QCA (version 2.6)

Interpret SOP expressions: translate, compute, findRows, sop: Functions to interpret or create a SOP expression

Description

These functions interpret an expression written in a SOP (sum of products) form, for both crisp and multivalue QCA. The function translate() translates the expression into a standard (canonical) SOP form using a matrix of implicants, while compute() uses the first to compute the scores based on a particular dataset.

The function findRows() takes a QCA expression written in SOP form, and applies it on a truth table to find all rows that match the pattern in the expression.

For crisp sets notation, upper case letters are considered the presence of that causal condition, and lower case letters are considered the absence of the respective causal condition. Tilde is recognized as a negation, even in combination with upper/lower letters.

Functions similar to translate() and compute() have initially been written by Jirka Lewandowski (2015) but the actual code in these functions has been completely re-written and expanded with more extensive functionality (see details and examples below).

The function sop() transforms any expression (most notably a POS product of sums) into a sum of products, minimizing it to the simplest equivalent logical expression.

Usage

translate(expression = "", snames = "", noflevels, data, ...)
compute(expression = "", data, separate = FALSE)
findRows(expression = "", ttobj, remainders = FALSE)
sop(expression = "", snames = "", noflevels, data)

Arguments

expression

String: a QCA expression written in sum of products form.

snames

A string containing the sets' names, separated by commas.

noflevels

Numerical vector containing the number of levels for each set.

data

A dataset with binary cs, mv and fs data.

separate

Logical, perform computations on individual, separate paths.

ttobj

A truth table, an object of class "tt".

remainders

Logical, find remainders only.

...

Other arguments, mainly for backwards compatibility.

Value

For the function translate(), a matrix containing the implicants on the rows and the set names on the columns, with the following codes:

0	absence of a causal condition
1	presence of a causal condition
-1	causal condition was eliminated

The matrix was also assigned a class "translate", to avoid printing the -1 codes. The mode of this matrix is character, to allow printing multiple levels in the same cell, such as "1,2".

Details

A SOP ("sum of products") is also known as a DNF ("disjunctive normal form"), or in other words a "union of intersections", for example A*D + B*c.

The same expression can be written in multivalue notation: A{1}*D{1} + B{1}*C{0}. Both types of expressions are valid, and yield the same result on the same dataset.

For multivalue notation, causal conditions are expected as upper case letters, and they will be converted to upper case by default. Expressions can contain multiple values to translate, separated by a comma. If B was a multivalue causal condition, an expression could be: A{1} + B{1,2}*C{0}.

In this example, all values in B equal to either 1 or 2 will be converted to 1, and the rest of the (multi)values will be converted to 0.

The function automatically detects the use of tilde "~" as a negation for a particular causal condition. ~A does two things: it identifies the presence of causal condition A (because it was specified as upper case) and it recognizes that it must be negated, because of the tilde. It works even combined with lower case names: ~a, which is interpreted as A.

To negate a multivalue condition using a tilde, the number of levels should be supplied (see examples below). Improvements in version 2.5 allow for intersections between multiple levels of the same condition. For a causal condition with 3 levels (0, 1 and 2) the following expression ~A{0,2}*A{1,2} is equivalent with A{1}, while A{0}*A{1} results in the empty set.

The number of levels, as well as the set names can be automatically detected from a dataset via the argument data. Arguments snames and noflevels have precedence over data, when specified.

The use of the product operator * is redundant when causal snames' names are single letters (for example AD + Bc), and is also redundant for multivalue data, where product terms can be separated by using the curly brackets notation.

When causal snames are binary and their names have multiple letters (for example AA + CC*bb), the use of the product operator * is preferable but the function manages to translate an expression even without it (AA + CCbb) by searching deep in the space of the conditions' names, at the cost of slowing down for a high number of causal conditions. For this reason, an arbitrary limit of 7 causal snames is imposed, to write an expression with.

References

Jirka Lewandowski (2015) QCAtools: Helper functions for QCA in R. R package version 0.1

Examples

# NOT RUN {
translate("A + B*C")

# same thing in multivalue notation
translate("A{1} + B{1}*C{1}")

# using upper/lower letters
translate("A + b*C")

# the negation with tilde is recognised
translate("~A + b*C")

# even in combination of upper/lower letters
translate("~A + ~b*C")

# and even for multivalue variables
# in multivalue notation, the product sign * is redundant
translate("C{1} + T{2} + T{1}V{0} + C{0}")

# negation of multivalue sets requires the number of levels
translate("~A{1} + ~B{0}*C{1}", noflevels = c(2, 2, 2))

# multiple values can be specified
translate("C{1} + T{1,2} + T{1}V{0} + C{0}")

# or even negated
translate("C{1} + ~T{1,2} + T{1}V{0} + C{0}", snames = "C, T, V", noflevels = c(2,3,2))

# if the expression does not contain the product sign *
# snames are required to complete the translation 
translate("AB + cD", snames = "A, B, C, D")

# otherwise snames are not required
translate("PER*FECT + str*ing")

# snames are required
translate("PERFECT + string", snames = "PER, FECT, STR, ING")

# it works even with overlapping columns
# SU overlaps with SUB in SUBER, but the result is still correct
translate("SUBER + subset", "SU, BER, SUB, SET")

# }
# NOT RUN {
# error because combinations of condition names clash
translate("SUPER + subset", "SUP, ER, SU, PER, SUB, SET")
# }
# NOT RUN {
# to print _all_ codes from the standard output matrix
(obj <- translate("A + b*C"))
print(obj, original = TRUE) # also prints the -1 code



# for compute()
data(CVF)
compute("natpride + GEOCON", data = CVF)

# calculating individual paths
compute("natpride + GEOCON", data = CVF, separate = TRUE)



# for findRows()
data(LC)
ttLC <- truthTable(LC, "SURV", show.cases = TRUE)

findRows("DEV*ind*STB", ttLC)

findRows("DEV*ind*STB", ttLC, remainders = TRUE)



# for sop()
sop("(A + B)(A + ~B)") # result is "A"

# to force a certain order of the set names
sop("(URB + LIT*~DEV)(~LIT + ~DEV)", snames = "DEV, URB, LIT")

# multilevel conditions can also be specified (and negated)
sop("(A{1} + ~B{0})(B{1} + C{0})", snames = "A, B, C", noflevels = c(2, 3, 2))
# }