x=\"74\" y=\"-172.8\" font-family=\"Lato\" font-size=\"14.00\">a & 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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"<text text-anchor=\"middle\" x=\"168\" y=\"-114.3\" font-family=\"Lato\" font-size=\"14.00\">4</text>\n",
|
|
"</g>\n",
|
|
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|
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"<g id=\"edge6\" class=\"edge\">\n",
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"<title>2->4</title>\n",
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|
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|
"</g>\n",
|
|
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|
|
"<g id=\"node7\" class=\"node\">\n",
|
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"<title>5</title>\n",
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|
"</g>\n",
|
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"<g id=\"edge7\" class=\"edge\">\n",
|
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"<title>2->5</title>\n",
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"</g>\n",
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"<g id=\"node8\" class=\"node\">\n",
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"<title>6</title>\n",
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"</g>\n",
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"<title>2->6</title>\n",
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"<path fill=\"none\" stroke=\"black\" d=\"M130.22,-290.8C137.48,-277.39 148.15,-257.67 156.3,-242.62\"/>\n",
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"<polygon fill=\"black\" stroke=\"black\" points=\"159.85,-236.06 159.29,-243.72 158.18,-239.14 156.52,-242.22 156.52,-242.22 156.52,-242.22 158.18,-239.14 153.75,-240.72 159.85,-236.06 159.85,-236.06\"/>\n",
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"<text text-anchor=\"start\" x=\"147\" y=\"-259.8\" 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=\"edge9\" class=\"edge\">\n",
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"<title>3->3</title>\n",
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"<path fill=\"none\" stroke=\"black\" d=\"M129.92,-26.63C140.15,-28.89 150,-26.01 150,-18 150,-11.87 144.23,-8.75 136.93,-8.63\"/>\n",
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"<polygon fill=\"black\" stroke=\"black\" points=\"129.92,-9.37 136.55,-5.5 133.4,-9 136.88,-8.63 136.88,-8.63 136.88,-8.63 133.4,-9 137.21,-11.77 129.92,-9.37 129.92,-9.37\"/>\n",
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"<text text-anchor=\"start\" x=\"161.5\" y=\"-21.8\" font-family=\"Lato\" font-size=\"14.00\">1</text>\n",
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"<text text-anchor=\"start\" x=\"150\" y=\"-7.8\" font-family=\"Lato\" font-size=\"14.00\" fill=\"#1f78b4\">⓿</text>\n",
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"<text text-anchor=\"start\" x=\"166\" y=\"-7.8\" font-family=\"Lato\" font-size=\"14.00\" fill=\"#ff4da0\">❶</text>\n",
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"</g>\n",
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"<!-- 4->3 -->\n",
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"<g id=\"edge10\" class=\"edge\">\n",
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"<title>4->3</title>\n",
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"<path fill=\"none\" stroke=\"black\" d=\"M159.56,-101.69C150.53,-85.29 136.12,-59.14 125.92,-40.64\"/>\n",
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"<polygon fill=\"black\" stroke=\"black\" points=\"122.38,-34.2 128.52,-38.82 124.07,-37.27 125.76,-40.34 125.76,-40.34 125.76,-40.34 124.07,-37.27 123,-41.86 122.38,-34.2 122.38,-34.2\"/>\n",
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"<text text-anchor=\"start\" x=\"159.5\" y=\"-71.8\" font-family=\"Lato\" font-size=\"14.00\">a</text>\n",
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"<text text-anchor=\"start\" x=\"147\" y=\"-57.8\" font-family=\"Lato\" font-size=\"14.00\" fill=\"#1f78b4\">⓿</text>\n",
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"<text text-anchor=\"start\" x=\"163\" y=\"-57.8\" 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=\"edge11\" class=\"edge\">\n",
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"<title>4->4</title>\n",
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"<path fill=\"none\" stroke=\"black\" d=\"M185.78,-122.49C195.31,-123.26 204,-121.76 204,-118 204,-115.24 199.32,-113.7 193.05,-113.38\"/>\n",
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"<polygon fill=\"black\" stroke=\"black\" points=\"185.78,-113.51 192.72,-110.23 189.28,-113.44 192.78,-113.38 192.78,-113.38 192.78,-113.38 189.28,-113.44 192.84,-116.53 185.78,-113.51 185.78,-113.51\"/>\n",
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"<text text-anchor=\"start\" x=\"204\" y=\"-121.8\" font-family=\"Lato\" font-size=\"14.00\">!a & !b</text>\n",
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"<text text-anchor=\"start\" x=\"216.5\" 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=\"edge12\" class=\"edge\">\n",
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"<title>4->4</title>\n",
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"<path fill=\"none\" stroke=\"black\" d=\"M185.02,-124.49C209.35,-129.88 245,-127.72 245,-118 245,-109.19 215.72,-106.59 192.12,-110.19\"/>\n",
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"<polygon fill=\"black\" stroke=\"black\" points=\"185.02,-111.51 191.33,-107.14 188.46,-110.87 191.9,-110.23 191.9,-110.23 191.9,-110.23 188.46,-110.87 192.47,-113.33 185.02,-111.51 185.02,-111.51\"/>\n",
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"<text text-anchor=\"start\" x=\"245\" y=\"-121.8\" font-family=\"Lato\" font-size=\"14.00\">!a & b</text>\n",
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"<text text-anchor=\"start\" x=\"247.5\" y=\"-107.8\" font-family=\"Lato\" font-size=\"14.00\" fill=\"#1f78b4\">⓿</text>\n",
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"<text text-anchor=\"start\" x=\"263.5\" 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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"<!-- 5->3 -->\n",
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"<g id=\"edge13\" class=\"edge\">\n",
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"<title>5->3</title>\n",
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"<path fill=\"none\" stroke=\"black\" d=\"M321.41,-107.19C317.46,-104.71 313.13,-102.14 309,-100 249.34,-69.1 175.15,-40.93 137.82,-27.42\"/>\n",
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"<polygon fill=\"black\" stroke=\"black\" points=\"130.86,-24.91 138.51,-24.32 134.16,-26.1 137.45,-27.28 137.45,-27.28 137.45,-27.28 134.16,-26.1 136.38,-30.25 130.86,-24.91 130.86,-24.91\"/>\n",
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"<text text-anchor=\"start\" x=\"278.5\" y=\"-71.8\" font-family=\"Lato\" font-size=\"14.00\">b</text>\n",
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"<text text-anchor=\"start\" x=\"267\" y=\"-57.8\" font-family=\"Lato\" font-size=\"14.00\" fill=\"#1f78b4\">⓿</text>\n",
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"<text text-anchor=\"start\" x=\"283\" y=\"-57.8\" font-family=\"Lato\" font-size=\"14.00\" fill=\"#ff4da0\">❶</text>\n",
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"</g>\n",
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"<!-- 5->5 -->\n",
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"<g id=\"edge14\" class=\"edge\">\n",
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"<title>5->5</title>\n",
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"<path fill=\"none\" stroke=\"black\" d=\"M351.92,-126.63C362.15,-128.89 372,-126.01 372,-118 372,-111.87 366.23,-108.75 358.93,-108.63\"/>\n",
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"<polygon fill=\"black\" stroke=\"black\" points=\"351.92,-109.37 358.55,-105.5 355.4,-109 358.88,-108.63 358.88,-108.63 358.88,-108.63 355.4,-109 359.21,-111.77 351.92,-109.37 351.92,-109.37\"/>\n",
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"<text text-anchor=\"start\" x=\"373.5\" y=\"-121.8\" font-family=\"Lato\" font-size=\"14.00\">!b</text>\n",
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"<text text-anchor=\"start\" x=\"372\" y=\"-106.8\" font-family=\"Lato\" font-size=\"14.00\" fill=\"#ff4da0\">❶</text>\n",
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"</g>\n",
|
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"<!-- 6->3 -->\n",
|
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"<g id=\"edge15\" class=\"edge\">\n",
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"<title>6->3</title>\n",
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"<path fill=\"none\" stroke=\"black\" d=\"M155.18,-206.97C140.26,-192.01 116.47,-164.89 107,-136 96.77,-104.81 102.21,-66.47 107.64,-42.38\"/>\n",
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"<polygon fill=\"black\" stroke=\"black\" points=\"109.3,-35.38 110.75,-42.92 108.49,-38.79 107.68,-42.19 107.68,-42.19 107.68,-42.19 108.49,-38.79 104.62,-41.46 109.3,-35.38 109.3,-35.38\"/>\n",
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"<text text-anchor=\"start\" x=\"107\" y=\"-121.8\" font-family=\"Lato\" font-size=\"14.00\">a & b</text>\n",
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"</g>\n",
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"<!-- 6->4 -->\n",
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"<g id=\"edge16\" class=\"edge\">\n",
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"<title>6->4</title>\n",
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"<path fill=\"none\" stroke=\"black\" d=\"M168,-201.58C168,-185.52 168,-161.52 168,-143.52\"/>\n",
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"<polygon fill=\"black\" stroke=\"black\" points=\"168,-136.22 171.15,-143.22 168,-139.72 168,-143.22 168,-143.22 168,-143.22 168,-139.72 164.85,-143.22 168,-136.22 168,-136.22\"/>\n",
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"<text text-anchor=\"start\" x=\"168\" y=\"-172.8\" font-family=\"Lato\" font-size=\"14.00\">!a & b</text>\n",
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"</g>\n",
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"<!-- 6->5 -->\n",
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"<g id=\"edge17\" class=\"edge\">\n",
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"<title>6->5</title>\n",
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"<path fill=\"none\" stroke=\"black\" d=\"M181.04,-207.53C197.69,-193.35 227.84,-169.26 257,-154 278.67,-142.66 286.95,-146.58 309,-136 311.1,-134.99 313.23,-133.86 315.34,-132.67\"/>\n",
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"<text text-anchor=\"start\" x=\"257\" y=\"-172.8\" font-family=\"Lato\" font-size=\"14.00\">a & !b</text>\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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"<path fill=\"none\" stroke=\"black\" d=\"M184.66,-227.38C194.62,-229.02 204,-226.56 204,-220 204,-215.08 198.73,-212.46 191.89,-212.16\"/>\n",
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"<polygon fill=\"black\" stroke=\"black\" points=\"184.66,-212.62 191.45,-209.03 188.16,-212.39 191.65,-212.17 191.65,-212.17 191.65,-212.17 188.16,-212.39 191.85,-215.31 184.66,-212.62 184.66,-212.62\"/>\n",
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"<text text-anchor=\"start\" x=\"204\" y=\"-216.3\" 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>"
|
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],
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"text/plain": [
|
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"<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",
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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": "3.7.5"
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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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