PN-matrices

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

In ct4l, we represent finite PN-matrices using the class ct4l::PNMatrix, which holds a pointer to a 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 \(\{0,1\}\) and you opt for this, then passing a vector having only the set \(\{1\}\) will produce a PN-matrix with \(D_1 = \{1\}\) and \(D_2 = \{0\}\). If the carrier of the algebra is \(\{0, 1, 2, 3\}\) and you give to the constructor a vector with sets \(\{0, 1\}\) and \(\{0\}\), then, internally, this vector will have four sets, in this sequence: \(\{0, 1\}, \{2, 3\}, \{0\}, \{1, 2, 3\}\). This is intended to help when we work with some notions of entailment associated to a PN-matrix. Check 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 \(n\)-dimensional) notions.

Example: Classical Logic matrix

Take the boolean algebra built in Example: Building the two-element Boolean algebra and assume we have a pointer to it stored in the variable algebra. The code below creates a PN-matrix based on such algebra:

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

API

template<typename T>
class ct4l::PNMatrix

Represents a non-deterministic matrix, allowing for empty sets in the interpretations, as well as multiple distinguished sets.

Author

Vitor Greati

Public Functions

PNMatrix(decltype(_algebra) algebra, const std::vector<std::set<T>> &dsets_domain, bool make_complements = true)

Constructor that accepts the algebra, the dsets and a flat indicating whether the complement of each dset should be added as a dset.

In case the flag is true, for each i, dset[2i]=D_i and dset[2i+1]=V\D_i.

Parameters

• algebra: the pointer to the algebra

• dsets: the designated sets in terms of the domain values

• make_complements: indicates whether the complements of dsets should be computed

inline decltype(_dsets) dsets() const

The distinguished sets in terms of the internal representation.

Return

distinguished sets in internal representation

inline decltype(_dsets_domain) dsets_domain() const

The distinguished sets in terms of the domain representation.

Return

distinguished sets in domain representation

inline decltype(_algebra) algebra() const

Return a pointer to the matrix algebra.

Return

a pointer to the algebra