Efficient Decomposition of Large Fuzzy Functions and Relations Marek Perkowski + Portland State University, Dept. Electrical Engineering, Portland, Oregon 97207, Tel. 503-725-5411, Fax (503) 725-4882, USA FUZZY LOGIC IN FUZZY LOGIC IN ROBOTICS ROBOTICS
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Efficient Decomposition of Large Fuzzy Functions and Relations Marek Perkowski + Portland State University, Dept. Electrical Engineering, Portland, Oregon.
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Efficient Decomposition of Large Fuzzy Functions and Relations
TransformationsSome transformations of fuzzy sets with examples
follow:x’b + xb = (x + x’)b bxb + xx’b = xb(1 + x’) = xbx’b + xx’b = x’b(1 + x) = x’ba + xa = a(1 + x) = aa + x’a = a(1 + x’) = aa + xx’a = aa + 0 = ax + 0 = xx * 0 = 0x + 1 = 1x * 1 = x Examples:Examples:a + xa + x’b + xx’b = a(1 + x) + x’b(1 + x) = a + x’b a + xa + x’a + xx’a = a(1 + x + x’ + xx’) = a
Differences Between Boolean Logic and Fuzzy Logic
Boolean logic the value of a variable and its inverse are always disjoint (X * X’ = 0) and (X + X’ = 1) because the values are either zero or one.
Fuzzy logic membership functions can be either disjoint or non-disjoint.
Example of a fuzzy non-linear and linear membership function X is shown (a) with its inverse membership function shown in (b).
Fuzzy Intersection and Union• From the membership
functions shown in the top in (a), and complement X’ (b) the intersection of fuzzy variable X and its complement X’ is shown bottom in (a).
• From the membership functions shown in the top in (a), and complement X’ (b) the union of fuzzy variable X and its complement X’ is shown bottom in (b).
Fuzzy map may be regarded as an extension of the Veitch diagram, which forms also the basis for the Karnauph map.
The functions shown in (a) and (b) are equivalent to f(x1, x2) = x’1 x2+ x1
g = a high pass filter whose acceptance threshold begins at
c > 1
Map of relation GMap of relation G
G
\ c
From CIG After induction
Cost FunctionCost Function
Decomposed Function Cardinalityis the total cost of all blocks.
Cost is defined for a single block in terms of the block’s n inputs and m outputs
Cost := m * 2n
Example of DFC calculationExample of DFC calculation
B1
B2
B3
Cost(B3) =22*1=4Cost(B1) =24*1=16
Cost(B2) =23*2=16
Total DFC = 16 + 16 + 4 = 36
Other cost functionsOther cost functions
Decomposition AlgorithmDecomposition Algorithm
• Find a set of partitions (Ai, Bi) of input variables (X) into free variables (A) and bound variables (B)
• For each partitioning, find decompositionF(X) = Hi(Gi(Bi), Ai) such that column multiplicity is minimal, and calculate DFC
• Repeat the process for all partitioning until the decomposition with minimum DFC is found.
Algorithm RequirementsAlgorithm Requirements
• Since the process is iterative, it is of high importance that minimization of the column multiplicity index is done as fast as possible.
• At the same time, for a given partitioning, it is important that the value of the column multiplicity is as close to the absolute minimumabsolute minimum value
Compatibility Checking and Correction for Relations Example
• Function that needs checked and corrected shown in a decomposition-map.
Compatibility Graph Show Cliques
• Cliques before checking and correction:
clique 0 = 0 1 2
clique 1 = 0 3
• Cliques after:
clique 0 = 0
clique 1 = 0 3
clique 2 = 1 2
• Compatibility graph and corrected cliques shown left
Discovering new concepts
• Discovering concepts useful for purchasing a carpurchasing a car
Variable orderingVariable ordering
Vacuous variables removingVacuous variables removing• Variables b and d
reduce uncertainty of y to 0 which means they provide all the information necessary for determination of the output y
• Variables a and c are vacuous
Example of removing inessential variables (a) original function (b)
variable a removed (c) variable b removed, variable c is no longer inessential.
Need to Decompose Multiple-valued Functions and Relations
• Multiple-valued and Inconsistent Data
• Ways to Create Relations– Decomposition Process to Create Relations– Program to Change Inconsistency data into
Relations
Decomposition Structure
• General flow chart of RMVGUD Program.
Results of Relation Decomposition
3
21
18
68
157
244
132
323
1066
4
10
10
hayes
flare 1
flare 2
No. of
Blocks
No. of
cubes
No. of
Cubes
No. of
Inputs
Output FileInput FileFile
CONCLUSION CONCLUSION • Fuzzy functions are efficiently realized as multi-level
networks of fuzzy operators
• The approach has been extended also to relations. Relation is decomposed to network of relations and functions, non-decomposable relations are realized using other methods as the simplest functions
• Advantages of the new approach to fuzzy logic decomposition
– Eliminates the need for time-consuming conversion to canonical form
– Eliminates the use of S-maps
– Entirely algorithmic
– Enables decomposition of large fuzzy expressions