5.6.2013
Institut für Politikwissenschaft
Schwerpunkt qualitative empirische Sozialforschung
Goethe-Universität Frankfurt
Fachbereich 03
Preparation for today
Jonas Buche
• Please download the following files
QCA and Fuzzy Sets
Research Applications and Software Tools
–
–
–
–
Truth Tables Algorithm II
5 June 2013
Krook 2010set.csv
Kim/Lee 2008set.csv
Emmenegger2011set.csv
Freitag/Schlicht2009set.csv
5.6.2013
Truth Tables Algorithm II
2
Today‘s outline
• Repetition
– 3 Steps of logical minimization, (home)work
– Logical minimization of truth tables by software
• Minimization and ‘logical remainders’
– in Tosmana
– in fsQCA 2.5
Repetition
LOGICAL MINIMIZATION
• Task for the next week / Outlook
5.6.2013
Truth Tables Algorithm II
3
Repitition on logical minimization
Quine-McCluskey algorithm in three steps
1. Make a Boolean expression of all truth table rows
connected with the outcome (primitive expressions)
Homework
2. Logically minimize this most complex expression of
sufficiency (matching similar conjunctions)
LOGICALLY MINIMIZE KROOK 2010
3. Detect logically redundant prime implicants
5.6.2013
Truth Tables Algorithm II
5
1
5.6.2013
Homework - Logically minimize Krook 2010
Conditions
Electoral
System
E
1
1
1
1
1
1
1
0
0
Quota
Q
1
0
1
0
1
1
1
0
1
Status
S
1
1
1
1
0
0
0
0
0
Movement
M
0
0
1
1
1
1
0
1
1
Logical minimization (Krook 2010)
Outcome
Cases
O
1
1
1
1
1
1
1
1
1
Country
id
SE
FI
NO,IS
DK
NL,BE
ES
AT
NZ
DE
Left Parties
L
0
0
1
1
1
0
1
1
1
EQS~M~L + E~QS~M~L + EQSML + E~QSML + EQ~SML +
EQ~SM~L + EQ~S~ML + ~E~Q~SML + ~EQ~SML  Y
E
1
1
1
1
1
1
1
0
0
Compare your result with the solution formula by Krook (2010: 896):
5.6.2013
Truth Tables Algorithm II
7
Q
1
0
1
0
1
1
1
0
1
Conditions
S
1
1
1
1
0
0
0
0
0
5.6.2013
Outcome
M
0
0
1
1
1
1
0
1
1
L
0
0
1
1
1
0
1
1
1
O
1
1
1
1
1
1
1
1
1
id
SE
FI
NO,IS
DK
NL,BE
ES
AT
NZ
DE
Truth Tables Algorithm II
8
Logical minimization (Krook 2010)
Logical minimization (Krook 2010)
EQS~M~L + E~QS~M~L + EQSML + E~QSML + EQ~SML +
EQ~SM~L + EQ~S~ML + ~E~Q~SML + ~EQ~SML  Y
EQS~M~L + E~QS~M~L + EQSML + E~QSML + EQ~SML +
EQ~SM~L + EQ~S~ML + ~E~Q~SML + ~EQ~SML  Y
ES~M~L + ESML + EQML + EQ~SM + EQ~SL + Q~SML + ~E~SML  Y
ES~M~L + ESML + EQML + EQ~SM + EQ~SL + Q~SML + ~E~SML  Y
Primitive expressions/Truth table rows
Prime
implicants
EQS~M~L
E~QS~M~L
EQSML
E~QSML
EQ~SML
EQ~SM~L
Primitive expressions/Truth table rows
EQ~S~ML
~E~Q~SML
~EQ~SML
Prime
implicants
EQS~M~L
E~QS~M~L
X
X
EQSML
E~QSML
X
EQ~SML
ES~M~L
ES~M~L
ESML
ESML
X
EQML
EQML
X
EQ~SM
EQ~SM
X
EQ~SL
EQ~SL
X
Q~SML
Q~SML
X
~E~SML
~E~SML
5.6.2013
Truth Tables Algorithm II
9
5.6.2013
EQ~SM~L
EQ~S~ML
~E~Q~SML
~EQ~SML
X
X
X
X
X
Truth Tables Algorithm II
Hypothetical logical minimization (Krook 2010)
Logical minimization by Tosmana (Krook 2010)
EQS~M~L + E~QS~M~L + EQSML + E~QSML + EQ~SML +
(Schneider/Wagemann
10)
EQ~SM~L
+ EQ~S~ML2010:
+ ~E~Q~SML
+ ~EQ~SML  Y
• How to do this with Tosmana?
X
10
ES~M~L + ESML + EQML + EQ~SM + EQ~SL + Q~SML + ~E~SML  Y
Primitive expressions/Truth table rows
EQSML
E~QSML
ESML
X
X
EQML
X
Prime
implicants
EQS~M~L
E~QS~M~L
X
X
ES~M~L
EQ~SML
EQ~SM~L
~E~Q~SML
~EQ~SML
X
EQ~SM
X
X
EQ~SL
X
X
Q~SML
X
~E~SML
5.6.2013
EQ~S~ML
X
X
X
Truth Tables Algorithm II
X
11
5.6.2013
Truth Tables Algorithm II
12
2
5.6.2013
Logical minimization by fsQCA2.5 (Krook 2010)
• How to do this with fsQCA 2.5?
Minimization and
LOGICAL REMAINDERS
5.6.2013
Truth Tables Algorithm II
13
Using logical remainders for minimization
Using logical remainders for minimization
• Types of logical remainders
• Conservative solution
– Arithmetic remainders
– Clustered remainders (female US president)
– Impossible remainders (pregnant man)
– No logical remainders are included
• Most parsimonious solution
– All simplifying logical remainders are included
• Why to use logical remainders for minimization?
– Reducing solution formula‘s complexity
 Just simplifying logical remainders are used!
5.6.2013
Truth Tables Algorithm II
15
5.6.2013
Truth Tables Algorithm II
16
Logical remainders in Krook 2010
Logical minimization with logical remainders
• Arithmetic remainders
• How to logically minimize Krook 2010 with Tosmana?
– 22 cases, 5 conditions
= at least 10 logical remainders
• Clustered remainders
– Just 14 truth table rows are covered by cases
= 18 logical remainders
• Impossible remainders
– Look at the conditions
- Electoral system, quotas, status, movements, left party
= no impossible remainder, still 18 logical remainders
5.6.2013
Truth Tables Algorithm II
17
5.6.2013
Truth Tables Algorithm II
18
3
5.6.2013
Logical minimization with logical remainders
Using logical remainders for minimization
• How to logically minimize Krook 2010 with Tosmana?
• Which ones to take into minimization?
– As we have logically equivalent prime implicants
(Tarik: competing supersets), we get two most
parsimonious solutions:
5.6.2013
Truth Tables Algorithm II
– Reducing solution formula‘s complexity
 Just simplifying logical remainders are used!
– 2 types of simplyfing assumptions:
 Easy counterfactuals vs. difficult counterfactuals
19
Easy vs. difficult counterfactuals
Truth Tables Algorithm II
20
Using logical remainders for minimization
• Definition
Criteria
5.6.2013
• Conservative solution
Easy counterfactuals
Difficult counterfactuals
Simplifying assumption


In line with empirical
evidence at hand


In line with theoretical
knowledge on the effect
of its components


– Includes no logical remainders
• Intermediate solution
– Includes only those simplifying logical remainders being in
line with theoretical knowledge (directional expectations)
• Most parsimonious solution
– Includes all simplifying logical remainders
‚directional expectations‘
5.6.2013
Truth Tables Algorithm II
21
5.6.2013
Truth Tables Algorithm II
Using logical remainders for minimization
Easy vs. difficult counterfactuals in Krook 2010
• Which ones to take into minimization?
• Which ones to take into minimization?
– In line with theoretical knowlegde (directional expectations)
• How to do that in practice?
– Look at all simplifying logical remainders
(all those used in the most parsimonious solution)
= drop those rows that contradict the directional expectations
5.6.2013
Truth Tables Algorithm II
23
22
– In line with empirical evidence
= all 18
– Simplifying remainders
= look at the most parsimonious solution (= 15 or 16)
– In line with theoretical knowlegde (directional expectations)
- Electoral System
- Quota
- Status
presence contributes the outcome
- Movements
- Left parties
5.6.2013
Truth Tables Algorithm II
24
4
5.6.2013
Using logical remainders for minimization (Krook 2010)
• Conservative solution
ES~M~L + ESML + EQ~SM + EQ~SL + ~E~SML  Y
ES + ESML + EQM + EQL + ~EML  Y
1. Compare conservative and most parsimonious
solution
• Intermediate solution
ES + EQM + EQL + ~EML  Y
• Most parsimonious solution
S + EM + QL + ~EL  Y
5.6.2013
Truth Tables Algorithm II
Generating intermediate solution
directional expectations
E= present
Q= present
S= present
M= present
25
L= present
Using logical remainders for minimization (Krook 2010)
• Conservative solution
ES~M~L + ESML + EQ~SM + EQ~SL + ~E~SML  Y
2. Eliminate those conditions from the conservative
solution formulas that
- contradrict the directional expectations
(but at the same time)
- do not contradict the most parsimonious solution
5.6.2013
Truth Tables Algorithm II
26
Logical minimization by fsQCA2.5 (Krook 2010)
• How to logically minimize Krook 2010 with fsQCA 2.5?
• Intermediate solution
• Most parsimonious solution
S + EM + QL + ML  Y
5.6.2013
Truth Tables Algorithm II
directional expectations
E= present
Q= present
S= present
M= present
27
L= present
5.6.2013
Truth Tables Algorithm II
Logical minimization – computer exercises
Outlook & Task for next week
• Tosmana: - Kim/Lee 2008 (y1, y2)
- Your csQCA text
Research lesson I – from idea to research question
• Homework
Send me your research question until Monday, 10 June
2013
• fsQCA2.5:- Kim/Lee 2008 (y1)
- Freitag/Schlicht 2009 (N.B.: fuzzy sets!)
- Emmenegger 2011
- Your csQCA and fsQCA text
5.6.2013
Truth Tables Algorithm II
28
• Logically minimize your texts and find
– Necessary conditions
– Solution formulas (conservative, most parsimonious, and
intermediate)
29
5.6.2013
Truth Tables Algorithm II
30
5