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decompose_strength: work with inherently weak SCCs

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* wrap/python/tests/decompose.ipynb: Adjust text.
* spot/twaalgos/strength.hh, spot/twaalgos/strength.cc:
Adjust to extract inherently weak SCCs instead of weak SCCs.  This gets
rids of the special handling for the "corner cases".
* spot/tests/strength.test: Adjust.
* NEWS: Mention it.
This commit is contained in:
Alexandre Duret-Lutz 2015-12-09 15:50:13 +01:00
parent 9bbcf85b3a
commit 2e15ed959d
5 changed files with 102 additions and 93 deletions
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115
wrap/python/tests/decompose.ipynb
View file

@ -18,7 +18,7 @@
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@ -48,14 +48,16 @@
"\n",
"# Basics\n",
"\n",
"Let's define the following strengths of SCCs:\n",
"Let's define the following strengths of accepting SCCs:\n",
"\n",
"- an accepting SCC is **weak** if all its transitions belong to the same acceptance sets\n",
"- 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",
"- an accepting SCC is **strictly weak** if it is *weak* and not complete (in other words: *weak* but not *terminal*)\n",
"- an accepting SCC is **strong** if it is not weak.\n",
"- an accepting SCC is **inherently weak** if it does not contain any rejecting cycle\n",
"- an accepting SCC is **inherently terminal** if it is *inherently weak* and complete (i.e. from any state, you can stay in the SCC by reading any word)\n",
"- an accepting SCC is **strictly inherently weak** if it is *inherently weak* and not complete (in other words: *weak* but not *terminal*)\n",
"- an accepting SCC is **strong** if it is not inherently weak.\n",
"\n",
"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."
"The strengths **strong**, **stricly inherently weak**, and **inherently 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.\n",
"\n",
"Note: the reason we use the word *inherently* is that the *weak* and *terminal* properties are usually defined syntactically: an accepting SCC would be weak if all its transitions belong to the same acceptance sets. This syntactic criterion is a sufficient condition for an accepting SCC to not have any rejecting cycle, but it is not necessary. Hence a *weak* SCC is *inherently weak*; but while an *inherently weak* SCC is not necessarily *weak*, it can be modified to be *weak* without alterning the langage."
]
},
{
@ -210,7 +212,7 @@
"</svg>\n"
],
"text": [
"<spot_impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f8dcc0f7a50> >"
"<spot_impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f7340048a20> >"
]
}
],
@ -224,11 +226,11 @@
"\n",
"The letters used for this specification are as follows:\n",
"\n",
"- `t`: terminal\n",
"- `w`: strictly weak\n",
"- `t`: (inherently) terminal\n",
"- `w`: (strictly inherently) weak\n",
"- `s`: strong\n",
"\n",
"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."
"For instance if we want to preserve only the **strictly inherently 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."
]
},
{
@ -327,7 +329,7 @@
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"<spot_impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f7340048990> >"
]
}
],
@ -337,7 +339,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"Similarly, we can extract all the behaviors captured by the **terminal** part of the automaton:"
"Similarly, we can extract all the behaviors captured by the **inherently terminal** part of the automaton:"
]
},
{
@ -468,7 +470,7 @@
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"<spot_impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f8dcc0f7990> >"
"<spot_impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f73400489c0> >"
]
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],
@ -478,7 +480,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"Here is the strong part:"
"Here is the **strong** part:"
]
},
{
@ -558,7 +560,7 @@
"</svg>\n"
],
"text": [
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]
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@ -635,7 +637,7 @@
"</svg>\n"
],
"text": [
"<spot_impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f8dcc11a3f0> >"
"<spot_impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f734006f3c0> >"
]
}
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@ -751,7 +753,7 @@
"</svg>\n"
],
"text": [
"<spot_impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f8dcc11a0f0> >"
"<spot_impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f734006f060> >"
]
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@ -875,7 +877,7 @@
"</svg>\n"
],
"text": [
"<spot_impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f8dcc11a420> >"
"<spot_impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f734006f3f0> >"
]
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{
@ -1018,7 +1020,7 @@
"</svg>\n"
],
"text": [
"<spot_impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f8dcc11a0f0> >"
"<spot_impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f734006f060> >"
]
}
],
@ -1030,7 +1032,7 @@
"source": [
"# Generalized acceptance\n",
"\n",
"There is (almost, see below) nothing that prevents the above decomposition to work with other types of acceptance.\n",
"There is nothing that prevents the above decomposition to work with other types of acceptance.\n",
"\n",
"## Rabin\n",
"\n",
@ -1423,7 +1425,7 @@
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],
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@ -1748,7 +1750,7 @@
"</svg>\n"
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@ -1945,7 +1947,7 @@
"</svg>\n"
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"text": [
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@ -2125,7 +2127,7 @@
"</svg>\n"
],
"text": [
"<spot_impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f8dcc11a0c0> >"
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@ -2137,7 +2139,7 @@
"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."
"When extracting multiple strengths and one of the strength is **strong**, we preserve the original acceptance. For instance extracting **strong** and **inherently terminal** gives the following automaton, where only **stricly inherently weak** SCCs have become rejecting."
]
},
{
@ -2460,7 +2462,7 @@
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],
"text": [
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"<spot_impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f734006fbd0> >"
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@ -2477,7 +2479,7 @@
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"collapsed": false,
"input": [
"for (name, opt) in (('terminal', 't'), ('strictly weak', 'w'), ('strong', 's')):\n",
"for (name, opt) in (('inherently terminal', 't'), ('strictly inherently weak', 'w'), ('strong', 's')):\n",
" a = spot.decompose_strength(aut, opt).postprocess('deterministic', 'SBAcc')\n",
" a.set_name(name)\n",
" display(a)"
@ -2500,7 +2502,7 @@
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"<text text-anchor=\"start\" x=\"177.37\" y=\"-148.8\" font-family=\"Lato\" font-size=\"14.00\">terminal</text>\n",
"<text text-anchor=\"start\" x=\"142.87\" y=\"-148.8\" font-family=\"Lato\" font-size=\"14.00\">inherently terminal</text>\n",
"<text text-anchor=\"start\" x=\"182.37\" y=\"-134.8\" font-family=\"Lato\" font-size=\"14.00\">Inf(</text>\n",
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@ -2604,7 +2606,7 @@
"</svg>\n"
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"text": [
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"<spot_impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f734006ff60> >"
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},
{
@ -2622,7 +2624,7 @@
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"<title>G</title>\n",
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"<text text-anchor=\"start\" x=\"164.87\" y=\"-335.8\" font-family=\"Lato\" font-size=\"14.00\">strictly weak</text>\n",
"<text text-anchor=\"start\" x=\"130.87\" y=\"-335.8\" font-family=\"Lato\" font-size=\"14.00\">strictly inherently weak</text>\n",
"<text text-anchor=\"start\" x=\"183.87\" y=\"-321.8\" font-family=\"Lato\" font-size=\"14.00\">Inf(</text>\n",
"<text text-anchor=\"start\" x=\"205.87\" y=\"-321.8\" font-family=\"Lato\" font-size=\"14.00\" fill=\"#5da5da\">\u24ff</text>\n",
"<text text-anchor=\"start\" x=\"221.87\" y=\"-321.8\" font-family=\"Lato\" font-size=\"14.00\">)</text>\n",
@ -2723,7 +2725,7 @@
"</svg>\n"
],
"text": [
"<spot_impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f8dcc11acf0> >"
"<spot_impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f734006fcc0> >"
]
},
{
@ -2903,7 +2905,7 @@
"</svg>\n"
],
"text": [
"<spot_impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f8dcc11a450> >"
"<spot_impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f734006f6c0> >"
]
}
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@ -3260,7 +3262,7 @@
"</svg>\n"
],
"text": [
"<spot_impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f8dcc0c2120> >"
"<spot_impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f733338d0f0> >"
]
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@ -3270,7 +3272,7 @@
"cell_type": "code",
"collapsed": false,
"input": [
"for (name, opt) in (('terminal', 't'), ('strictly weak', 'w'), ('strong', 's')):\n",
"for (name, opt) in (('inherently terminal', 't'), ('strictly inherently weak', 'w'), ('strong', 's')):\n",
" a = spot.decompose_strength(aut, opt)\n",
" a.set_name(name)\n",
" display(a)"
@ -3293,7 +3295,7 @@
"<g id=\"graph0\" class=\"graph\" transform=\"scale(0.688337 0.688337) rotate(0) translate(4 519)\">\n",
"<title>G</title>\n",
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"<text text-anchor=\"start\" x=\"461.48\" y=\"-500.8\" font-family=\"Lato\" font-size=\"14.00\">terminal</text>\n",
"<text text-anchor=\"start\" x=\"426.98\" y=\"-500.8\" font-family=\"Lato\" font-size=\"14.00\">inherently terminal</text>\n",
"<text text-anchor=\"start\" x=\"466.48\" y=\"-486.8\" font-family=\"Lato\" font-size=\"14.00\">Inf(</text>\n",
"<text text-anchor=\"start\" x=\"488.48\" y=\"-486.8\" font-family=\"Lato\" font-size=\"14.00\" fill=\"#5da5da\">\u24ff</text>\n",
"<text text-anchor=\"start\" x=\"504.48\" y=\"-486.8\" font-family=\"Lato\" font-size=\"14.00\">)</text>\n",
@ -3550,7 +3552,7 @@
"</svg>\n"
],
"text": [
"<spot_impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f8dcc0c20c0> >"
"<spot_impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f733338d090> >"
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@ -3568,7 +3570,7 @@
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"<title>G</title>\n",
"<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-392 587.22,-392 587.22,4 -4,4\"/>\n",
"<text text-anchor=\"start\" x=\"251.61\" y=\"-373.8\" font-family=\"Lato\" font-size=\"14.00\">strictly weak</text>\n",
"<text text-anchor=\"start\" x=\"217.61\" y=\"-373.8\" font-family=\"Lato\" font-size=\"14.00\">strictly inherently weak</text>\n",
"<text text-anchor=\"start\" x=\"270.61\" y=\"-359.8\" font-family=\"Lato\" font-size=\"14.00\">Inf(</text>\n",
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@ -3702,7 +3704,7 @@
"</svg>\n"
],
"text": [
"<spot_impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f8dcc11a450> >"
"<spot_impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f734006f6c0> >"
]
},
{
@ -3839,7 +3841,7 @@
"</svg>\n"
],
"text": [
"<spot_impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f8dcc0c20c0> >"
"<spot_impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f733338d090> >"
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@ -3851,7 +3853,7 @@
"source": [
"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",
"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."
"This is easy to understand using an example. In the following extraction of the **strong** and **inherently terminal** parts, the rejecting SCCs (that were either rejecting or strictly inherently weak originally) have been labeled by the same acceptance sets, to ensure that they are rejected."
]
},
{
@ -4142,7 +4144,7 @@
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],
"text": [
"<spot_impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f8dcc0c2210> >"
"<spot_impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f733338d1e0> >"
]
}
],
@ -4154,15 +4156,13 @@
"source": [
"## Corner cases\n",
"\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",
"When the acceptance condition is always satisfiable, all non-trivial SCCs are accepting, and inherently weak.\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:"
"This occur frequently whant translating LTL formulas that are safety properties:"
]
},
{
@ -4255,7 +4255,7 @@
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@ -4347,7 +4347,7 @@
"</svg>\n"
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"text": [
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{
@ -4413,7 +4413,7 @@
"</svg>\n"
],
"text": [
"<spot_impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f8dcc0c23f0> >"
"<spot_impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f733338d3c0> >"
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}
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@ -4499,7 +4499,7 @@
"</svg>\n"
],
"text": [
"<spot_impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f8dcc0c2450> >"
"<spot_impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f733338d420> >"
]
}
],
@ -4518,7 +4518,7 @@
"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."
"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 SCCs that are *inherently weak* but not *weak*."
]
},
{
@ -4654,7 +4654,7 @@
"</svg>\n"
],
"text": [
"<spot_impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f8dcc0c24e0> >"
"<spot_impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f733338d4b0> >"
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],
@ -4664,10 +4664,7 @@
"cell_type": "markdown",
"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."
"By our definitions, SCC $\\{0\\}$ and $\\{1,2\\}$ are inherently weak, and SCC $\\{3\\}$ is terminal."
]
},
{
@ -4786,7 +4783,7 @@
"</svg>\n"
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"text": [
"<spot_impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f8dcc0c2d50> >"
"<spot_impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f734006f6c0> >"
]
},
{
@ -4874,7 +4871,7 @@
"</svg>\n"
],
"text": [
"<spot_impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f8dcc11a450> >"
"<spot_impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f734006fc60> >"
]
},
{
@ -4991,7 +4988,7 @@
"</svg>\n"
],
"text": [
"<spot_impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f8dcc11a450> >"
"<spot_impl.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7f734006fc60> >"
]
}
],