# Negate Boolean expressions

## Description

Functions to negate a DNF/SOP expression, or to invert SOP to POS or POS to SOP.

## Usage

    negate(input, snames = "", noflevels, simplify = TRUE)
    invert(input, snames = "", noflevels)

## Arguments

`input` |  |  |  | A string representing a SOP expresson, or a minimization object of class `"QCA_min"`.
`snames` |  |  |  | A string containing the sets' names, separated by commas.
`noflevels` |  |  |  | Numerical vector containing the number of levels for each set.
`simplify` |  |  |  | Logical, allow users to choose between the raw negation or its simplest form.

## Details

In Boolean algebra, there are two transformation rules named after the British mathematician Augustus De Morgan. These rules state that:

1. The complement of the union of two sets is the intersection of their complements.

2. The complement of the intersection of two sets is the union of their complements.

In "normal" language, these would be written as:

1. `not (A and B) = (not A) or (not B)`

2. `not (A or B) = (not A) and (not B)`

Based on these two laws, any Boolean expression written in disjunctive normal form can be transformed into its negation.

It is also possible to negate all models and solutions from the result of a Boolean minimization from function `minimize()`. The resulting object, of class `"qca"`, is automatically recognised by this function.

In a SOP expression, the products should normally be split by using a star `*` sign, otherwise the sets' names will be considered the individual letters in alphabetical order, unless they are specified via `snames`.

To negate multilevel expressions, the argument `noflevels` is required.

It is entirely possible to obtain multiple negations of a single expression, since the result of the negation is passed to function `simplify()`.

## Value

A character vector when the input is a SOP expresison, or a named list for minimization input objects, each component containing all possible negations of the model(s).

## References

Ragin, Charles C. 1987. The Comparative Method: Moving beyond Qualitative and Quantitative Strategies. Berkeley: University of California Press.

## Examples

    # example from Ragin (1987, p.99)
    negate("AC + B~C", simplify = FALSE)

    N1: ~A~B + ~AC + ~B~C

    # the simplified, logically equivalent negation
      negate("AC + B~C")

    N1: ~AC + ~B~C

    # with different intersection operators
    negate("AB*EF + ~CD*EF")

    N1: ~EF + ~AB*CD

    # invert to POS
    invert("a*b + ~c*d")

    N1: (~a + ~b)*(c + ~d)

    # using an object of class "qca" produced with minimize()
    cLC <- minimize(LC, outcome = "SURV")

    negate(cLC)

    M1: DEV*~URB*LIT*STB + DEV*LIT*IND*STB
      N1: ~DEV + ~LIT + ~STB + URB*~IND

    # parsimonious solution
    pLC <- minimize(LC, outcome = "SURV", include = "?")

    negate(pLC)

    M1: DEV*STB
      N1: ~DEV + ~STB

## Author

Adrian Dusa

## See also

`minimize`