# Minimize a truth table

## 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

    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 intermediate solutions.
`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 solutions.
`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 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 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 default method (when `all.sol`), 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 (2017) that out of the entire set of non redundant prime implicants, only a subset actually contribute to solving the PI chart. The search depth can be 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.

Depending on the complexity of the PI chart, it sometimes may take a very long time to identify all possible irredundant (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.

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.

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 \bold{\code{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(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.
`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, no.2, pp.357-370, DOI: 10.32614/RJ-2018-080.

Dusa, A. (2019) QCA with R. A Comprehensive Resource. Springer International Publishing.

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

Ragin, C.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.

## Examples

    # -----
    # Lipset binary crisp data

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

      OUT: outcome value
        n: number of cases in configuration
     incl: sufficiency inclusion score

        DEV URB LIT IND STB OUT n  incl  PRI
    32   1   1   1   1   1   1  4  1.000 1.000
    22   1   0   1   0   1   1  2  1.000 1.000
    24   1   0   1   1   1   1  2  1.000 1.000
     1   0   0   0   0   0   0  3  0.000 0.000
     2   0   0   0   0   1   0  2  0.000 0.000
     5   0   0   1   0   0   0  2  0.000 0.000
     6   0   0   1   0   1   0  1  0.000 0.000
    23   1   0   1   1   0   0  1  0.000 0.000
    31   1   1   1   1   0   0  1  0.000 0.000

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

    M1: DEV*urb*LIT*STB + DEV*LIT*IND*STB <=> SURV

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

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

    n OUT = 1/0/C: 8/10/0
      Total      : 18

    Number of multiple-covered cases: 2

    M1: DEV*~URB*LIT*STB + DEV*LIT*IND*STB <=> SURV

                         inclS   PRI   covS   covU   cases
    -------------------------------------------------------------------
    1  DEV*~URB*LIT*STB  1.000  1.000  0.500  0.250  FI,IE; FR,SE
    2  DEV*LIT*IND*STB   1.000  1.000  0.750  0.500  FR,SE; BE,CZ,NL,UK
    -------------------------------------------------------------------
       M1                1.000  1.000  1.000

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

    M1: ~DEV*~URB*~IND + DEV*LIT*IND*~STB <=> ~SURV

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

    n OUT = 1/0/C: 8/10/0
      Total      : 18

    Number of multiple-covered cases: 0

    M1: DEV*STB <=> SURV

                inclS  PRI    covS   covU   cases
    -----------------------------------------------------------------
    1  DEV*STB  1.000  1.000  1.000    -    FI,IE; FR,SE; BE,CZ,NL,UK
    -----------------------------------------------------------------
       M1       1.000  1.000  1.000

    # the associated simplifying assumptions
    pLC$SA

    $M1
       DEV URB LIT IND STB
    18   1   0   0   0   1
    20   1   0   0   1   1
    26   1   1   0   0   1
    28   1   1   0   1   1
    30   1   1   1   0   1

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

    n OUT = 1/0/C: 10/8/0
      Total      : 18

    Number of multiple-covered cases: 5

    M1: ~DEV + ~STB <=> ~SURV

             inclS   PRI   covS   covU   cases
    ---------------------------------------------------------------
    1  ~DEV  1.000  1.000  0.800  0.300  GR,PT,ES; IT,RO; HU,PL; EE
    2  ~STB  1.000  1.000  0.700  0.200  GR,PT,ES; HU,PL; AU; DE
    ---------------------------------------------------------------
       M1    1.000  1.000  1.000

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

    # 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)

    n OUT = 1/0/C: 7/11/0
      Total      : 18

    Number of multiple-covered cases: 0

    M1: DEV[2]*LIT[1]*IND[1] + DEV[1]*URB[0]*LIT[1]*IND[0] => SURV[1]

                                    inclS  PRI    covS   covU   cases
    ---------------------------------------------------------------------------
    1  DEV[2]*LIT[1]*IND[1]         1.000  1.000  0.625  0.625  FR,SE; BE,NL,UK
    2  DEV[1]*URB[0]*LIT[1]*IND[0]  1.000  1.000  0.250  0.250  FI,IE
    ---------------------------------------------------------------------------
       M1                           1.000  1.000  0.875

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

    n OUT = 1/0/C: 7/11/0
      Total      : 18

    Number of multiple-covered cases: 0

    M1: DEV[2] + DEV[1]*IND[0] => SURV[1]

                      inclS  PRI    covS   covU   cases
    -------------------------------------------------------------
    1  DEV[2]         1.000  1.000  0.625  0.625  FR,SE; BE,NL,UK
    2  DEV[1]*IND[0]  1.000  1.000  0.250  0.250  FI,IE
    -------------------------------------------------------------
       M1             1.000  1.000  0.875

    # 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, show.cases = TRUE)

    n OUT = 1/0/C: 9/9/0
      Total      : 18

    Number of multiple-covered cases: 0

    M1: DEV[0]*URB[0]*IND[0] + DEV[1]*URB[0]*LIT[1]*IND[1] => ~SURV[1]

                                    inclS  PRI    covS   covU   cases
    ------------------------------------------------------------------------------------
    1  DEV[0]*URB[0]*IND[0]         1.000  1.000  0.800  0.800  GR,IT,PT,RO,ES; EE,HU,PL
    2  DEV[1]*URB[0]*LIT[1]*IND[1]  1.000  1.000  0.100  0.100  AU
    ------------------------------------------------------------------------------------
       M1                           1.000  1.000  0.900

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

    n OUT = 1/0/C: 9/9/0
      Total      : 18

    Number of multiple-covered cases: 0

    M1: DEV[0] + DEV[1]*URB[0]*IND[1] => ~SURV[1]

                             inclS  PRI    covS   covU   cases
    -----------------------------------------------------------------------------
    1  DEV[0]                1.000  1.000  0.800  0.800  GR,IT,PT,RO,ES; EE,HU,PL
    2  DEV[1]*URB[0]*IND[1]  1.000  1.000  0.100  0.100  AU
    -----------------------------------------------------------------------------
       M1                    1.000  1.000  0.900

    # -----
    # Lipset fuzzy sets data (Ragin 2009, p.112)

    ttLF <- truthTable(LF, "SURV", incl.cut = 0.8, show.cases = TRUE, sort.by="incl")

    # conservative solution
    minimize(ttLF, details = TRUE)

    n OUT = 1/0/C: 6/12/0
      Total      : 18

    Number of multiple-covered cases: 0

    M1: DEV*URB*LIT*IND*STB + DEV*~URB*LIT*~IND*STB => SURV

                              inclS   PRI   covS   covU   cases
    -----------------------------------------------------------------
    1  DEV*URB*LIT*IND*STB    0.904  0.886  0.454  0.393  BE,CZ,NL,UK
    2  DEV*~URB*LIT*~IND*STB  0.804  0.719  0.265  0.204  FI,IE
    -----------------------------------------------------------------
       M1                     0.870  0.843  0.658

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

    n OUT = 1/0/C: 6/12/0
      Total      : 18

    Number of multiple-covered cases: 0

    M1: DEV*~IND + URB*STB => SURV

                 inclS   PRI   covS   covU   cases
    ----------------------------------------------------
    1  DEV*~IND  0.815  0.721  0.284  0.194  FI,IE
    2  URB*STB   0.874  0.845  0.520  0.430  BE,CZ,NL,UK
    ----------------------------------------------------
       M1        0.850  0.819  0.714

    # intermediate solution
    minimize(ttLF, include = "?", details = TRUE,
                 dir.exp = "DEV, URB, LIT, IND, STB")

    n OUT = 1/0/C: 6/12/0
      Total      : 18

    From C1P1:

    Number of multiple-covered cases: 0

    M1:    DEV*URB*LIT*STB + DEV*LIT*~IND*STB => SURV

                         inclS   PRI   covS   covU   cases
    ------------------------------------------------------------
    1  DEV*URB*LIT*STB   0.901  0.879  0.468  0.378  BE,CZ,NL,UK
    2  DEV*LIT*~IND*STB  0.814  0.721  0.282  0.191  FI,IE
    ------------------------------------------------------------
       M1                0.866  0.839  0.660

    # directional expectations can also be specified using a sequence
    minimize(ttLF, include = "?", details = TRUE, dir.exp = DEV:STB)

    n OUT = 1/0/C: 6/12/0
      Total      : 18

    From C1P1:

    Number of multiple-covered cases: 0

    M1:    DEV*URB*LIT*STB + DEV*LIT*~IND*STB => SURV

                         inclS   PRI   covS   covU   cases
    ------------------------------------------------------------
    1  DEV*URB*LIT*STB   0.901  0.879  0.468  0.378  BE,CZ,NL,UK
    2  DEV*LIT*~IND*STB  0.814  0.721  0.282  0.191  FI,IE
    ------------------------------------------------------------
       M1                0.866  0.839  0.660

    # URB as a don't care condition (left unspecified) and
    # conjunctural directional expectations
    minimize(ttLF, include = "?", details = TRUE, show.cases = TRUE,
             dir.exp = "DEV, STB, ~LIT*IND")

    n OUT = 1/0/C: 6/12/0
      Total      : 18

    From C1P1:

    Number of multiple-covered cases: 0

    M1:    DEV*URB*STB + DEV*~IND*STB => SURV

                     inclS   PRI   covS   covU   cases
    --------------------------------------------------------
    1  DEV*URB*STB   0.901  0.879  0.468  0.378  BE,CZ,NL,UK
    2  DEV*~IND*STB  0.814  0.721  0.282  0.191  FI,IE
    --------------------------------------------------------
       M1            0.866  0.839  0.660

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

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

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

    n OUT = 1/0/C: 13/16/0
      Total      : 29

    Number of multiple-covered cases: 5

    M1: ~NATPRIDE + DEMOC*GEOCON*POLDIS + (~DEMOC*ETHFRACT*POLDIS + DEMOC*ETHFRACT*GEOCON) => PROTEST
    M2: ~NATPRIDE + DEMOC*GEOCON*POLDIS + (~DEMOC*ETHFRACT*POLDIS + DEMOC*ETHFRACT*~POLDIS) => PROTEST
    M3: ~NATPRIDE + DEMOC*GEOCON*POLDIS + (DEMOC*ETHFRACT*GEOCON + ETHFRACT*GEOCON*POLDIS) => PROTEST
    M4: ~NATPRIDE + DEMOC*GEOCON*POLDIS + (DEMOC*ETHFRACT*~POLDIS + ETHFRACT*GEOCON*POLDIS) => PROTEST

                                                    ---------------------------------
                               inclS   PRI   covS   covU   (M1)   (M2)   (M3)   (M4)
    ---------------------------------------------------------------------------------
    1 ~NATPRIDE               0.899  0.807  0.597  0.121  0.132  0.122  0.136  0.126
    2 DEMOC*GEOCON*POLDIS     0.906  0.805  0.342  0.065  0.065  0.070  0.065  0.065
    ---------------------------------------------------------------------------------
    3  ~DEMOC*ETHFRACT*POLDIS  0.842  0.718  0.299  0.000  0.040  0.040
    4  DEMOC*ETHFRACT*GEOCON   0.935  0.826  0.480  0.000  0.085         0.085
    5  DEMOC*ETHFRACT*~POLDIS  0.932  0.773  0.417  0.000         0.085         0.085
    6  ETHFRACT*GEOCON*POLDIS  0.869  0.786  0.365  0.005                0.045  0.045
    ---------------------------------------------------------------------------------
       M1                      0.877  0.777  0.805
       M2                      0.877  0.777  0.805
       M3                      0.879  0.782  0.810
       M4                      0.879  0.782  0.810

                               cases
    --------------------------------
    1  ~NATPRIDE               CrimRussiansUkr,RussiansUkraine; HungariansYugo,KosovoAlbanians; RussiansLatvia;
                                BasquesSpain; AlbaniansFYROM
    2  DEMOC*GEOCON*POLDIS     HungariansRom,CatholicsNIreland; AlbaniansFYROM; RussiansEstonia
    --------------------------------
    3  ~DEMOC*ETHFRACT*POLDIS  HungariansYugo,KosovoAlbanians; GagauzMoldova
    4  DEMOC*ETHFRACT*GEOCON   BasquesSpain; SerbsFYROM,CatalansSpain; AlbaniansFYROM; RussiansEstonia
    5  DEMOC*ETHFRACT*~POLDIS  BasquesSpain; SerbsFYROM,CatalansSpain
    6  ETHFRACT*GEOCON*POLDIS  HungariansYugo,KosovoAlbanians; GagauzMoldova; AlbaniansFYROM; RussiansEstonia
    --------------------------------

    # inspect the PI chart
    pCVF$PIchart

                             5  15 16 24 27 29 30 31 32
    ~NATPRIDE                x  x  -  -  x  x  -  x  -
    ~DEMOC*ETHFRACT*POLDIS   -  x  x  -  -  -  -  -  -
    DEMOC*ETHFRACT*GEOCON    -  -  -  -  -  x  x  x  x
    DEMOC*ETHFRACT*~POLDIS   -  -  -  -  -  x  x  -  -
    DEMOC*GEOCON*POLDIS      -  -  -  x  -  -  -  x  x
    ETHFRACT*GEOCON*POLDIS   -  x  x  -  -  -  -  x  x

    # 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, show.cases = 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)

      OUT: outcome value
        n: number of cases in configuration
     incl: sufficiency inclusion score

         P  E  A  EBA S    OUT    n  incl  PRI
     5   0  0  0   -  0     0     3  0.000 0.000
    17   0  1  0   -  0     0     1  0.000 0.000
    20   0  1  1   0  1     1     1  1.000 1.000
    29   1  0  0   -  0     0     1  0.000 0.000
    30   1  0  0   -  1     0     3  0.000 0.000
    36   1  0  1   -  1     0     2  0.000 0.000
    42   1  1  0   -  1     1     1  1.000 1.000
    44   1  1  1   0  1     1     1  1.000 1.000
    45   1  1  1   1  0     1     2  1.000 1.000
    46   1  1  1   1  1     1     2  1.000 1.000

    n OUT = 1/0/C: 7/10/0
      Total      : 17

    Number of multiple-covered cases: 3

    M1: P*E*S + P*E*A*EBA + E*A*~EBA*S <=> REC

                   inclS   PRI   covS   covU
    -----------------------------------------
    1  P*E*S       1.000  1.000  0.571  0.143
    2  P*E*A*EBA   1.000  1.000  0.571  0.286
    3  E*A*~EBA*S  1.000  1.000  0.286  0.143
    -----------------------------------------
       M1          1.000  1.000  1.000

## Author

Adrian Dusa

## See also

`truthTable`, `factorize`