version 3.23

# 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

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    minimize(input, include = "", dir.exp = NULL, details = FALSE, pi.cons = 0,
        sol.cons = 0, all.sol = FALSE, row.dom = FALSE, first.min = FALSE,
        max.comb = 0, use.labels = 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.

                first.min

                Logical, to return only the very first minimal solution (see
                    Details).

                max.comb

                Numeric real, to limit the size of the PI chart (see Details).

                use.labels

                Logical, use category labels if present (See Examples).

                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 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.

        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

    # -----
    # employ category labels

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

    minimize(ttLF, include = "?", use.labels = TRUE)

    M1: Developed*Not industrialised + High urbanisation*Stability -> Survival

    # or

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

    minimize(ttLF, include = "?")

    M1: Developed*Not industrialised + High urbanisation*Stability -> Survival

        Author

    Adrian Dusa

        See also
        truthTable,
        factorize