ltlfsynt
Table of Contents
Reactive synthesis is the problem of synthesizing a controller that produces output signals based on a history of input signals, under the constraint of some specification that relates the input and output signals over time.
ltlfsynt solves this problem when the specification is an LTLf
formula (i.e., LTL interpreted over finite traces). It can either
decide realizability (i.e., decide if a controller satisfying the
formula exist) or synthesize a reactive controller as an And-Inverter
Graph (AIG).
Consider a set \(I\) of input atomic propositions, a set \(O\) of output atomic propositions, and an LTLf formula φ over the propositions in \(I \cup O\). This tool can build controllers using two different semantics:
- Mealy controller
- A Mealy controller is a function \(c: (2^{I})^+\to 2^O\). A Mealy controller realizes \(\varphi\) if for any ω-word \((u_i)_{i \in \mathbb{N}} \in (2^I)^\omega\) over the input propositions, there exists some position \(k>0\) such that the (nonempty) word \((u_i \cup c(u_0 \dots u_{i-1}, u_i))_{0 \le i \le k}\in (2^{I\cup O})^k\) satisfies φ.
- Moore controller
- A Moore controller is a function \(c: (2^{I})^*\to 2^O\). A Moore controller realizes \(\varphi\) if for any ω-word \((u_i)_{i \in \mathbb{N}} \in (2^I)^\omega\) over the input propositions, there exists some position \(k>0\) such that the (nonempty) word \((u_i \cup c(u_0 \dots u_{i-1}))_{0 \le i \le k}\in (2^{I\cup O})^k\) satisfies φ.
In other words, a Mealy controller should decide the outputs for the current instant based on the history of all inputs as well as the current inputs. A Moore controller should decide outputs without knowing the current inputs, it only knows about past inputs. In either semantics, the controller should ensure that φ is satisfied after a finite number of steps. Once φ has been satisfied, the output of the controller is unconstrained.
Example
Here is an example LTLf specification generated by genltl:
genltl --tv-counter-mealy=2 >counter2.ltlf
sed 's/(*G/\n&/g' counter2.ltlf
(init0 <-> ob0) & (init1 <-> ob1) & G(inc <-> oc0) & G((X[!]ob0 -> (ob0 xor oc0)) & (X[!]!ob0 -> (ob0 <-> oc0))) & G(oc1 <-> (ob0 & oc0)) & G((X[!]ob1 -> (ob1 xor oc1)) & (X[!]!ob1 -> (ob1 <-> oc1))) & (G(!inc -> X[!]inc) -> F(!ob0 & !ob1))
Output variables ob0 and ob1 encode a 2-bit counter. Input
variables init0 and init1 specify the initial value of this
counter, as encoded on the first line.
When the input variable inc is set, the counter should be
incremented by one on the next step. This is encoded on the next four
lines, by using to extra output variables to represent carries
(oc0 and oc1), and ensuring that, if a next step exists, the next
value of the counter (X[!]ob0 and X[!]ob1) will be set according
to its current value and to that of the carries.
X[!] is Spot's notation for the strong next operator of LTLf. The
weak next is simply noted X. Since we are working with finite
words, a strong next ensures that the current instant is not the last
one. A weak next is always satisfied on the last instant.
The last implication is something we would like to confirm under the
above rules, and regardless of the initial value of the counter.
Assuming the input variable inc is raised at least every other step
(G(!inc -> X[!]inc)), we claim that the counter will eventually wrap
back to zero F(!ob0 & !ob1).
We show that the answer is positive by passing this formula to
ltlfsynt. (Notice that in this formula, input and output variables
start with i and o respectively, so ltlfsynt can tell them apart
automatically.)
ltlfsynt --semantics=Mealy -F counter2.ltlf --realizability
REALIZABLE
This specification is realizable.
Without the --realizability switch, ltlfsynt will additionally
print a Mealy machine representing the controller, in the HOA format.
ltlfsynt --semantics=Mealy -F counter2.ltlf
REALIZABLE HOA: v1 States: 5 Start: 0 AP: 7 "init0" "init1" "inc" "ob0" "ob1" "oc1" "oc0" acc-name: all Acceptance: 0 t properties: trans-labels explicit-labels state-acc deterministic controllable-AP: 3 4 5 6 --BODY-- State: 0 [!0&!1&!2&!3&!4&!5&!6 | !0&!1&2&!3&!4&!5&6 | !0&1&!2&!3&4&!5&!6 | 0&!1&!2&3&!4&!5&!6 | 0&1&!2&3&4&!5&!6] 1 [!0&1&2&!3&4&!5&6] 2 [0&!1&2&3&!4&5&6] 3 [0&1&2&3&4&5&6] 4 State: 1 [!2&!6 | 2&6] 1 State: 2 [!2&3&4&!5&!6] 1 [2&3&4&5&6] 4 State: 3 [!2&!3&4&!5&!6] 1 [2&!3&4&!5&6] 2 State: 4 [!2&!3&!4&!5&!6 | 2&!3&!4&!5&6] 1 --END--
We can hide the REALIZABLE line with --hide, and print the Mealy
machine in GraphViz's format as follows:
ltlfsynt --semantics=Mealy -F counter2.ltlf --hide --dot=strategy
Remember that unlike ltlsynt, we are working with finite semantics
here. The controller will read a sequence of input assignments of
infinite size, but that sequence should have at least one position
such that the combination of inputs and outputs up to that point
satisfy the specification. Once the specification has been satisfied,
the controller is free to output anything. This is why the values of
ob0, ob1, oc1 are not specified on the transition leaving the
rightmost state: that state is reached either when the counter wrapped
to zero, or when the input signals do not satisfy our assumption
(G(!inc -> X[!]inc)), so at that point the specification has been
satisfied, and the output variable can be chosen freely.
The reason we still force the value of oc0 to be equal to that of
inc on the last state is because of an optimization in ltlfsynt:
since the specification contains G(inc <-> oc0), the formula can be
simplified by replacing all occurrences of oc0 by inc. ltlfsynt
does this to reduce the number of variables it has to deal with, and
only adds the oc0 variable back at the very end, before presenting
results.
The controller may be output as an And-Inverter-Graph in AIGER format
using the --aiger option:
ltlfsynt --semantics=Mealy -F counter2.ltlf --aiger
REALIZABLE aag 59 3 3 4 53 2 4 6 8 83 10 109 12 119 25 35 45 6 14 10 13 16 9 14 18 9 13 20 2 18 22 3 16 24 21 23 26 8 14 28 19 27 30 4 29 32 5 14 34 31 33 36 6 16 38 6 18 40 2 38 42 3 36 44 41 43 46 10 12 48 8 13 50 9 47 52 49 51 54 11 13 56 8 54 58 9 11 60 57 59 62 6 61 64 7 53 66 63 65 68 11 12 70 9 68 72 57 71 74 6 73 76 65 75 78 4 77 80 5 67 82 79 81 84 6 26 86 9 54 88 27 87 90 6 89 92 4 90 94 5 84 96 93 95 98 4 84 100 5 90 102 99 101 104 2 103 106 3 97 108 105 107 110 4 38 112 5 36 114 111 113 116 2 115 118 43 117 i0 init0 i1 init1 i2 inc o0 ob0 o1 ob1 o2 oc1 o3 oc0
Here is a graphical representation of the And-Inverter Graph for this controller:
ltlfsynt --semantics=Mealy -F counter2.ltlf --hide --dot=aig
The input signals are shown at the bottom in blue triangles. The
output signals are shown at that top in red triangles. Three latches
(1-bit registers), represented with orange rectangles are used to
remember the state of the controller. Latches are all false
initially. The L*_out nodes emit the current latch value, and the
L*_in receive the value of the latch for the next step. All round
nodes correspond to binary AND gates. Black dots invert a signal.
Invoking ltlfsynt
The specification can be supplied in three ways:
- as an LTLf formula, supplied on the command-line:
-f formula. - as an LTLf formula, stored in a file:
-F FILENAME. - as a file in the TLSF v1.2 format:
--tlsf FILENAME.
When receiving an LTLf formula with -f or -F, ltlfilt needs to
know which atomic propositions are input or output. This can be
specified with options --ins, --outs, or with --part-file. See
ltlfsynt's documentation of those options. Without any such option,
ltlfsynt assumes that variables starting with i are inputs, and
those starting with o are output.
Similarly, the semantics also have to be specified using
--semantics=Mealy (the default) or --semantics=Moore. When
comparing ltlfsynt to other tools, beware that they may not
have the same semantics by default.
When the specification is passed as a TLSF file with the --tlsf
option, the syfco tool is automatically invoked to transform the
specification into LTLf, to retrieve the lists of input/output
variables, and to obtain the semantics to use.
Fine-tuning
ltlfsynt operates in several phases as follows:
- Preprocess the specification.
- Translate the LTLf specification into two-player game.
- Solve the two-player game.
- Extract a winning strategy as a Mealy machine.
- Encode the Mealy machine in AIG.
Steps 2 & 3 are optionally merged together and performed on-the-fly.
Preprocessing
Various optimizations are enabled by default.
- Variables that always appear with the same polarity in the
specification can be changed into constants and removed from the
specification. This optimization can be disabled with
--polarity=no. - Group of variables that are specified as equivalent can be replaced
by a unique variable of that group. This can be disabled with
--global-equivalence=no. - A specification can be decomposed into multiple output-disjoint
specifications that can be solved independently. This decomposition
can be disabled with
--decompose=no. (Currently this optimization has been restricted to X-free specification, because of issue #610.) - The formula is simplified with some cheap rewriting rules. This is
especially important when decomposition is used, because the
decomposition sometimes need to distribute subformulas to favor
decompositions. This can be disabled with
--simplify=no.
The easiest way to track what is going on with these options is to run
ltlfsynt with the --verbose options.
ltlfsynt --semantics=Mealy -F counter2.ltlf --verbose --realizability
using Mealy semantics the following signals can be temporarily removed: oc0 := inc new formula: (init0 <-> ob0) & (init1 <-> ob1) & G((X[!]ob1 -> (ob1 xor oc1)) & (X[!]!ob1 -> (ob1 <-> oc1))) & (G(!inc -> X[!]inc) -> F(!ob0 & !ob1)) & G((X[!]ob0 -> (inc xor ob0)) & (X[!]!ob0 -> (inc <-> ob0))) & G(oc1 <-> (inc & ob0)) decomposition not attempted because the formula uses X formula simplified to (init0 <-> ob0) & (init1 <-> ob1) & (G(inc | X[!]inc) -> F(!ob0 & !ob1)) & G((oc1 <-> (inc & ob0)) & ((ob1 xor oc1) | X!ob1) & ((ob1 <-> oc1) | Xob1) & ((inc xor ob0) | X!ob0) & ((inc <-> ob0) | Xob0)) starting on-the-fly translation with node-based backpropagation, with BFS order, with one-step preprocess translation to MTDFA (1 roots, 6 APs) took 0.00305169 seconds MTDFA game was solved during translation REALIZABLE
Here we see that the oc0 was replaced by inc (such replacement
occur because of --global-equiv is enabled by default), that the
formula could not be decomposed (--decompose=yes is the default),
and the result of --simplify=yes (the default) is shown before
starting the translation.
Translation & Game solving
By default, those two steps are combined and performed on-the-fly.
This gives the best results. ltlfilt supports alternative
approaches. In all approaches, the DFA that is constructed is
represented using multi-terminal binary decision diagrams (MTBDD).
Off-line approaches
In these approaches, first a DFA is built, then it is interpreted
as a game and solved. The why the DFA is built is controlled by
the --translation option:
--translation=full: a full DFA is created for the input specification; this is a direct translation from LTLf to MTBDD-based DFA.--translation=compositional: a full DFA is created for the input specification, but the translation is compositional. The specification will be split on top-level Boolean operators, then subformulas will be translated separately, and finally the resulting automata will be combined.--translation=restrictedbuilds a restricted version of the DFA in which accepting states are replaced by accepting sinks. This is a direct translation as well.
The restricted translation is always faster than the full
translation. The compositional translation is usually faster than
the full translation as well (provided intermediate automata are
minimized). restricted is usually better than compositional on
meaningful specifications, and compositional is often than
restricted or specifications built from random combination
specifications.
The MTDFA (that Spot's name for MTBDD-encoded DFA) resulting from
these translations, and that will be used directly as a reachability
game, can be displayed with --dot=game. Here is the output for the
full translation (the compositional translation would give the same
output on this example). See this page if you need an introduction to
MTDFAs. The MTDFA is interpreted as a game in which round node are
player by the environment, diamond nodes are played by the controller,
and the controller want to force the game to reach an accepting state.
ltlfsynt --trans=full --semantics=Mealy -F counter2.ltlf --hide --dot=game
For comparison, here is MTDFA output by the restricted translation.
Note how all accepting states have been merged into the true BDD node,
and the game have been simplified already.
ltlfsynt --trans=restrict --semantics=Mealy -F counter2.ltlf --hide --dot=game
In the above case, we are lucky that the result of this translation is already a strategy: each diamond node has exactly one child that is the false BDD node, so the strategy is to always take the other child. Generally, the result of the translation may not be a strategy; that will be computed in a second step.
The automaton that has been built can be minimized. This is controlled with
the --minimize=yes and --minimize=no options. By default,
minimization is only turned on for --translation=compositional,
because minimization is what makes the compositional translation fast.
For other approaches, minimization is not recommended, because solving
the unminimized automaton is faster than minimizing it.
The MTBDD-based DFA is interpreted as a game that can be solved in two ways:
--backprop=nodes: solve the game by backpropagation at the level of the MTBDD nodes. This is a linear algorithm.--backprop=states: solve the game by backpropagation but at the level of the MTDFA states. This uses less memory, but the algorithm may require quadratic time.
On-the-fly approaches
The DFA can be built on-the-fly. This builds the equivalent of the "restricted" automaton from above and solves the game as it is constructed.
The order in which states are explored can be controlled with
--translation=dfs or --translation=bfs.
The BFS on-the-fly translation is the default, because it had a very slight advantage over DFS in a benchmark we did.
An extra optimization that is enabled by default in on-the-fly
approaches is one-step (un)realizability where the formula α
for the current step is rewritten into two formulas Boolean formulas
\(\alpha_r\) and \(\alpha_u\) such that if \(\alpha_r\) is realizable then
\(\alpha\) is realizable, and if \(\alpha_u\) is unrealizable, then
\(\alpha\) is unrealizable. There are some specifications where this
optimization is a huge help. Conversely, for some specifications
where this does not help, this is just an unnecessary overhead. You
may disable it with --one-step-preprocess=no.
The option --dot=game still works with on-the-fly approaches, but in
that case it can only display the strategy that results from creating
the game and solving it on-the-fly. If this is combined with
--realizability (discussed below), the strategy returned by the
on-the-fly approach is simplified to true or false, so there is not
much to see.
Realizability
Use the --realizability to interrupts the process once the game has
been solved.
In addition to our test-suite, we have run ltlfsynt --realizability
on benchmarks from the LTLf tracks of the Synthesis Competition, and
ensured that all results where consistent between various
configurations of ltlfsynt and with several other LTLf realizability
tools.
In contrast, the generation of controllers in any form (HOA, AIG) has not yet been thoroughly tested and should be regarded as highly experimental.
AIG encoding
ltlfsynt shares its AIG encoder with ltlsynt, so see ltlsynt's –aiger option.