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Fuzzy logic
By zaid darsquoood
Fuzzy Logic Introduction1IntroductionFuzzy Logic was initiated in 1965 [1] [2] [3] by Lotfi A Zadeh professor for computer science at the University of California in Berkeley
Basically Fuzzy Logic (FL) is a multivalued logic that allows intermediate values to be defined between conventional evaluations like truefalseyesno highlow etc Notions like rather tall or very fast can be formulatedmathematically and processed by computers in order to apply a more humanminuslike way of thinking in the programming of computers[4]
Fuzzy systems is an alternative to traditional notions of set membership
Fuzzy systems is an alternative to traditional notions of set membership and logic that has its origins in ancient Greek philosophy The precision of mathematics owes its success in large part to the efforts of Aristotle and the philosophers who preceded him In their efforts to devise a concise theory of logic and later a the sominuscalled Laws of Thought were posited[5]
One of these the Law of the Excluded Middle states that every proposition must either be True or
False Even when Parminedes proposed the first version of this law (around 400 BC) there were strong and immediate objections for example
Heraclitus proposed that things could be simultaneously True and not True It was Plato who
laid the foundation for what would become fuzzy logic indicating that there was a third region
(beyond True and False) where these opposites tumbled about Other more modern philosophers echoed his sentiments notably Hegel Marx and
Engels But it was Lukasiewicz who first proposed a systematic alternative to the biminusvalued logic of
Aristotle [6]
Even in the present time some Greeks are still outstanding examples for fussiness and fuzziness (note the connection to logic got lost somewhere
during the last 2 mileniums [7])Fuzzy Logic has emerged as a a profitable tool for
the controlling and steering of systems and complex industrial processes as well as for household and
entertainment electronics as well as for other expert systems and applications like the classification of
SAR data
Fuzzy SetsIn contrast to crisp sets fuzzy sets have
membership Degrees That means in addition to the values 1 (belongs to) and 0 (does not belong to) an element can have any value in between (ldquokind of belongs tordquo) Formally speaking the
membership function μA(x) for a fuzzy set A maps elements x to any value in the interval [0 1]
μA X rarr [0 1]
Comparing Fuzzy SetsEqualityCrisp sets two sets A and B are equal if theycontain the same elementsFuzzy sets all membership degrees have to beequal ie μA(x) = μB(x) for all x isin XContainmentCrisp sets A is contained in B (A sube B) if allelements in A are also elements of BFuzzy sets again membership degrees have to
beconsidered ie A sube B hArr μA(x) le μB(x) for all x isin
X
Fuzzy Operators
Complement (logical NOT) μA(u) = 1 minus μA(u)
Union (logical OR) μA[B(u) = max(μA(u) μB(u))
Intersection(logical AND)μAB(u)=min(μA(u)μB(u))
Example Build a Fuzzy ControllerFROM Fuzzy Thinkring Bart Kosko
Goal Design a motor speed controller for airConditionerStep 1 assign input and output variables Let X be the temperature in Fahrenheit Let Y be the motor speed of the air conditioner
Step 2 Pick fuzzy setsDefine linguistic terms of the linguistic variablestemperature (X) and motor speed (Y) andassociate them with fuzzy setsFor example 5 linguistic terms fuzzysets on Xbull Cold Cool Just Right Warm and HotSay 5 linguistic terms fuzzy sets on Ybull Stop Slow Medium Fast and Blast
Input Fuzzy sets
Output Fuzzy sets
Step 3 Assign a motor speed set to eachtemperature setbull If temperature is cold then motor speed is stopbull If temperature is cool then motor speed is slowbull If temperature is just right then motor speed is Mediumbull If temperature is warm then motor speed is fastbull If temperature is hot then motor speed is blast
A Fuzzy Relationexpressed by a
rule
A Fuzzy controller with 5 patches
If temperature is just right then motor speed is medium
t = 63 degree F
Summary of Fuzzy Logic
bull Rather than ealing with probability (00-10) Fuzzy Logic deals with fuzzy set membership ndash an entity can be in two or more sets at the same time to differentdegrees
bull Key concept is thus the degree of set membership (0-1) The degree of set membership can also be referred to as degree of truth of the proposition X is in set Y
bull Degree of membership often calculated algebraically eg if height= 185 then degree = (185-170) (200-170) = 05
bull Degree can also be given in tables (which would give a stepped membership graph)
OPERATIONSbull Subset A fuzzy set A is a subset of fuzzy set B if for all elements of A the degreeof membership in A is less than in Bbull Complement 2 fuzzy sets are complementary if for all elements of A the degree ofmembership in B is 1-A(s)bull Union The union of two fuzzy sets will assign degree of membership equal to thehighest of the two setsbull Intersection The intersection of two fuzzy sets will assign degree of membershipequal to the lowest of the two sets
MEMBERSHIP IN COMPLEX SETS
bull To what degree is X in SET1 and SET2 ndash take the lowest of the truth values (egas in traditional logic AampB is false if any one is false)
bull To what degree is X in SET1 or SET2 ndash take the highest of the truth values (eg as in traditional logic AvB is true if any one is true)
NATURAL LANGUAGE MODIFIERS
References
[1]LA Zadeh Fuzzy Sets Information and Control 1965[2]LA Zadeh Outline of A New Approach to the Analysis of of Complex Systems and Decision Processes 1973[3] LA Zadeh Fuzzy algorithms Info amp Ctl Vol 12 1968 pp 941048576 102[4] LA Zadeh Making computers think like people IEEE Spectrum 81984pp 261048576 32[5] S Korner Laws of thought Encyclopedia of Philosophy Vol 4 MacMillanNY 1967 pp 4141048576 417[6] C Lejewski Jan Lukasiewicz Encyclopedia of Philosophy Vol 5MacMillan NY 1967 pp 1041048576 107[7] A Reigber My life with Kostas unpublished report Neverending StoryPress 1999
THANK YOU
Fuzzy Logic Introduction1IntroductionFuzzy Logic was initiated in 1965 [1] [2] [3] by Lotfi A Zadeh professor for computer science at the University of California in Berkeley
Basically Fuzzy Logic (FL) is a multivalued logic that allows intermediate values to be defined between conventional evaluations like truefalseyesno highlow etc Notions like rather tall or very fast can be formulatedmathematically and processed by computers in order to apply a more humanminuslike way of thinking in the programming of computers[4]
Fuzzy systems is an alternative to traditional notions of set membership
Fuzzy systems is an alternative to traditional notions of set membership and logic that has its origins in ancient Greek philosophy The precision of mathematics owes its success in large part to the efforts of Aristotle and the philosophers who preceded him In their efforts to devise a concise theory of logic and later a the sominuscalled Laws of Thought were posited[5]
One of these the Law of the Excluded Middle states that every proposition must either be True or
False Even when Parminedes proposed the first version of this law (around 400 BC) there were strong and immediate objections for example
Heraclitus proposed that things could be simultaneously True and not True It was Plato who
laid the foundation for what would become fuzzy logic indicating that there was a third region
(beyond True and False) where these opposites tumbled about Other more modern philosophers echoed his sentiments notably Hegel Marx and
Engels But it was Lukasiewicz who first proposed a systematic alternative to the biminusvalued logic of
Aristotle [6]
Even in the present time some Greeks are still outstanding examples for fussiness and fuzziness (note the connection to logic got lost somewhere
during the last 2 mileniums [7])Fuzzy Logic has emerged as a a profitable tool for
the controlling and steering of systems and complex industrial processes as well as for household and
entertainment electronics as well as for other expert systems and applications like the classification of
SAR data
Fuzzy SetsIn contrast to crisp sets fuzzy sets have
membership Degrees That means in addition to the values 1 (belongs to) and 0 (does not belong to) an element can have any value in between (ldquokind of belongs tordquo) Formally speaking the
membership function μA(x) for a fuzzy set A maps elements x to any value in the interval [0 1]
μA X rarr [0 1]
Comparing Fuzzy SetsEqualityCrisp sets two sets A and B are equal if theycontain the same elementsFuzzy sets all membership degrees have to beequal ie μA(x) = μB(x) for all x isin XContainmentCrisp sets A is contained in B (A sube B) if allelements in A are also elements of BFuzzy sets again membership degrees have to
beconsidered ie A sube B hArr μA(x) le μB(x) for all x isin
X
Fuzzy Operators
Complement (logical NOT) μA(u) = 1 minus μA(u)
Union (logical OR) μA[B(u) = max(μA(u) μB(u))
Intersection(logical AND)μAB(u)=min(μA(u)μB(u))
Example Build a Fuzzy ControllerFROM Fuzzy Thinkring Bart Kosko
Goal Design a motor speed controller for airConditionerStep 1 assign input and output variables Let X be the temperature in Fahrenheit Let Y be the motor speed of the air conditioner
Step 2 Pick fuzzy setsDefine linguistic terms of the linguistic variablestemperature (X) and motor speed (Y) andassociate them with fuzzy setsFor example 5 linguistic terms fuzzysets on Xbull Cold Cool Just Right Warm and HotSay 5 linguistic terms fuzzy sets on Ybull Stop Slow Medium Fast and Blast
Input Fuzzy sets
Output Fuzzy sets
Step 3 Assign a motor speed set to eachtemperature setbull If temperature is cold then motor speed is stopbull If temperature is cool then motor speed is slowbull If temperature is just right then motor speed is Mediumbull If temperature is warm then motor speed is fastbull If temperature is hot then motor speed is blast
A Fuzzy Relationexpressed by a
rule
A Fuzzy controller with 5 patches
If temperature is just right then motor speed is medium
t = 63 degree F
Summary of Fuzzy Logic
bull Rather than ealing with probability (00-10) Fuzzy Logic deals with fuzzy set membership ndash an entity can be in two or more sets at the same time to differentdegrees
bull Key concept is thus the degree of set membership (0-1) The degree of set membership can also be referred to as degree of truth of the proposition X is in set Y
bull Degree of membership often calculated algebraically eg if height= 185 then degree = (185-170) (200-170) = 05
bull Degree can also be given in tables (which would give a stepped membership graph)
OPERATIONSbull Subset A fuzzy set A is a subset of fuzzy set B if for all elements of A the degreeof membership in A is less than in Bbull Complement 2 fuzzy sets are complementary if for all elements of A the degree ofmembership in B is 1-A(s)bull Union The union of two fuzzy sets will assign degree of membership equal to thehighest of the two setsbull Intersection The intersection of two fuzzy sets will assign degree of membershipequal to the lowest of the two sets
MEMBERSHIP IN COMPLEX SETS
bull To what degree is X in SET1 and SET2 ndash take the lowest of the truth values (egas in traditional logic AampB is false if any one is false)
bull To what degree is X in SET1 or SET2 ndash take the highest of the truth values (eg as in traditional logic AvB is true if any one is true)
NATURAL LANGUAGE MODIFIERS
References
[1]LA Zadeh Fuzzy Sets Information and Control 1965[2]LA Zadeh Outline of A New Approach to the Analysis of of Complex Systems and Decision Processes 1973[3] LA Zadeh Fuzzy algorithms Info amp Ctl Vol 12 1968 pp 941048576 102[4] LA Zadeh Making computers think like people IEEE Spectrum 81984pp 261048576 32[5] S Korner Laws of thought Encyclopedia of Philosophy Vol 4 MacMillanNY 1967 pp 4141048576 417[6] C Lejewski Jan Lukasiewicz Encyclopedia of Philosophy Vol 5MacMillan NY 1967 pp 1041048576 107[7] A Reigber My life with Kostas unpublished report Neverending StoryPress 1999
THANK YOU
Fuzzy systems is an alternative to traditional notions of set membership and logic that has its origins in ancient Greek philosophy The precision of mathematics owes its success in large part to the efforts of Aristotle and the philosophers who preceded him In their efforts to devise a concise theory of logic and later a the sominuscalled Laws of Thought were posited[5]
One of these the Law of the Excluded Middle states that every proposition must either be True or
False Even when Parminedes proposed the first version of this law (around 400 BC) there were strong and immediate objections for example
Heraclitus proposed that things could be simultaneously True and not True It was Plato who
laid the foundation for what would become fuzzy logic indicating that there was a third region
(beyond True and False) where these opposites tumbled about Other more modern philosophers echoed his sentiments notably Hegel Marx and
Engels But it was Lukasiewicz who first proposed a systematic alternative to the biminusvalued logic of
Aristotle [6]
Even in the present time some Greeks are still outstanding examples for fussiness and fuzziness (note the connection to logic got lost somewhere
during the last 2 mileniums [7])Fuzzy Logic has emerged as a a profitable tool for
the controlling and steering of systems and complex industrial processes as well as for household and
entertainment electronics as well as for other expert systems and applications like the classification of
SAR data
Fuzzy SetsIn contrast to crisp sets fuzzy sets have
membership Degrees That means in addition to the values 1 (belongs to) and 0 (does not belong to) an element can have any value in between (ldquokind of belongs tordquo) Formally speaking the
membership function μA(x) for a fuzzy set A maps elements x to any value in the interval [0 1]
μA X rarr [0 1]
Comparing Fuzzy SetsEqualityCrisp sets two sets A and B are equal if theycontain the same elementsFuzzy sets all membership degrees have to beequal ie μA(x) = μB(x) for all x isin XContainmentCrisp sets A is contained in B (A sube B) if allelements in A are also elements of BFuzzy sets again membership degrees have to
beconsidered ie A sube B hArr μA(x) le μB(x) for all x isin
X
Fuzzy Operators
Complement (logical NOT) μA(u) = 1 minus μA(u)
Union (logical OR) μA[B(u) = max(μA(u) μB(u))
Intersection(logical AND)μAB(u)=min(μA(u)μB(u))
Example Build a Fuzzy ControllerFROM Fuzzy Thinkring Bart Kosko
Goal Design a motor speed controller for airConditionerStep 1 assign input and output variables Let X be the temperature in Fahrenheit Let Y be the motor speed of the air conditioner
Step 2 Pick fuzzy setsDefine linguistic terms of the linguistic variablestemperature (X) and motor speed (Y) andassociate them with fuzzy setsFor example 5 linguistic terms fuzzysets on Xbull Cold Cool Just Right Warm and HotSay 5 linguistic terms fuzzy sets on Ybull Stop Slow Medium Fast and Blast
Input Fuzzy sets
Output Fuzzy sets
Step 3 Assign a motor speed set to eachtemperature setbull If temperature is cold then motor speed is stopbull If temperature is cool then motor speed is slowbull If temperature is just right then motor speed is Mediumbull If temperature is warm then motor speed is fastbull If temperature is hot then motor speed is blast
A Fuzzy Relationexpressed by a
rule
A Fuzzy controller with 5 patches
If temperature is just right then motor speed is medium
t = 63 degree F
Summary of Fuzzy Logic
bull Rather than ealing with probability (00-10) Fuzzy Logic deals with fuzzy set membership ndash an entity can be in two or more sets at the same time to differentdegrees
bull Key concept is thus the degree of set membership (0-1) The degree of set membership can also be referred to as degree of truth of the proposition X is in set Y
bull Degree of membership often calculated algebraically eg if height= 185 then degree = (185-170) (200-170) = 05
bull Degree can also be given in tables (which would give a stepped membership graph)
OPERATIONSbull Subset A fuzzy set A is a subset of fuzzy set B if for all elements of A the degreeof membership in A is less than in Bbull Complement 2 fuzzy sets are complementary if for all elements of A the degree ofmembership in B is 1-A(s)bull Union The union of two fuzzy sets will assign degree of membership equal to thehighest of the two setsbull Intersection The intersection of two fuzzy sets will assign degree of membershipequal to the lowest of the two sets
MEMBERSHIP IN COMPLEX SETS
bull To what degree is X in SET1 and SET2 ndash take the lowest of the truth values (egas in traditional logic AampB is false if any one is false)
bull To what degree is X in SET1 or SET2 ndash take the highest of the truth values (eg as in traditional logic AvB is true if any one is true)
NATURAL LANGUAGE MODIFIERS
References
[1]LA Zadeh Fuzzy Sets Information and Control 1965[2]LA Zadeh Outline of A New Approach to the Analysis of of Complex Systems and Decision Processes 1973[3] LA Zadeh Fuzzy algorithms Info amp Ctl Vol 12 1968 pp 941048576 102[4] LA Zadeh Making computers think like people IEEE Spectrum 81984pp 261048576 32[5] S Korner Laws of thought Encyclopedia of Philosophy Vol 4 MacMillanNY 1967 pp 4141048576 417[6] C Lejewski Jan Lukasiewicz Encyclopedia of Philosophy Vol 5MacMillan NY 1967 pp 1041048576 107[7] A Reigber My life with Kostas unpublished report Neverending StoryPress 1999
THANK YOU
One of these the Law of the Excluded Middle states that every proposition must either be True or
False Even when Parminedes proposed the first version of this law (around 400 BC) there were strong and immediate objections for example
Heraclitus proposed that things could be simultaneously True and not True It was Plato who
laid the foundation for what would become fuzzy logic indicating that there was a third region
(beyond True and False) where these opposites tumbled about Other more modern philosophers echoed his sentiments notably Hegel Marx and
Engels But it was Lukasiewicz who first proposed a systematic alternative to the biminusvalued logic of
Aristotle [6]
Even in the present time some Greeks are still outstanding examples for fussiness and fuzziness (note the connection to logic got lost somewhere
during the last 2 mileniums [7])Fuzzy Logic has emerged as a a profitable tool for
the controlling and steering of systems and complex industrial processes as well as for household and
entertainment electronics as well as for other expert systems and applications like the classification of
SAR data
Fuzzy SetsIn contrast to crisp sets fuzzy sets have
membership Degrees That means in addition to the values 1 (belongs to) and 0 (does not belong to) an element can have any value in between (ldquokind of belongs tordquo) Formally speaking the
membership function μA(x) for a fuzzy set A maps elements x to any value in the interval [0 1]
μA X rarr [0 1]
Comparing Fuzzy SetsEqualityCrisp sets two sets A and B are equal if theycontain the same elementsFuzzy sets all membership degrees have to beequal ie μA(x) = μB(x) for all x isin XContainmentCrisp sets A is contained in B (A sube B) if allelements in A are also elements of BFuzzy sets again membership degrees have to
beconsidered ie A sube B hArr μA(x) le μB(x) for all x isin
X
Fuzzy Operators
Complement (logical NOT) μA(u) = 1 minus μA(u)
Union (logical OR) μA[B(u) = max(μA(u) μB(u))
Intersection(logical AND)μAB(u)=min(μA(u)μB(u))
Example Build a Fuzzy ControllerFROM Fuzzy Thinkring Bart Kosko
Goal Design a motor speed controller for airConditionerStep 1 assign input and output variables Let X be the temperature in Fahrenheit Let Y be the motor speed of the air conditioner
Step 2 Pick fuzzy setsDefine linguistic terms of the linguistic variablestemperature (X) and motor speed (Y) andassociate them with fuzzy setsFor example 5 linguistic terms fuzzysets on Xbull Cold Cool Just Right Warm and HotSay 5 linguistic terms fuzzy sets on Ybull Stop Slow Medium Fast and Blast
Input Fuzzy sets
Output Fuzzy sets
Step 3 Assign a motor speed set to eachtemperature setbull If temperature is cold then motor speed is stopbull If temperature is cool then motor speed is slowbull If temperature is just right then motor speed is Mediumbull If temperature is warm then motor speed is fastbull If temperature is hot then motor speed is blast
A Fuzzy Relationexpressed by a
rule
A Fuzzy controller with 5 patches
If temperature is just right then motor speed is medium
t = 63 degree F
Summary of Fuzzy Logic
bull Rather than ealing with probability (00-10) Fuzzy Logic deals with fuzzy set membership ndash an entity can be in two or more sets at the same time to differentdegrees
bull Key concept is thus the degree of set membership (0-1) The degree of set membership can also be referred to as degree of truth of the proposition X is in set Y
bull Degree of membership often calculated algebraically eg if height= 185 then degree = (185-170) (200-170) = 05
bull Degree can also be given in tables (which would give a stepped membership graph)
OPERATIONSbull Subset A fuzzy set A is a subset of fuzzy set B if for all elements of A the degreeof membership in A is less than in Bbull Complement 2 fuzzy sets are complementary if for all elements of A the degree ofmembership in B is 1-A(s)bull Union The union of two fuzzy sets will assign degree of membership equal to thehighest of the two setsbull Intersection The intersection of two fuzzy sets will assign degree of membershipequal to the lowest of the two sets
MEMBERSHIP IN COMPLEX SETS
bull To what degree is X in SET1 and SET2 ndash take the lowest of the truth values (egas in traditional logic AampB is false if any one is false)
bull To what degree is X in SET1 or SET2 ndash take the highest of the truth values (eg as in traditional logic AvB is true if any one is true)
NATURAL LANGUAGE MODIFIERS
References
[1]LA Zadeh Fuzzy Sets Information and Control 1965[2]LA Zadeh Outline of A New Approach to the Analysis of of Complex Systems and Decision Processes 1973[3] LA Zadeh Fuzzy algorithms Info amp Ctl Vol 12 1968 pp 941048576 102[4] LA Zadeh Making computers think like people IEEE Spectrum 81984pp 261048576 32[5] S Korner Laws of thought Encyclopedia of Philosophy Vol 4 MacMillanNY 1967 pp 4141048576 417[6] C Lejewski Jan Lukasiewicz Encyclopedia of Philosophy Vol 5MacMillan NY 1967 pp 1041048576 107[7] A Reigber My life with Kostas unpublished report Neverending StoryPress 1999
THANK YOU
Even in the present time some Greeks are still outstanding examples for fussiness and fuzziness (note the connection to logic got lost somewhere
during the last 2 mileniums [7])Fuzzy Logic has emerged as a a profitable tool for
the controlling and steering of systems and complex industrial processes as well as for household and
entertainment electronics as well as for other expert systems and applications like the classification of
SAR data
Fuzzy SetsIn contrast to crisp sets fuzzy sets have
membership Degrees That means in addition to the values 1 (belongs to) and 0 (does not belong to) an element can have any value in between (ldquokind of belongs tordquo) Formally speaking the
membership function μA(x) for a fuzzy set A maps elements x to any value in the interval [0 1]
μA X rarr [0 1]
Comparing Fuzzy SetsEqualityCrisp sets two sets A and B are equal if theycontain the same elementsFuzzy sets all membership degrees have to beequal ie μA(x) = μB(x) for all x isin XContainmentCrisp sets A is contained in B (A sube B) if allelements in A are also elements of BFuzzy sets again membership degrees have to
beconsidered ie A sube B hArr μA(x) le μB(x) for all x isin
X
Fuzzy Operators
Complement (logical NOT) μA(u) = 1 minus μA(u)
Union (logical OR) μA[B(u) = max(μA(u) μB(u))
Intersection(logical AND)μAB(u)=min(μA(u)μB(u))
Example Build a Fuzzy ControllerFROM Fuzzy Thinkring Bart Kosko
Goal Design a motor speed controller for airConditionerStep 1 assign input and output variables Let X be the temperature in Fahrenheit Let Y be the motor speed of the air conditioner
Step 2 Pick fuzzy setsDefine linguistic terms of the linguistic variablestemperature (X) and motor speed (Y) andassociate them with fuzzy setsFor example 5 linguistic terms fuzzysets on Xbull Cold Cool Just Right Warm and HotSay 5 linguistic terms fuzzy sets on Ybull Stop Slow Medium Fast and Blast
Input Fuzzy sets
Output Fuzzy sets
Step 3 Assign a motor speed set to eachtemperature setbull If temperature is cold then motor speed is stopbull If temperature is cool then motor speed is slowbull If temperature is just right then motor speed is Mediumbull If temperature is warm then motor speed is fastbull If temperature is hot then motor speed is blast
A Fuzzy Relationexpressed by a
rule
A Fuzzy controller with 5 patches
If temperature is just right then motor speed is medium
t = 63 degree F
Summary of Fuzzy Logic
bull Rather than ealing with probability (00-10) Fuzzy Logic deals with fuzzy set membership ndash an entity can be in two or more sets at the same time to differentdegrees
bull Key concept is thus the degree of set membership (0-1) The degree of set membership can also be referred to as degree of truth of the proposition X is in set Y
bull Degree of membership often calculated algebraically eg if height= 185 then degree = (185-170) (200-170) = 05
bull Degree can also be given in tables (which would give a stepped membership graph)
OPERATIONSbull Subset A fuzzy set A is a subset of fuzzy set B if for all elements of A the degreeof membership in A is less than in Bbull Complement 2 fuzzy sets are complementary if for all elements of A the degree ofmembership in B is 1-A(s)bull Union The union of two fuzzy sets will assign degree of membership equal to thehighest of the two setsbull Intersection The intersection of two fuzzy sets will assign degree of membershipequal to the lowest of the two sets
MEMBERSHIP IN COMPLEX SETS
bull To what degree is X in SET1 and SET2 ndash take the lowest of the truth values (egas in traditional logic AampB is false if any one is false)
bull To what degree is X in SET1 or SET2 ndash take the highest of the truth values (eg as in traditional logic AvB is true if any one is true)
NATURAL LANGUAGE MODIFIERS
References
[1]LA Zadeh Fuzzy Sets Information and Control 1965[2]LA Zadeh Outline of A New Approach to the Analysis of of Complex Systems and Decision Processes 1973[3] LA Zadeh Fuzzy algorithms Info amp Ctl Vol 12 1968 pp 941048576 102[4] LA Zadeh Making computers think like people IEEE Spectrum 81984pp 261048576 32[5] S Korner Laws of thought Encyclopedia of Philosophy Vol 4 MacMillanNY 1967 pp 4141048576 417[6] C Lejewski Jan Lukasiewicz Encyclopedia of Philosophy Vol 5MacMillan NY 1967 pp 1041048576 107[7] A Reigber My life with Kostas unpublished report Neverending StoryPress 1999
THANK YOU
Fuzzy SetsIn contrast to crisp sets fuzzy sets have
membership Degrees That means in addition to the values 1 (belongs to) and 0 (does not belong to) an element can have any value in between (ldquokind of belongs tordquo) Formally speaking the
membership function μA(x) for a fuzzy set A maps elements x to any value in the interval [0 1]
μA X rarr [0 1]
Comparing Fuzzy SetsEqualityCrisp sets two sets A and B are equal if theycontain the same elementsFuzzy sets all membership degrees have to beequal ie μA(x) = μB(x) for all x isin XContainmentCrisp sets A is contained in B (A sube B) if allelements in A are also elements of BFuzzy sets again membership degrees have to
beconsidered ie A sube B hArr μA(x) le μB(x) for all x isin
X
Fuzzy Operators
Complement (logical NOT) μA(u) = 1 minus μA(u)
Union (logical OR) μA[B(u) = max(μA(u) μB(u))
Intersection(logical AND)μAB(u)=min(μA(u)μB(u))
Example Build a Fuzzy ControllerFROM Fuzzy Thinkring Bart Kosko
Goal Design a motor speed controller for airConditionerStep 1 assign input and output variables Let X be the temperature in Fahrenheit Let Y be the motor speed of the air conditioner
Step 2 Pick fuzzy setsDefine linguistic terms of the linguistic variablestemperature (X) and motor speed (Y) andassociate them with fuzzy setsFor example 5 linguistic terms fuzzysets on Xbull Cold Cool Just Right Warm and HotSay 5 linguistic terms fuzzy sets on Ybull Stop Slow Medium Fast and Blast
Input Fuzzy sets
Output Fuzzy sets
Step 3 Assign a motor speed set to eachtemperature setbull If temperature is cold then motor speed is stopbull If temperature is cool then motor speed is slowbull If temperature is just right then motor speed is Mediumbull If temperature is warm then motor speed is fastbull If temperature is hot then motor speed is blast
A Fuzzy Relationexpressed by a
rule
A Fuzzy controller with 5 patches
If temperature is just right then motor speed is medium
t = 63 degree F
Summary of Fuzzy Logic
bull Rather than ealing with probability (00-10) Fuzzy Logic deals with fuzzy set membership ndash an entity can be in two or more sets at the same time to differentdegrees
bull Key concept is thus the degree of set membership (0-1) The degree of set membership can also be referred to as degree of truth of the proposition X is in set Y
bull Degree of membership often calculated algebraically eg if height= 185 then degree = (185-170) (200-170) = 05
bull Degree can also be given in tables (which would give a stepped membership graph)
OPERATIONSbull Subset A fuzzy set A is a subset of fuzzy set B if for all elements of A the degreeof membership in A is less than in Bbull Complement 2 fuzzy sets are complementary if for all elements of A the degree ofmembership in B is 1-A(s)bull Union The union of two fuzzy sets will assign degree of membership equal to thehighest of the two setsbull Intersection The intersection of two fuzzy sets will assign degree of membershipequal to the lowest of the two sets
MEMBERSHIP IN COMPLEX SETS
bull To what degree is X in SET1 and SET2 ndash take the lowest of the truth values (egas in traditional logic AampB is false if any one is false)
bull To what degree is X in SET1 or SET2 ndash take the highest of the truth values (eg as in traditional logic AvB is true if any one is true)
NATURAL LANGUAGE MODIFIERS
References
[1]LA Zadeh Fuzzy Sets Information and Control 1965[2]LA Zadeh Outline of A New Approach to the Analysis of of Complex Systems and Decision Processes 1973[3] LA Zadeh Fuzzy algorithms Info amp Ctl Vol 12 1968 pp 941048576 102[4] LA Zadeh Making computers think like people IEEE Spectrum 81984pp 261048576 32[5] S Korner Laws of thought Encyclopedia of Philosophy Vol 4 MacMillanNY 1967 pp 4141048576 417[6] C Lejewski Jan Lukasiewicz Encyclopedia of Philosophy Vol 5MacMillan NY 1967 pp 1041048576 107[7] A Reigber My life with Kostas unpublished report Neverending StoryPress 1999
THANK YOU
Comparing Fuzzy SetsEqualityCrisp sets two sets A and B are equal if theycontain the same elementsFuzzy sets all membership degrees have to beequal ie μA(x) = μB(x) for all x isin XContainmentCrisp sets A is contained in B (A sube B) if allelements in A are also elements of BFuzzy sets again membership degrees have to
beconsidered ie A sube B hArr μA(x) le μB(x) for all x isin
X
Fuzzy Operators
Complement (logical NOT) μA(u) = 1 minus μA(u)
Union (logical OR) μA[B(u) = max(μA(u) μB(u))
Intersection(logical AND)μAB(u)=min(μA(u)μB(u))
Example Build a Fuzzy ControllerFROM Fuzzy Thinkring Bart Kosko
Goal Design a motor speed controller for airConditionerStep 1 assign input and output variables Let X be the temperature in Fahrenheit Let Y be the motor speed of the air conditioner
Step 2 Pick fuzzy setsDefine linguistic terms of the linguistic variablestemperature (X) and motor speed (Y) andassociate them with fuzzy setsFor example 5 linguistic terms fuzzysets on Xbull Cold Cool Just Right Warm and HotSay 5 linguistic terms fuzzy sets on Ybull Stop Slow Medium Fast and Blast
Input Fuzzy sets
Output Fuzzy sets
Step 3 Assign a motor speed set to eachtemperature setbull If temperature is cold then motor speed is stopbull If temperature is cool then motor speed is slowbull If temperature is just right then motor speed is Mediumbull If temperature is warm then motor speed is fastbull If temperature is hot then motor speed is blast
A Fuzzy Relationexpressed by a
rule
A Fuzzy controller with 5 patches
If temperature is just right then motor speed is medium
t = 63 degree F
Summary of Fuzzy Logic
bull Rather than ealing with probability (00-10) Fuzzy Logic deals with fuzzy set membership ndash an entity can be in two or more sets at the same time to differentdegrees
bull Key concept is thus the degree of set membership (0-1) The degree of set membership can also be referred to as degree of truth of the proposition X is in set Y
bull Degree of membership often calculated algebraically eg if height= 185 then degree = (185-170) (200-170) = 05
bull Degree can also be given in tables (which would give a stepped membership graph)
OPERATIONSbull Subset A fuzzy set A is a subset of fuzzy set B if for all elements of A the degreeof membership in A is less than in Bbull Complement 2 fuzzy sets are complementary if for all elements of A the degree ofmembership in B is 1-A(s)bull Union The union of two fuzzy sets will assign degree of membership equal to thehighest of the two setsbull Intersection The intersection of two fuzzy sets will assign degree of membershipequal to the lowest of the two sets
MEMBERSHIP IN COMPLEX SETS
bull To what degree is X in SET1 and SET2 ndash take the lowest of the truth values (egas in traditional logic AampB is false if any one is false)
bull To what degree is X in SET1 or SET2 ndash take the highest of the truth values (eg as in traditional logic AvB is true if any one is true)
NATURAL LANGUAGE MODIFIERS
References
[1]LA Zadeh Fuzzy Sets Information and Control 1965[2]LA Zadeh Outline of A New Approach to the Analysis of of Complex Systems and Decision Processes 1973[3] LA Zadeh Fuzzy algorithms Info amp Ctl Vol 12 1968 pp 941048576 102[4] LA Zadeh Making computers think like people IEEE Spectrum 81984pp 261048576 32[5] S Korner Laws of thought Encyclopedia of Philosophy Vol 4 MacMillanNY 1967 pp 4141048576 417[6] C Lejewski Jan Lukasiewicz Encyclopedia of Philosophy Vol 5MacMillan NY 1967 pp 1041048576 107[7] A Reigber My life with Kostas unpublished report Neverending StoryPress 1999
THANK YOU
Fuzzy Operators
Complement (logical NOT) μA(u) = 1 minus μA(u)
Union (logical OR) μA[B(u) = max(μA(u) μB(u))
Intersection(logical AND)μAB(u)=min(μA(u)μB(u))
Example Build a Fuzzy ControllerFROM Fuzzy Thinkring Bart Kosko
Goal Design a motor speed controller for airConditionerStep 1 assign input and output variables Let X be the temperature in Fahrenheit Let Y be the motor speed of the air conditioner
Step 2 Pick fuzzy setsDefine linguistic terms of the linguistic variablestemperature (X) and motor speed (Y) andassociate them with fuzzy setsFor example 5 linguistic terms fuzzysets on Xbull Cold Cool Just Right Warm and HotSay 5 linguistic terms fuzzy sets on Ybull Stop Slow Medium Fast and Blast
Input Fuzzy sets
Output Fuzzy sets
Step 3 Assign a motor speed set to eachtemperature setbull If temperature is cold then motor speed is stopbull If temperature is cool then motor speed is slowbull If temperature is just right then motor speed is Mediumbull If temperature is warm then motor speed is fastbull If temperature is hot then motor speed is blast
A Fuzzy Relationexpressed by a
rule
A Fuzzy controller with 5 patches
If temperature is just right then motor speed is medium
t = 63 degree F
Summary of Fuzzy Logic
bull Rather than ealing with probability (00-10) Fuzzy Logic deals with fuzzy set membership ndash an entity can be in two or more sets at the same time to differentdegrees
bull Key concept is thus the degree of set membership (0-1) The degree of set membership can also be referred to as degree of truth of the proposition X is in set Y
bull Degree of membership often calculated algebraically eg if height= 185 then degree = (185-170) (200-170) = 05
bull Degree can also be given in tables (which would give a stepped membership graph)
OPERATIONSbull Subset A fuzzy set A is a subset of fuzzy set B if for all elements of A the degreeof membership in A is less than in Bbull Complement 2 fuzzy sets are complementary if for all elements of A the degree ofmembership in B is 1-A(s)bull Union The union of two fuzzy sets will assign degree of membership equal to thehighest of the two setsbull Intersection The intersection of two fuzzy sets will assign degree of membershipequal to the lowest of the two sets
MEMBERSHIP IN COMPLEX SETS
bull To what degree is X in SET1 and SET2 ndash take the lowest of the truth values (egas in traditional logic AampB is false if any one is false)
bull To what degree is X in SET1 or SET2 ndash take the highest of the truth values (eg as in traditional logic AvB is true if any one is true)
NATURAL LANGUAGE MODIFIERS
References
[1]LA Zadeh Fuzzy Sets Information and Control 1965[2]LA Zadeh Outline of A New Approach to the Analysis of of Complex Systems and Decision Processes 1973[3] LA Zadeh Fuzzy algorithms Info amp Ctl Vol 12 1968 pp 941048576 102[4] LA Zadeh Making computers think like people IEEE Spectrum 81984pp 261048576 32[5] S Korner Laws of thought Encyclopedia of Philosophy Vol 4 MacMillanNY 1967 pp 4141048576 417[6] C Lejewski Jan Lukasiewicz Encyclopedia of Philosophy Vol 5MacMillan NY 1967 pp 1041048576 107[7] A Reigber My life with Kostas unpublished report Neverending StoryPress 1999
THANK YOU
Example Build a Fuzzy ControllerFROM Fuzzy Thinkring Bart Kosko
Goal Design a motor speed controller for airConditionerStep 1 assign input and output variables Let X be the temperature in Fahrenheit Let Y be the motor speed of the air conditioner
Step 2 Pick fuzzy setsDefine linguistic terms of the linguistic variablestemperature (X) and motor speed (Y) andassociate them with fuzzy setsFor example 5 linguistic terms fuzzysets on Xbull Cold Cool Just Right Warm and HotSay 5 linguistic terms fuzzy sets on Ybull Stop Slow Medium Fast and Blast
Input Fuzzy sets
Output Fuzzy sets
Step 3 Assign a motor speed set to eachtemperature setbull If temperature is cold then motor speed is stopbull If temperature is cool then motor speed is slowbull If temperature is just right then motor speed is Mediumbull If temperature is warm then motor speed is fastbull If temperature is hot then motor speed is blast
A Fuzzy Relationexpressed by a
rule
A Fuzzy controller with 5 patches
If temperature is just right then motor speed is medium
t = 63 degree F
Summary of Fuzzy Logic
bull Rather than ealing with probability (00-10) Fuzzy Logic deals with fuzzy set membership ndash an entity can be in two or more sets at the same time to differentdegrees
bull Key concept is thus the degree of set membership (0-1) The degree of set membership can also be referred to as degree of truth of the proposition X is in set Y
bull Degree of membership often calculated algebraically eg if height= 185 then degree = (185-170) (200-170) = 05
bull Degree can also be given in tables (which would give a stepped membership graph)
OPERATIONSbull Subset A fuzzy set A is a subset of fuzzy set B if for all elements of A the degreeof membership in A is less than in Bbull Complement 2 fuzzy sets are complementary if for all elements of A the degree ofmembership in B is 1-A(s)bull Union The union of two fuzzy sets will assign degree of membership equal to thehighest of the two setsbull Intersection The intersection of two fuzzy sets will assign degree of membershipequal to the lowest of the two sets
MEMBERSHIP IN COMPLEX SETS
bull To what degree is X in SET1 and SET2 ndash take the lowest of the truth values (egas in traditional logic AampB is false if any one is false)
bull To what degree is X in SET1 or SET2 ndash take the highest of the truth values (eg as in traditional logic AvB is true if any one is true)
NATURAL LANGUAGE MODIFIERS
References
[1]LA Zadeh Fuzzy Sets Information and Control 1965[2]LA Zadeh Outline of A New Approach to the Analysis of of Complex Systems and Decision Processes 1973[3] LA Zadeh Fuzzy algorithms Info amp Ctl Vol 12 1968 pp 941048576 102[4] LA Zadeh Making computers think like people IEEE Spectrum 81984pp 261048576 32[5] S Korner Laws of thought Encyclopedia of Philosophy Vol 4 MacMillanNY 1967 pp 4141048576 417[6] C Lejewski Jan Lukasiewicz Encyclopedia of Philosophy Vol 5MacMillan NY 1967 pp 1041048576 107[7] A Reigber My life with Kostas unpublished report Neverending StoryPress 1999
THANK YOU
Step 2 Pick fuzzy setsDefine linguistic terms of the linguistic variablestemperature (X) and motor speed (Y) andassociate them with fuzzy setsFor example 5 linguistic terms fuzzysets on Xbull Cold Cool Just Right Warm and HotSay 5 linguistic terms fuzzy sets on Ybull Stop Slow Medium Fast and Blast
Input Fuzzy sets
Output Fuzzy sets
Step 3 Assign a motor speed set to eachtemperature setbull If temperature is cold then motor speed is stopbull If temperature is cool then motor speed is slowbull If temperature is just right then motor speed is Mediumbull If temperature is warm then motor speed is fastbull If temperature is hot then motor speed is blast
A Fuzzy Relationexpressed by a
rule
A Fuzzy controller with 5 patches
If temperature is just right then motor speed is medium
t = 63 degree F
Summary of Fuzzy Logic
bull Rather than ealing with probability (00-10) Fuzzy Logic deals with fuzzy set membership ndash an entity can be in two or more sets at the same time to differentdegrees
bull Key concept is thus the degree of set membership (0-1) The degree of set membership can also be referred to as degree of truth of the proposition X is in set Y
bull Degree of membership often calculated algebraically eg if height= 185 then degree = (185-170) (200-170) = 05
bull Degree can also be given in tables (which would give a stepped membership graph)
OPERATIONSbull Subset A fuzzy set A is a subset of fuzzy set B if for all elements of A the degreeof membership in A is less than in Bbull Complement 2 fuzzy sets are complementary if for all elements of A the degree ofmembership in B is 1-A(s)bull Union The union of two fuzzy sets will assign degree of membership equal to thehighest of the two setsbull Intersection The intersection of two fuzzy sets will assign degree of membershipequal to the lowest of the two sets
MEMBERSHIP IN COMPLEX SETS
bull To what degree is X in SET1 and SET2 ndash take the lowest of the truth values (egas in traditional logic AampB is false if any one is false)
bull To what degree is X in SET1 or SET2 ndash take the highest of the truth values (eg as in traditional logic AvB is true if any one is true)
NATURAL LANGUAGE MODIFIERS
References
[1]LA Zadeh Fuzzy Sets Information and Control 1965[2]LA Zadeh Outline of A New Approach to the Analysis of of Complex Systems and Decision Processes 1973[3] LA Zadeh Fuzzy algorithms Info amp Ctl Vol 12 1968 pp 941048576 102[4] LA Zadeh Making computers think like people IEEE Spectrum 81984pp 261048576 32[5] S Korner Laws of thought Encyclopedia of Philosophy Vol 4 MacMillanNY 1967 pp 4141048576 417[6] C Lejewski Jan Lukasiewicz Encyclopedia of Philosophy Vol 5MacMillan NY 1967 pp 1041048576 107[7] A Reigber My life with Kostas unpublished report Neverending StoryPress 1999
THANK YOU
Input Fuzzy sets
Output Fuzzy sets
Step 3 Assign a motor speed set to eachtemperature setbull If temperature is cold then motor speed is stopbull If temperature is cool then motor speed is slowbull If temperature is just right then motor speed is Mediumbull If temperature is warm then motor speed is fastbull If temperature is hot then motor speed is blast
A Fuzzy Relationexpressed by a
rule
A Fuzzy controller with 5 patches
If temperature is just right then motor speed is medium
t = 63 degree F
Summary of Fuzzy Logic
bull Rather than ealing with probability (00-10) Fuzzy Logic deals with fuzzy set membership ndash an entity can be in two or more sets at the same time to differentdegrees
bull Key concept is thus the degree of set membership (0-1) The degree of set membership can also be referred to as degree of truth of the proposition X is in set Y
bull Degree of membership often calculated algebraically eg if height= 185 then degree = (185-170) (200-170) = 05
bull Degree can also be given in tables (which would give a stepped membership graph)
OPERATIONSbull Subset A fuzzy set A is a subset of fuzzy set B if for all elements of A the degreeof membership in A is less than in Bbull Complement 2 fuzzy sets are complementary if for all elements of A the degree ofmembership in B is 1-A(s)bull Union The union of two fuzzy sets will assign degree of membership equal to thehighest of the two setsbull Intersection The intersection of two fuzzy sets will assign degree of membershipequal to the lowest of the two sets
MEMBERSHIP IN COMPLEX SETS
bull To what degree is X in SET1 and SET2 ndash take the lowest of the truth values (egas in traditional logic AampB is false if any one is false)
bull To what degree is X in SET1 or SET2 ndash take the highest of the truth values (eg as in traditional logic AvB is true if any one is true)
NATURAL LANGUAGE MODIFIERS
References
[1]LA Zadeh Fuzzy Sets Information and Control 1965[2]LA Zadeh Outline of A New Approach to the Analysis of of Complex Systems and Decision Processes 1973[3] LA Zadeh Fuzzy algorithms Info amp Ctl Vol 12 1968 pp 941048576 102[4] LA Zadeh Making computers think like people IEEE Spectrum 81984pp 261048576 32[5] S Korner Laws of thought Encyclopedia of Philosophy Vol 4 MacMillanNY 1967 pp 4141048576 417[6] C Lejewski Jan Lukasiewicz Encyclopedia of Philosophy Vol 5MacMillan NY 1967 pp 1041048576 107[7] A Reigber My life with Kostas unpublished report Neverending StoryPress 1999
THANK YOU
Output Fuzzy sets
Step 3 Assign a motor speed set to eachtemperature setbull If temperature is cold then motor speed is stopbull If temperature is cool then motor speed is slowbull If temperature is just right then motor speed is Mediumbull If temperature is warm then motor speed is fastbull If temperature is hot then motor speed is blast
A Fuzzy Relationexpressed by a
rule
A Fuzzy controller with 5 patches
If temperature is just right then motor speed is medium
t = 63 degree F
Summary of Fuzzy Logic
bull Rather than ealing with probability (00-10) Fuzzy Logic deals with fuzzy set membership ndash an entity can be in two or more sets at the same time to differentdegrees
bull Key concept is thus the degree of set membership (0-1) The degree of set membership can also be referred to as degree of truth of the proposition X is in set Y
bull Degree of membership often calculated algebraically eg if height= 185 then degree = (185-170) (200-170) = 05
bull Degree can also be given in tables (which would give a stepped membership graph)
OPERATIONSbull Subset A fuzzy set A is a subset of fuzzy set B if for all elements of A the degreeof membership in A is less than in Bbull Complement 2 fuzzy sets are complementary if for all elements of A the degree ofmembership in B is 1-A(s)bull Union The union of two fuzzy sets will assign degree of membership equal to thehighest of the two setsbull Intersection The intersection of two fuzzy sets will assign degree of membershipequal to the lowest of the two sets
MEMBERSHIP IN COMPLEX SETS
bull To what degree is X in SET1 and SET2 ndash take the lowest of the truth values (egas in traditional logic AampB is false if any one is false)
bull To what degree is X in SET1 or SET2 ndash take the highest of the truth values (eg as in traditional logic AvB is true if any one is true)
NATURAL LANGUAGE MODIFIERS
References
[1]LA Zadeh Fuzzy Sets Information and Control 1965[2]LA Zadeh Outline of A New Approach to the Analysis of of Complex Systems and Decision Processes 1973[3] LA Zadeh Fuzzy algorithms Info amp Ctl Vol 12 1968 pp 941048576 102[4] LA Zadeh Making computers think like people IEEE Spectrum 81984pp 261048576 32[5] S Korner Laws of thought Encyclopedia of Philosophy Vol 4 MacMillanNY 1967 pp 4141048576 417[6] C Lejewski Jan Lukasiewicz Encyclopedia of Philosophy Vol 5MacMillan NY 1967 pp 1041048576 107[7] A Reigber My life with Kostas unpublished report Neverending StoryPress 1999
THANK YOU
Step 3 Assign a motor speed set to eachtemperature setbull If temperature is cold then motor speed is stopbull If temperature is cool then motor speed is slowbull If temperature is just right then motor speed is Mediumbull If temperature is warm then motor speed is fastbull If temperature is hot then motor speed is blast
A Fuzzy Relationexpressed by a
rule
A Fuzzy controller with 5 patches
If temperature is just right then motor speed is medium
t = 63 degree F
Summary of Fuzzy Logic
bull Rather than ealing with probability (00-10) Fuzzy Logic deals with fuzzy set membership ndash an entity can be in two or more sets at the same time to differentdegrees
bull Key concept is thus the degree of set membership (0-1) The degree of set membership can also be referred to as degree of truth of the proposition X is in set Y
bull Degree of membership often calculated algebraically eg if height= 185 then degree = (185-170) (200-170) = 05
bull Degree can also be given in tables (which would give a stepped membership graph)
OPERATIONSbull Subset A fuzzy set A is a subset of fuzzy set B if for all elements of A the degreeof membership in A is less than in Bbull Complement 2 fuzzy sets are complementary if for all elements of A the degree ofmembership in B is 1-A(s)bull Union The union of two fuzzy sets will assign degree of membership equal to thehighest of the two setsbull Intersection The intersection of two fuzzy sets will assign degree of membershipequal to the lowest of the two sets
MEMBERSHIP IN COMPLEX SETS
bull To what degree is X in SET1 and SET2 ndash take the lowest of the truth values (egas in traditional logic AampB is false if any one is false)
bull To what degree is X in SET1 or SET2 ndash take the highest of the truth values (eg as in traditional logic AvB is true if any one is true)
NATURAL LANGUAGE MODIFIERS
References
[1]LA Zadeh Fuzzy Sets Information and Control 1965[2]LA Zadeh Outline of A New Approach to the Analysis of of Complex Systems and Decision Processes 1973[3] LA Zadeh Fuzzy algorithms Info amp Ctl Vol 12 1968 pp 941048576 102[4] LA Zadeh Making computers think like people IEEE Spectrum 81984pp 261048576 32[5] S Korner Laws of thought Encyclopedia of Philosophy Vol 4 MacMillanNY 1967 pp 4141048576 417[6] C Lejewski Jan Lukasiewicz Encyclopedia of Philosophy Vol 5MacMillan NY 1967 pp 1041048576 107[7] A Reigber My life with Kostas unpublished report Neverending StoryPress 1999
THANK YOU
A Fuzzy Relationexpressed by a
rule
A Fuzzy controller with 5 patches
If temperature is just right then motor speed is medium
t = 63 degree F
Summary of Fuzzy Logic
bull Rather than ealing with probability (00-10) Fuzzy Logic deals with fuzzy set membership ndash an entity can be in two or more sets at the same time to differentdegrees
bull Key concept is thus the degree of set membership (0-1) The degree of set membership can also be referred to as degree of truth of the proposition X is in set Y
bull Degree of membership often calculated algebraically eg if height= 185 then degree = (185-170) (200-170) = 05
bull Degree can also be given in tables (which would give a stepped membership graph)
OPERATIONSbull Subset A fuzzy set A is a subset of fuzzy set B if for all elements of A the degreeof membership in A is less than in Bbull Complement 2 fuzzy sets are complementary if for all elements of A the degree ofmembership in B is 1-A(s)bull Union The union of two fuzzy sets will assign degree of membership equal to thehighest of the two setsbull Intersection The intersection of two fuzzy sets will assign degree of membershipequal to the lowest of the two sets
MEMBERSHIP IN COMPLEX SETS
bull To what degree is X in SET1 and SET2 ndash take the lowest of the truth values (egas in traditional logic AampB is false if any one is false)
bull To what degree is X in SET1 or SET2 ndash take the highest of the truth values (eg as in traditional logic AvB is true if any one is true)
NATURAL LANGUAGE MODIFIERS
References
[1]LA Zadeh Fuzzy Sets Information and Control 1965[2]LA Zadeh Outline of A New Approach to the Analysis of of Complex Systems and Decision Processes 1973[3] LA Zadeh Fuzzy algorithms Info amp Ctl Vol 12 1968 pp 941048576 102[4] LA Zadeh Making computers think like people IEEE Spectrum 81984pp 261048576 32[5] S Korner Laws of thought Encyclopedia of Philosophy Vol 4 MacMillanNY 1967 pp 4141048576 417[6] C Lejewski Jan Lukasiewicz Encyclopedia of Philosophy Vol 5MacMillan NY 1967 pp 1041048576 107[7] A Reigber My life with Kostas unpublished report Neverending StoryPress 1999
THANK YOU
A Fuzzy controller with 5 patches
If temperature is just right then motor speed is medium
t = 63 degree F
Summary of Fuzzy Logic
bull Rather than ealing with probability (00-10) Fuzzy Logic deals with fuzzy set membership ndash an entity can be in two or more sets at the same time to differentdegrees
bull Key concept is thus the degree of set membership (0-1) The degree of set membership can also be referred to as degree of truth of the proposition X is in set Y
bull Degree of membership often calculated algebraically eg if height= 185 then degree = (185-170) (200-170) = 05
bull Degree can also be given in tables (which would give a stepped membership graph)
OPERATIONSbull Subset A fuzzy set A is a subset of fuzzy set B if for all elements of A the degreeof membership in A is less than in Bbull Complement 2 fuzzy sets are complementary if for all elements of A the degree ofmembership in B is 1-A(s)bull Union The union of two fuzzy sets will assign degree of membership equal to thehighest of the two setsbull Intersection The intersection of two fuzzy sets will assign degree of membershipequal to the lowest of the two sets
MEMBERSHIP IN COMPLEX SETS
bull To what degree is X in SET1 and SET2 ndash take the lowest of the truth values (egas in traditional logic AampB is false if any one is false)
bull To what degree is X in SET1 or SET2 ndash take the highest of the truth values (eg as in traditional logic AvB is true if any one is true)
NATURAL LANGUAGE MODIFIERS
References
[1]LA Zadeh Fuzzy Sets Information and Control 1965[2]LA Zadeh Outline of A New Approach to the Analysis of of Complex Systems and Decision Processes 1973[3] LA Zadeh Fuzzy algorithms Info amp Ctl Vol 12 1968 pp 941048576 102[4] LA Zadeh Making computers think like people IEEE Spectrum 81984pp 261048576 32[5] S Korner Laws of thought Encyclopedia of Philosophy Vol 4 MacMillanNY 1967 pp 4141048576 417[6] C Lejewski Jan Lukasiewicz Encyclopedia of Philosophy Vol 5MacMillan NY 1967 pp 1041048576 107[7] A Reigber My life with Kostas unpublished report Neverending StoryPress 1999
THANK YOU
If temperature is just right then motor speed is medium
t = 63 degree F
Summary of Fuzzy Logic
bull Rather than ealing with probability (00-10) Fuzzy Logic deals with fuzzy set membership ndash an entity can be in two or more sets at the same time to differentdegrees
bull Key concept is thus the degree of set membership (0-1) The degree of set membership can also be referred to as degree of truth of the proposition X is in set Y
bull Degree of membership often calculated algebraically eg if height= 185 then degree = (185-170) (200-170) = 05
bull Degree can also be given in tables (which would give a stepped membership graph)
OPERATIONSbull Subset A fuzzy set A is a subset of fuzzy set B if for all elements of A the degreeof membership in A is less than in Bbull Complement 2 fuzzy sets are complementary if for all elements of A the degree ofmembership in B is 1-A(s)bull Union The union of two fuzzy sets will assign degree of membership equal to thehighest of the two setsbull Intersection The intersection of two fuzzy sets will assign degree of membershipequal to the lowest of the two sets
MEMBERSHIP IN COMPLEX SETS
bull To what degree is X in SET1 and SET2 ndash take the lowest of the truth values (egas in traditional logic AampB is false if any one is false)
bull To what degree is X in SET1 or SET2 ndash take the highest of the truth values (eg as in traditional logic AvB is true if any one is true)
NATURAL LANGUAGE MODIFIERS
References
[1]LA Zadeh Fuzzy Sets Information and Control 1965[2]LA Zadeh Outline of A New Approach to the Analysis of of Complex Systems and Decision Processes 1973[3] LA Zadeh Fuzzy algorithms Info amp Ctl Vol 12 1968 pp 941048576 102[4] LA Zadeh Making computers think like people IEEE Spectrum 81984pp 261048576 32[5] S Korner Laws of thought Encyclopedia of Philosophy Vol 4 MacMillanNY 1967 pp 4141048576 417[6] C Lejewski Jan Lukasiewicz Encyclopedia of Philosophy Vol 5MacMillan NY 1967 pp 1041048576 107[7] A Reigber My life with Kostas unpublished report Neverending StoryPress 1999
THANK YOU
t = 63 degree F
Summary of Fuzzy Logic
bull Rather than ealing with probability (00-10) Fuzzy Logic deals with fuzzy set membership ndash an entity can be in two or more sets at the same time to differentdegrees
bull Key concept is thus the degree of set membership (0-1) The degree of set membership can also be referred to as degree of truth of the proposition X is in set Y
bull Degree of membership often calculated algebraically eg if height= 185 then degree = (185-170) (200-170) = 05
bull Degree can also be given in tables (which would give a stepped membership graph)
OPERATIONSbull Subset A fuzzy set A is a subset of fuzzy set B if for all elements of A the degreeof membership in A is less than in Bbull Complement 2 fuzzy sets are complementary if for all elements of A the degree ofmembership in B is 1-A(s)bull Union The union of two fuzzy sets will assign degree of membership equal to thehighest of the two setsbull Intersection The intersection of two fuzzy sets will assign degree of membershipequal to the lowest of the two sets
MEMBERSHIP IN COMPLEX SETS
bull To what degree is X in SET1 and SET2 ndash take the lowest of the truth values (egas in traditional logic AampB is false if any one is false)
bull To what degree is X in SET1 or SET2 ndash take the highest of the truth values (eg as in traditional logic AvB is true if any one is true)
NATURAL LANGUAGE MODIFIERS
References
[1]LA Zadeh Fuzzy Sets Information and Control 1965[2]LA Zadeh Outline of A New Approach to the Analysis of of Complex Systems and Decision Processes 1973[3] LA Zadeh Fuzzy algorithms Info amp Ctl Vol 12 1968 pp 941048576 102[4] LA Zadeh Making computers think like people IEEE Spectrum 81984pp 261048576 32[5] S Korner Laws of thought Encyclopedia of Philosophy Vol 4 MacMillanNY 1967 pp 4141048576 417[6] C Lejewski Jan Lukasiewicz Encyclopedia of Philosophy Vol 5MacMillan NY 1967 pp 1041048576 107[7] A Reigber My life with Kostas unpublished report Neverending StoryPress 1999
THANK YOU
Summary of Fuzzy Logic
bull Rather than ealing with probability (00-10) Fuzzy Logic deals with fuzzy set membership ndash an entity can be in two or more sets at the same time to differentdegrees
bull Key concept is thus the degree of set membership (0-1) The degree of set membership can also be referred to as degree of truth of the proposition X is in set Y
bull Degree of membership often calculated algebraically eg if height= 185 then degree = (185-170) (200-170) = 05
bull Degree can also be given in tables (which would give a stepped membership graph)
OPERATIONSbull Subset A fuzzy set A is a subset of fuzzy set B if for all elements of A the degreeof membership in A is less than in Bbull Complement 2 fuzzy sets are complementary if for all elements of A the degree ofmembership in B is 1-A(s)bull Union The union of two fuzzy sets will assign degree of membership equal to thehighest of the two setsbull Intersection The intersection of two fuzzy sets will assign degree of membershipequal to the lowest of the two sets
MEMBERSHIP IN COMPLEX SETS
bull To what degree is X in SET1 and SET2 ndash take the lowest of the truth values (egas in traditional logic AampB is false if any one is false)
bull To what degree is X in SET1 or SET2 ndash take the highest of the truth values (eg as in traditional logic AvB is true if any one is true)
NATURAL LANGUAGE MODIFIERS
References
[1]LA Zadeh Fuzzy Sets Information and Control 1965[2]LA Zadeh Outline of A New Approach to the Analysis of of Complex Systems and Decision Processes 1973[3] LA Zadeh Fuzzy algorithms Info amp Ctl Vol 12 1968 pp 941048576 102[4] LA Zadeh Making computers think like people IEEE Spectrum 81984pp 261048576 32[5] S Korner Laws of thought Encyclopedia of Philosophy Vol 4 MacMillanNY 1967 pp 4141048576 417[6] C Lejewski Jan Lukasiewicz Encyclopedia of Philosophy Vol 5MacMillan NY 1967 pp 1041048576 107[7] A Reigber My life with Kostas unpublished report Neverending StoryPress 1999
THANK YOU
bull Rather than ealing with probability (00-10) Fuzzy Logic deals with fuzzy set membership ndash an entity can be in two or more sets at the same time to differentdegrees
bull Key concept is thus the degree of set membership (0-1) The degree of set membership can also be referred to as degree of truth of the proposition X is in set Y
bull Degree of membership often calculated algebraically eg if height= 185 then degree = (185-170) (200-170) = 05
bull Degree can also be given in tables (which would give a stepped membership graph)
OPERATIONSbull Subset A fuzzy set A is a subset of fuzzy set B if for all elements of A the degreeof membership in A is less than in Bbull Complement 2 fuzzy sets are complementary if for all elements of A the degree ofmembership in B is 1-A(s)bull Union The union of two fuzzy sets will assign degree of membership equal to thehighest of the two setsbull Intersection The intersection of two fuzzy sets will assign degree of membershipequal to the lowest of the two sets
MEMBERSHIP IN COMPLEX SETS
bull To what degree is X in SET1 and SET2 ndash take the lowest of the truth values (egas in traditional logic AampB is false if any one is false)
bull To what degree is X in SET1 or SET2 ndash take the highest of the truth values (eg as in traditional logic AvB is true if any one is true)
NATURAL LANGUAGE MODIFIERS
References
[1]LA Zadeh Fuzzy Sets Information and Control 1965[2]LA Zadeh Outline of A New Approach to the Analysis of of Complex Systems and Decision Processes 1973[3] LA Zadeh Fuzzy algorithms Info amp Ctl Vol 12 1968 pp 941048576 102[4] LA Zadeh Making computers think like people IEEE Spectrum 81984pp 261048576 32[5] S Korner Laws of thought Encyclopedia of Philosophy Vol 4 MacMillanNY 1967 pp 4141048576 417[6] C Lejewski Jan Lukasiewicz Encyclopedia of Philosophy Vol 5MacMillan NY 1967 pp 1041048576 107[7] A Reigber My life with Kostas unpublished report Neverending StoryPress 1999
THANK YOU
bull Degree can also be given in tables (which would give a stepped membership graph)
OPERATIONSbull Subset A fuzzy set A is a subset of fuzzy set B if for all elements of A the degreeof membership in A is less than in Bbull Complement 2 fuzzy sets are complementary if for all elements of A the degree ofmembership in B is 1-A(s)bull Union The union of two fuzzy sets will assign degree of membership equal to thehighest of the two setsbull Intersection The intersection of two fuzzy sets will assign degree of membershipequal to the lowest of the two sets
MEMBERSHIP IN COMPLEX SETS
bull To what degree is X in SET1 and SET2 ndash take the lowest of the truth values (egas in traditional logic AampB is false if any one is false)
bull To what degree is X in SET1 or SET2 ndash take the highest of the truth values (eg as in traditional logic AvB is true if any one is true)
NATURAL LANGUAGE MODIFIERS
References
[1]LA Zadeh Fuzzy Sets Information and Control 1965[2]LA Zadeh Outline of A New Approach to the Analysis of of Complex Systems and Decision Processes 1973[3] LA Zadeh Fuzzy algorithms Info amp Ctl Vol 12 1968 pp 941048576 102[4] LA Zadeh Making computers think like people IEEE Spectrum 81984pp 261048576 32[5] S Korner Laws of thought Encyclopedia of Philosophy Vol 4 MacMillanNY 1967 pp 4141048576 417[6] C Lejewski Jan Lukasiewicz Encyclopedia of Philosophy Vol 5MacMillan NY 1967 pp 1041048576 107[7] A Reigber My life with Kostas unpublished report Neverending StoryPress 1999
THANK YOU
OPERATIONSbull Subset A fuzzy set A is a subset of fuzzy set B if for all elements of A the degreeof membership in A is less than in Bbull Complement 2 fuzzy sets are complementary if for all elements of A the degree ofmembership in B is 1-A(s)bull Union The union of two fuzzy sets will assign degree of membership equal to thehighest of the two setsbull Intersection The intersection of two fuzzy sets will assign degree of membershipequal to the lowest of the two sets
MEMBERSHIP IN COMPLEX SETS
bull To what degree is X in SET1 and SET2 ndash take the lowest of the truth values (egas in traditional logic AampB is false if any one is false)
bull To what degree is X in SET1 or SET2 ndash take the highest of the truth values (eg as in traditional logic AvB is true if any one is true)
NATURAL LANGUAGE MODIFIERS
References
[1]LA Zadeh Fuzzy Sets Information and Control 1965[2]LA Zadeh Outline of A New Approach to the Analysis of of Complex Systems and Decision Processes 1973[3] LA Zadeh Fuzzy algorithms Info amp Ctl Vol 12 1968 pp 941048576 102[4] LA Zadeh Making computers think like people IEEE Spectrum 81984pp 261048576 32[5] S Korner Laws of thought Encyclopedia of Philosophy Vol 4 MacMillanNY 1967 pp 4141048576 417[6] C Lejewski Jan Lukasiewicz Encyclopedia of Philosophy Vol 5MacMillan NY 1967 pp 1041048576 107[7] A Reigber My life with Kostas unpublished report Neverending StoryPress 1999
THANK YOU
MEMBERSHIP IN COMPLEX SETS
bull To what degree is X in SET1 and SET2 ndash take the lowest of the truth values (egas in traditional logic AampB is false if any one is false)
bull To what degree is X in SET1 or SET2 ndash take the highest of the truth values (eg as in traditional logic AvB is true if any one is true)
NATURAL LANGUAGE MODIFIERS
References
[1]LA Zadeh Fuzzy Sets Information and Control 1965[2]LA Zadeh Outline of A New Approach to the Analysis of of Complex Systems and Decision Processes 1973[3] LA Zadeh Fuzzy algorithms Info amp Ctl Vol 12 1968 pp 941048576 102[4] LA Zadeh Making computers think like people IEEE Spectrum 81984pp 261048576 32[5] S Korner Laws of thought Encyclopedia of Philosophy Vol 4 MacMillanNY 1967 pp 4141048576 417[6] C Lejewski Jan Lukasiewicz Encyclopedia of Philosophy Vol 5MacMillan NY 1967 pp 1041048576 107[7] A Reigber My life with Kostas unpublished report Neverending StoryPress 1999
THANK YOU
NATURAL LANGUAGE MODIFIERS
References
[1]LA Zadeh Fuzzy Sets Information and Control 1965[2]LA Zadeh Outline of A New Approach to the Analysis of of Complex Systems and Decision Processes 1973[3] LA Zadeh Fuzzy algorithms Info amp Ctl Vol 12 1968 pp 941048576 102[4] LA Zadeh Making computers think like people IEEE Spectrum 81984pp 261048576 32[5] S Korner Laws of thought Encyclopedia of Philosophy Vol 4 MacMillanNY 1967 pp 4141048576 417[6] C Lejewski Jan Lukasiewicz Encyclopedia of Philosophy Vol 5MacMillan NY 1967 pp 1041048576 107[7] A Reigber My life with Kostas unpublished report Neverending StoryPress 1999
THANK YOU
References
[1]LA Zadeh Fuzzy Sets Information and Control 1965[2]LA Zadeh Outline of A New Approach to the Analysis of of Complex Systems and Decision Processes 1973[3] LA Zadeh Fuzzy algorithms Info amp Ctl Vol 12 1968 pp 941048576 102[4] LA Zadeh Making computers think like people IEEE Spectrum 81984pp 261048576 32[5] S Korner Laws of thought Encyclopedia of Philosophy Vol 4 MacMillanNY 1967 pp 4141048576 417[6] C Lejewski Jan Lukasiewicz Encyclopedia of Philosophy Vol 5MacMillan NY 1967 pp 1041048576 107[7] A Reigber My life with Kostas unpublished report Neverending StoryPress 1999
THANK YOU
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