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spot/tests/python/satmin.ipynb
Florian Renkin 6dc11b4715 notebooks: correction of typos 
* tests/python/_partitioned_relabel.ipynb,
  tests/python/_product_weak.ipynb,
  tests/python/acc_cond.ipynb,
  tests/python/aliases.ipynb,
  tests/python/automata.ipynb,
  tests/python/cav22-figs.ipynb,
  tests/python/contains.ipynb,
  tests/python/decompose.ipynb,
  tests/python/formulas.ipynb,
  tests/python/games.ipynb,
  tests/python/highlighting.ipynb,
  tests/python/ltsmin-dve.ipynb,
  tests/python/parity.ipynb,
  tests/python/product.ipynb,
  tests/python/satmin.ipynb,
  tests/python/stutter-inv.ipynb,
  tests/python/synthesis.ipynb,
  tests/python/twagraph-internals.ipynb,
  tests/python/word.ipynb,
  tests/python/zlktree.ipynb: here
2023-09-26 11:56:26 +02:00

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{
"cells": [
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"cell_type": "markdown",
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"source": [
"# SAT-based minimization of deterministic ω-automata\n",
"\n",
"The `spot.sat_minimize()` Python function is the main entry point for minimizing any **deterministic** ω-automaton. This notebook demonstrates how to use that function.\n",
"\n",
"**Warning:** while the automata used in this notebook are quite small, working with large automata can require a lot of RAM and take huge amount of time. In its most straightforward variant, `sat_minimize()` takes a input automaton (called *reference*) and then makes a loop to ask a SAT-solver for an equivalent automaton (called *candidate*) with 1 fewer state at each iteration. If the reference has size ($n_i$, $s_i$), i.e. $n_i$ states, $s_i$ acceptance sets, and the candidate has size $(n_o, s_o)$, the SAT encoding uses $\\mathrm{O}(n_i^2\\times n_o^2\\times 2^{s_i+s_o})$ variables and $\\mathrm{O}(n_i^2 \\times n_o^3\\times 2^{s_i+2s_o}\\times |\\Sigma|)$ clauses. Reducing the number of acceptance set the therefore the most important way to simplify a problem."
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"import spot\n",
"spot.setup(show_default='.b')\n",
"from IPython.display import display"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Minimizing DBA\n",
"\n",
"Let's take a simple formula and translate it into a DBA:"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\mathsf{G} \\mathsf{F} (a \\leftrightarrow \\mathsf{X} \\mathsf{X} b)$"
],
"text/plain": [
"spot.formula(\"GF(a <-> XXb)\")"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"f = spot.formula('GF(a <-> XXb)'); f"
]
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{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
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"source": [
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"source": [
"The above automaton is not minimal and is easily reduced by `sat_minimize()`:"
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"source": [
"## Minimizing deterministic ω-automata with arbitrary acceptance condition\n",
"\n",
"Now let's look at examples with more complicated acceptance conditions. \n",
"The following Rabin automaton was produced using ltl2dstar 0.5.4 and spot 2.5.2 with\n",
"```\n",
"ltlfilt --lbt -f '(FGa | Fb) & FGc' | ltl2dstar -H --ltl2nba=spin:ltl2tgba@-Ds - -\n",
"```\n",
"however we hardcode it so that the notebook can be used even with `ltl2dstar` installed."
]
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"text/plain": [
"<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7fe1382323f0> >"
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}
],
"source": [
"large = spot.automaton('''\n",
"HOA: v1 States: 6 properties: implicit-labels trans-labels no-univ-branch\n",
"deterministic complete stutter-invariant tool: \"ltl2dstar\" \"0.5.4\"\n",
"name: \"& | F G a F b F G c\" comment: \"Safra[NBA=4]\" acc-name: Rabin 2\n",
"Acceptance: 4 (Fin(0)&Inf(1))|(Fin(2)&Inf(3)) Start: 5 AP: 3 \"a\" \"b\"\n",
"\"c\" --BODY-- State: 0 {1 3} 4 4 4 4 1 0 1 0 State: 1 {0 3} 4 4 4 4 1 1\n",
"1 1 State: 2 {1 2} 4 4 4 4 2 2 2 2 State: 3 {1 2} 5 5 4 4 5 3 4 0 State:\n",
"4 {0 2} 4 4 4 4 2 2 2 2 State: 5 {0 2} 5 5 4 4 5 3 2 2 --END--''')\n",
"large.merge_edges()\n",
"large"
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},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"It can be minimized as a 2-state transition-based Rabin automaton:"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
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"But do we really need 2 Rabin pairs? Let's ask if we can get an equivalent with only one pair. (Note that reducing the number of pairs might require more state, but the `sat_minimize()` function will never attempt to add state unless explicitly instructed to do so. In this case we are therefore looking for a state-based Rabin-1 automaton with at most 4 states.)"
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"Using the `display_log` option, we can have a hint of what is going on under the hood. Each line in the table shows one call to the SAT solver. The column labeled `target.states` gives the size of the equivalent automaton we ask the SAT-solver to produce, but some of these states may actually be unreachable in the result. The `variables` and `clauses` columns give an indication of the size of the SAT problem. The `enc.*` and `sat.*` columns give the user and system time taken to encode and solve the SAT problem (the unit is \"ticks\", which usually is 1/100 of seconds).\n",
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"How did the procedure look for a complete automaton of size 5 when the input had only 2 states? It's because the input uses transition-based acceptance: to estimate an upper bound of the size of the state-based output, the `sat_minimize()` procedure converted its transition-based input to state-based acceptance (using the `spot.sbacc()` function) and counted the number of states in the result.\n",
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"## Variant for incremental SAT solving\n",
"\n",
"Using `sat_incr=1`, we encode the problem of finding an equivalent automaton with $n$ states, and add 6 additional variables and some additional constraints to the problem:\n",
"\n",
"| variable | implied constraints |\n",
"| :--- | ---: |\n",
"| $v_1$ | transitions to state $(n-1)$ must not be used | \n",
"| $v_2$ | $v_1\\land{}$ transitions to state $(n-2)$ must not be used |\n",
"| ... | |\n",
"| $v_6$ | $v_5\\land{}$ transitions to state $(n-5)$ must not be used |\n",
"\n",
"Now using `assume` directives on variable $v_i$ amounts to testing whether the problem is solved with $n-i$ states, but we do not have to reencode the problem for each test, and the solver can (probably) reuse some of the knowledge it gathered during a previous attempt. We do a binary search on these 6 assumptions, to find some $i$ such that the problem is satisfiable with assumption $v_i$ but not with $v_{i+1}$. If such cast exists, we have found the minimal automaton. If assumption $v_6$ is satisfiable, we re-encode the problem with $n-7$ states and start over. Watch how the number of variables and clauses do not change in the following log.\n",
"\n",
"The number of assumption variables to use in a one encoding can be set with the `sat_incr_steps` argument. Its default value of 6 was chosen empirically by benchmarking different values."
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"Another incremental variant consists is the equivalent of forcing $v_1$, $v_2$, ... in order. But to do that we do not need to use any assumption. We just add the constraints that transitions going to state $n-i$ are forbidden. This variant is enabled by option `sat_incr=2`. As in the previous case, we do a few of those incremental steps (2 by default, but that can be changed with the `sat_incr_steps` parameter) and then we reencode the problem to reduce its size.\n",
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"## Miscellaneous options\n",
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"### `return_log`\n",
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" input.states target.states reachable.states edges transitions variables \\\n",
"0 6 6 5 13 40 2742 \n",
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" automaton \n",
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"source": [
"aut, log = spot.sat_minimize(large, acc='co-Buchi', sat_incr=2, state_based=True, return_log=True)\n",
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{
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"source": [
"Here is how we can extract the automata from that log:"
]
},
{
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"metadata": {},
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{
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"text": [
"automaton from line 0:\n"
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"text/plain": [
"<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7fe120526cc0> >"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"for line, data in log.iterrows():\n",
" if type(data.automaton) is str:\n",
" print(\"automaton from line {}:\".format(line))\n",
" display(spot.automaton(data.automaton + \"\\n\"))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## `sat_langmap`\n",
"\n",
"When using the default binary search approach, the `sat_langmap=True` can help refine the lower bound by first testing the language-equivalence of all states in the automaton. This allows to form equivalence classes of states, and clearly the minimal automaton needs at least as many states as the number of equivalence states.\n",
"\n",
"For instance in the `large` automaton we use as example, the 6 states correspond to only two different languages. This can be seen with the `highlight_language()` function, which colors states with identical languages. This information can be used by the minimization function to search a minimal automaton between 2 and 6 states."
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {},
"outputs": [
{
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"source": [
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"source": [
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"execution_count": 19,
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"source": [
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"source": [
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"### `states`\n",
"\n",
"Sometimes we do not want a minimization loop, we just want to generate an equivalent automaton with a given number of states. In that case, we use the `states` option. However there is no constraint that all states should be reachable, so in the end, you could end with an automaton with fewer states than requested."
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"text/plain": [
"<spot.twa_graph; proxy of <Swig Object of type 'std::shared_ptr< spot::twa_graph > *' at 0x7fe120556f00> >"
]
},
"execution_count": 21,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"spot.sat_minimize(small, acc=\"co-Buchi\", states=7, state_based=True, display_log=True)"
]
}
],
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