Grammatical inference is the research field that is concerned with learning the set of rules that describe the behavior of a system, such as a network protocol, a piece of software, or a (formal) language.

One of the best studied problems in grammatical inference is that of finding a deterministic finite automaton (DFA) of minimal size that accepts a given set of positive strings and rejects a given set of negative strings.

This problem is can be very hard, as it has been shown to be NP-complete.

Z3GI provides different ways of solving this (and similar) problem(s) using satisfiability modulo theories (SMT).

Installing with pip (recommended)

---------------------------------

The recommended way of installing Z3GI is with `pip`:

Consider a deterministic finite automaton (DFA) that accepts strings of `0`s and `1`s in which the number of `0`s minus twice the number of `1`s is either 1 or 3 more than a multiple of 5 (such a DFA is described [here][dfa]).

Consider a deterministic finite automaton (DFA) corresponding to the regex (ab)*.

[dfa]:http://abbadingo.cs.nuim.ie/dfa.html

The training file at `resources\traces\regex_example` for this DFA reads:

A training file `train.txt` for this DFA could read (if you have the sources of this package, this file can be found at `docs/train.txt`):

```

1 a b

1 a b a b

0 a

0 b

0 a b a

0 a a

0 a b b

0 a b a a

```

Each line represents an observation comprising elements separated by spaces.

For a DFA, the observation has the syntax:

```

16 2

1 4 1 0 0 0

1 4 0 1 0 0

1 4 0 0 1 0

1 5 1 0 1 1 1

1 6 1 1 1 1 0 1

1 6 0 1 0 0 0 0

1 6 1 0 0 0 0 0

1 7 0 0 0 1 1 0 1

1 7 0 0 0 0 1 0 1

0 3 1 0 1

0 4 0 0 0 0

0 4 1 1 0 1

0 5 0 0 0 0 0

0 5 0 0 1 0 1

0 6 0 1 0 1 1 1

0 7 1 0 0 0 1 1 1

label symbol1 symbol2 ... symboln

```

In this [Abbadingo file][abbadingo], the first line is a header, giving the number of strings in the file (16) and the number of symbols (2).

Each line after the header has the format

The symols form a string, the label indicates that the DFA accepts the string (if its value is `1`) or

that it rejects the string (if its value is `0`).

In the current version, we require that the first observation introduces all inputs in the alphabet.

We can use Z3GI to learn a model for the observations in `train.txt` as follows:

We can apply Z3GI to learn simulations of .dot models. In particular, we can

learn the Mealy machine models obtained by various case-studies which are included

in the `resources\models` directory.

Say we want to learn the biometric passport. We then need to run:

```

Learned model:

[state3 = 3,

start = 0,

state0 = 0,

n = 5,

state2 = 2,

state4 = 4,

1 = INPUT!val!0,

state1 = 1,

0 = INPUT!val!1,

out = [3 -> True, 4 -> True, else -> False],

trans = [(0, INPUT!val!0) -> 3,

(0, INPUT!val!1) -> 4,

(4, INPUT!val!0) -> 2,

(3, INPUT!val!0) -> 4,

(3, INPUT!val!1) -> 2,

(2, INPUT!val!0) -> 1,

(1, INPUT!val!1) -> 3,

(4, INPUT!val!1) -> 1,

else -> 0]]

$ python z3gi -m dot -a MealyMachine -f resources\models\biometric.dot

```

We can interpret this learned model as follows.

-`0 = INPUT!val!1` and `1 = INPUT!val!0` provide identifiers for `0` and `1` (notice that the values in the identifiers and the actual inputs are different!)

-`n = 5` indicates that the learned model has 5 states

-`state0 = 0` through `state4 = 4` provide the identifiers for these states

-`out` describes an output function for these states (`True` if it is accepting and `False` if it is rejecting)

-`trans` desribes a transition function for states and symbols to states (e.g. `(0, INPUT!val!0) -> 3)` describes a transition from `state0` with `1` to `state3`)

Using z3gi in Python

--------------------

Let's learn the same model (from `train.txt`) in Python:

1. Open your Python interpreter:

```

$ python

```

2. Let's use a different encoder this time:

```

>>> from z3gi.encoders import expressive

>>> encoder = expressive.Encoder()

```

3. Create a sample:

```

>>> from z3gi.sample import Sample

>>> sample = Sample(encoder)

```

4. Add constraints for strings in `train.txt` to the sample:

```

>>> from z3gi.parsers import abbadingo

>>> for string, label in abbadingo.read(open('train.txt', 'r'), header=1):

... sample[string] = label

...

```

5. Obtain the model!

```

>>> model = sample.model()

>>> print(model)

```

\ No newline at end of file

For bigger models which are more difficult to test, we may require an external

test algorithm. We provide a Windows binary for the Yannakakis test algorithm.

You can activate this algorithm using the `-y` option. By using `-m dotnorst`,

Z3GI will attempt to learn without using resets.

Our implementation currently only supports Mealy machines, though functionality

for other formalisms will be added in the future. Note there is currently no

standard .dot format for register automata. Also notethat the `yannakakis` test

algorithm only works on Mealy machines.

Learning randomly generated Mealy machines without reset

-----------------------

Z3GI can be used to learn randomly generated MealyMachines without reset with the

property that they are strongly connected (though not necessarily minimal).

Say we want to learn a randomly generated Mealy machine with 2 inputs, 2 outputs