# Working with ImplicationSets

# Introduction

In this vignette, we present the main functionalities and data structures of the `fcaR` package when working with implications in FCA.

We load the `fcaR` package by:

    library(fcaR)

# Datasets

We are going to work with two datasets, a crisp one and a fuzzy one.

The classical (binary) dataset is the well-known `planets` formal context, presented in

> Wille R (1982). “Restructuring Lattice Theory: An Approach Based on Hierarchies of Concepts.” In Ordered Sets, pp. 445–470. Springer.

    knitr::kable(planets, format = "html", booktabs = TRUE)

| small | medium | large | near | far | moon | no_moon
--- | --- | --- | --- | --- | --- | --- | ---
Mercury | 1 | 0 | 0 | 1 | 0 | 0 | 1
Venus | 1 | 0 | 0 | 1 | 0 | 0 | 1
Earth | 1 | 0 | 0 | 1 | 0 | 1 | 0
Mars | 1 | 0 | 0 | 1 | 0 | 1 | 0
Jupiter | 0 | 0 | 1 | 0 | 1 | 1 | 0
Saturn | 0 | 0 | 1 | 0 | 1 | 1 | 0
Uranus | 0 | 1 | 0 | 0 | 1 | 1 | 0
Neptune | 0 | 1 | 0 | 0 | 1 | 1 | 0
Pluto | 1 | 0 | 0 | 0 | 1 | 1 | 0

The other formal context is fuzzy and is defined by the following matrix I:

    knitr::kable(I, format = "html", booktabs = TRUE)

| P1 | P2 | P3 | P4 | P5 | P6
--- | --- | --- | --- | --- | --- | ---
O1 | 0.0 | 0.0 | 0.5 | 0.5 | 1.0 | 0
O2 | 1.0 | 1.0 | 1.0 | 0.0 | 0.0 | 0
O3 | 0.5 | 0.5 | 0.0 | 0.0 | 0.0 | 1
O4 | 0.0 | 0.0 | 0.0 | 1.0 | 0.5 | 0
O5 | 0.0 | 0.0 | 1.0 | 0.5 | 0.0 | 0
O6 | 0.5 | 0.5 | 0.0 | 0.0 | 0.0 | 1

# Working with ImplicationSets

Although `ImplicationSet` objects can be created ad hoc, the usual way to get implications is by the application of the NextClosure algorithm to a `FormalContext` object.

Thus, let us create different formal contexts with the previous datasets:

    fc_planets <- FormalContext$new(planets)
    fc_I <- FormalContext$new(I)

## Extraction of the canonical basis of implications

The function `find_implications()` use the NextClosure algorithm in a formal context to extract the canonical basis of implications:

    fc_planets$find_implications()
    fc_I$find_implications()
    #> LinCbO is only available for binary contexts. Falling back to NextClosure.FALSE

We can inspect the implications by doing:

    fc_planets$implications
    #> Implication set with 10 implications.
    #> Rule 1: {no_moon} -> {small, near}
    #> Rule 2: {far} -> {moon}
    #> Rule 3: {near} -> {small}
    #> Rule 4: {large} -> {far, moon}
    #> Rule 5: {medium} -> {far, moon}
    #> Rule 6: {medium, large, far, moon} -> {small, near, no_moon}
    #> Rule 7: {small, near, moon, no_moon} -> {medium, large, far}
    #> Rule 8: {small, near, far, moon} -> {medium, large, no_moon}
    #> Rule 9: {small, large, far, moon} -> {medium, near, no_moon}
    #> Rule 10: {small, medium, far, moon} -> {large, near, no_moon}
    fc_I$implications
    #> Implication set with 12 implications.
    #> Rule 1: {P6 [0.5]} -> {P1 [0.5], P2 [0.5], P6}
    #> Rule 2: {P5 [0.5]} -> {P4 [0.5]}
    #> Rule 3: {P3 [0.5], P4 [0.5], P5 [0.5]} -> {P5}
    #> Rule 4: {P3 [0.5], P4} -> {P3}
    #> Rule 5: {P3, P4 [0.5], P5} -> {P4}
    #> Rule 6: {P2 [0.5]} -> {P1 [0.5]}
    #> Rule 7: {P1 [0.5]} -> {P2 [0.5]}
    #> Rule 8: {P1 [0.5], P2 [0.5], P4 [0.5]} -> {P1, P2, P3, P4, P5, P6}
    #> Rule 9: {P1 [0.5], P2 [0.5], P3 [0.5]} -> {P1, P2, P3}
    #> Rule 10: {P1 [0.5], P2} -> {P1}
    #> Rule 11: {P1, P2 [0.5]} -> {P2}
    #> Rule 12: {P1, P2, P3, P6} -> {P4, P5}

Internally, an `ImplicationSet` consists of two matrices (one for the left-hand sides and the other for the right-hand sides of the rules). We can get these (sparse) matrices as:

    fc_planets$implications$get_LHS_matrix()
    #> 7 x 10 sparse Matrix of class "dgCMatrix"
    #>   [[ suppressing 10 column names '1', '2', '3' ... ]]
    #>
    #> small   . . . . . . 1 1 1 1
    #> medium  . . . . 1 1 . . . 1
    #> large   . . . 1 . 1 . . 1 .
    #> near    . . 1 . . . 1 1 . .
    #> far     . 1 . . . 1 . 1 1 1
    #> moon    . . . . . 1 1 1 1 1
    #> no_moon 1 . . . . . 1 . . .
    fc_planets$implications$get_RHS_matrix()
    #> 7 x 10 sparse Matrix of class "dgCMatrix"
    #>   [[ suppressing 10 column names '1', '2', '3' ... ]]
    #>
    #> small   1 . 1 . . 1 . . . .
    #> medium  . . . . . . 1 1 1 .
    #> large   . . . . . . 1 1 . 1
    #> near    1 . . . . 1 . . 1 1
    #> far     . . . 1 1 . 1 . . .
    #> moon    . 1 . 1 1 . . . . .
    #> no_moon . . . . . 1 . 1 1 1

The main practical use of an `ImplicationSet` is to compute the closure of a set of attributes, by using the `closure()` function:

    # Let us build a set of attributes
    S <- Set$new(attributes = fc_planets$attributes)
    S$assign(large = 1, far = 1)
    S
    #> {large, far}

    fc_planets$implications$closure(S)$closure
    #> {large, far, moon}

## Validity of implications

We can check if an `ImplicationSet` holds in a `FormalContext` by using the `%holds_in%` operator:

    # Let us clone the implication basis
    imps <- fc_planets$implications$clone()
    imps %holds_in% fc_planets
    #>  [1] TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE

Each component of this vector represents whether the corresponding implication holds in the formal context. In this case, as the `ImplicationSet` is the Duquenne-Guigues basis for the `FormalContext`, all implications hold.

Conversely, we can check if a list of attribute sets (or a formal context) respects an `ImplicationSet`, via the `%respects%` operator:

    fc_planets %respects% imps
    #> 9 x 10 sparse Matrix of class "lgCMatrix"
    #>   [[ suppressing 10 column names 'imp_01', 'imp_02', 'imp_03' ... ]]
    #>
    #> set_1 | | | | | | | | | |
    #> set_2 | | | | | | | | | |
    #> set_3 | | | | | | | | | |
    #> set_4 | | | | | | | | | |
    #> set_5 | | | | | | | | | |
    #> set_6 | | | | | | | | | |
    #> set_7 | | | | | | | | | |
    #> set_8 | | | | | | | | | |
    #> set_9 | | | | | | | | | |

The result is a matrix where each row correspond to a attribute set and each column to an implication. An element is `TRUE` if the corresponding set respects the corresponding implication. If the first argument is a `FormalContext`, the function will consider the set of attributes of each object.

## Cardinality, size and support of the implication set

Some quantities can be computed for an `ImplicationSet`:

- Cardinality: the number of implications in the set

    fc_planets$implications$cardinality()
    #> [1] 10

- Size: The number of attributes in the LHS and the RHS of each implication

    sizes <- fc_planets$implications$size()
    # Total number of attributes in the LHS and the RHS
    colSums(sizes)
    #> LHS RHS
    #>  25  23

- Support: The proportion of objects in the formal context whose attributes contain the LHS of a particular rule

    fc_planets$implications$support()
    #>  [1] 0.2222222 0.5555556 0.4444444 0.2222222 0.2222222 0.0000000 0.0000000
    #>  [8] 0.0000000 0.0000000 0.0000000

## Export to LaTeX

A nice feature is the ability to export an `ImplicationSet` to LaTeX format:

    fc_planets$implications$to_latex()
    #> \begin{longtable*}{rrcl}
    #> 1: &\left\{\mathrm{no\_moon}\right\}&\ensuremath{\Rightarrow}&\left\{\mathrm{small}, \mathrm{near}\right\}\\
    #> 2: &\left\{\mathrm{far}\right\}&\ensuremath{\Rightarrow}&\left\{\mathrm{moon}\right\}\\
    #> 3: &\left\{\mathrm{near}\right\}&\ensuremath{\Rightarrow}&\left\{\mathrm{small}\right\}\\
    #> 4: &\left\{\mathrm{large}\right\}&\ensuremath{\Rightarrow}&\left\{\mathrm{far}, \mathrm{moon}\right\}\\
    #> 5: &\left\{\mathrm{medium}\right\}&\ensuremath{\Rightarrow}&\left\{\mathrm{far}, \mathrm{moon}\right\}\\
    #> 6: &\left\{\mathrm{medium}, \mathrm{large}, \mathrm{far}, \mathrm{moon}\right\}&\ensuremath{\Rightarrow}&\left\{\mathrm{small}, \mathrm{near}, \mathrm{no\_moon}\right\}\\
    #> 7: &\left\{\mathrm{small}, \mathrm{near}, \mathrm{moon}, \mathrm{no\_moon}\right\}&\ensuremath{\Rightarrow}&\left\{\mathrm{medium}, \mathrm{large}, \mathrm{far}\right\}\\
    #> 8: &\left\{\mathrm{small}, \mathrm{near}, \mathrm{far}, \mathrm{moon}\right\}&\ensuremath{\Rightarrow}&\left\{\mathrm{medium}, \mathrm{large}, \mathrm{no\_moon}\right\}\\
    #> 9: &\left\{\mathrm{small}, \mathrm{large}, \mathrm{far}, \mathrm{moon}\right\}&\ensuremath{\Rightarrow}&\left\{\mathrm{medium}, \mathrm{near}, \mathrm{no\_moon}\right\}\\
    #> 10: &\left\{\mathrm{small}, \mathrm{medium}, \mathrm{far}, \mathrm{moon}\right\}&\ensuremath{\Rightarrow}&\left\{\mathrm{large}, \mathrm{near}, \mathrm{no\_moon}\right\}\\
    #> \end{longtable*}

## Filtering of implications

Sometimes it is needed to work with a subset of the implications, using only the implications that fulfill certain conditions:

    # Implications with P1 and P2 in the LHS and P5 in the RHS
    fc_I$implications$filter(
      lhs = c("P1", "P2"),
      rhs = "P5"
    )
    #> Implication set with 2 implications.
    #> Rule 1: {P1 [0.5], P2 [0.5], P4 [0.5]} -> {P1, P2, P3, P4, P5, P6}
    #> Rule 2: {P1, P2, P3, P6} -> {P4, P5}

## Simplification Logic

In this package, we have implemented logic tools to operate on the implications.

First, some simplification rules have been developed, named reduction, composition, generalization and simplification, that can be applied using the `apply_rules()` function:

    fc_I$implications$apply_rules(rules = c(
      "composition",
      "simplification"
    ))

This enables the reduction of the cardinality and/or the size of the `ImplicationSet`.

In addition, the “simplification” rule to remove redundancies can be used in the computation of the closure of a set, to provide a reduced set of implications that is inferred from the set of attributes:

    # Let us build a set of attributes
    S <- Set$new(attributes = fc_planets$attributes)
    S$assign(large = 1, far = 1)
    S
    #> {large, far}

    fc_planets$implications$closure(S, reduce = TRUE)
    #> $closure
    #> {large, far, moon}
    #>
    #> $implications
    #> Implication set with 4 implications.
    #> Rule 1: {medium} -> {small, near, no_moon}
    #> Rule 2: {small} -> {medium, near, no_moon}
    #> Rule 3: {no_moon} -> {small, near}
    #> Rule 4: {near} -> {small}

## Entailment and equivalence of implications

We can check if a given `ImplicationSet` follows from another one:

    # imps is the basis
    imps <- fc_planets$implications$clone()
    imps2 <- imps$clone()
    # imps2 is an equivalent set of implications
    # where we have removed redundancies
    imps2$apply_rules(c("simp", "rsimp"))
    # Any implication in imps2 follows from imps
    imps %entails% imps2
    #>      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
    #> [1,] TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE  TRUE
    # And viceversa
    imps2 %entails% imps
    #>      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
    #> [1,] TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE  TRUE

We can also check if the two sets of implications are equivalent:

    imps %~% imps2
    #> [1] TRUE
    # If we remove any implication from imps2,
    # they will not be equivalent
    imps %~% imps2[1:9]
    #> [1] FALSE

## Standard Context

The `ImplicationSet` allows us to recover the underlying lattice structure (the lattice of closed sets) by computing the standard context. This context has the meet-irreducible closed sets of the implication set as objects, and the original attributes as attributes:

    # Get the standard context from the implications of the planets context
    sc <- fc_planets$implications$get_standard_context()

    # We can inspect the new context
    sc
    #> FormalContext with 5 objects and 7 attributes.
    #>     small  medium  large  near  far  moon  no_moon
    #>  C1                  X           X     X
    #>  C2           X                  X     X
    #>  C3   X                          X     X
    #>  C4   X                     X                 X
    #>  C5   X                     X          X

The concept lattice of this standard context is isomorphic to the lattice of closed sets of the original implication set.

## Recommendation systems

One of the final applications of an `ImplicationSet` is the easy development of a recommendation system where, from an attribute set, the system would infer the value to other attribute. This is done by the `recommend()` function, which internally computes the closure of the attribute set:

    S <- Set$new(attributes = fc_I$attributes)
    S$assign(P1 = 1, P4 = 0.5)

    fc_I$implications$recommend(S, attribute_filter = c("P3", "P5"))
    #> P3 P5
    #>  1  1