minimize: Minimize a truth table

Description Usage Arguments Details Value Author(s) References See Also Examples

Description

This function performs the QCA minimization of an input truth table, or if the input is a dataset the minimization it minimizes a set of causal conditions with respect to an outcome. Three minimization methods are available: the classical Quine-McCluskey, the enhanced Quine-McCluskey and the latest Consistency Cubes algorithm that is built for performance.

All algorithms return the same, exact solutions, see Dusa (2017) and Dusa and Thiem (2015).

Usage

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minimize(input, include = "", exclude = NULL, dir.exp = "",
         pi.cons = 0, pi.depth = 0, sol.cons = 0, sol.cov = 1, sol.depth = 0,
         min.pin = FALSE, row.dom = FALSE, all.sol = FALSE,
         details = FALSE, use.tilde = FALSE, method = "CCubes", ...)

Arguments

input

A truth table object (preferred) or a data frame containing calibrated causal conditions and an outcome.

include

A vector of other output values to include in the minimization process.

exclude

A vector of row numbers from the truth table, or a matrix of causal combinations to exclude from the minimization process.

dir.exp

A vector of directional expectations to derive intermediate solutions.

pi.cons

Numerical fuzzy value between 0 and 1, minimal consistency threshold for a prime implicant to be declared as sufficient.

pi.depth

Integer, a maximum number of causal conditions to be used when searching for conjunctive prime implicants.

sol.cons

Numerical fuzzy value between 0 and 1, minimal consistency threshold for a model to be declared as sufficient.

sol.cov

Numerical fuzzy value between 0 and 1, minimal coverage threshold for a model to be declared as necessary.

sol.depth

Integer, a maximum number of prime implicants to be used when searching for disjunctive solutions.

min.pin

Logical, terminate the search at the depth where newly found prime implicants do not contribute to minimally solving the PI chart.

row.dom

Logical, perform row dominance in the prime implicants' chart to eliminate redundant prime implicants.

all.sol

Logical, search for all possible solutions even of not minimal.

details

Logical, print more details about the solution.

use.tilde

Logical, use tilde to signal the absence of conditions.

method

Minimization method, one of "CCubes" (default), or "QMC" the classical Quine-McCluskey, or "eQMC" the enhanced Quine-McCluskey.

...

Other arguments to be passed to function truthTable.

Details

Most of the times, this function takes a truth table object as the input for the minimization procedure, but the same argument can refer to a data frame containing calibrated columns.

For the later case, the function minimize() originally had some additional formal arguments which were sent to the function truthTable(): outcome, conditions, n.cut, incl.cut, show.cases, use.letters and inf.test.

All of these parameters are still possible with function minimize(), but since they are sent to the truthTable() function anyway, it is unnecessary to duplicate their explanation here. The only situation which does need an additional description relates to the argument outcome, where unlike truthTable() which accepts a single one, the function minimize() accepts multiple outcomes and performs a minimization for each of them (a situation when all columns are considered causal conditions).

The argument include specifies which other truth table rows are included in the minimization process. Most often, the remainders are included but any value accepted in the argument explain is also accepted in the argument include.

The argument exclude is used to exclude truth table rows from the minimization process, from the positive configurations and/or from the remainders. It can be specified as a vector of truth table line numbers, or as a matrix of causal combinations.

The argument dir.exp is used to specify directional expectations, as described by Ragin (2003). They can be specified as a single string, with values separated by commas. For multi-value directional expectations, they are specified together, separated by semicolons. The total length of the directional expectations must match the number of causal conditions specified in the analysis, using a dash "-" if there are no particular expectations for a specific causal condition.

Activating the details argument has the effect of printing parameters of fit for each prime implicant and each overall solution, the essential prime implicants being listed in the top part of the table. It also prints the truth table, in case the argument input has been provided as a data frame instead of a truth table object.

The argument use.tilde signals the absence of a causal condition, in a sufficiency relation with the outcome, using a tilde sign "~". It is ignored if the data is multivalent.

By default, the package QCA employes a different search algorithm based on Consistency Cubes (Dusa, 2017), analysing all possible combinations of causal conditions and all possible combinations of their respective levels. The structure of the input dataset (number of causal conditions, number of levels, number of unique rows in the truth table) has a direct implication on the search time, as all of those characteristics become entry parameters when calculating all possible combinations.

Consequently, two kinds of depth arguments are provided:

pi.depth the maximum number of causal conditions needed to construct a prime
implicant, the complexity level where the search can be stopped, as long as the
PI chart can be solved.
sol.depth the maximum number of prime implicants needed to find a solution
(to cover all initial positive output configurations)

These arguments introduce a possible new way of deriving prime implicants and solutions, that can lead to different results (i.e. even more parsimonious) compared to the classical Quine-McCluskey. When either of them is modified from the default value of 0, the minimization method is automatically set to "CCubes" and the remainders are automatically included in the minimization.

The search time is larger the higher these depths, or inversely the search time can be significantly shorter if these depths are smaller. Irrespective of how large pi.depth is, the algorithm will always stop at a maximum complexity level where no new, non-redundant prime implicants are found. The argument sol.depth is relevant only when activating the argument all.sol to solve the PI chart.

The default method (when all.sol = FALSE), is to find the minimal number (k) of prime implicants needed to cover all initial positive output configurations, then it exhaustively searches through all possible disjunctions of k prime implicants which do cover those configurations.

The argument min.pin introduces an additional parameter to control when to stop the search for prime implicants. It is based on the observation by Dusa (2017) that out of the entire set of non redundant prime implicants, only a subset actually contribute to solving the chart with disjunctions of k PIs. The search depth can be shortened at the level where the next subset of PIs do not contribute to solving the PI chart, thus avoiding to spend unnecessary time on finding the maximal number of non-redundant PIs. Instead, it finds the set of minimal ("min") number of PIs ("pin") necessary to obtain exactly the same solutions, with a dramatically improved overall performance.

Once the PI chart is constructed using the prime implicants found in the previous stages, the argument row.dom can be used to further eliminate irrelevant prime implicants when solving the PI chart, applying the principle of row dominance: if a prime implicant A covers the same (intial) positive output configurations as another prime implicant B and in the same time covers other configurations which B does not cover, then B is irrelevant and eliminated.

The argument all.sol automatically deactivates the argument min.pin, because it aims to exhaustively identify all possible non-redundant disjunctions of n prime implicants that solve the PI chart, where n >= k, with an inflated number of possible solutions. Depending on the complexity of the PI chart, sometimes it may take a very long time to identify all possible non-redundant (disjunctions that are not subsets of previously found) disjunctive solutions. In such a situation, the number of combinations of all possible numbers of prime implicants is potentially too large to be solved in a polynomial time and if not otherwise specified, the depth for the disjunctive solutions is automatically bounded to 5 prime implicants.

The task of solving the PI chart depends on its size, with prime implicants on the rows and the positive output configurations on the columns. Since the columns are fixed, another possible way to reduce the solving time is to eliminate redundant rows, by activating the argument row.dom.

If minimizing a dataset instead of a truth table, unless otherwise specified the argument incl.cut is automatically set to the minimum value between pi.cons and sol.cons, then passed to the function truthTable().

The argument sol.cons introduces another possibility to change the method of solving the PI chart. Normally, once the solutions are found among all possible combinations of k prime implicants, consistencies and coverages are subsequently calculated. When sol.cons is lower than 1, then solutions are searched based on their consistencies, which should be at least equal to this threshold.

Value

An object of class "qca" when using a single outcomes, or class "mqca" when using multiple outcomes. These objects are lists having the following components:

tt

The truth table object.

options

Values for the various options used in the function (including defaults).

negatives

The line number(s) of the negative configuration(s).

initials

The initial positive configuration(s).

PIchart

A list containing the PI chart(s).

primes

The prime implicant(s).

solution

A list of solution(s).

essential

A list of essential PI(s).

pims

A list of PI membership scores.

IC

The matrix containing the inclusion and coverage scores for the solution(s).

SA

A list of simplifying assumptions.

i.sol

A list of components specific to intermediate solution(s), each having a PI chart, prime implicant membership scores, (non-simplifying) easy counterfactuals and difficult counterfactuals.

call

The user's command which produced all these objects and result(s).

Author(s)

Adrian Dusa

References

Cebotari, V.; Vink, M.P. (2013) “A Configurational Analysis of Ethnic Protest in Europe”. International Journal of Comparative Sociology vol.54, no.4, pp.298-324.

Cebotari, V.; Vink, M.P. (2015) “Replication Data for: A configurational analysis of ethnic protest in Europe”, Harvard Dataverse, V2, DOI: 10.7910/DVN/PT2IB9,

Cronqvist, L.; Berg-Schlosser, D. (2009) “Multi-Value QCA (mvQCA)”, in Rihoux, B.; Ragin, C. (eds.) Configurational Comparative Methods. Qualitative Comparative Analysis (QCA) and Related Techniques, SAGE.

Dusa, A.; Thiem, A. (2015) “Enhancing the Minimization of Boolean and Multivalue Output Functions With eQMC” Journal of Mathematical Sociology vol.39, no.2, pp.92-108,
DOI: 10.1080/0022250X.2014.897949.

Dusa, A. (2017) “Consistency Cubes: A Fast, Efficient Method for Boolean Minimization”, unpublished working paper on ResearchGate, DOI: RG.2.2.36359.19361/1

Ragin, C. (2003) Recent Advances in Fuzzy-Set Methods and Their Application to Policy Questions. WP 2003-9, COMPASSS.

Ragin, C. (2009) “Qualitative Comparative Analysis Using Fuzzy-Sets (fsQCA)”, in Rihoux, B.; Ragin, C. (eds.) Configurational Comparative Methods. Qualitative Comparative Analysis (QCA) and Related Techniques, SAGE.

Ragin, C.C.; Strand, S.I. (2008) “Using Qualitative Comparative Analysis to Study Causal Order: Comment on Caren and Panofsky (2005).” Sociological Methods & Research vol.36, no.4, pp.431-441.

Rihoux, B.; De Meur, G. (2009) “Crisp Sets Qualitative Comparative Analysis (mvQCA)”, in Rihoux, B.; Ragin, C. (eds.) Configurational Comparative Methods. Qualitative Comparative Analysis (QCA) and Related Techniques, SAGE.

See Also

truthTable, factorize

Examples

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# -----
# Lipset binary crisp data
data(LC)

# the associated truth table
ttLC <- truthTable(LC, "SURV", sort.by = "incl, n", show.cases = TRUE)
ttLC

# conservative solution (Rihoux & De Meur 2009, p.57)
cLC <- minimize(ttLC)
cLC

# view the Venn diagram for the associated truth table
library(venn)
venn(cLC)

# add details and case names
minimize(ttLC, details = TRUE)

# negating the outcome
ttLCn <- truthTable(LC, "~SURV", sort.by = "incl, n", show.cases = TRUE)
minimize(ttLCn)

# using a tilde instead of upper/lower case names
minimize(ttLCn, use.tilde = TRUE)

# parsimonious solution, positive output
pLC <- minimize(ttLC, include = "?", details = TRUE)
pLC

# the associated simplifying assumptions
pLC$SA

# parsimonious solution, negative output
pLCn <- minimize(ttLCn, include = "?", details = TRUE)
pLCn


# -----
# Lipset multi-value crisp data (Cronqvist & Berg-Schlosser 2009, p.80)
data(LM)

# truth table 
ttLM <- truthTable(LM, "SURV", conditions = "DEV, URB, LIT, IND",
        sort.by = "incl", show.cases = TRUE)

# conservative solution, positive output
minimize(ttLM, details = TRUE)

# parsimonious solution, positive output
minimize(ttLM, include = "?", details = TRUE)

# negate the outcome
ttLMn <- truthTable(LM, "~SURV", conditions = "DEV, URB, LIT, IND",
         sort.by = "incl", show.cases = TRUE)

# conservative solution, negative output
minimize(ttLMn, details = TRUE)

# parsimonious solution, positive output
minimize(ttLMn, include = "?", details = TRUE)

# -----
# Lipset fuzzy sets data (Ragin 2009, p.112)
data(LF)
# truth table using a very low inclusion cutoff
ttLF <- truthTable(LF, "SURV", incl.cut = 0.7,
        sort.by = "incl", show.cases = TRUE)

# conservative solution
minimize(ttLF, details = TRUE)

# parsimonious solution
minimize(ttLF, include = "?", details = TRUE)

# intermediate solution using directional expectations
iLF <- minimize(ttLF, include = "?", details = TRUE,
                dir.exp = "1,1,1,1,1")
iLF



# -----
# Cebotari & Vink (2013, 2015)
data(CVF) 

ttCVF <- truthTable(CVF, outcome = "PROTEST", incl.cut = 0.8,
                    sort.by = "incl, n", show.cases = TRUE)

pCVF <- minimize(ttCVF, include = "?", details = TRUE)
pCVF

# inspect the PI chart
pCVF$PIchart

# DEMOC*ETHFRACT*poldis is dominated by DEMOC*ETHFRACT*GEOCON
# using row dominance to solve the PI chart
pCVFrd <- minimize(ttCVF, include = "?", row.dom = TRUE, details = TRUE)

# plot the prime implicants on the outcome
pims <- pCVFrd$pims

par(mfrow = c(2, 2))
for(i in 1:4) {
    XYplot(pims[, i], CVF$PROTEST, cex.axis = 0.6)
}



# -----
# temporal QCA (Ragin & Strand 2008) serving the input as a dataset,
# which will automatically be passed to truthTable() as an intermediary
# step before the minimization

data(RS)
minimize(RS, outcome = "REC", details = TRUE)

QCA documentation built on April 1, 2018, 12:12 p.m.

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