# minimize: Minimize a truth table In QCA: Qualitative Comparative Analysis

## 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 (2018) and Dusa and Thiem (2015).

## Usage

 ```1 2 3``` ```minimize(input, include = "", dir.exp = NULL, details = FALSE, pi.cons = 0, sol.cons = 0, all.sol = FALSE, row.dom = FALSE, min.pin = FALSE, max.comb = 0, first.min = 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. `dir.exp` Character, a vector of directional expectations to derive the intermediate solution. `details` Logical, print more details about the solution. `pi.cons` Numerical fuzzy value between 0 and 1, minimal consistency threshold for a prime implicant to be declared as sufficient. `sol.cons` Numerical fuzzy value between 0 and 1, minimal consistency threshold for a model to be declared as sufficient. `all.sol` Logical, search for all possible models, including the non-minimal. `row.dom` Logical, perform row dominance in the prime implicants' chart to eliminate redundant prime implicants. `min.pin` Logical, terminate the search at the depth where newly found prime implicants do not contribute to minimally solving the PI chart. `max.comb` Numeric real, to limit the size of the PI chart (see Details). `first.min` Logical, to return only the very first minimal solution (see Details). `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 using SOP (sum of products) expressions, which opens up the possibility to experiment with conjunctural directional expectations. "Don't care" conditions are simply left unspecified.

Activating the `details` argument has the effect of printing parameters of fit for each prime implicant and each overall model, 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.

By default, the package QCA employes a different search algorithm based on Consistency Cubes (Dusa, 2018), 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 conjunctively construct a prime implicant; it is 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 disjunctively build a solution model that covers all initial positive output configurations.

These arguments introduce a possible new way of deriving prime implicants and solution models, 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 higher these depths, the higher the search time. Connversely, 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 (minterms), then exhaustively search 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 especially useful for very large truth tables and uses an observation by Dusa (2018) that out of the entire set of non redundant prime implicants, only a subset actually contribute to minimally solving the PI chart. The search depth is shortened when the PIs found at the next complexity level do not decrease the minimum `k`, thus producing absolute minimal models in both number of disjunctive prime implicants, and their depth level.

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.

A large number of causal conditions (i.e. over 15), combined with a large number of cases (i.e. hundreds) usually produce a very large number of prime implicants, resulting in a huge and extremely complex PI chart with sometimes thousands of rows and hundreds of columns.

For such a complex PI chart, even finding a minimum is a formidable task, and exhaustively solving it is very likely impossible in polynomial time. For this reason, after each level of complexity the CCubes algorithm determines if the PI chart is too difficult, by calculating the total number of combinations of minimum `k` PIs necessary to cover all columns.

The argument `max.comb` controls this maximum number of combinations. It is a rational number counted in (fractions of) billions, defaulted at zero to signal searching to the maximum possible extent. If the total number of combinations exceeds a positive value of `max.comb`, the PI chart is determined as too complex, the search is stopped and CCubes attempts to return all possible models using the PIs from the previous levels of complexity, when the PI chart was still not too complex.

In the extreme situation even this is not feasible, the argument `first.min` controls returning only one (the very first found) minimal model, if at all possible.

## Value

An object of class `"qca"` when using a single outcome, 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 model(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 model(s). `SA` A list of simplifying assumptions. `i.sol` A list of components specific to intermediate model(s), each having a PI chart, prime implicant membership scores, (non-simplifying) easy counterfactuals and difficult counterfactuals. `complex` Flag solutions from a too complex PI chart `call` The user's command which produced all these objects and result(s).

## 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, doi: 10.1177/0020715213508567.

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. (2018) “Consistency Cubes: A Fast, Efficient Method for Boolean Minimization”, R Journal vol.10, issue 2, pp. 357-370, doi: 10.32614/RJ-2018-080

Dusa, A. (2019) QCA with R. A Comprehensive Resource. Springer International Publishing, doi: 10.1007/978-3-319-75668-4.

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

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, doi: 10.1177/0049124107313903.

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.

`truthTable`, `factorize`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119``` ``` # ----- # Lipset binary crisp data # 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) # 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) # truth table, conditions all columns from DEV to IND # note the sequence operator ":" ttLM <- truthTable(LM, SURV, conditions = DEV: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: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) # truth table using a very low inclusion cutoff ttLF <- truthTable(LF, SURV, incl.cut = 0.8, sort.by = "incl", show.cases = TRUE) # conservative solution minimize(ttLF, details = TRUE) # parsimonious solution minimize(ttLF, include = "?", details = TRUE) # intermediate solution using directional expectations minimize(ttLF, include = "?", details = TRUE, dir.exp = c(DEV, URB, LIT, IND, STB)) # directional expectations can also be specified usin a sequence minimize(ttLF, include = "?", details = TRUE, dir.exp = DEV:STB) # URB as a don't care condition (left unspecified) and # experimentally, conjunctural directional expectations minimize(ttLF, include = "?", details = TRUE, dir.exp = c(DEV, STB, ~LIT*IND)) # ----- # Cebotari & Vink (2013, 2015) 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 minimize(RS, outcome = REC, details = TRUE) ```