104a372c41
* wrap/python/tests/decompose.ipynb: New file. * wrap/python/tests/Makefile.am: Add it. * src/twaalgos/strength.cc: Fix corner cases. * src/tests/strength.test: Adjust corner case. * NEWS, doc/org/tut.org: Mention the notebook.
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4800 lines
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"input": [
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"from IPython.display import display\n",
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"import spot\n",
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"spot.setup()"
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],
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"language": "python",
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"This notebook demonstrates how to use the `strength_decompose()` function to split an automaton in up to three automata capturing different behaviors. This is based on the paper [Strength-based decomposition of the property B\u00fcchi automaton for faster model checking](https://www.lrde.epita.fr/~adl/dl/adl/renault.13.tacas.pdf) (TACAS'13).\n",
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"\n",
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"# Basics\n",
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"\n",
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"Let's define the following strengths of SCCs:\n",
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"- an accepting SCC is **weak** if all its transitions belong to the same acceptance sets\n",
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"- an accepting SCC is **terminal** if it is *weak* and complete (i.e. from any state, you can stay in the SCC by reading any word)\n",
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"- an accepting SCC is **strictly weak** if it is *weak* and not complete (in other words: *weak* but not *terminal*)\n",
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"- an accepting SCC is **strong** if it is not weak.\n",
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"\n",
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"The strengths **strong**, **stricly weak**, and **terminal** define a partition of all accepting SCCs. The following B\u00fcchi automaton has 4 SCCs, and its 3 accepting SCCs show an example of each strength."
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"The `strength_decompose()` function takes an automaton, and a string specifying which strength to preserve. \n",
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"\n",
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"The letters used for this specification are as follows:\n",
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"- `t`: terminal\n",
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"- `w`: strictly weak\n",
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"- `s`: strong\n",
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"For instance if we want to preserve only the **strictly weak** part of this automaton, we should get only the SCC with the self-loop on $b$, and the SCC above it so that we can reach it. However the SCC above is not stricly weak, so it should not accept any word in the new automaton."
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"source": [
|
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"Similarly, we can extract all the behaviors captured by the **terminal** part of the automaton:"
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]
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"term = spot.decompose_strength(aut, 't')\n",
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"term.show('.sa')"
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],
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"language": "python",
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"metadata": {},
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"outputs": [
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{
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"metadata": {},
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"svg": [
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"</g>\n",
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"<!-- 1->2 -->\n",
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"<g class=\"edge\" id=\"edge7\"><title>1->2</title>\n",
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"</g>\n",
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"<g class=\"edge\" id=\"edge8\"><title>1->3</title>\n",
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],
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"text": [
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"<IPython.core.display.SVG object>"
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]
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}
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],
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
|
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"Here is the strong part:"
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|
]
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"strong = spot.decompose_strength(aut, 's')\n",
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"strong.show('.sa')"
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],
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"language": "python",
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|
"metadata": {},
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"outputs": [
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{
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"metadata": {},
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"output_type": "pyout",
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"prompt_number": 5,
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"svg": [
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"The union of these three automata recognize the same language as the original automaton.\n",
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"\n",
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"\n",
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"The application proposed in the aforementioned TACAS'13 paper is for parallelizing model checking. Instead of testing the emptiness of the product between `aut` and a system, one could test the emptiness **3** products in parallel: each with a sub-automaton of different strength. Model checking using weak and terminal automata can be done with much more simpler emptiness check procedures than needed for the general case. So in effect, we have isolated the \"hard\" (i.e. strong) part of the original automaton in a smaller automaton, and we only need to use a full-fledged emptiness check for this case.\n",
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"\n",
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"An additional bonus is that it is possible that the simplification algorithms will do a better job at simplifying the sub-automata than at simplifying the original `aut`. For instance here the `strong` automaton can be further simplified:"
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"# Multi-strength extraction\n",
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"\n",
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"The string passed to `decompose_strength()` can include multiple letters to extract multiple strengths at once."
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]
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"for opt in ('sw', 'st', 'wt'):\n",
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" a = spot.decompose_strength(aut, opt)\n",
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" a.set_name(\"option: \" + opt)\n",
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" display(a.show('.asn'))"
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"</g>\n",
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"<g class=\"edge\" id=\"edge5\"><title>1->0</title>\n",
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"</svg>"
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],
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"text": [
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]
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},
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{
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"metadata": {},
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"svg": [
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"<g class=\"edge\" id=\"edge1\"><title>I->0</title>\n",
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"</g>\n",
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"</svg>"
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],
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"text": [
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"<IPython.core.display.SVG object>"
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]
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}
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],
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"prompt_number": 7
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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": [
|
|
"# Generalized acceptance\n",
|
|
"\n",
|
|
"There is (almost, see below) nothing that prevents the above decomposition to work with other types of acceptance.\n",
|
|
"\n",
|
|
"## Rabin\n",
|
|
"\n",
|
|
"The following Rabin automaton was generated with\n",
|
|
"\n",
|
|
" ltldo -f '(Ga -> Gb) W c' 'ltl2dstar --ltl2nba=spin:ltl2tgba@-Ds' -H | autfilt -H --merge-transitions\n",
|
|
" \n",
|
|
"(The `autfilt -H --merge-transitions` pass is just here to reduce the size of the file and make the automaton more readable.)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"collapsed": false,
|
|
"input": [
|
|
"aut = spot.automaton(\"\"\"\n",
|
|
"HOA: v1\n",
|
|
"States: 9\n",
|
|
"Start: 2\n",
|
|
"AP: 3 \"a\" \"b\" \"c\"\n",
|
|
"acc-name: Rabin 2\n",
|
|
"Acceptance: 4 (Fin(0) & Inf(1)) | (Fin(2) & Inf(3))\n",
|
|
"properties: trans-labels explicit-labels state-acc complete\n",
|
|
"properties: deterministic\n",
|
|
"--BODY--\n",
|
|
"State: 0 {2}\n",
|
|
"[0&!2] 0\n",
|
|
"[0&2] 1\n",
|
|
"[!0&!2] 5\n",
|
|
"[!0&2] 6\n",
|
|
"State: 1 {2}\n",
|
|
"[0] 1\n",
|
|
"[!0] 6\n",
|
|
"State: 2 {2}\n",
|
|
"[0&!1&!2] 3\n",
|
|
"[0&1&!2] 4\n",
|
|
"[!0&!2] 5\n",
|
|
"[2] 6\n",
|
|
"State: 3 {1 2}\n",
|
|
"[0&!2] 0\n",
|
|
"[0&2] 1\n",
|
|
"[!0&!2] 5\n",
|
|
"[!0&2] 6\n",
|
|
"State: 4 {1 2}\n",
|
|
"[0&!1&!2] 0\n",
|
|
"[0&!1&2] 1\n",
|
|
"[!0&!2] 5\n",
|
|
"[!0&2] 6\n",
|
|
"[0&1&!2] 7\n",
|
|
"[0&1&2] 8\n",
|
|
"State: 5 {1 2}\n",
|
|
"[0&!1&!2] 0\n",
|
|
"[!0&!2] 5\n",
|
|
"[2] 6\n",
|
|
"[0&1&!2] 7\n",
|
|
"State: 6 {1 2}\n",
|
|
"[t] 6\n",
|
|
"State: 7 {3}\n",
|
|
"[0&!1&!2] 0\n",
|
|
"[0&!1&2] 1\n",
|
|
"[!0&!2] 5\n",
|
|
"[!0&2] 6\n",
|
|
"[0&1&!2] 7\n",
|
|
"[0&1&2] 8\n",
|
|
"State: 8 {3}\n",
|
|
"[0&!1] 1\n",
|
|
"[!0] 6\n",
|
|
"[0&1] 8\n",
|
|
"--END--\n",
|
|
"\"\"\")\n",
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|
"aut.show(\".as\")"
|
|
],
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"language": "python",
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"metadata": {},
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"outputs": [
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{
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"metadata": {},
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"output_type": "pyout",
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"prompt_number": 8,
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"svg": [
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"<g class=\"graph\" id=\"graph0\" transform=\"scale(0.576923 0.576923) rotate(0) translate(4 620)\">\n",
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"<!-- I->2 -->\n",
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"<g class=\"edge\" id=\"edge1\"><title>I->2</title>\n",
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"<!-- 2->5 -->\n",
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"<g class=\"edge\" id=\"edge10\"><title>2->5</title>\n",
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"<!-- 2->3 -->\n",
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"<g class=\"edge\" id=\"edge8\"><title>2->3</title>\n",
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"<path d=\"M886.779,-270.297C901.227,-279.719 920.436,-292.245 936.826,-302.933\" fill=\"none\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"908.725\" y=\"-294.8\">!a</text>\n",
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"</g>\n",
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"<!-- 1->1 -->\n",
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"<g class=\"edge\" id=\"edge6\"><title>1->1</title>\n",
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"<path d=\"M854.562,-281.37C853.921,-291.924 857.019,-300.87 863.855,-300.87 869.089,-300.87 872.131,-295.626 872.982,-288.431\" fill=\"none\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"860.355\" y=\"-304.67\">a</text>\n",
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"</g>\n",
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"<!-- 8 -->\n",
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"<g class=\"node\" id=\"node5\"><title>8</title>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"859.355\" y=\"-46.8\">8</text>\n",
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"<text fill=\"#b276b2\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"855.855\" y=\"-31.8\">\u2778</text>\n",
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"</g>\n",
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"<!-- 8->6 -->\n",
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"<g class=\"edge\" id=\"edge34\"><title>8->6</title>\n",
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"<path d=\"M885.194,-59.8794C896.985,-70.7349 911.152,-85.8048 919.725,-102 950.677,-160.469 962.917,-237.273 967.692,-283.79\" fill=\"none\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"908.725\" y=\"-105.8\">!a</text>\n",
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"</g>\n",
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"<!-- 8->1 -->\n",
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"<g class=\"edge\" id=\"edge33\"><title>8->1</title>\n",
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"<path d=\"M847.171,-64.7811C827.002,-92.619 794.184,-143.137 781.485,-192.5 779.824,-198.956 778.176,-201.712 781.485,-207.5 792.194,-226.234 813.276,-238.449 831.429,-245.935\" fill=\"none\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"769.485\" y=\"-196.3\">a & !b</text>\n",
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"</g>\n",
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"<!-- 8->8 -->\n",
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"<g class=\"edge\" id=\"edge35\"><title>8->8</title>\n",
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"<path d=\"M854.562,-68.37C853.921,-78.9238 857.019,-87.8701 863.855,-87.8701 869.089,-87.8701 872.131,-82.6259 872.982,-75.4312\" fill=\"none\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"846.855\" y=\"-91.6701\">a & b</text>\n",
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"</g>\n",
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"<!-- 0 -->\n",
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"<g class=\"node\" id=\"node6\"><title>0</title>\n",
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"<ellipse cx=\"390.492\" cy=\"-317\" fill=\"#ffffaa\" rx=\"26.7407\" ry=\"26.7407\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"385.992\" y=\"-320.8\">0</text>\n",
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"<text fill=\"#faa43a\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"382.492\" y=\"-305.8\">\u2777</text>\n",
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"</g>\n",
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"<!-- 0->6 -->\n",
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"<g class=\"edge\" id=\"edge5\"><title>0->6</title>\n",
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"<path d=\"M405.588,-339.236C430.449,-375.408 485.801,-443 552.304,-443 552.304,-443 552.304,-443 864.855,-443 908.58,-443 938.792,-397.534 955.419,-363.046\" fill=\"none\" stroke=\"black\"/>\n",
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"<polygon fill=\"black\" points=\"958.563,-356.312 958.456,-363.987 957.082,-359.483 955.602,-362.655 955.602,-362.655 955.602,-362.655 957.082,-359.483 952.747,-361.322 958.563,-356.312 958.563,-356.312\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"694.115\" y=\"-446.8\">!a & c</text>\n",
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"</g>\n",
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"<!-- 0->1 -->\n",
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"<g class=\"edge\" id=\"edge3\"><title>0->1</title>\n",
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"<path d=\"M414.107,-330.764C420.892,-334.795 428.385,-339.151 435.362,-343 472.087,-363.257 478.59,-376.171 519.362,-386 548.692,-393.071 557.636,-391.793 587.245,-386 684.636,-366.944 787.277,-306.018 835.186,-274.812\" fill=\"none\" stroke=\"black\"/>\n",
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"<polygon fill=\"black\" points=\"841.303,-270.795 837.181,-277.27 838.377,-272.716 835.452,-274.637 835.452,-274.637 835.452,-274.637 838.377,-272.716 833.722,-272.004 841.303,-270.795 841.303,-270.795\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"620.245\" y=\"-385.8\">a & c</text>\n",
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"</g>\n",
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"<!-- 0->0 -->\n",
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"<g class=\"edge\" id=\"edge2\"><title>0->0</title>\n",
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"<path d=\"M376.663,-340.087C374.75,-351.6 379.36,-361.87 390.492,-361.87 399.19,-361.87 403.906,-355.602 404.641,-347.369\" fill=\"none\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"372.492\" y=\"-365.67\">a & !c</text>\n",
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"</g>\n",
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"<!-- 0->5 -->\n",
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"<g class=\"edge\" id=\"edge4\"><title>0->5</title>\n",
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"<path d=\"M416.844,-310.208C422.87,-308.876 429.308,-307.681 435.362,-307 460.857,-304.131 489.667,-304.824 512.312,-306.262\" fill=\"none\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"448.362\" y=\"-310.8\">!a & !c</text>\n",
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"</g>\n",
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"<!-- 5->6 -->\n",
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"<g class=\"edge\" id=\"edge24\"><title>5->6</title>\n",
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"<path d=\"M585.837,-319.983C592.202,-321.623 598.89,-323.085 605.245,-324 734.36,-342.596 768.306,-330.718 898.725,-328 909.083,-327.784 920.274,-327.394 930.673,-326.964\" fill=\"none\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"784.485\" y=\"-337.8\">c</text>\n",
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"</g>\n",
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"<!-- 5->0 -->\n",
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"<g class=\"edge\" id=\"edge22\"><title>5->0</title>\n",
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"<path d=\"M520.438,-318.696C514.161,-320.071 507.59,-321.276 501.362,-322 472.225,-325.386 464.649,-323.664 435.362,-322 431.818,-321.799 428.12,-321.507 424.441,-321.163\" fill=\"none\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"435.362\" y=\"-327.8\">a & !b & !c</text>\n",
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"</g>\n",
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"<!-- 5->5 -->\n",
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"<g class=\"edge\" id=\"edge23\"><title>5->5</title>\n",
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"<path d=\"M537.989,-340.41C537.308,-352.168 542.413,-361.941 553.304,-361.941 561.812,-361.941 566.789,-355.976 568.235,-347.76\" fill=\"none\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"533.304\" y=\"-365.741\">!a & !c</text>\n",
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"</g>\n",
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"<!-- 7 -->\n",
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"<g class=\"node\" id=\"node8\"><title>7</title>\n",
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"<ellipse cx=\"712.115\" cy=\"-252\" fill=\"#ffffaa\" rx=\"26.7407\" ry=\"26.7407\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"707.615\" y=\"-255.8\">7</text>\n",
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"<text fill=\"#b276b2\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"704.115\" y=\"-240.8\">\u2778</text>\n",
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"</g>\n",
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"<!-- 5->7 -->\n",
|
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"<g class=\"edge\" id=\"edge25\"><title>5->7</title>\n",
|
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"<path d=\"M587.223,-307.418C610.467,-304.627 641.865,-298.86 667.245,-287 673.865,-283.906 680.372,-279.621 686.263,-275.108\" fill=\"none\" stroke=\"black\"/>\n",
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"<polygon fill=\"black\" points=\"691.931,-270.555 688.446,-277.395 689.202,-272.747 686.473,-274.939 686.473,-274.939 686.473,-274.939 689.202,-272.747 684.501,-272.484 691.931,-270.555 691.931,-270.555\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"605.245\" y=\"-308.8\">a & b & !c</text>\n",
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"</g>\n",
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"<!-- 7->6 -->\n",
|
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"<g class=\"edge\" id=\"edge30\"><title>7->6</title>\n",
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"<path d=\"M723.736,-227.525C730.987,-214.105 742.01,-198.676 756.985,-191 813.214,-162.179 846.198,-166.882 898.725,-202 928.236,-221.73 947.267,-257.988 958.294,-286.045\" fill=\"none\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"845.855\" y=\"-205.8\">!a & c</text>\n",
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"</g>\n",
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"<!-- 7->1 -->\n",
|
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"<g class=\"edge\" id=\"edge28\"><title>7->1</title>\n",
|
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"<path d=\"M739.198,-252.697C764.35,-253.369 802.514,-254.388 829.787,-255.117\" fill=\"none\" stroke=\"black\"/>\n",
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|
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"756.985\" y=\"-258.8\">a & !b & c</text>\n",
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"</g>\n",
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"<!-- 7->8 -->\n",
|
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"<g class=\"edge\" id=\"edge32\"><title>7->8</title>\n",
|
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"<path d=\"M720.839,-226.238C728.129,-204.294 740.415,-172.516 756.985,-148 779.396,-114.841 813.352,-83.4366 836.734,-63.7617\" fill=\"none\" stroke=\"black\"/>\n",
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|
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"758.985\" y=\"-151.8\">a & b & c</text>\n",
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"</g>\n",
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"<!-- 7->0 -->\n",
|
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"<g class=\"edge\" id=\"edge27\"><title>7->0</title>\n",
|
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"<path d=\"M685.502,-248.089C648.268,-243.276 577.308,-237.402 519.362,-252 482.63,-261.254 444.405,-282.459 419.429,-298.146\" fill=\"none\" stroke=\"black\"/>\n",
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|
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"520.304\" y=\"-255.8\">a & !b & !c</text>\n",
|
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"</g>\n",
|
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"<!-- 7->5 -->\n",
|
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"<g class=\"edge\" id=\"edge29\"><title>7->5</title>\n",
|
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"<path d=\"M685.23,-252.094C663.105,-253.05 630.877,-256.561 605.245,-268 597.558,-271.43 590.089,-276.349 583.346,-281.619\" fill=\"none\" stroke=\"black\"/>\n",
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"<polygon fill=\"black\" points=\"577.863,-286.105 581.287,-279.234 580.572,-283.888 583.281,-281.672 583.281,-281.672 583.281,-281.672 580.572,-283.888 585.276,-284.11 577.863,-286.105 577.863,-286.105\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"616.245\" y=\"-271.8\">!a & !c</text>\n",
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"</g>\n",
|
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"<!-- 7->7 -->\n",
|
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"<g class=\"edge\" id=\"edge31\"><title>7->7</title>\n",
|
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"<path d=\"M698.579,-275.546C696.907,-286.872 701.419,-296.87 712.115,-296.87 720.471,-296.87 725.052,-290.768 725.86,-282.698\" fill=\"none\" stroke=\"black\"/>\n",
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"<polygon fill=\"black\" points=\"725.651,-275.546 729.004,-282.451 725.753,-279.045 725.856,-282.543 725.856,-282.543 725.856,-282.543 725.753,-279.045 722.707,-282.636 725.651,-275.546 725.651,-275.546\" stroke=\"black\"/>\n",
|
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"681.115\" y=\"-300.67\">a & b & !c</text>\n",
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"</g>\n",
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"<!-- 3->6 -->\n",
|
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"<g class=\"edge\" id=\"edge15\"><title>3->6</title>\n",
|
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"<path d=\"M248.804,-445.016C276.447,-478.728 329.604,-532 389.492,-532 389.492,-532 389.492,-532 864.855,-532 938.544,-532 960.944,-426.71 967.732,-366.314\" fill=\"none\" stroke=\"black\"/>\n",
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"<polygon fill=\"black\" points=\"968.485,-359.182 970.882,-366.474 968.117,-362.663 967.75,-366.143 967.75,-366.143 967.75,-366.143 968.117,-362.663 964.617,-365.813 968.485,-359.182 968.485,-359.182\" stroke=\"black\"/>\n",
|
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"618.245\" y=\"-535.8\">!a & c</text>\n",
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"</g>\n",
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"<!-- 3->1 -->\n",
|
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"<g class=\"edge\" id=\"edge13\"><title>3->1</title>\n",
|
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"<path d=\"M259.066,-431.446C342.79,-466.322 581.561,-550.984 746.985,-462 812.598,-426.705 843.426,-337.849 855.806,-289.164\" fill=\"none\" stroke=\"black\"/>\n",
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|
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"537.304\" y=\"-505.8\">a & c</text>\n",
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"</g>\n",
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"<!-- 3->0 -->\n",
|
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"<g class=\"edge\" id=\"edge12\"><title>3->0</title>\n",
|
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"<path d=\"M257.061,-400.176C286.355,-381.777 331.797,-353.237 361.097,-334.834\" fill=\"none\" stroke=\"black\"/>\n",
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"<polygon fill=\"black\" points=\"367.093,-331.069 362.84,-337.459 364.129,-332.93 361.165,-334.792 361.165,-334.792 361.165,-334.792 364.129,-332.93 359.489,-332.124 367.093,-331.069 367.093,-331.069\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"294.622\" y=\"-389.8\">a & !c</text>\n",
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"</g>\n",
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"<!-- 3->5 -->\n",
|
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"<g class=\"edge\" id=\"edge14\"><title>3->5</title>\n",
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"<path d=\"M261.614,-415.181C285.137,-412.895 317.472,-409.195 345.622,-404 418.726,-390.509 436.998,-380.193 501.362,-343 506.991,-339.748 512.864,-336.147 518.538,-332.554\" fill=\"none\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"370.492\" y=\"-405.8\">!a & !c</text>\n",
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"</g>\n",
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"<!-- 4->6 -->\n",
|
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"<g class=\"edge\" id=\"edge19\"><title>4->6</title>\n",
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"<path d=\"M259.868,-162.119C349.094,-132.667 612.268,-57.5477 818.985,-118 859.362,-129.808 869.773,-138.478 898.725,-169 929.959,-201.928 949.481,-250.832 960.178,-285.084\" fill=\"none\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"618.245\" y=\"-99.8\">!a & c</text>\n",
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"</g>\n",
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"<!-- 4->1 -->\n",
|
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"<g class=\"edge\" id=\"edge17\"><title>4->1</title>\n",
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"<path d=\"M261.724,-171.805C344.713,-169.603 568.065,-168.64 746.985,-213 776.744,-220.378 809.383,-232.917 832.414,-242.557\" fill=\"none\" stroke=\"black\"/>\n",
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"text": [
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]
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}
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],
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
|
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"Let's decompose it into three strengths:"
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]
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
|
|
"for (name, opt) in (('terminal', 't'), ('strictly weak', 'w'), ('strong', 's')):\n",
|
|
" a = spot.decompose_strength(aut, opt)\n",
|
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" a.set_name(name)\n",
|
|
" display(a.show('.asn'))"
|
|
],
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|
"language": "python",
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"metadata": {},
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"outputs": [
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{
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"metadata": {},
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"output_type": "display_data",
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"svg": [
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"<svg height=\"360pt\" viewBox=\"0.00 0.00 543.00 360.00\" width=\"543pt\" 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 class=\"edge\" id=\"edge1\"><title>I->0</title>\n",
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"<g class=\"edge\" id=\"edge2\"><title>0->1</title>\n",
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"</g>\n",
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"<g class=\"edge\" id=\"edge32\"><title>7->4</title>\n",
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"<path d=\"M787.419,-247.669C800.184,-256.229 819.066,-268.891 834.089,-278.966\" fill=\"none\" stroke=\"black\"/>\n",
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"<g class=\"edge\" id=\"edge31\"><title>7->7</title>\n",
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"text": [
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}
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],
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},
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{
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"source": [
|
|
"Note how the two weak automata (i.e., stricly weak and terminal) are now using a B\u00fcchi acceptance condition (because that is sufficient for weak automata) while the strong automaton inherited the original acceptance condition.\n",
|
|
"\n",
|
|
"When extracting multiple strengths and one of the strength is **strong**, we preserve the original acceptance. For instance extracting **strong** and **terminal** gives the following automaton, where only **stricly weak** SCCs have become rejecting."
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]
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},
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{
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],
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"metadata": {},
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"outputs": [
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{
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"</g>\n",
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"<g class=\"cluster\" id=\"clust5\"><title>cluster_4</title>\n",
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"<polygon fill=\"none\" points=\"168,-375 168,-427 220,-427 220,-375 168,-375\" stroke=\"black\"/>\n",
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"</g>\n",
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"<g class=\"cluster\" id=\"clust6\"><title>cluster_5</title>\n",
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"<polygon fill=\"none\" points=\"168,-130 168,-182 220,-182 220,-130 168,-130\" stroke=\"black\"/>\n",
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"</g>\n",
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"<g class=\"cluster\" id=\"clust7\"><title>cluster_6</title>\n",
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"<polygon fill=\"none\" points=\"30,-266 30,-318 82,-318 82,-266 30,-266\" stroke=\"black\"/>\n",
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"</g>\n",
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"<!-- I -->\n",
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"<!-- 0 -->\n",
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"<g class=\"node\" id=\"node2\"><title>0</title>\n",
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"<ellipse cx=\"56\" cy=\"-292\" fill=\"#ffffaa\" rx=\"18\" ry=\"18\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"middle\" x=\"56\" y=\"-288.3\">0</text>\n",
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"</g>\n",
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"<!-- I->0 -->\n",
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"<g class=\"edge\" id=\"edge1\"><title>I->0</title>\n",
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"<path d=\"M1.15491,-292C2.79388,-292 17.1543,-292 30.6317,-292\" fill=\"none\" stroke=\"black\"/>\n",
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"</g>\n",
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"<!-- 4 -->\n",
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"<g class=\"node\" id=\"node3\"><title>4</title>\n",
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"<ellipse cx=\"904.304\" cy=\"-308\" fill=\"#ffffaa\" rx=\"33.8824\" ry=\"33.8824\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"899.804\" y=\"-311.8\">4</text>\n",
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"<text fill=\"#f17cb0\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"888.304\" y=\"-297.8\">\u2776</text>\n",
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"<text fill=\"#faa43a\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"904.304\" y=\"-297.8\">\u2777</text>\n",
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"</g>\n",
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"<!-- 0->4 -->\n",
|
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"<g class=\"edge\" id=\"edge5\"><title>0->4</title>\n",
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"<path d=\"M57.667,-310.068C60.9454,-370.381 80.2273,-561 193,-561 193,-561 193,-561 806.362,-561 835.275,-561 838.164,-540.186 852.362,-515 882.062,-462.32 894.602,-392.555 899.788,-348.978\" fill=\"none\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"500.181\" y=\"-564.8\">c</text>\n",
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"</g>\n",
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"<!-- 3 -->\n",
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"<g class=\"node\" id=\"node7\"><title>3</title>\n",
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"<ellipse cx=\"503.681\" cy=\"-293\" fill=\"#ffffaa\" rx=\"33.8824\" ry=\"33.8824\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"499.181\" y=\"-296.8\">3</text>\n",
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"<text fill=\"#f17cb0\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"487.681\" y=\"-282.8\">\u2776</text>\n",
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"<text fill=\"#faa43a\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"503.681\" y=\"-282.8\">\u2777</text>\n",
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"</g>\n",
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"<!-- 0->3 -->\n",
|
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"<g class=\"edge\" id=\"edge4\"><title>0->3</title>\n",
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"<path d=\"M73.3993,-285.969C103.776,-275.155 171.076,-252.862 230,-245 314.358,-233.745 412.459,-261.017 464.944,-278.909\" fill=\"none\" stroke=\"black\"/>\n",
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"<polygon fill=\"black\" points=\"471.732,-281.258 464.087,-281.946 468.425,-280.113 465.117,-278.969 465.117,-278.969 465.117,-278.969 468.425,-280.113 466.147,-275.992 471.732,-281.258 471.732,-281.258\" stroke=\"black\"/>\n",
|
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"243\" y=\"-248.8\">!a & !c</text>\n",
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"</g>\n",
|
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"<!-- 1 -->\n",
|
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"<g class=\"node\" id=\"node9\"><title>1</title>\n",
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"<ellipse cx=\"194\" cy=\"-401\" fill=\"#ffffaa\" rx=\"18\" ry=\"18\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"middle\" x=\"194\" y=\"-397.3\">1</text>\n",
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"</g>\n",
|
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"<!-- 0->1 -->\n",
|
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"<g class=\"edge\" id=\"edge2\"><title>0->1</title>\n",
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"<path d=\"M70.5374,-302.85C94.8747,-322.355 146.073,-363.389 173.942,-385.726\" fill=\"none\" stroke=\"black\"/>\n",
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"<polygon fill=\"black\" points=\"179.517,-390.194 172.085,-388.274 176.786,-388.005 174.055,-385.816 174.055,-385.816 174.055,-385.816 176.786,-388.005 176.025,-383.358 179.517,-390.194 179.517,-390.194\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"92\" y=\"-375.8\">a & !b & !c</text>\n",
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"</g>\n",
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"<!-- 2 -->\n",
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"<g class=\"node\" id=\"node10\"><title>2</title>\n",
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"<ellipse cx=\"194\" cy=\"-156\" fill=\"#ffffaa\" rx=\"18\" ry=\"18\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"middle\" x=\"194\" y=\"-152.3\">2</text>\n",
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"</g>\n",
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"<!-- 0->2 -->\n",
|
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"<g class=\"edge\" id=\"edge3\"><title>0->2</title>\n",
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"<path d=\"M61.7076,-274.613C66.9079,-258.007 76.6402,-233.054 92,-216 113.894,-191.69 148.005,-174.288 170.427,-164.682\" fill=\"none\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"94\" y=\"-219.8\">a & b & !c</text>\n",
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"</g>\n",
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"<!-- 4->4 -->\n",
|
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"<g class=\"edge\" id=\"edge20\"><title>4->4</title>\n",
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"<path d=\"M893.804,-340.463C893.804,-351.284 897.304,-359.941 904.304,-359.941 909.663,-359.941 912.971,-354.867 914.227,-347.632\" fill=\"none\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"middle\" x=\"904.304\" y=\"-363.741\">1</text>\n",
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"</g>\n",
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"<g class=\"node\" id=\"node4\"><title>7</title>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"middle\" x=\"805.362\" y=\"-234.3\">7</text>\n",
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"</g>\n",
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"<!-- 7->4 -->\n",
|
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"<g class=\"edge\" id=\"edge32\"><title>7->4</title>\n",
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"<path d=\"M820.666,-248.329C833.613,-257.677 853.176,-271.803 870.018,-283.965\" fill=\"none\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"841.362\" y=\"-274.8\">!a</text>\n",
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"</g>\n",
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"<!-- 7->7 -->\n",
|
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"<g class=\"edge\" id=\"edge31\"><title>7->7</title>\n",
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"<path d=\"M798.331,-254.664C796.769,-264.625 799.112,-274 805.362,-274 810.05,-274 812.54,-268.727 812.833,-261.888\" fill=\"none\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"801.862\" y=\"-277.8\">a</text>\n",
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"</g>\n",
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|
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"<g class=\"node\" id=\"node5\"><title>8</title>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"middle\" x=\"805.362\" y=\"-30.3\">8</text>\n",
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"</g>\n",
|
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"<!-- 8->4 -->\n",
|
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"<g class=\"edge\" id=\"edge34\"><title>8->4</title>\n",
|
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"<path d=\"M819.681,-44.9366C830.138,-54.2488 844.244,-68.5454 852.362,-84 883.259,-142.817 895.516,-219.939 900.309,-266.637\" fill=\"none\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"841.362\" y=\"-87.8\">!a</text>\n",
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"</g>\n",
|
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"<!-- 8->7 -->\n",
|
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"<g class=\"edge\" id=\"edge33\"><title>8->7</title>\n",
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"<path d=\"M794.478,-48.5917C776.969,-73.0775 743.258,-124.665 730.862,-174.5 729.253,-180.97 727.685,-183.639 730.862,-189.5 741.513,-209.143 763.946,-222.08 781.427,-229.614\" fill=\"none\" stroke=\"black\"/>\n",
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|
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"719.862\" y=\"-178.3\">a & !b</text>\n",
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"</g>\n",
|
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"<!-- 8->8 -->\n",
|
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"<g class=\"edge\" id=\"edge35\"><title>8->8</title>\n",
|
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"<path d=\"M798.331,-50.6641C796.769,-60.625 799.112,-70 805.362,-70 810.05,-70 812.54,-64.7266 812.833,-57.8876\" fill=\"none\" stroke=\"black\"/>\n",
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|
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"788.362\" y=\"-73.8\">a & b</text>\n",
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"</g>\n",
|
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"<!-- 5 -->\n",
|
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"<g class=\"node\" id=\"node6\"><title>5</title>\n",
|
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"<ellipse cx=\"340.87\" cy=\"-300\" fill=\"#ffffaa\" rx=\"26.7407\" ry=\"26.7407\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"336.37\" y=\"-303.8\">5</text>\n",
|
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"<text fill=\"#faa43a\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"332.87\" y=\"-288.8\">\u2777</text>\n",
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"</g>\n",
|
|
"<!-- 5->4 -->\n",
|
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"<g class=\"edge\" id=\"edge24\"><title>5->4</title>\n",
|
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"<path d=\"M355.965,-322.236C380.826,-358.408 436.179,-426 502.681,-426 502.681,-426 502.681,-426 806.362,-426 848.245,-426 875.351,-381.069 889.969,-346.658\" fill=\"none\" stroke=\"black\"/>\n",
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|
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"644.492\" y=\"-429.8\">!a & c</text>\n",
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"</g>\n",
|
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"<!-- 5->7 -->\n",
|
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"<g class=\"edge\" id=\"edge22\"><title>5->7</title>\n",
|
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"<path d=\"M364.485,-313.764C371.27,-317.795 378.763,-322.151 385.74,-326 422.464,-346.257 428.968,-359.171 469.74,-369 499.07,-376.071 508.014,-374.793 537.622,-369 637.518,-349.454 741.871,-282.226 784.476,-252.422\" fill=\"none\" stroke=\"black\"/>\n",
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|
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"570.622\" y=\"-368.8\">a & c</text>\n",
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"</g>\n",
|
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"<!-- 5->5 -->\n",
|
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"<g class=\"edge\" id=\"edge21\"><title>5->5</title>\n",
|
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"<path d=\"M327.041,-323.087C325.128,-334.6 329.737,-344.87 340.87,-344.87 349.568,-344.87 354.284,-338.602 355.018,-330.369\" fill=\"none\" stroke=\"black\"/>\n",
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"<polygon fill=\"black\" points=\"354.699,-323.087 358.153,-329.942 354.852,-326.583 355.006,-330.08 355.006,-330.08 355.006,-330.08 354.852,-326.583 351.859,-330.218 354.699,-323.087 354.699,-323.087\" stroke=\"black\"/>\n",
|
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"322.87\" y=\"-348.67\">a & !c</text>\n",
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"</g>\n",
|
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"<!-- 5->3 -->\n",
|
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"<g class=\"edge\" id=\"edge23\"><title>5->3</title>\n",
|
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"<path d=\"M366.143,-290.787C372.433,-288.807 379.262,-287.009 385.74,-286 411.218,-282.033 440.142,-283.712 462.849,-286.418\" fill=\"none\" stroke=\"black\"/>\n",
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"<polygon fill=\"black\" points=\"469.991,-287.326 462.649,-289.568 466.519,-286.885 463.047,-286.443 463.047,-286.443 463.047,-286.443 466.519,-286.885 463.445,-283.318 469.991,-287.326 469.991,-287.326\" stroke=\"black\"/>\n",
|
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"398.74\" y=\"-289.8\">!a & !c</text>\n",
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"</g>\n",
|
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"<!-- 3->4 -->\n",
|
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"<g class=\"edge\" id=\"edge18\"><title>3->4</title>\n",
|
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"<path d=\"M536.205,-303.058C542.57,-304.693 549.261,-306.132 555.622,-307 665.904,-322.04 797.699,-315.933 862.8,-311.339\" fill=\"none\" stroke=\"black\"/>\n",
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"<polygon fill=\"black\" points=\"870.024,-310.816 863.27,-314.463 866.533,-311.069 863.042,-311.322 863.042,-311.322 863.042,-311.322 866.533,-311.069 862.815,-308.18 870.024,-310.816 870.024,-310.816\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"734.862\" y=\"-320.8\">c</text>\n",
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"</g>\n",
|
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"<!-- 3->5 -->\n",
|
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"<g class=\"edge\" id=\"edge16\"><title>3->5</title>\n",
|
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"<path d=\"M470.816,-301.696C464.539,-303.071 457.968,-304.276 451.74,-305 422.603,-308.386 415.026,-306.664 385.74,-305 382.196,-304.799 378.498,-304.507 374.818,-304.163\" fill=\"none\" stroke=\"black\"/>\n",
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"<polygon fill=\"black\" points=\"367.81,-303.45 375.093,-301.025 371.292,-303.804 374.774,-304.159 374.774,-304.159 374.774,-304.159 371.292,-303.804 374.455,-307.292 367.81,-303.45 367.81,-303.45\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"385.74\" y=\"-310.8\">a & !b & !c</text>\n",
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"</g>\n",
|
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"<!-- 3->3 -->\n",
|
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"<g class=\"edge\" id=\"edge17\"><title>3->3</title>\n",
|
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"<path d=\"M488.366,-323.41C487.686,-335.168 492.791,-344.941 503.681,-344.941 512.19,-344.941 517.167,-338.976 518.612,-330.76\" fill=\"none\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"483.681\" y=\"-348.741\">!a & !c</text>\n",
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"</g>\n",
|
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"<!-- 6 -->\n",
|
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"<g class=\"node\" id=\"node8\"><title>6</title>\n",
|
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"<ellipse cx=\"662.492\" cy=\"-235\" fill=\"#ffffaa\" rx=\"26.7407\" ry=\"26.7407\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"657.992\" y=\"-238.8\">6</text>\n",
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"<text fill=\"#b276b2\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"654.492\" y=\"-223.8\">\u2778</text>\n",
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"</g>\n",
|
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"<!-- 3->6 -->\n",
|
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"<g class=\"edge\" id=\"edge19\"><title>3->6</title>\n",
|
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"<path d=\"M537.601,-290.418C560.844,-287.627 592.242,-281.86 617.622,-270 624.243,-266.906 630.749,-262.621 636.641,-258.108\" fill=\"none\" stroke=\"black\"/>\n",
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"<polygon fill=\"black\" points=\"642.308,-253.555 638.824,-260.395 639.58,-255.747 636.851,-257.939 636.851,-257.939 636.851,-257.939 639.58,-255.747 634.878,-255.484 642.308,-253.555 642.308,-253.555\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"555.622\" y=\"-291.8\">a & b & !c</text>\n",
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"</g>\n",
|
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"<!-- 6->4 -->\n",
|
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"<g class=\"edge\" id=\"edge28\"><title>6->4</title>\n",
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"<path d=\"M673.849,-210.452C681.063,-196.743 692.138,-180.878 707.362,-173 731.836,-160.337 742.142,-168.718 769.362,-173 797.965,-177.499 808.523,-175.204 831.362,-193 857.027,-212.997 875.943,-244.903 887.92,-270.041\" fill=\"none\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"787.362\" y=\"-196.8\">!a & c</text>\n",
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"</g>\n",
|
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"<!-- 6->7 -->\n",
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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 weak automata seem to be good candidates for further simplification. Let's add a call to `postprocess()` to out decomposition loop, trying to preserve the determinism and state-based acceptance of the original automaton."
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]
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"for (name, opt) in (('terminal', 't'), ('strictly weak', 'w'), ('strong', 's')):\n",
|
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" a = spot.decompose_strength(aut, opt).postprocess('deterministic', 'SBAcc')\n",
|
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" a.set_name(name)\n",
|
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" display(a.show('.asn'))"
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],
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"language": "python",
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"metadata": {},
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"outputs": [
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"254\" y=\"-110.8\">!a & c</text>\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>"
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],
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"text": [
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"<IPython.core.display.SVG object>"
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],
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{
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"svg": [
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{
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"metadata": {},
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"source": [
|
|
"## Streett\n",
|
|
"\n",
|
|
"Since this notebook also serves as a test suite, let's try a Streett automaton. This one was generated with\n",
|
|
"\n",
|
|
" ltldo -f '(Ga -> Gb) W c' 'ltl2dstar --automata=streett --ltl2nba=spin:ltl2tgba@-Ds' -H | \n",
|
|
" autfilt -H --merge-transitions"
|
|
]
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|
},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
|
|
"aut = spot.automaton(\"\"\"\n",
|
|
"HOA: v1\n",
|
|
"States: 8\n",
|
|
"Start: 7\n",
|
|
"AP: 3 \"a\" \"b\" \"c\"\n",
|
|
"acc-name: Streett 2\n",
|
|
"Acceptance: 4 (Fin(0) | Inf(1)) & (Fin(2) | Inf(3))\n",
|
|
"properties: trans-labels explicit-labels state-acc complete\n",
|
|
"properties: deterministic\n",
|
|
"--BODY--\n",
|
|
"State: 0 {2}\n",
|
|
"[0&1] 0\n",
|
|
"[0&!1] 3\n",
|
|
"[!0] 4\n",
|
|
"State: 1 {2}\n",
|
|
"[0&1&2] 0\n",
|
|
"[0&1&!2] 1\n",
|
|
"[0&!1&!2] 2\n",
|
|
"[0&!1&2] 3\n",
|
|
"[!0&2] 4\n",
|
|
"[!0&!2] 7\n",
|
|
"State: 2 {2}\n",
|
|
"[0&1&!2] 1\n",
|
|
"[0&!1&!2] 2\n",
|
|
"[0&2] 3\n",
|
|
"[!0&2] 4\n",
|
|
"[!0&!2] 7\n",
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"State: 3 {0 3}\n",
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"[0] 3\n",
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"[!0] 4\n",
|
|
"State: 4 {1 3}\n",
|
|
"[t] 4\n",
|
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"State: 5 {3}\n",
|
|
"[0&!1] 3\n",
|
|
"[!0] 4\n",
|
|
"[0&1] 5\n",
|
|
"State: 6 {3}\n",
|
|
"[0&!1&!2] 2\n",
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|
"[0&!1&2] 3\n",
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"[!0&2] 4\n",
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|
"[0&1&2] 5\n",
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"[0&1&!2] 6\n",
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"[!0&!2] 7\n",
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|
"State: 7 {3}\n",
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|
"[0&!1&!2] 2\n",
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"[2] 4\n",
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|
"[0&1&!2] 6\n",
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"[!0&!2] 7\n",
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"--END--\n",
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"\"\"\")\n",
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"aut.show(\".as\")"
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],
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"language": "python",
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"metadata": {},
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"outputs": [
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"metadata": {},
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"output_type": "pyout",
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"prompt_number": 12,
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"svg": [
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"</g>\n",
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"<!-- I->7 -->\n",
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"<g class=\"edge\" id=\"edge1\"><title>I->7</title>\n",
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"<path d=\"M1.04557,-219C1.94668,-219 16.0699,-219 30.6965,-219\" fill=\"none\" stroke=\"black\"/>\n",
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"<polygon fill=\"black\" points=\"37.8616,-219 30.8617,-222.15 34.3616,-219 30.8616,-219 30.8616,-219 30.8616,-219 34.3616,-219 30.8616,-215.85 37.8616,-219 37.8616,-219\" stroke=\"black\"/>\n",
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"</g>\n",
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"<!-- 7->7 -->\n",
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"<g class=\"edge\" id=\"edge31\"><title>7->7</title>\n",
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"<path d=\"M56.6704,-244.821C56.2072,-255.178 58.9404,-263.87 64.8701,-263.87 69.3173,-263.87 71.9666,-258.981 72.8178,-252.167\" fill=\"none\" stroke=\"black\"/>\n",
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"<polygon fill=\"black\" points=\"73.0697,-244.821 75.9779,-251.925 72.9497,-248.319 72.8297,-251.817 72.8297,-251.817 72.8297,-251.817 72.9497,-248.319 69.6816,-251.709 73.0697,-244.821 73.0697,-244.821\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"44.8701\" y=\"-267.67\">!a & !c</text>\n",
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"</g>\n",
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"<!-- 4 -->\n",
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"<g class=\"node\" id=\"node3\"><title>4</title>\n",
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"<ellipse cx=\"953.304\" cy=\"-135\" fill=\"#ffffaa\" rx=\"33.8824\" ry=\"33.8824\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"948.804\" y=\"-138.8\">4</text>\n",
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"<text fill=\"#f17cb0\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"937.304\" y=\"-124.8\">\u2776</text>\n",
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"<text fill=\"#b276b2\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"953.304\" y=\"-124.8\">\u2778</text>\n",
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"</g>\n",
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"<!-- 7->4 -->\n",
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"<g class=\"edge\" id=\"edge29\"><title>7->4</title>\n",
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"<path d=\"M71.4782,-192.583C85.5575,-134.336 128.171,-0 221.74,-0 221.74,-0 221.74,-0 839.421,-0 889.962,-0 922.321,-56.3742 938.892,-96.2636\" fill=\"none\" stroke=\"black\"/>\n",
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"<polygon fill=\"black\" points=\"941.61,-103.007 936.071,-97.6919 940.301,-99.7606 938.993,-96.5144 938.993,-96.5144 938.993,-96.5144 940.301,-99.7606 941.914,-95.3368 941.61,-103.007 941.61,-103.007\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"540.98\" y=\"-3.8\">c</text>\n",
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"</g>\n",
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"<!-- 2 -->\n",
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"<g class=\"node\" id=\"node7\"><title>2</title>\n",
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"<ellipse cx=\"384.61\" cy=\"-233\" fill=\"#ffffaa\" rx=\"26.7407\" ry=\"26.7407\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"380.11\" y=\"-236.8\">2</text>\n",
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"<text fill=\"#faa43a\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"376.61\" y=\"-221.8\">\u2777</text>\n",
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"</g>\n",
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"<!-- 7->2 -->\n",
|
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"<g class=\"edge\" id=\"edge28\"><title>7->2</title>\n",
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"<path d=\"M91.8721,-220.146C149.269,-222.675 286.739,-228.732 350.318,-231.533\" fill=\"none\" stroke=\"black\"/>\n",
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"<polygon fill=\"black\" points=\"357.695,-231.858 350.563,-234.697 354.198,-231.704 350.701,-231.55 350.701,-231.55 350.701,-231.55 354.198,-231.704 350.84,-228.403 357.695,-231.858 357.695,-231.858\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"189.74\" y=\"-231.8\">a & !b & !c</text>\n",
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"</g>\n",
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"<!-- 6 -->\n",
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"<g class=\"node\" id=\"node9\"><title>6</title>\n",
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"<ellipse cx=\"222.74\" cy=\"-305\" fill=\"#ffffaa\" rx=\"26.7407\" ry=\"26.7407\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"218.24\" y=\"-308.8\">6</text>\n",
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"<text fill=\"#b276b2\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"214.74\" y=\"-293.8\">\u2778</text>\n",
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"</g>\n",
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"<!-- 7->6 -->\n",
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"<g class=\"edge\" id=\"edge30\"><title>7->6</title>\n",
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"<path d=\"M76.0877,-243.444C83.3639,-257.804 94.553,-275.04 109.74,-285 133.091,-300.313 164.809,-304.813 188.53,-305.772\" fill=\"none\" stroke=\"black\"/>\n",
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"<polygon fill=\"black\" points=\"195.617,-305.964 188.534,-308.923 192.118,-305.869 188.62,-305.774 188.62,-305.774 188.62,-305.774 192.118,-305.869 188.705,-302.625 195.617,-305.964 195.617,-305.964\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"109.74\" y=\"-307.8\">a & b & !c</text>\n",
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"</g>\n",
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"<!-- 4->4 -->\n",
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"<g class=\"edge\" id=\"edge18\"><title>4->4</title>\n",
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"<path d=\"M942.804,-167.463C942.804,-178.284 946.304,-186.941 953.304,-186.941 958.663,-186.941 961.971,-181.867 963.227,-174.632\" fill=\"none\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"middle\" x=\"953.304\" y=\"-190.741\">1</text>\n",
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"</g>\n",
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"<!-- 3 -->\n",
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"<g class=\"node\" id=\"node4\"><title>3</title>\n",
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"<ellipse cx=\"838.421\" cy=\"-254\" fill=\"#ffffaa\" rx=\"33.8824\" ry=\"33.8824\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"833.921\" y=\"-257.8\">3</text>\n",
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"<text fill=\"#5da5da\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"822.421\" y=\"-243.8\">\u24ff</text>\n",
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"<text fill=\"#b276b2\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"838.421\" y=\"-243.8\">\u2778</text>\n",
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"</g>\n",
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"<!-- 3->4 -->\n",
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"<g class=\"edge\" id=\"edge17\"><title>3->4</title>\n",
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"<path d=\"M862.537,-229.631C880.419,-210.781 905.254,-184.6 924.269,-164.554\" fill=\"none\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"890.362\" y=\"-202.8\">!a</text>\n",
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"</g>\n",
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"<!-- 3->3 -->\n",
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"<g class=\"edge\" id=\"edge16\"><title>3->3</title>\n",
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"<path d=\"M827.921,-286.463C827.921,-297.284 831.421,-305.941 838.421,-305.941 843.781,-305.941 847.088,-300.867 848.345,-293.632\" fill=\"none\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"834.921\" y=\"-309.741\">a</text>\n",
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"</g>\n",
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"<!-- 0 -->\n",
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"<g class=\"node\" id=\"node5\"><title>0</title>\n",
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"<ellipse cx=\"700.48\" cy=\"-195\" fill=\"#ffffaa\" rx=\"26.7407\" ry=\"26.7407\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"695.98\" y=\"-198.8\">0</text>\n",
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"<text fill=\"#faa43a\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"692.48\" y=\"-183.8\">\u2777</text>\n",
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"</g>\n",
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"<!-- 0->4 -->\n",
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"<g class=\"edge\" id=\"edge4\"><title>0->4</title>\n",
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"<path d=\"M726.911,-188.917C770.803,-178.417 861.061,-156.826 913.059,-144.388\" fill=\"none\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"832.921\" y=\"-175.8\">!a</text>\n",
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"</g>\n",
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"<!-- 0->3 -->\n",
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"<g class=\"edge\" id=\"edge3\"><title>0->3</title>\n",
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"<path d=\"M725.471,-205.412C746.233,-214.423 776.59,-227.599 800.456,-237.957\" fill=\"none\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"749.48\" y=\"-235.8\">a & !b</text>\n",
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"</g>\n",
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"<!-- 0->0 -->\n",
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"<g class=\"edge\" id=\"edge2\"><title>0->0</title>\n",
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"<path d=\"M688.35,-219.005C687.027,-230.141 691.07,-239.87 700.48,-239.87 707.832,-239.87 711.908,-233.932 712.709,-226.024\" fill=\"none\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"683.48\" y=\"-243.67\">a & b</text>\n",
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"</g>\n",
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"<!-- 5 -->\n",
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"<g class=\"node\" id=\"node6\"><title>5</title>\n",
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"<ellipse cx=\"700.48\" cy=\"-376\" fill=\"#ffffaa\" rx=\"26.7407\" ry=\"26.7407\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"695.98\" y=\"-379.8\">5</text>\n",
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"<text fill=\"#b276b2\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"692.48\" y=\"-364.8\">\u2778</text>\n",
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"</g>\n",
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"<!-- 5->4 -->\n",
|
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"<g class=\"edge\" id=\"edge20\"><title>5->4</title>\n",
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"<path d=\"M727.515,-377.74C766.507,-378.815 839.814,-374.655 880.362,-333 922.411,-289.804 940.136,-220.153 947.429,-176.019\" fill=\"none\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"832.921\" y=\"-375.8\">!a</text>\n",
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"</g>\n",
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"<!-- 5->3 -->\n",
|
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"<g class=\"edge\" id=\"edge19\"><title>5->3</title>\n",
|
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"<path d=\"M725.442,-365.273C747.026,-355.314 776.803,-340.747 786.48,-332 799.383,-320.338 810.434,-304.711 818.912,-290.507\" fill=\"none\" stroke=\"black\"/>\n",
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|
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"749.48\" y=\"-356.8\">a & !b</text>\n",
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"</g>\n",
|
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"<!-- 5->5 -->\n",
|
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"<g class=\"edge\" id=\"edge21\"><title>5->5</title>\n",
|
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"<path d=\"M688.35,-400.005C687.027,-411.141 691.07,-420.87 700.48,-420.87 707.832,-420.87 711.908,-414.932 712.709,-407.024\" fill=\"none\" stroke=\"black\"/>\n",
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|
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"683.48\" y=\"-424.67\">a & b</text>\n",
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"</g>\n",
|
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"<!-- 2->7 -->\n",
|
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"<g class=\"edge\" id=\"edge15\"><title>2->7</title>\n",
|
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"<path d=\"M358.349,-226.562C332.707,-220.382 291.764,-211.488 255.74,-208 200.897,-202.689 136.874,-208.976 98.9438,-213.997\" fill=\"none\" stroke=\"black\"/>\n",
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|
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"202.74\" y=\"-211.8\">!a & !c</text>\n",
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"</g>\n",
|
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"<!-- 2->4 -->\n",
|
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"<g class=\"edge\" id=\"edge14\"><title>2->4</title>\n",
|
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"<path d=\"M393.7,-207.391C411.658,-156.176 459.992,-46 543.48,-46 543.48,-46 543.48,-46 839.421,-46 876.544,-46 908.638,-76.0486 929.013,-101.226\" fill=\"none\" stroke=\"black\"/>\n",
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"<polygon fill=\"black\" points=\"933.432,-106.84 926.627,-103.288 931.268,-104.09 929.103,-101.34 929.103,-101.34 929.103,-101.34 931.268,-104.09 931.578,-99.3911 933.432,-106.84 933.432,-106.84\" stroke=\"black\"/>\n",
|
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"682.48\" y=\"-49.8\">!a & c</text>\n",
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"</g>\n",
|
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"<!-- 2->3 -->\n",
|
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"<g class=\"edge\" id=\"edge13\"><title>2->3</title>\n",
|
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"<path d=\"M409.904,-242.508C416.194,-244.636 423.018,-246.657 429.48,-248 562.836,-275.71 599.419,-275.621 735.35,-267 755.811,-265.702 778.44,-262.955 797.181,-260.32\" fill=\"none\" stroke=\"black\"/>\n",
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"<polygon fill=\"black\" points=\"804.425,-259.279 797.944,-263.392 800.961,-259.777 797.496,-260.274 797.496,-260.274 797.496,-260.274 800.961,-259.777 797.049,-257.156 804.425,-259.279 804.425,-259.279\" stroke=\"black\"/>\n",
|
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"606.48\" y=\"-276.8\">a & c</text>\n",
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"</g>\n",
|
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"<!-- 2->2 -->\n",
|
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"<g class=\"edge\" id=\"edge12\"><title>2->2</title>\n",
|
|
"<path d=\"M370.781,-256.087C368.868,-267.6 373.477,-277.87 384.61,-277.87 393.308,-277.87 398.024,-271.602 398.758,-263.369\" fill=\"none\" stroke=\"black\"/>\n",
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"<polygon fill=\"black\" points=\"398.439,-256.087 401.893,-262.942 398.593,-259.583 398.746,-263.08 398.746,-263.08 398.746,-263.08 398.593,-259.583 395.599,-263.218 398.439,-256.087 398.439,-256.087\" stroke=\"black\"/>\n",
|
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"351.61\" y=\"-281.67\">a & !b & !c</text>\n",
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"</g>\n",
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"<!-- 1 -->\n",
|
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"<g class=\"node\" id=\"node8\"><title>1</title>\n",
|
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"<ellipse cx=\"544.48\" cy=\"-185\" fill=\"#ffffaa\" rx=\"26.7407\" ry=\"26.7407\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"539.98\" y=\"-188.8\">1</text>\n",
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"<text fill=\"#faa43a\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"536.48\" y=\"-173.8\">\u2777</text>\n",
|
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"</g>\n",
|
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"<!-- 2->1 -->\n",
|
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"<g class=\"edge\" id=\"edge11\"><title>2->1</title>\n",
|
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"<path d=\"M411.628,-231.237C434.35,-229.061 467.915,-224.292 495.48,-214 502.321,-211.446 509.273,-207.899 515.685,-204.161\" fill=\"none\" stroke=\"black\"/>\n",
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"<polygon fill=\"black\" points=\"521.892,-200.388 517.546,-206.716 518.901,-202.206 515.91,-204.024 515.91,-204.024 515.91,-204.024 518.901,-202.206 514.274,-201.332 521.892,-200.388 521.892,-200.388\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"431.48\" y=\"-232.8\">a & b & !c</text>\n",
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"</g>\n",
|
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"<!-- 1->7 -->\n",
|
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"<g class=\"edge\" id=\"edge10\"><title>1->7</title>\n",
|
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"<path d=\"M517.587,-183.085C510.429,-182.636 502.657,-182.224 495.48,-182 466.161,-181.085 458.798,-181.06 429.48,-182 287.111,-186.566 249.308,-175.527 109.74,-204 105.597,-204.845 101.307,-205.975 97.105,-207.237\" fill=\"none\" stroke=\"black\"/>\n",
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"<polygon fill=\"black\" points=\"90.1574,-209.46 95.865,-204.327 93.491,-208.393 96.8246,-207.327 96.8246,-207.327 96.8246,-207.327 93.491,-208.393 97.7843,-210.327 90.1574,-209.46 90.1574,-209.46\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"286.74\" y=\"-187.8\">!a & !c</text>\n",
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"</g>\n",
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"<!-- 1->4 -->\n",
|
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"<g class=\"edge\" id=\"edge9\"><title>1->4</title>\n",
|
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"<path d=\"M559.853,-162.77C568.242,-151.796 579.855,-139.536 593.48,-133 699.699,-82.0446 844.795,-107.463 913.368,-124.225\" fill=\"none\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"749.98\" y=\"-108.8\">!a & c</text>\n",
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"</g>\n",
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"<!-- 1->3 -->\n",
|
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"<g class=\"edge\" id=\"edge8\"><title>1->3</title>\n",
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"<path d=\"M567.697,-171.029C604.064,-150.066 677.989,-115.516 735.35,-141 754.546,-149.528 789.239,-191.008 812.794,-221.246\" fill=\"none\" stroke=\"black\"/>\n",
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"</g>\n",
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"<g class=\"edge\" id=\"edge5\"><title>1->0</title>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"593.48\" y=\"-195.8\">a & b & c</text>\n",
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"</g>\n",
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"<!-- 1->2 -->\n",
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"<g class=\"edge\" id=\"edge7\"><title>1->2</title>\n",
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"</g>\n",
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"<!-- 1->1 -->\n",
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"<g class=\"edge\" id=\"edge6\"><title>1->1</title>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"513.48\" y=\"-233.67\">a & b & !c</text>\n",
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"</g>\n",
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"<g class=\"edge\" id=\"edge27\"><title>6->7</title>\n",
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"</g>\n",
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"<!-- 6->4 -->\n",
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"<g class=\"edge\" id=\"edge24\"><title>6->4</title>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"604.48\" y=\"-467.8\">!a & c</text>\n",
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"</g>\n",
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"<!-- 6->3 -->\n",
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"<g class=\"edge\" id=\"edge23\"><title>6->3</title>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"513.48\" y=\"-324.8\">a & !b & c</text>\n",
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"</g>\n",
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"<!-- 6->5 -->\n",
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"<g class=\"edge\" id=\"edge25\"><title>6->5</title>\n",
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"<path d=\"M245.113,-320.686C253.614,-326.137 263.741,-331.706 273.74,-335 413.861,-381.157 592.916,-380.365 666.077,-377.696\" fill=\"none\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"433.48\" y=\"-377.8\">a & b & c</text>\n",
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"</g>\n",
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"<!-- 6->2 -->\n",
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"<g class=\"edge\" id=\"edge22\"><title>6->2</title>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"273.74\" y=\"-285.8\">a & !b & !c</text>\n",
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"</g>\n",
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"<!-- 6->6 -->\n",
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"<g class=\"edge\" id=\"edge26\"><title>6->6</title>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"191.74\" y=\"-353.67\">a & b & !c</text>\n",
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"</g>\n",
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"</g>\n",
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"</svg>"
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],
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"text": [
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"<IPython.core.display.SVG object>"
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]
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}
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],
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"prompt_number": 12
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},
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{
|
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"cell_type": "code",
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"collapsed": false,
|
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"input": [
|
|
"for (name, opt) in (('terminal', 't'), ('strictly weak', 'w'), ('strong', 's')):\n",
|
|
" a = spot.decompose_strength(aut, opt)\n",
|
|
" a.set_name(name)\n",
|
|
" display(a.show('.asn'))"
|
|
],
|
|
"language": "python",
|
|
"metadata": {},
|
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"outputs": [
|
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{
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"metadata": {},
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"output_type": "display_data",
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"svg": [
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"<svg height=\"360pt\" viewBox=\"0.00 0.00 675.63 360.00\" width=\"676pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
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"<g class=\"graph\" id=\"graph0\" transform=\"scale(0.715706 0.715706) rotate(0) translate(4 499)\">\n",
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"<title>G</title>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"442\" y=\"-480.8\">terminal</text>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"447\" y=\"-466.8\">Inf(</text>\n",
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"<text fill=\"#5da5da\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"469\" y=\"-466.8\">\u24ff</text>\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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"<!-- I -->\n",
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"<!-- 0 -->\n",
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"<g class=\"node\" id=\"node2\"><title>0</title>\n",
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"<ellipse cx=\"60\" cy=\"-215\" fill=\"none\" rx=\"22\" ry=\"22\" stroke=\"black\"/>\n",
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"</g>\n",
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"<!-- I->0 -->\n",
|
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"<g class=\"edge\" id=\"edge1\"><title>I->0</title>\n",
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"<path d=\"M1.16638,-215C2.84121,-215 16.884,-215 30.7112,-215\" fill=\"none\" stroke=\"black\"/>\n",
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"</g>\n",
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"<!-- 0->0 -->\n",
|
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"<g class=\"edge\" id=\"edge5\"><title>0->0</title>\n",
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"<path d=\"M52.6831,-235.991C51.7798,-246.087 54.2188,-255 60,-255 64.3359,-255 66.7918,-249.987 67.3677,-243.22\" fill=\"none\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"40\" y=\"-258.8\">!a & !c</text>\n",
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"</g>\n",
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|
"<g class=\"node\" id=\"node3\"><title>2</title>\n",
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"<ellipse cx=\"898\" cy=\"-135\" fill=\"none\" rx=\"22\" ry=\"22\" stroke=\"black\"/>\n",
|
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"middle\" x=\"898\" y=\"-131.3\">2</text>\n",
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"</g>\n",
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|
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"<g class=\"edge\" id=\"edge3\"><title>0->2</title>\n",
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"<path d=\"M65.2774,-193.599C77.5204,-139.217 118.038,-0 212,-0 212,-0 212,-0 808,-0 858.477,-0 881.953,-66.6001 891.456,-106.243\" fill=\"none\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"521.5\" y=\"-3.8\">c</text>\n",
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"</g>\n",
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"<g class=\"node\" id=\"node7\"><title>1</title>\n",
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"<ellipse cx=\"370\" cy=\"-228\" fill=\"none\" rx=\"22\" ry=\"22\" stroke=\"black\"/>\n",
|
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"middle\" x=\"370\" y=\"-224.3\">1</text>\n",
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"</g>\n",
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|
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"<g class=\"edge\" id=\"edge2\"><title>0->1</title>\n",
|
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"<path d=\"M82.1701,-215.894C136.239,-218.176 279.455,-224.221 340.639,-226.803\" fill=\"none\" stroke=\"black\"/>\n",
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|
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"180\" y=\"-226.8\">a & !b & !c</text>\n",
|
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"</g>\n",
|
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"<!-- 3 -->\n",
|
|
"<g class=\"node\" id=\"node9\"><title>3</title>\n",
|
|
"<ellipse cx=\"213\" cy=\"-295\" fill=\"#ffffaa\" rx=\"18\" ry=\"18\" stroke=\"black\"/>\n",
|
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"<ellipse cx=\"213\" cy=\"-295\" fill=\"none\" rx=\"22\" ry=\"22\" stroke=\"black\"/>\n",
|
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"middle\" x=\"213\" y=\"-291.3\">3</text>\n",
|
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"</g>\n",
|
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"<!-- 0->3 -->\n",
|
|
"<g class=\"edge\" id=\"edge4\"><title>0->3</title>\n",
|
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"<path d=\"M67.8974,-235.769C73.9503,-250.589 84.2768,-269.659 100,-280 124.807,-296.315 159.427,-298.582 183.587,-297.709\" fill=\"none\" stroke=\"black\"/>\n",
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|
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"100\" y=\"-300.8\">a & b & !c</text>\n",
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"</g>\n",
|
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"<!-- 2->2 -->\n",
|
|
"<g class=\"edge\" id=\"edge11\"><title>2->2</title>\n",
|
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"<path d=\"M889.994,-155.581C888.886,-165.845 891.555,-175 898,-175 902.834,-175 905.544,-169.85 906.129,-162.945\" fill=\"none\" stroke=\"black\"/>\n",
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|
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"middle\" x=\"898\" y=\"-178.8\">1</text>\n",
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"</g>\n",
|
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"<!-- 4 -->\n",
|
|
"<g class=\"node\" id=\"node4\"><title>4</title>\n",
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"<ellipse cx=\"807\" cy=\"-246\" fill=\"#ffffaa\" rx=\"18\" ry=\"18\" stroke=\"black\"/>\n",
|
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"<ellipse cx=\"807\" cy=\"-246\" fill=\"none\" rx=\"22\" ry=\"22\" stroke=\"black\"/>\n",
|
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"middle\" x=\"807\" y=\"-242.3\">4</text>\n",
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|
"</g>\n",
|
|
"<!-- 4->2 -->\n",
|
|
"<g class=\"edge\" id=\"edge19\"><title>4->2</title>\n",
|
|
"<path d=\"M821.517,-229.141C836.808,-210.071 861.787,-178.918 878.787,-157.715\" fill=\"none\" stroke=\"black\"/>\n",
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"<polygon fill=\"black\" points=\"883.356,-152.017 881.434,-159.449 881.166,-154.748 878.977,-157.478 878.977,-157.478 878.977,-157.478 881.166,-154.748 876.519,-155.508 883.356,-152.017 883.356,-152.017\" stroke=\"black\"/>\n",
|
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"847\" y=\"-201.8\">!a</text>\n",
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"</g>\n",
|
|
"<!-- 4->4 -->\n",
|
|
"<g class=\"edge\" id=\"edge18\"><title>4->4</title>\n",
|
|
"<path d=\"M798.994,-266.581C797.886,-276.845 800.555,-286 807,-286 811.834,-286 814.544,-280.85 815.129,-273.945\" fill=\"none\" stroke=\"black\"/>\n",
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"<polygon fill=\"black\" points=\"815.006,-266.581 818.273,-273.527 815.065,-270.08 815.123,-273.58 815.123,-273.58 815.123,-273.58 815.065,-270.08 811.973,-273.632 815.006,-266.581 815.006,-266.581\" stroke=\"black\"/>\n",
|
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"803.5\" y=\"-289.8\">a</text>\n",
|
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"</g>\n",
|
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"<!-- 7 -->\n",
|
|
"<g class=\"node\" id=\"node5\"><title>7</title>\n",
|
|
"<ellipse cx=\"681\" cy=\"-190\" fill=\"#ffffaa\" rx=\"18\" ry=\"18\" stroke=\"black\"/>\n",
|
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"<ellipse cx=\"681\" cy=\"-190\" fill=\"none\" rx=\"22\" ry=\"22\" stroke=\"black\"/>\n",
|
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"middle\" x=\"681\" y=\"-186.3\">7</text>\n",
|
|
"</g>\n",
|
|
"<!-- 7->2 -->\n",
|
|
"<g class=\"edge\" id=\"edge31\"><title>7->2</title>\n",
|
|
"<path d=\"M702.615,-184.726C741.543,-174.768 825.873,-153.195 869.546,-142.023\" fill=\"none\" stroke=\"black\"/>\n",
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|
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"801.5\" y=\"-169.8\">!a</text>\n",
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"</g>\n",
|
|
"<!-- 7->4 -->\n",
|
|
"<g class=\"edge\" id=\"edge30\"><title>7->4</title>\n",
|
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"<path d=\"M701.375,-198.75C722.577,-208.325 756.67,-223.722 780.143,-234.323\" fill=\"none\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"730\" y=\"-231.8\">a & !b</text>\n",
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"</g>\n",
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"<!-- 7->7 -->\n",
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"<g class=\"edge\" id=\"edge29\"><title>7->7</title>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"193\" y=\"-205.8\">!a & !c</text>\n",
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"</g>\n",
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],
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"metadata": {},
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"source": [
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"The subtlety of Streett acceptance is that if a path that does not visit any accepting set infinitely often *is* accepting. So when disabling SCCs, we must be careful to label them with a combination of rejecting acceptance sets.\n",
|
|
"\n",
|
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"This is easy to understand using an example. In the following extraction of the **strong** and **terminal** parts, the rejecting SCCs (that were either rejecting or strictly weak originally) have been labeled by the same acceptance sets, to ensure that they are rejected."
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]
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},
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"spot.decompose_strength(aut, 'st').show('.as')"
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],
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"language": "python",
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"metadata": {},
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"outputs": [
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{
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"## Corner cases\n",
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|
"\n",
|
|
"Here is the reason we said there is almost nothing preventing the decomposition to work for any acceptance condition.\n",
|
|
"To disable SCCs like above, the `decompose_strength()` must be able to find a set of acceptance sets to put those SCCs\n",
|
|
"in such that they will be rejected. This is impossible if the acceptance condition is always satisifiable.\n",
|
|
"\n",
|
|
"This include acceptances like `Acceptance: 0 t`, but also trickier ones like `Acceptance: 1 Inf(0) | Fin(0)` that you can make as complex as you fancy.\n",
|
|
"\n",
|
|
"### `Acceptance: 0 t`\n",
|
|
"\n",
|
|
"This is the least problematic case. Because automata with this acceptance are necessary weak, there is reason to extract the **strong** strength, and extracting the other two strengths will work as expected:"
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]
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},
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{
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"input": [
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"aut = spot.translate('(Gb|c) R a', 'any'); aut.show('.as')"
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],
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"language": "python",
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"metadata": {},
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"outputs": [
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{
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"svg": [
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"<path d=\"M1.15491,-41C2.79388,-41 17.1543,-41 30.6317,-41\" fill=\"none\" stroke=\"black\"/>\n",
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"</g>\n",
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"</svg>"
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],
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]
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}
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],
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
|
|
"# There is no strong part for this automaton\n",
|
|
"assert spot.decompose_strength(aut, 's') is None"
|
|
],
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"language": "python",
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|
"metadata": {},
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"outputs": [],
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"prompt_number": 16
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
|
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"for opt in ('w', 't'):\n",
|
|
" display(spot.decompose_strength(aut, opt).show('.as'))"
|
|
],
|
|
"language": "python",
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|
"metadata": {},
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"outputs": [
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|
{
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"metadata": {},
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"output_type": "display_data",
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"svg": [
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"<svg height=\"140pt\" viewBox=\"0.00 0.00 248.00 140.00\" width=\"248pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
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"<g class=\"graph\" id=\"graph0\" transform=\"scale(1 1) rotate(0) translate(4 136)\">\n",
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"<!-- 0 -->\n",
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"<g class=\"edge\" id=\"edge1\"><title>I->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>"
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],
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"text": [
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"<IPython.core.display.SVG object>"
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]
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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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"svg": [
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"<svg height=\"140pt\" viewBox=\"0.00 0.00 210.00 140.00\" width=\"210pt\" 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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"<!-- I -->\n",
|
|
"<!-- 0 -->\n",
|
|
"<g class=\"node\" id=\"node2\"><title>0</title>\n",
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"<ellipse cx=\"56\" cy=\"-38\" fill=\"#ffffaa\" rx=\"18\" ry=\"18\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"middle\" x=\"56\" y=\"-34.3\">0</text>\n",
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"</g>\n",
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"<!-- I->0 -->\n",
|
|
"<g class=\"edge\" id=\"edge1\"><title>I->0</title>\n",
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"<path d=\"M1.15491,-38C2.79388,-38 17.1543,-38 30.6317,-38\" fill=\"none\" stroke=\"black\"/>\n",
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"</g>\n",
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"<ellipse cx=\"164\" cy=\"-38\" fill=\"none\" rx=\"22\" ry=\"22\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"middle\" x=\"164\" y=\"-34.3\">1</text>\n",
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"</g>\n",
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"<g class=\"edge\" id=\"edge3\"><title>0->1</title>\n",
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|
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"92\" y=\"-41.8\">a & c</text>\n",
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"</g>\n",
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"<!-- 1->1 -->\n",
|
|
"<g class=\"edge\" id=\"edge4\"><title>1->1</title>\n",
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"<path d=\"M153.933,-57.7575C152.223,-68.3499 155.578,-78 164,-78 170.448,-78 173.926,-72.3433 174.435,-64.9379\" fill=\"none\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"middle\" x=\"164\" y=\"-81.8\">1</text>\n",
|
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"</g>\n",
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"</g>\n",
|
|
"</svg>"
|
|
],
|
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"text": [
|
|
"<IPython.core.display.SVG object>"
|
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]
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}
|
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],
|
|
"prompt_number": 17
|
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},
|
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{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"If we try to extract multiple strengths and include the (empty) strong part, this request will simply be ignored:"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"collapsed": false,
|
|
"input": [
|
|
"spot.decompose_strength(aut, 'st').show('.as')"
|
|
],
|
|
"language": "python",
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"metadata": {},
|
|
"output_type": "pyout",
|
|
"prompt_number": 18,
|
|
"svg": [
|
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"<svg height=\"140pt\" viewBox=\"0.00 0.00 210.00 140.00\" width=\"210pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
|
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"<g class=\"graph\" id=\"graph0\" transform=\"scale(1 1) rotate(0) translate(4 136)\">\n",
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"<title>G</title>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"80\" y=\"-117.8\">Inf(</text>\n",
|
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"<text fill=\"#5da5da\" font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"102\" y=\"-117.8\">\u24ff</text>\n",
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"</g>\n",
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"<!-- I -->\n",
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"<!-- 0 -->\n",
|
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"<g class=\"node\" id=\"node2\"><title>0</title>\n",
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"<ellipse cx=\"56\" cy=\"-38\" fill=\"#ffffaa\" rx=\"18\" ry=\"18\" stroke=\"black\"/>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"middle\" x=\"56\" y=\"-34.3\">0</text>\n",
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"</g>\n",
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"<!-- I->0 -->\n",
|
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"<g class=\"edge\" id=\"edge1\"><title>I->0</title>\n",
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"<path d=\"M1.15491,-38C2.79388,-38 17.1543,-38 30.6317,-38\" fill=\"none\" stroke=\"black\"/>\n",
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"</g>\n",
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"<g class=\"edge\" id=\"edge2\"><title>0->0</title>\n",
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"<path d=\"M49.6208,-55.0373C48.3189,-64.8579 50.4453,-74 56,-74 60.166,-74 62.4036,-68.8576 62.7128,-62.1433\" fill=\"none\" stroke=\"black\"/>\n",
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"</svg>"
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],
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"text": [
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"<IPython.core.display.SVG object>"
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]
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}
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],
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"prompt_number": 18
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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": [
|
|
"Note that the above is exactly the output of `decompose_strength(aut, 't')`. The `'s'` flag was actively ignored. If `'s'` had not been ignored an the automaton processed as if its strong part had to be preserved, the original acceptance conditions would have been used, and this would have prevented the disabling of the initial SCC."
|
|
]
|
|
},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
|
|
"### `Acceptance: 1 Inf(0) | Fin(0)`\n",
|
|
"\n",
|
|
"This acceptance could be replaced by `Acceptance: 0 t` without altering the language of the automaton. However its use of acceptance sets allows us to define some automata with strong SCCs."
|
|
]
|
|
},
|
|
{
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|
"cell_type": "code",
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|
"collapsed": false,
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|
"input": [
|
|
"aut = spot.automaton(\"\"\"\n",
|
|
"HOA: v1\n",
|
|
"States: 4\n",
|
|
"Start: 0\n",
|
|
"AP: 1 \"a\"\n",
|
|
"Acceptance: 1 Inf(0) | Fin(0)\n",
|
|
"--BODY--\n",
|
|
"State: 0\n",
|
|
"[0] 0 \n",
|
|
"[!0] 1\n",
|
|
"State: 1\n",
|
|
"[0] 1 \n",
|
|
"[!0] 2 {0}\n",
|
|
"State: 2\n",
|
|
"[0] 1\n",
|
|
"[!0] 3\n",
|
|
"State: 3\n",
|
|
"[t] 3 {0}\n",
|
|
"--END--\n",
|
|
"\"\"\")\n",
|
|
"aut.show('.as')"
|
|
],
|
|
"language": "python",
|
|
"metadata": {},
|
|
"outputs": [
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|
{
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"metadata": {},
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"output_type": "pyout",
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"prompt_number": 19,
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"svg": [
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"<svg height=\"149pt\" viewBox=\"0.00 0.00 352.00 149.00\" width=\"352pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
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"<g class=\"graph\" id=\"graph0\" transform=\"scale(1 1) rotate(0) translate(4 145)\">\n",
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"<title>G</title>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"161\" y=\"-126.8\">) | Fin(</text>\n",
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"</g>\n",
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"<!-- I->0 -->\n",
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"<g class=\"edge\" id=\"edge1\"><title>I->0</title>\n",
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"</g>\n",
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"</g>\n",
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"</svg>"
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],
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"text": [
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"<IPython.core.display.SVG object>"
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]
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}
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],
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"prompt_number": 19
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},
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{
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"cell_type": "markdown",
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"metadata": {},
|
|
"source": [
|
|
"By our definitions, SCC $\\{0\\}$ is stricly weak, SCC $\\{1,2\\}$ is strong, and SCC $\\{3\\}$ is terminal.\n",
|
|
"\n",
|
|
"However with this acceptance condition, we would be unable to extract only the strong behaviors: because that would require a way to disable the SCC $\\{0\\}$, and we cannot do that with this acceptance condition. This could be solved by adding a new acceptance set, but if we are about the modify the accepance condition, it seems wiser to simply it.\n",
|
|
"Our solution to this problem is that whenever an acceptance condition is always satisfiable, we consider the entire automaton as weak, and ignore all requests to extract the strong part. As a consequence, the output will always use B\u00fcchi acceptance."
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"collapsed": false,
|
|
"input": [
|
|
"for (name, opt) in (('terminal', 't'), ('strictly weak', 'w'), ('strong', 's'), ('all strengths', 'swt')):\n",
|
|
" a = spot.decompose_strength(aut, opt)\n",
|
|
" if a:\n",
|
|
" a.set_name(name)\n",
|
|
" display(a.show('.asn'))\n",
|
|
" else:\n",
|
|
" print(\"no output for \" + name)"
|
|
],
|
|
"language": "python",
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
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"metadata": {},
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"output_type": "display_data",
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"svg": [
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"<svg height=\"177pt\" viewBox=\"0.00 0.00 347.00 177.00\" width=\"347pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
|
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"<title>G</title>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"143.5\" y=\"-154.8\">terminal</text>\n",
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"<text font-family=\"Lato\" font-size=\"14.00\" text-anchor=\"start\" x=\"148.5\" y=\"-140.8\">Inf(</text>\n",
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"</g>\n",
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"<polygon fill=\"none\" points=\"30,-9 30,-94 82,-94 82,-9 30,-9\" stroke=\"red\"/>\n",
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"<g class=\"node\" id=\"node2\"><title>0</title>\n",
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"</g>\n",
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"<!-- I->0 -->\n",
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"<g class=\"edge\" id=\"edge1\"><title>I->0</title>\n",
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"</g>\n",
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"</g>\n",
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