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# `ltlsynt`

## Table of Contents

- Basic usage
- TLSF
- Internal details
- Other useful options
- References

## Basic usage

This tool synthesizes reactive controllers from LTL/PSL formulas.

Consider a set \(I\) of input atomic propositions, a set \(O\) of output atomic propositions, and a PSL formula φ over the propositions in \(I \cup O\). A reactive controller realizing φ is a function \(c: (2^{I})^\star \times 2^I \mapsto 2^O\) such that, for every ω-word \((u_i)_{i \in N} \in (2^I)^\omega\) over the input propositions, the word \((u_i \cup c(u_0 \dots u_{i-1}, u_i))_{i \in N}\) satisfies φ.

If a reactive controller exists, then one with finite memory exists. Such controllers are easily represented as automata (or more specifically as Mealy machines). In the automaton representing the controller, the acceptance condition is irrelevant and trivially true.

`ltlsynt` has three mandatory options:

- `--ins`: a comma-separated list of input atomic propositions;
- `--outs`: a comma-separated list of output atomic propositions;
- `--formula` or `--file`: a specification in LTL or PSL.

One of `--ins` or `--outs` may be omitted, as any atomic proposition not listed as input can be assumed to be output and vice-versa.

The following example illustrates the synthesis of a controller ensuring that input `i1` and `i2` are both true initially if and only if eventually output `o1` will go from true to false at some point. Note that this is an equivalence, not an implication.

    ltlsynt --ins=i1,i2 -f '(i1 & i2) <-> F(o1 & X(!o1))'

    REALIZABLE
    HOA: v1
    States: 3
    Start: 0
    AP: 3 "i1" "i2" "o1"
    acc-name: all
    Acceptance: 0 t
    properties: trans-labels explicit-labels state-acc deterministic
    controllable-AP: 2
    --BODY--
    State: 0
    [0&1&2] 1
    [!0&2 | !1&2] 2
    State: 1
    [!2] 1
    State: 2
    [2] 2
    --END--

The output is composed of two parts:

- The first one is a single line `REALIZABLE` or `UNREALIZABLE`; the presence of this line, required by the SyntComp competition, can be disabled with option `--hide-status`.
- The second one, only present in the `REALIZABLE` case, is an automaton describing the controller.

The controller contains the line `controllable-AP: 2`, which means that this automaton should be interpreted as a Mealy machine where `o0` is part of the output. Using the `--dot` option, makes it easier to visualize this machine.

    ltlsynt --ins=i1,i2 -f '(i1 & i2) <-> F(o1 & X(!o1))' --hide-status --dot

The following example illustrates the case of an unrealizable specification. As `a` is an input proposition, there is no way to guarantee that it will eventually hold.

    ltlsynt --ins=a -f 'F a'

    UNREALIZABLE

By default, the controller is output in HOA format, but it can be output as an And-Inverter-Graph in AIGER format using the `--aiger` flag. This is the output format required for the SYNTCOMP competition.

    ltlsynt --ins=i1,i2 -f '(i1 & i2) <-> F(o1 & X(!o1))' --aiger

    REALIZABLE
    aag 18 2 2 1 14
    2
    4
    6 23
    8 37
    7
    10 6 9
    12 4 9
    14 5 10
    16 13 15
    18 2 17
    20 3 10
    22 19 21
    24 7 8
    26 4 24
    28 5 7
    30 27 29
    32 2 31
    34 3 7
    36 33 35
    i0 i1
    i1 i2
    o0 o1

The above format is not very human friendly. Again, by passing both `--aiger` and `--dot`, one can display the And-Inverter-Graph representing the controller:

    ltlsynt --ins=i1,i2 -f '(i1 & i2) <-> F(o1 & X(!o1))' --hide-status --aiger --dot

In the above diagram, round nodes represent AND gates. Small black circles represent inversions (or negations), colored triangles are used to represent input signals (at the bottom) and output signals (at the top), and finally rectangles represent latches. A latch is a one bit register that delays the signal by one step. Initially, all latches are assumed to contain `false`, and them emit their value from the `L0_out` and `L1_out` rectangles at the bottom. Their input value, to be emitted at the next step, is received via the `L0_in` and `L1_in` boxes at the top. In `ltlsynt`'s encoding, the set of latches is used to keep track of the current state of the Mealy machine.

The generation of a controller can be disabled with the flag `--realizability`. In this case, `ltlsynt`'s output is limited to `REALIZABLE` or `UNREALIZABLE`.

## TLSF

`ltlsynt` was made with the SYNTCOMP competition in mind, and more specifically the TLSF track of this competition. TLSF is a high-level specification language created for the purpose of this competition. Fortunately, the SYNTCOMP organizers also provide a tool called `syfco` which can translate a TLSF specification to an LTL formula.

The following line shows how a TLSF specification called `FILE` can be synthesized using `syfco` and `ltlsynt`:

    ltlsynt --tlsf FILE

The above `--tlsf` option will call `syfco` to perform the conversion and extract output signals, as if you had used:

    LTL=$(syfco -f ltlxba -m fully FILE)
    OUT=$(syfco --print-output-signals FILE)
    ltlsynt --formula="$LTL" --outs="$OUT"

## Internal details

The tool reduces the synthesis problem to a parity game, and solves the parity game using Zielonka's recursive algorithm. The process can be pictured as follows.

LTL decomposition consist in splitting the specification into multiple smaller constraints on disjoint subsets of the output values (as described by Finkbeiner, Geier, and Passing), solve those constraints separately, and then combine them while encoding the AIGER circuit. This is enabled by default, but can be disabled by passing option `--decompose=no`.

The ad hoc construction on the top is just a shortcut for some type of constraints that can be solved directly by converting the constraint into a DBA.

Otherwise, conversion to parity game (represented by the blue zone) is done using one of several algorithms specified by the `--algo` option. The game is then solved, producing a strategy if the game is realizable.

If `ltlsynt` is in `--realizability` mode, the process stops here

In synthesis mode, the strategy is first simplified. How this is done can be fine-tuned with option `--simplify`:

    --simplify=no|bisim|bwoa|sat|bisim-sat|bwoa-sat
                           simplification to apply to the controller (no)
                           nothing, (bisim) bisimulation-based reduction,
                           (bwoa) bisimulation-based reduction with output
                           assignment, (sat) SAT-based minimization,
                           (bisim-sat) SAT after bisim, (bwoa-sat) SAT after
                           bwoa.  Defaults to 'bwoa'.

Finally, the strategy is encoded into AIGER. The `--aiger` option can take an argument to specify a type of encoding to use: by default it is `ite` for if-then-else, because it follows the structure of BDD used to encode the conditions in the strategy. An alternative encoding is `isop` where condition are first put into irredundant-sum-of-product, or `both` if both encodings should be tried. Additionally, these optiosn can accept the suffix `+ud` (use dual) to attempt to encode each condition and its negation and keep the smallest one, `+dc` (don't care) to take advantage of don't care values in the output, and one of `+sub0`, `+sub1`, or `+sub2` to test various grouping of variables in the encoding. Multiple encodings can be tried by separating them using commas. For instance `--aiger=isop,isop+dc,isop+ud` will try three different encodings.

## Other useful options

You can also ask `ltlsynt` to print to obtained parity game into PGSolver format, with the flag `--print-pg`, or in the HOA format, using `--print-game-hoa`. These flag deactivate the resolution of the parity game. Note that if any of those flag is used with `--dot`, the game will be printed in the Dot format instead:

    ltlsynt --ins=i1,i2 -f '(i1 & i2) <-> F(o1 & X(!o1))' --print-game-hoa --dot

For benchmarking purpose, the `--csv` option can be used to record intermediate statistics about the resolution.

The `--verify` option requests that the produced strategy or aiger circuit are compatible with the specification. This is done by ensuring that they do not intersect the negation of the specification.

## References

The initial reduction from LTL to parity game is described in the following paper:

- Reactive Synthesis from LTL Specification with Spot, Thibaud Michaud, Maximilien Colange. Presented in SYNT@CAV'18. (pdf | bib)

Further improvements are described in the following paper:

- Improvements to `ltlsynt`, Florian Renkin, Philipp Schlehuber, Alexandre Duret-Lutz, and Adrien Pommellet. Presented at the SYNT'21 workshop. (pdf | bib)

Simplification of Mealy machines is discussed in:

- Effective reductions of Mealy machines, Florian Renkin, Philipp Schlehuber-Caissier, Alexandre Duret-Lutz, and Adrien Pommellet. Presented at FORTE'22. (pdf | bib)