1{ 2 "cells": [ 3 { 4 "cell_type": "code", 5 "execution_count": 1, 6 "metadata": {}, 7 "outputs": [], 8 "source": [ 9 "import spot\n", 10 "from spot.jupyter import display_inline\n", 11 "spot.setup(show_default='.bav')" 12 ] 13 }, 14 { 15 "cell_type": "markdown", 16 "metadata": {}, 17 "source": [ 18 "# Support for alternating automata" 19 ] 20 }, 21 { 22 "cell_type": "markdown", 23 "metadata": {}, 24 "source": [ 25 "The following automata are what we will use as examples." 26 ] 27 }, 28 { 29 "cell_type": "code", 30 "execution_count": 2, 31 "metadata": {}, 32 "outputs": [], 33 "source": [ 34 "aut1, aut2, aut3, aut4, aut5 = spot.automata('''\n", 35 "HOA: v1 tool: \"ltl3ba\" \"1.1.3\" name: \"VWAA for FGa && GFb\" States: 6\n", 36 "Start: 0 acc-name: co-Buchi Acceptance: 1 Fin(0) AP: 2 \"a\" \"b\" properties:\n", 37 "trans-labels explicit-labels state-acc univ-branch very-weak --BODY--\n", 38 "State: 0 \"(FG(a) && GF(b))\" [t] 3&1 State: 1 \"GF(b)\" [(1)] 1 [(!1)]\n", 39 "2&1 State: 2 \"F(b)\" {0} [(1)] 5 [(!1)] 2 State: 3 \"FG(a)\" {0} [(0)] 4\n", 40 "[t] 3 State: 4 \"G(a)\" [(0)] 4 State: 5 \"t\" [t] 5 --END--\n", 41 "/* Example from ADL's PSL2TGBA talk. */\n", 42 "HOA: v1 States: 3 Start: 0 acc-name: co-Buchi Acceptance: 1 Fin(0) AP:\n", 43 "3 \"a\" \"b\" \"p\" --BODY-- State: 0 \"(a;a*;b)*\" {0} [0] 1 [!0] 2 State: 1\n", 44 "\"a*;b;(a;a*;b)*\" {0} [0&1&2] 0&1 [!1&2] 1 [!0&!1] 2 [!0&1&2] 0 State:\n", 45 "2 [t] 2 --END--\n", 46 "HOA: v1 States: 5 Start: 3 acc-name: co-Buchi Acceptance: 1 Fin(0) AP:\n", 47 "3 \"a\" \"b\" \"p\" --BODY-- State: 0 \"(a;a*;b)*\" {0} [0] 1 [!0] 2 State: 1\n", 48 "\"a*;b;(a;a*;b)*\" {0} [0&1&2] 0&1 [!1&2] 1 [!0&!1] 2 [!0&1&2] 0 State:\n", 49 "2 [t] 2 State: 3 [0] 4&0 State: 4 [t] 3 --END--\n", 50 "HOA: v1 States: 3 Start: 0 acc-name: co-Buchi Acceptance: 1 Fin(0) AP:\n", 51 "3 \"a\" \"b\" \"p\" --BODY-- State: 0 \"(a;a*;b)*\" {0} [0] 1 [!0] 2 State: 1\n", 52 "\"a*;b;(a;a*;b)*\" {0} [0&1&2] 0&1 [!1&2] 1 [!0&!1] 2 [!0&1&2] 0 State:\n", 53 "2 [t] 2 --END--\n", 54 "HOA: v1 tool: \"ltl3dra\" \"0.2.2\" name: \"VWAA for GFa\" States: 3 Start: 0\n", 55 "acc-name: co-Buchi Acceptance: 1 Fin(0) AP: 1 \"a\" properties: trans-labels\n", 56 "explicit-labels state-acc univ-branch very-weak --BODY-- State: 0 \"GF(a)\"\n", 57 "[t] 1&0 State: 1 \"F(a)\" {0} [(0)] 2 [t] 1 State: 2 \"t\" [t] 2 --END--\n", 58 "''')" 59 ] 60 }, 61 { 62 "cell_type": "markdown", 63 "metadata": {}, 64 "source": [ 65 "## Various display options" 66 ] 67 }, 68 { 69 "cell_type": "markdown", 70 "metadata": {}, 71 "source": [ 72 "Here is the default output, using the `bav` options as set by default in the first cell. " 73 ] 74 }, 75 { 76 "cell_type": "code", 77 "execution_count": 3, 78 "metadata": {}, 79 "outputs": [ 80 { 81 "data": { 82 "text/html": [ 83 "<div style='vertical-align:text-top;display:inline-block;'><?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n", 84 "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n", 85 " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n", 86 "<!-- Generated by graphviz version 2.43.0 (0)\n", 87 " -->\n", 88 "<!-- Title: VWAA for FGa && GFb Pages: 1 -->\n", 89 "<svg width=\"129pt\" height=\"360pt\"\n", 90 " viewBox=\"0.00 0.00 129.40 360.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n", 91 "<g id=\"graph0\" class=\"graph\" transform=\"scale(0.7246376811594204 0.7246376811594204) rotate(0) translate(4 492.6)\">\n", 92 "<title>VWAA for FGa && GFb</title>\n", 93 "<polygon fill=\"white\" stroke=\"transparent\" points=\"-4,4 -4,-492.6 174.5,-492.6 174.5,4 -4,4\"/>\n", 94 "<text text-anchor=\"start\" 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"</svg>\n", 242 "</div><div style='vertical-align:text-top;display:inline-block;'><?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n", 243 "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n", 244 " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n", 245 "<!-- Generated by graphviz version 2.43.0 (0)\n", 246 " -->\n", 247 "<!-- Pages: 1 -->\n", 248 "<svg width=\"201pt\" height=\"296pt\"\n", 249 " viewBox=\"0.00 0.00 200.64 296.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n", 250 "<g id=\"graph0\" class=\"graph\" transform=\"scale(1.0 1.0) rotate(0) translate(4 292)\">\n", 251 "<polygon fill=\"white\" stroke=\"transparent\" points=\"-4,4 -4,-292 196.64,-292 196.64,4 -4,4\"/>\n", 252 "<text text-anchor=\"start\" x=\"74.82\" y=\"-273.8\" font-family=\"Lato\" font-size=\"14.00\">Fin(</text>\n", 253 "<text text-anchor=\"start\" x=\"97.82\" y=\"-273.8\" font-family=\"Lato\" font-size=\"14.00\" fill=\"#1f78b4\">⓿</text>\n", 254 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id=\"edge1\" class=\"edge\">\n", 1966 "<title>I->0</title>\n", 1967 "<path fill=\"none\" stroke=\"black\" d=\"M26,-254.45C26,-252.81 26,-238.45 26,-224.97\"/>\n", 1968 "<polygon fill=\"black\" stroke=\"black\" points=\"26,-217.66 29.15,-224.66 26,-221.16 26,-224.66 26,-224.66 26,-224.66 26,-221.16 22.85,-224.66 26,-217.66 26,-217.66\"/>\n", 1969 "</g>\n", 1970 "<!-- -1 -->\n", 1971 "<g id=\"node3\" class=\"node\">\n", 1972 "<title>-1</title>\n", 1973 "<ellipse fill=\"#ffffaa\" stroke=\"black\" cx=\"26\" cy=\"-128.8\" rx=\"1.8\" ry=\"1.8\"/>\n", 1974 "</g>\n", 1975 "<!-- 0->-1 -->\n", 1976 "<g id=\"edge2\" class=\"edge\">\n", 1977 "<title>0->-1</title>\n", 1978 "<path fill=\"none\" stroke=\"black\" d=\"M18.1,-181.38C14.69,-171.69 12.07,-159.41 15,-148.6 16.19,-144.21 18.69,-139.8 21,-136.35\"/>\n", 1979 "<polygon fill=\"none\" stroke=\"black\" points=\"22.97,-137.81 25.2,-130.73 19.05,-134.87 22.97,-137.81\"/>\n", 1980 "<text text-anchor=\"start\" x=\"15\" y=\"-152.4\" font-family=\"Lato\" font-size=\"14.00\">1</text>\n", 1981 "</g>\n", 1982 "<!-- -1->0 -->\n", 1983 "<g id=\"edge3\" class=\"edge\">\n", 1984 "<title>-1->0</title>\n", 1985 "<path fill=\"none\" stroke=\"black\" d=\"M26,-131.19C26,-136.7 26,-157.51 26,-174.57\"/>\n", 1986 "<polygon fill=\"black\" stroke=\"black\" points=\"26,-181.59 22.85,-174.59 26,-178.09 26,-174.59 26,-174.59 26,-174.59 26,-178.09 29.15,-174.59 26,-181.59 26,-181.59\"/>\n", 1987 "</g>\n", 1988 "<!-- 1 -->\n", 1989 "<g id=\"node4\" class=\"node\">\n", 1990 "<title>1</title>\n", 1991 "<g id=\"a_node4\"><a xlink:title=\"F(a)\">\n", 1992 "<path fill=\"#ffffaa\" stroke=\"black\" d=\"M32,-90C32,-90 20,-90 20,-90 14,-90 8,-84 8,-78 8,-78 8,-64 8,-64 8,-58 14,-52 20,-52 20,-52 32,-52 32,-52 38,-52 44,-58 44,-64 44,-64 44,-78 44,-78 44,-84 38,-90 32,-90\"/>\n", 1993 "<text text-anchor=\"start\" x=\"21.5\" y=\"-74.8\" font-family=\"Lato\" font-size=\"14.00\">1</text>\n", 1994 "<text text-anchor=\"start\" x=\"18\" y=\"-59.8\" 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font-size=\"14.00\">1</text>\n", 2010 "</g>\n", 2011 "<!-- T2T1 -->\n", 2012 "<!-- 1->T2T1 -->\n", 2013 "<g id=\"edge5\" class=\"edge\">\n", 2014 "<title>1->T2T1</title>\n", 2015 "<path fill=\"none\" stroke=\"black\" d=\"M26,-51.87C26,-37.64 26,-18.56 26,-8.29\"/>\n", 2016 "<polygon fill=\"black\" stroke=\"black\" points=\"26,-1.09 29.15,-8.09 26,-4.59 26,-8.09 26,-8.09 26,-8.09 26,-4.59 22.85,-8.09 26,-1.09 26,-1.09\"/>\n", 2017 "<text text-anchor=\"start\" x=\"26\" y=\"-22.8\" font-family=\"Lato\" font-size=\"14.00\">a</text>\n", 2018 "</g>\n", 2019 "</g>\n", 2020 "</svg>\n", 2021 "</div>" 2022 ], 2023 "text/plain": [ 2024 "<IPython.core.display.HTML object>" 2025 ] 2026 }, 2027 "metadata": {}, 2028 "output_type": "display_data" 2029 } 2030 ], 2031 "source": [ 2032 "display_inline(aut1, aut2, aut3, aut4, aut5, show='.bav1u')" 2033 ] 2034 }, 2035 { 2036 "cell_type": "markdown", 2037 "metadata": {}, 2038 "source": [ 2039 "Let's make sure that option `u` and `s` (to display SCCs) work well together:" 2040 ] 2041 }, 2042 { 2043 "cell_type": "code", 2044 "execution_count": 6, 2045 "metadata": {}, 2046 "outputs": [ 2047 { 2048 "data": { 2049 "text/html": [ 2050 "<div style='vertical-align:text-top;display:inline-block;'><?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n", 2051 "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n", 2052 " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n", 2053 "<!-- Generated by graphviz version 2.43.0 (0)\n", 2054 " -->\n", 2055 "<!-- Title: VWAA for FGa && GFb Pages: 1 -->\n", 2056 "<svg width=\"147pt\" height=\"360pt\"\n", 2057 " viewBox=\"0.00 0.00 147.40 360.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n", 2058 "<g id=\"graph0\" class=\"graph\" transform=\"scale(0.78125 0.78125) rotate(0) translate(4 457.6)\">\n", 2059 "<title>VWAA for FGa && GFb</title>\n", 2060 "<polygon fill=\"white\" stroke=\"transparent\" points=\"-4,4 -4,-457.6 185,-457.6 185,4 -4,4\"/>\n", 2061 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} 2702 ], 2703 "source": [ 2704 "display_inline(aut1, aut2, aut3, aut4, aut5, show='.bav1us')" 2705 ] 2706 }, 2707 { 2708 "cell_type": "markdown", 2709 "metadata": {}, 2710 "source": [ 2711 "## Alternation removal" 2712 ] 2713 }, 2714 { 2715 "cell_type": "markdown", 2716 "metadata": {}, 2717 "source": [ 2718 "The `remove_alternation()` function works on any alternating automaton that is weak (not necessarily very weak), i.e., in each SCC all transition should belong to the same accepting sets.\n", 2719 "\n", 2720 "The second argument of `remove_alternation()`, set to `True` below, simply asks for states to be labeled to help debugging. 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class=\"edge\">\n", 3165 "<title>2->4</title>\n", 3166 "<path fill=\"none\" stroke=\"black\" d=\"M55.32,-188.99C63.75,-179.15 74.75,-166.65 85.03,-156 89.35,-151.52 94.1,-146.85 98.7,-142.44\"/>\n", 3167 "<polygon fill=\"black\" stroke=\"black\" points=\"104.02,-137.41 101.1,-144.51 101.48,-139.82 98.93,-142.23 98.93,-142.23 98.93,-142.23 101.48,-139.82 96.77,-139.94 104.02,-137.41 104.02,-137.41\"/>\n", 3168 "<text text-anchor=\"start\" x=\"85.03\" y=\"-159.8\" font-family=\"Lato\" font-size=\"14.00\">a & b</text>\n", 3169 "</g>\n", 3170 "<!-- 3->3 -->\n", 3171 "<g id=\"edge11\" class=\"edge\">\n", 3172 "<title>3->3</title>\n", 3173 "<path fill=\"none\" stroke=\"black\" d=\"M79.91,-27.32C89.49,-27.84 97.53,-24.74 97.53,-18 97.53,-13.05 93.19,-10.06 87.09,-9.03\"/>\n", 3174 "<polygon fill=\"black\" stroke=\"black\" points=\"79.91,-8.68 87.05,-5.88 83.4,-8.85 86.9,-9.02 86.9,-9.02 86.9,-9.02 83.4,-8.85 86.75,-12.17 79.91,-8.68 79.91,-8.68\"/>\n", 3175 "<text text-anchor=\"start\" 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x=\"166.03\" y=\"-108.8\" font-family=\"Lato\" font-size=\"14.00\" fill=\"#1f78b4\">⓿</text>\n", 3199 "</g>\n", 3200 "</g>\n", 3201 "</svg>\n", 3202 "</div>" 3203 ], 3204 "text/plain": [ 3205 "<IPython.core.display.HTML object>" 3206 ] 3207 }, 3208 "metadata": {}, 3209 "output_type": "display_data" 3210 } 3211 ], 3212 "source": [ 3213 "nba1t = spot.remove_alternation(aut1, True)\n", 3214 "nba1 = spot.scc_filter(nba1t, True)\n", 3215 "display_inline(aut1.show('.bav1u'), nba1t, nba1)" 3216 ] 3217 }, 3218 { 3219 "cell_type": "markdown", 3220 "metadata": {}, 3221 "source": [ 3222 "Let's apply this process to the other 4 automata (which are not very-weak, unlike `aut1`). The states marked with `~` are part of a break-point construction." 3223 ] 3224 }, 3225 { 3226 "cell_type": "code", 3227 "execution_count": 8, 3228 "metadata": {}, 3229 "outputs": [ 3230 { 3231 "data": { 3232 "text/html": [ 3233 "<div style='vertical-align:text-top;display:inline-block;'><?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n", 3234 "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n", 3235 " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n", 3236 "<!-- Generated by graphviz version 2.43.0 (0)\n", 3237 " -->\n", 3238 "<!-- Pages: 1 -->\n", 3239 "<svg width=\"230pt\" height=\"360pt\"\n", 3240 " viewBox=\"0.00 0.00 229.87 360.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n", 3241 "<g id=\"graph0\" class=\"graph\" transform=\"scale(0.9523809523809523 0.9523809523809523) rotate(0) translate(4 375)\">\n", 3242 "<polygon fill=\"white\" stroke=\"transparent\" points=\"-4,4 -4,-375 238,-375 238,4 -4,4\"/>\n", 3243 "<text 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explicit-labels state-acc\n", 4263 "semi-deterministic --BODY-- State: 0 [0] 1 [!0] 2 [!0] 3 State: 1 [!1]\n", 4264 "1 [1] 4 State: 2 {0} [!1] 2 State: 3 [!0] 3 [!0] 5 State: 4 {0} [t] 4\n", 4265 "State: 5 {0} [!0&!1] 5 --END--\"\"\")\n", 4266 "altneg = spot.dualize(pos)\n", 4267 "neg = spot.remove_alternation(altneg)\n", 4268 "display_inline(pos, altneg.show('.bvu'), neg)" 4269 ] 4270 }, 4271 { 4272 "cell_type": "code", 4273 "execution_count": 10, 4274 "metadata": {}, 4275 "outputs": [], 4276 "source": [ 4277 "# Issue #382.\n", 4278 "w = spot.parse_word('cycle{!a&b}').as_automaton()\n", 4279 "assert pos.intersects(w) != neg.intersects(w)" 4280 ] 4281 } 4282 ], 4283 "metadata": { 4284 "kernelspec": { 4285 "display_name": "Python 3", 4286 "language": "python", 4287 "name": "python3" 4288 }, 4289 "language_info": { 4290 "codemirror_mode": { 4291 "name": "ipython", 4292 "version": 3 4293 }, 4294 "file_extension": ".py", 4295 "mimetype": "text/x-python", 4296 "name": "python", 4297 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