Interpret SOP/DNF expressions: compute, simplify

Description

These functions interpret an expression written in a SOP - sum of products (or in canonical DNF - disjunctive normal form), for both crisp and multivalue QCA. The function compute() tcalculates set membership scores based on a SOP expression applied to a calibrated data set.

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.

A function similar to compute() was initially written by 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 simplify() transforms any expression (most notably a POS product of sums) into a simpler sum of products, minimizing it to the simplest equivalent logical expression. It provides a software implementation of the intersection examples presented by Ragin (1987: 144-147), and extended to multi-value sets.

Usage

compute(expression = "", data, separate = FALSE)
simplify(expression = "", snames = "", noflevels, use.tilde = FALSE)

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.
use.tilde Logical, use tilde to negate bivalent conditions.
data A dataset with binary cs, mv and fs data.
separate Logical, perform computations on individual, separate paths.

Details

An expression written in SOP - sum of products, is a "union of intersections", for example A*B + B*c.The DNF - disjunctive normal form is also a sum of products, with the restriction that each product has to contain all literals. The equivalent expression is: A*B*c + A*B*c + a*B*c

The same expression can be written in multivalue notation: A{1}*B{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.

These functions 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 the set 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 conditions 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.

For the function simplify(), if a tilde is present in the expression, the argument use.tilde is automatically activated. For Boolean expressions, the simplest equivalent logical expression can result in the empty set, if the conditions cancel each other out.

Value

For the function compute(), a vector of set membership values. For function simplify(), a character expression.

References

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

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

Examples

# for compute() compute("DEV*ind + URB*STB", data = LF)
[1] 0.27 0.89 0.91 0.16 0.58 0.19 0.31 0.09 0.13 0.72 0.34 0.99 0.02 0.01 0.03 [16] 0.20 0.33 0.98
data(CVF) compute("DEV*ind + URB*STB", data = LF, separate = TRUE)
DEV*ind URB*STB 1 0.27 0.12 2 0.00 0.89 3 0.10 0.91 4 0.16 0.07 5 0.58 0.03 6 0.19 0.03 7 0.04 0.31 8 0.04 0.09 9 0.07 0.13 10 0.72 0.05 11 0.34 0.10 12 0.06 0.99 13 0.02 0.00 14 0.01 0.01 15 0.01 0.03 16 0.03 0.20 17 0.33 0.13 18 0.00 0.98
# for simplify() simplify("(A + B)(A + ~B)")
S1: A
# to force a certain order of the set names simplify("(URB + LIT*~DEV)(~LIT + ~DEV)", snames = "DEV, URB, LIT")
S1: ~DEV*LIT + URB*~LIT
# multilevel conditions can also be specified (and negated) simplify("(A{1} + ~B{0})(B{1} + C{0})", snames = "A, B, C", noflevels = c(2, 3, 2))
S1: B{1} + A{1}C{0} + B{2}C{0}
# in Ragin's (1987) book, the equation E = SG + LW is the result # of the Boolean minimization for the ethnic political mobilization. # intersecting the reactive ethnicity perspective (R = lw) # with the equation E (page 144) simplify("lw(SG + LW)", snames = "S, L, W, G")
S1: SlwG
# resources for size and wealth (C = SW) with E (page 145) simplify("SW(SG + LW)", snames = "S, L, W, G")
S1: SLW + SWG
# and factorized factorize(simplify("SW(SG + LW)", snames = "S, L, W, G"))
F1: SW(G + L)
# developmental perspective (D = Lg) and E (page 146) simplify("Lg(SG + LW)", snames = "S, L, W, G", use.tilde = TRUE)
S1: LW~G
# subnations that exhibit ethnic political mobilization (E) but were # not hypothesized by any of the three theories (page 147) # ~H = ~(lw + SW + Lg) = GLs + GLw + GsW + lsW simplify("(GLs + GLw + GsW + lsW)(SG + LW)", snames = "S, L, W, G")
S1: sLWG + SLwG

Author

Adrian Dusa