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# SAT-based Minimization of Deterministic ω-Automata

## Table of Contents

- How to change the SAT solver used
- Enabling SAT-based minimization in `ltl2tgba` or `dstar2tgba`
- More acceptance sets
- Low-level details
- Using `autfilt --sat-minimize` to minimize any deterministic ω-automaton
- Logging statistics
- Python interface

This page explains how to use `ltl2tgba`, `dstar2tgba`, or `autfilt` to minimize deterministic automata using a SAT solver.

Let us first state a few facts about this minimization procedure.

1. The procedure works only on deterministic Büchi automata: any recurrence property can be converted into a deterministic Büchi automaton, and sometimes there are several ways of doing so.
2. Spot actually implements two SAT-based minimization procedures: one that builds a deterministic transition-based Büchi automaton (DTBA), and one that builds a deterministic transition-based ω-automaton with arbitrary acceptance condition (DTωA). In `ltl2tgba` and `dstar2tgba`, the latter procedure is restricted to TGBA. In `autfilt` it can use different and acceptance conditions for input and output, so you could for instance input a Rabin automaton, and produce a Streett automaton.
3. These two procedures can optionally constrain their output to use state-based acceptance. (They simply restrict all the outgoing transitions of a state to belong to the same acceptance sets.)
4. Spot is built using PicoSAT 965. This solver was chosen for its performance, simplicity of integration and license compatibility. However, it is still possible to use an external SAT solver (as described below).
5. `ltl2tgba` and `dstar2tgba` will always try to output an automaton. If they fail to determinize the property, they will simply output a nondeterministic automaton, if they managed to obtain a deterministic automaton but failed to minimize it (e.g., the requested number of states in the final automaton is too low), they will return that "unminimized" deterministic automaton. There are only two cases where these tool will abort without returning an automaton: when the number of clauses output by Spot (and to be fed to the SAT solver) exceeds \(2^{31}\), or when the SAT-solver was killed by a signal. `autfilt --sat-minimize` will only output an automaton if the SAT-based minimization was successful.
6. Our FORTE'14 paper describes the SAT encoding for the minimization of deterministic BA and TGBA. Our LPAR'15 paper describes the generalization of the SAT encoding to deal with deterministic TωA with any acceptance condition.

## How to change the SAT solver used

By default, Spot uses PicoSAT 965, this SAT-solver is built into the Spot library, so that no temporary files are used to store the problem.

The environment variable `SPOT_SATSOLVER` can be used to change the SAT solver used by Spot. This variable should describe a shell command to run the SAT-solver on an input file called `%I` so that a model satisfying the formula will be written in `%O`. For instance to use Glucose 3.0, instead of the builtin version of PicoSAT, define

    export SPOT_SATSOLVER='glucose -verb=0 -model %I >%O'

We assume the SAT solver follows the input/output conventions of the SAT competition

## Enabling SAT-based minimization in `ltl2tgba` or `dstar2tgba`

Both tools follow the same interface, because they use the same post-processing steps internally (i.e., the `spot::postprocessor` class).

First, option `-D` should be used to declare that you are looking for more determinism. This will tweak the translation algorithm used by `ltl2tgba` to improve determinism, and will also instruct the post-processing routine used by both tools to prefer a deterministic automaton over a smaller equivalent nondeterministic automaton.

However, `-D` is not a guarantee to obtain a deterministic automaton, even if one exists. For instance, `-D` fails to produce a deterministic automaton for `a U X(b | GF!b)`. Instead, we get a 4-state non-deterministic automaton.

    ltl2tgba -D 'a U X(b | GF!b)' --stats='states=%s, det=%d'

    states=4, det=0

Option `-x tba-det` enables an additional determinization procedure, that would otherwise not be used by `-D` alone. This procedure will work on any automaton that can be represented by a DTBA; if the automaton to process use multiple acceptance conditions, it will be degeneralized first.

On our example, `-x tba-det` successfully produces a deterministic TBA, but a non-minimal one:

    ltl2tgba -D -x tba-det 'a U X(b | GF!b)' --stats='states=%s, det=%d'

    states=8, det=1

Option `-x sat-minimize` will turn-on SAT-based minimization. It also implies `-x tba-det`, so there is no need to supply both options.

    ltl2tgba -D -x sat-minimize 'a U X(b | GF!b)' --stats='states=%s, det=%d'

    states=5, det=1

We can draw it:

    ltl2tgba -D -x sat-minimize 'a U X(b | GF!b)' -d

Clearly this automaton benefits from the transition-based acceptance. If we want a traditional Büchi automaton, with state-based acceptance, we only need to add the `-B` option. The result will of course be slightly bigger.

    ltl2tgba -BD -x sat-minimize 'a U X(b | GF!b)' -d

There are cases where `ltl2tgba`'s `tba-det` algorithm fails to produce a deterministic automaton. In that case, SAT-based minimization is simply skipped. For instance:

    ltl2tgba -D -x sat-minimize 'G(F(!b & (X!a M (F!a & F!b))) U !b)' --stats='states=%s, det=%d'

    states=5, det=0

The question, of course, is whether there exist a deterministic automaton for this formula, in other words: is this a recurrence property? There are two ways to answer this question using Spot and some help from `ltl2dstar`.

The first is purely syntactic. If a formula belongs to the class of "syntactic recurrence formulas", it expresses a syntactic property. (Of course there are formulas that expresses syntactic properties without being syntactic recurrences.) `ltlfilt` can be instructed to print only formulas that are syntactic recurrences:

    ltlfilt --syntactic-recurrence -f 'G(F(!b & (X!a M (F!a & F!b))) U !b)'

    G(F(!b & (X!a M (F!a & F!b))) U !b)

Since our input formula was output, it expresses a recurrence property.

The second way to check whether a formula is a recurrence is by converting a deterministic Rabin automaton using `dstar2tgba`. The output is guaranteed to be deterministic if and only if the input DRA expresses a recurrence property.

    ltlfilt --remove-wm -f 'G(F(!b & (X!a M (F!a & F!b))) U !b)' -l |
    ltl2dstar --ltl2nba=spin:ltl2tgba@-Ds - - |
    dstar2tgba -D --stats='input(states=%S) output(states=%s, acc-sets=%a, det=%d)'

    input(states=11) output(states=8, acc-sets=1, det=1)

In the above command, `ltldo` is used to convert the LTL formula into `ltl2dstar`'s syntax. Then `ltl2dstar` creates a deterministic Rabin automaton (using `ltl2tgba` as an LTL to BA translator), and the resulting 11-state DRA is converted into a 8-state DTBA by `dstar2tgba`. Since that result is deterministic, we can conclude that the formula was a recurrence.

As far as SAT-based minimization goes, `dstar2tgba` will take the same options as `ltl2tgba`. For instance we can see that the smallest DTBA has 4 states:

    ltlfilt --remove-wm -f 'G(F(!b & (X!a M (F!a & F!b))) U !b)' -l |
    ltl2dstar --ltl2nba=spin:ltl2tgba@-Ds - - |
    dstar2tgba -D -x sat-minimize --stats='input(states=%S) output(states=%s, acc-sets=%a, det=%d)'

    input(states=11) output(states=4, acc-sets=1, det=1)

## More acceptance sets

The formula "`G(F(!b & (X!a M (F!a & F!b))) U !b)`" can in fact be minimized into an even smaller automaton if we use multiple acceptance sets.

Unfortunately because `dstar2tgba` does not know the formula being translated, and it always converts a DRA into a DBA (with a single acceptance set) before further processing, it does not know if using more acceptance sets could be useful to further minimize it. This number of acceptance sets can however be specified on the command-line with option `-x sat-acc=M`. For instance:

    ltlfilt --remove-wm -f 'G(F(!b & (X!a M (F!a & F!b))) U !b)' -l |
    ltl2dstar --ltl2nba=spin:ltl2tgba@-Ds - - |
    dstar2tgba -D -x sat-minimize,sat-acc=2 --stats='input(states=%S) output(states=%s, acc-sets=%a, det=%d)'

    input(states=11) output(states=3, acc-sets=2, det=1)

Beware that the size of the SAT problem is exponential in the number of acceptance sets (adding one acceptance set, in the input automaton or in the output automaton, will double the size of the problem).

The case of `ltl2tgba` is slightly different because it can remember the number of acceptance sets used by the translation algorithm, and reuse that for SAT-minimization even if the automaton had to be degeneralized in the meantime for the purpose of determinization.

## Low-level details

The following figure (from our FORTE'14 paper) gives an overview of the processing chains that can be used to turn an LTL formula into a minimal DBA/DTBA/DTGBA. The blue area at the top describes `ltl2tgba -D -x sat-minimize`, while the purple area at the bottom corresponds to `dstar2tgba -D -x stat-minimize` (but `autfilt` support similar options).

The picture is slightly inaccurate in the sense that both `ltl2tgba` and `dstar2tgba` are actually using the same post-processing chain: only the initial translation to TGBA or conversion to DBA differs, the rest is the same. However, in the case of `dstar2tgba`, no degeneration or determinization are needed.

Also, the picture does not show what happens when `-B` is used: any DTBA is degeneralized into a DBA, before being sent to "DTBA SAT minimization", with a special option to request state-based output.

The WDBA-minimization boxes are able to produce minimal Weak DBA from any TGBA representing an obligation property. In that case using transition-based or generalized acceptance will not allow further reduction. This minimal WDBA is always used when `-D` is given (otherwise, for the default `--small` option, the minimal WDBA is only used if it is smaller than the nondeterministic automaton it has been built from).

The "simplify" boxes are actually simulation-based reductions, and SCC-based simplifications.

The red boxes "not in TCONG" or "not a recurrence" correspond to situations where the tools will produce non-deterministic automata.

The following options can be used to fine-tune this procedure:

`-x tba-det`
  attempt a powerset construction and check if there exists an acceptance set such that the resulting DTBA is equivalent to the input.
`-x sat-minimize`
  enable SAT-based minimization. It is the same as `-x sat-minimize=1` (which is the default value). It performs a binary search to find the correct automaton size. This option implies `-x tba-det`.
`-x sat-minimize=2|3`

  enable SAT-based minimization. Let us consider each intermediate automaton as a `step` towards the minimal automaton and assume `N` as the size of the starting automaton. `2` and `3` have been implemented with the aim of not restarting the encoding from scratch at each step. To do so, both restart the encoding after `N-1-(sat-incr-steps)` states have been won. Now, where is the difference? They both start by encoding the research of the `N-1` step and then:

  - `2` uses PicoSAT assumptions. It adds `sat-incr-steps` assumptions (each of them removing one more state) and then checks directly the `N-1-(sat-incr-steps)` step. If such automaton is found, the process is restarted. Otherwise, a binary search begins between `N-1` and `N-1-sat-incr-steps`. If not provided, `sat-incr-steps` default value is 6.
  - `3` checks incrementally each `N-(2+i)` step, `i` ranging from `0` to `sat-incr-steps`. This process is fully repeated until the minimal automaton is found. The last SAT problem solved correspond to the minimal automaton. `sat-incr-steps` defaults to 2.

  Both implies `-x tba-det`.

`-x sat-minimize=4`
  enable SAT-based minimization. It tries to reduce the size of the automaton one state at a time. This option implies `-x tba-det`.
`-x sat-incr-steps=N`
  set the value of `sat-incr-steps` to N. It does not make sense to use it without `-x sat-minimize=2` or `-x sat-minimize=3`.
`-x sat-acc=$m$`
  attempt to build a minimal DTGBA with \(m\) acceptance sets. This option implies `-x sat-minimize`.
`-x sat-states=$n$`
  attempt to build an equivalent DTGBA with \(n\) states. This also implies `-x sat-minimize` but won't perform any loop to lower the number of states. Note that \(n\) should be the number of states in a complete automaton, while `ltl2tgba` and `dstar2tgba` both remove sink states in their output by default (use option `--complete` to output a complete automaton). Also note that even with the `--complete` option, the output automaton may appear to have fewer states because the other are unreachable.
`-x state-based`
  for all outgoing transition of each state to belong to the same acceptance sets.
`-x !wdba-minimize`
  disable WDBA minimization.

When options `-B` and `-x sat-minimize` are both used, `-x state-based` and `-x sat-acc=1` are implied. Similarly, when option `-S` and `-x sat-minimize` are both used, then option `-x state-based` is implied.

## Using `autfilt --sat-minimize` to minimize any deterministic ω-automaton

This interface is new in Spot 1.99 and allows minimizing any deterministic ω-automaton, regardless of the acceptance condition used. By default, the procedure will try to use the same acceptance condition (or any inferior one) and produce transition-based acceptance.

For our example, let us first generate a deterministic Rabin automaton with `ltl2dstar`.

    ltlfilt -f 'FGa | FGb' -l |
    ltl2dstar --ltl2nba=spin:ltl2tgba@-Ds --output-format=hoa - - > output.hoa

Let's draw it:

    autfilt output.hoa --dot

So this is a state-based Rabin automaton with two pairs. If we call `autfilt` with the `--sat-minimize` option, we can get the following transition-based version (the output may change depending on the SAT solver used):

    autfilt --sat-minimize output.hoa --dot

We can also attempt to build a state-based version with

    autfilt -S --sat-minimize output.hoa --dot

This is clearly smaller than the input automaton. In this example the acceptance condition did not change. The SAT-based minimization only tries to minimize the number of states, but sometimes the simplifications algorithms that are run before we attempt SAT-solving will simplify the acceptance, because even removing a single acceptance set can halve the run time.

You can however force a specific acceptance to be used as output. Let's try with generalized co-Büchi for instance:

    autfilt -S --sat-minimize='acc="generalized-co-Buchi 2"' output.hoa --dot

Note that instead of naming the acceptance condition, you can actually give an acceptance formula in the HOA syntax. For example, we can attempt to create a co-Büchi automaton with

    autfilt -S --sat-minimize='acc="Fin(0)"' output.hoa --dot

When forcing an acceptance condition, you should keep in mind that the SAT-based minimization algorithm will look for automata that have fewer states than the original automaton (after preliminary simplifications). This is not always reasonable. For instance constructing a Streett automaton from a Rabin automaton might require more states. An upper bound on the number of state can be passed using a `max-states=123` argument to `--sat-minimize`.

If the input automaton is transition-based, but option `-S` is used to produce a state-based automaton, then the original automaton is temporarily converted into an automaton with state-based acceptance to obtain an upper bound on the number of states if you haven't specified `max-state`. This upper bound might be larger than the one you would specify by hand.

Here is an example demonstrating the case where the input automaton is smaller than the output. Let's take this small TGBA as input:

    ltl2tgba 'GFa & GFb' >output2.hoa
    autfilt output2.hoa --dot

If we attempt to minimize it into a transition-based Büchi automaton, with fewer states, it will fail, output no result, and return with a non-zero exit code (because no automata were output).

    autfilt --sat-minimize='acc="Buchi"' output2.hoa
    echo $?

    1

However, if we allow more states, it will work:

    autfilt --sat-minimize='acc="Buchi",max-states=3' output2.hoa --dot

By default, the SAT-based minimization tries to find a smaller automaton by performing a binary search starting from `N/2` (N being the size of the starting automaton). After various benchmarks, this algorithm proves to be the best. However, in some cases, other rather similar methods might be better. The algorithm to execute, and some other parameters can be set thanks to the `--sat-minimize` option.

The `--sat-minimize` option takes a comma separated list of arguments that can be any of the following:

`acc=DOUBLEQUOTEDSTRING`
  where the `DOUBLEQUOTEDSTRING` is an acceptance formula in the HOA syntax, or a parameterized acceptance name (the different `acc-name:` options from HOA).
`max-states=N`
  where `N` is an upper-bound on the maximum number of states of the constructed automaton.
`states=M`
  where `M` is a fixed number of states to use in the result (all the states needs not be accessible in the result, so the output might be smaller nonetheless). If this option is used the SAT-based procedure is just used once to synthesize one automaton, and no further minimization is attempted.
`sat-incr`[1|2]=

  `1` and `2` correspond respectively to `-x sat-minimize=2` and `-x sat-minimize=3` options. They have been implemented with the aim of not restarting the encoding from scratch at each step - a step is a minimized intermediate automaton. To do so, both restart the encoding after `N-1-(sat-incr-steps)` states have been won - `N` being the size of the starting automaton. Now, where is the difference? They both start by encoding the research of the `N-1` step and then:

  - `1` uses PicoSAT assumptions. It adds `steps` assumptions (each of them removing one more state) and then checks directly the `N-1-(sat-incr-steps)` step. If such automaton is found, the process is restarted. Otherwise, a binary search begins between `N-1` and `N-1-sat-incr-steps`. If not provided, `sat-incr-steps` defaults to 6.
  - `2` checks incrementally each `N-(2+i)` step, `i` ranging from `0` to `sat-incr-steps`. This process is fully repeated until the minimal automaton is found. The last SAT problem solved correspond to the minimal automaton. `sat-incr-steps` defaults to 2.

  Both implies `-x tba-det`.

`sat-incr-steps=N`
  set the value of `sat-incr-steps` to `N`. This is used by `sat-incr` option.
`sat-naive`
  use the `naive` algorithm to find a smaller automaton. It starts from `N` and then checks `N-1`, `N-2`, etc. until the last successful check.
`sat-langmap`
  Find the lower bound of default sat-minimize procedure. This relies on the fact that the size of the minimal automaton is at least equal to the total number of different languages recognized by the automaton's states.
`colored`
  force all transitions (or all states if `-S` is used) to belong to exactly one acceptance condition.

The `colored` option is useful when used in conjunction with a parity acceptance condition. Indeed, the parity acceptance condition by itself does not require that the acceptance sets form a partition of the automaton (which is expected from parity automata).

Compare the following, where parity acceptance is used, but the automaton is not colored:

    autfilt -S --sat-minimize='acc="parity max even 3"' output2.hoa --dot

… to the following, where the automaton is colored, i.e., each state belongs to exactly one acceptance set:

    autfilt -S --sat-minimize='acc="parity max even 3",colored' output2.hoa --dot

## Logging statistics

If the environment variable `SPOT_SATLOG` is set to the name of a file, the minimization function will append statistics about each of its iterations in this file.

    rm -f stats.csv
    export SPOT_SATLOG=stats.csv
    ltlfilt -f 'Ga R (F!b & (c U b))' -l |
    ltl2dstar --ltl2nba=spin:ltl2tgba@-Ds - - |
    dstar2tgba -D -x sat-minimize,sat-acc=2 --stats='input(states=%S) output(states=%s, acc-sets=%a, det=%d)'

    input(states=11) output(states=5, acc-sets=2, det=1)

Here are the contents of the `stats.csv` file:

input.states | target.states | reachable.states | edges | transitions | variables | clauses | enc.user | enc.sys | sat.user | sat.sys | automaton
--- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | ---
8 | 4 |  |  |  | 5120 | 446320 | 14 | 0 | 13 | 0 |
8 | 6 | 6 | 29 | 48 | 11520 | 1515749 | 41 | 3 | 166 | 0 | HOA:…
8 | 5 |  |  |  | 8000 | 874992 | 25 | 0 | 51 | 0 |

The generated CSV file use the following columns:

- `input.states`: the number of states of the reference automaton at this step
- `target.states`: the n passed to the SAT-based minimization algorithm (it means the input automaton had n+1 states)
- `reachable.states`: number of reachable states in the output of the minimization (with any luck this can be smaller than `target.states`)
- `edges`, `transitions`: number of edges or transitions in the output
- `variables`, `clauses`: size of the SAT problem
- `enc.user`, `enc.sys`: user and system time for encoding the SAT problem
- `sat.user`, `sat.sys`: user and system time for solving the SAT problem
- `automaton`: the automaton produced in HOA format.

Times are measured with the times() function, and expressed in ticks (usually: 1/100 of seconds). The encoding of the automaton in the CSV file follows RFC4180 in escaping double-quote by doubling them.

In the above example, the DRA produced by `ltl2dstar` had 11 states. In the first line of the `stats.csv` file, you can see the minimization function had an 8-state input, which means that `dstar2tgba` first reduced the 11-state (complete) DRA into an 8-state (complete) DBA before calling the SAT-based minimization (the fact that the input was reduced to a DBA is not very obvious from this trace), This first line shows the SAT-based minimization for a (complete) 5-state DTGBA and failing to find one. Then on the next line it looks for a 6-state solution, finds one. Finally, it looks for a 5-state solution, and cannot find one. The reason the 5-state attempt uses the 8-state automaton as input rather than the 6-state automaton is because the 8-state automaton uses 1 acceptance states (it's a DBA) while the 6-state automaton uses 2.

The final output is reported with 5 states, because by default we output trim automata. If the `--complete` option had been given, the useless sink state would have been kept and the output automaton would have 6 states.

## Python interface

See the `satmin.ipynb` notebook.