===========
PN-matrices
===========

Given a propositional signature :math:`\Sigma`,
a PN-matrix over :math:`\Sigma` is a structure
:math:`\mathbb{M} = \langle \mathbf{M}, \{D_i\}_{i \in I} \rangle`,
where :math:`\mathbf{M}` is a multi-algebra over :math:`\Sigma`
and :math:`D_i \subseteq M` is called a *distinguished set*, for each :math:`i \in I`.

In *ct4l*, we represent finite PN-matrices using the class 
:cpp:class:`ct4l::PNMatrix`, which holds a pointer to
a :cpp:class:`ct4l::MultiAlgebra` object and 
a vector of subsets of the carrier of the multi-algebra.
We also store internally the translation of such subsets to
the domain values.

When building a PN-matrix, you may specify whether you want the
constructor to compute the complements of each distinguished set 
and store it as a distinguished set too. For example, if the carrier of the algebra is :math:`\{0,1\}` and you 
opt for this, then passing a vector having only the set :math:`\{1\}`
will produce a PN-matrix with :math:`D_1 = \{1\}` and 
:math:`D_2 = \{0\}`. If the carrier of the algebra is :math:`\{0, 1, 2, 3\}` and you give to the constructor a vector
with sets :math:`\{0, 1\}` and :math:`\{0\}`, then, internally,
this vector will have four sets, in this sequence:
:math:`\{0, 1\}, \{2, 3\}, \{0\}, \{1, 2, 3\}`. This is intended to help 
when we work with some notions of entailment associated to a PN-matrix.
Check :cpp:func:`ct4l::PNMatrix::PNMatrix()` to see how to disable this option.

.. note::

    Notice that the notion of PN-matrix we are using is a generalization
    of the usual one, which accepts only a single designated set. By taking
    a family, as we implemented, we can work more easily with many different notions
    of entailment, including two-dimensional (and :math:`n`-dimensional) notions.

Example: Classical Logic matrix
---------------------------------------

Take the boolean algebra built in :ref:`boolean algebra example`
and assume we have a pointer to it stored in the variable `algebra`. 
The code below creates a PN-matrix based on such algebra:

.. code-block::

    ct4l::PNMatrix matrix {algebra, {{"T"}}};


API
---

.. doxygenclass:: ct4l::PNMatrix
    :members: