Fuzzy set- key concept fuzzy logic. Let E- universal set, X- element E, a R is some property. Regular (crisp) subset A universal set E, whose elements satisfy the property R is defined as the set of ordered pairs
A = ( μA(x) / x},
Where μ A (x) —characteristic function, taking the value 1 if X satisfies property R, and 0 otherwise.
Fuzzy subset is different from regular topics, which is for elements X from E there is no clear yes-no answer regarding property R. In this regard, the fuzzy subset A universal set E is defined as the set of ordered pairs
A = ( μA(x) / x},
Where μ A (x) — characteristic membership function(or simply membership function), taking values in some completely ordered set M(For example, M = ).
The membership function indicates the degree (or level) of membership of an element X subset A. A bunch of M called a set of accessories. If M= (0, 1), then the fuzzy subset A can be considered as an ordinary or crisp set.
Examples of writing a fuzzy set
Let E = {x 1 , x 2 , x z,x 4 , x 5 ), M = ; A is a fuzzy set for which μ A ( x 1 )= 0.3; μ A ( x 2)= 0; μ A ( X 3) = 1; μ A (x 4) = 0.5; μ A ( x 5)= 0,9.
Then A can be represented in the form
A ={0,3/x 1 ; 0/X 2 ; 1/X 3 ; 0,5/X 4 ; 0,9/X 5 } ,
or
A={0,3/x 1 +0/X 2 +1/X 3 +0,5/X 4 +0,9/X 5 },
or
Comment. Here the “+” sign does not denote the operation of addition, but has the meaning of union.
Basic characteristics of fuzzy sets
Let M= and A— fuzzy set with elements from the universal set E and many accessories M.
The quantity is called height fuzzy set A. Fuzzy set It's okay if its height is 1, i.e. upper limit its membership function is 1 (= 1). At< 1нечеткое множество называется subnormal.
Fuzzy set empty if ∀ xϵ E μ A ( x) = 0. A non-empty subnormal set can be normalized using the formula
Fuzzy set unimodal, If μ A ( x) = 1 on only one X from E.
. Carrier fuzzy set A is an ordinary subset with the property μ A ( x)>0, i.e. carrier A = {x/x ϵ E, μ A ( x)>0}.
Elements xϵ E, for which μ A ( x) = 0,5 , are called transition points sets A.
Examples of fuzzy sets
1. Let E = {0, 1, 2, . . ., 10}, M =. Fuzzy set"Several" can be defined as follows:
“Several” = 0.5/3 + 0.8/4 + 1/5 + 1/6 + 0.8/7 + 0.5/8; its characteristics:height = 1, carrier = {3, 4, 5, 6, 7, 8}, transition points — {3, 8}.
2. Let E = {0, 1, 2, 3,…, n,… ). The fuzzy set “Small” can be defined:
3. Let E= (1, 2, 3,..., 100) and corresponds to the concept “Age”, then the fuzzy set “Young” can be defined using
Fuzzy set “Young” on the universal set E"= (IVANOV, PETROV, SIDOROV,...) is specified using the membership function μ Young ( x) on E =(1, 2, 3, ..., 100) (age), called in relation to E" compatibility function, while:
Where X— SIDOROV’s age.
4. Let E= (ZAPOROZHETS, ZHIGULI, MERCEDES,...) - a set of car brands, and E"= is the universal set “Cost”, then on E" we can define fuzzy sets of the type:
Rice. 1.1. Examples of membership functions
“For the poor”, “For the middle class”, “Prestigious”, with affiliation functions like Fig. 1.1.
Having these functions and knowing the cost of cars from E V this moment time, we will thereby determine E" fuzzy sets with the same names.
So, for example, the fuzzy set “For the poor”, defined on the universal set E =(ZAPOROZHETZ, ZHIGULI, MERCEDES,...), looks as shown in Fig. 1.2.
Rice. 1.2. An example of specifying a fuzzy set
Similarly, you can define the fuzzy set “High-speed”, “Medium”, “Slow-speed”, etc.
5. Let E- set of integers:
E= {-8, -5, -3, 0, 1, 2, 4, 6, 9}.
Then the fuzzy subset of numbers, according to absolute value close to zero, can be defined, for example, like this:
A ={0/-8 + 0,5/-5 + 0,6/-3 +1/0 + 0,9/1 + 0,8/2 + 0,6/4 + 0,3/6 + 0/9}.
On methods for constructing membership functions of fuzzy sets
The above examples used straight methods when an expert either simply sets for each X ϵ E meaning μ A (x), or defines a compatibility function. As a rule, direct methods for specifying the membership function are used for measurable concepts such as speed, time, distance, pressure, temperature, etc., or when polar values are distinguished.
In many problems, when characterizing an object, it is possible to select a set of features and for each of them determine polar values corresponding to the values of the membership function, 0 or 1.
For example, in the task of face recognition, we can distinguish the scales given in table. 1.1.
Table 1.1. Scales in the face recognition task
|
x 1 |
forehead height |
||
|
x 2 |
nose profile |
snub |
hunchbacked |
|
nose length |
short |
||
|
x 4 |
eye shape |
||
|
eye color |
|||
|
chin shape |
pointed |
square |
|
|
x 7 |
lip thickness |
||
|
complexion |
|||
|
face outline |
oval |
square |
For a specific personAthe expert, based on the given scale, setsμ A(x)ϵ, forming the vector membership function (μ A(x 1) , μ A(x 2),…, μ A(x 9)}.
With direct methods, group direct methods are also used, when, for example, a group of experts is presented with a specific person and everyone must give one of two answers: “this person is bald” or “this person is not bald”, then the number of affirmative answers divided on total number experts, gives meaning μ bald ( of this person). (In this example, you can act through the compatibility function, but then you will have to count the number of hairs on the head of each person presented to the expert.)
Indirect methods for determining the values of the membership function are used in cases where there are no elementary measurable properties through which the fuzzy set of interest to us is determined. As a rule, these are pairwise comparison methods. If the values of the membership functions were known to us, for example, μ A(X-i) = ω i , i= 1, 2, ..., n, then pairwise comparisons can be represented by a matrix of relations A= ( a ij ), where a ij= ωi/ ω j(division operation).
In practice, the expert himself forms the matrix A, in this case it is assumed that the diagonal elements are equal to 1, and for elements that are symmetrical with respect to the diagonal a ij = 1/a ij , i.e. if one element evaluates to α times stronger than the other, then this latter must be 1/α times stronger than the first. IN general case the problem reduces to finding a vector ω that satisfies an equation of the form Aw= λmax w, where λ max is the largest eigenvalue of the matrix A. Since the matrix A is positive by construction, a solution to this problem exists and is positive.
Two more approaches can be noted:
- use of standard forms curves for specifying membership functions (in the form of (L-R)-Type - see below) with clarification of their parameters in accordance with experimental data;
- use of relative frequenciesaccording to the experiment as membership values.
Fuzzy set is a collection of elements of an arbitrary nature, regarding which it is impossible to say with complete certainty whether one or another element of the collection under consideration belongs to a given set or not. In other words, a fuzzy set differs from an ordinary set in that for all or part of its elements there is no unambiguous answer to the question: “Whether this or that element belongs or does not belong to the fuzzy set under consideration.”
For building fuzzy models systems itself concept of fuzzy sets should be strictly defined in order to eliminate ambiguity in the interpretation of certain of its properties. The most natural and intuitive way is to specify the range of values of such a function as the interval of real numbers between 0 and 1 (including these values themselves).
Mathematical definition of a fuzzy set. Formally, a fuzzy set is defined as a set of ordered pairs or tuples of the form: 
, Where 
is an element of some universal set, or universe 
, A 
– membership function that assigns each of the elements 
some real number from the interval 
, i.e. this function is defined in display form:
In this case, the value 
for some 
means that the element 
definitely belongs to the fuzzy set 
, and the value 
means that the element 
definitely does not belong to the fuzzy set 
.
Formally, a finite fuzzy set in the general case has the form:
Universe 
is a set containing all possible elements within a certain context. Formally, it is convenient to assume that the membership function of the universe as a fuzzy set is identically equal to one for all elements without exception: 
.
Empty fuzzy set, or a set that does not contain a single element, is denoted 
and is formally defined as a fuzzy set whose membership function is identically equal to zero for all elements without exception: 
The formal definition of a fuzzy set does not impose any restrictions on the choice of a specific membership function for its representation. However, in practice it is convenient to use those of them that allow analytical representation in the form of some simple mathematical function. This simplifies not only the corresponding numerical calculations, but also reduces the computational resources required to store the individual values of these membership functions.
Membership function– a mathematical function that determines the degree to which the elements of a certain set belong to a given fuzzy set. This function assigns each element of the fuzzy set a real number from the interval 
To define a specific fuzzy set means to determine the corresponding membership function.
When constructing membership functions for fuzzy sets, one should adhere to certain rules, which are predetermined by the nature of the uncertainty that occurs when constructing specific fuzzy models.
From a practical point of view, it is convenient to associate with each fuzzy set a certain property that characterizes the considered set of objects of the universe. Moreover, by analogy with classical sets, the property under consideration can generate a certain predicate, which is quite natural to be called a fuzzy predicate. This fuzzy predicate can take not one of two truth values (“true” or “false”), but a whole continuum of truth values, which for convenience are selected from the interval 
In this case, the value “true” still corresponds to the number 1, and the value “false” still corresponds to the number 0.
In essence, this means the following: the more an element 
has the property under consideration, the more close to 1 the truth value of the corresponding fuzzy predicate should be. And vice versa, the less the element 
has the property in question, the more close to 0 the truth value of this fuzzy predicate should be. If element 
definitely does not have the property in question, then the corresponding fuzzy predicate takes the value “false” (or the number 0). If the element 
definitely possesses the property in question, then the corresponding fuzzy predicate takes the value “true” (or the number 1).
Then, in the general case, defining a fuzzy set using a special property is equivalent to specifying a membership function that meaningfully represents the degree of truth of the corresponding one-place fuzzy predicate.
Concept fuzzy relationship along with the concept of the fuzzy set itself should be attributed to fundamentals the entire theory of fuzzy sets. Based on fuzzy relationships, a whole series of additional concepts, used to build fuzzy models of complex systems.
In the general case, a fuzzy relation defined on sets (universes) 
, is some fixed fuzzy subset of the Cartesian product of these universes. In other words, if we denote an arbitrary fuzzy relation through 
, then by definition, where 
- membership function of a given fuzzy relation, which is defined as a mapping. Through 
denotes a tuple of 
elements, each of which is selected from its own universe:
Fuzzy logic, which serves as the basis for the implementation of fuzzy control methods, more naturally describes the nature of human thinking and the course of its reasoning than traditional formal logical systems. That is why the study and use of mathematical tools to represent fuzzy initial information allows us to build models that most adequately reflect various aspects of the uncertainty that is constantly present in the reality around us.
Fuzzy logic is intended to formalize human abilities for imprecise or approximate reasoning, which allows us to more adequately describe situations with uncertainty. Classical logic inherently ignores the problem of uncertainty, since all statements and reasoning in formal logical systems can only have the value “truth” ( AND,1) or false ( L,0). In contrast, in fuzzy logic, the truth of reasoning is assessed to some extent, which can take other different
meanings. Fuzzy logic uses the basic concepts of fuzzy set theory to formalize imprecise knowledge and perform approximate reasoning in a particular subject area.
In the version of fuzzy logic proposed by L. Zade, the set of truth values of statements is generalized to the interval of real values 
, which allows the statement to take any truth value from this interval. This numerical value is quantitative assessment the degree of truth of a statement regarding which it is impossible to conclude with complete certainty whether it is true or false. Using an interval as a set of truth values 
allows you to build a logical system within which it turned out to be possible to reason with uncertainty and evaluate the truth of statements.
The initial concept of fuzzy logic is the concept of an elementary fuzzy statement.
Elementary fuzzy statement is a declarative sentence that expresses a complete thought about which we can judge its truth or falsity only with some degree of certainty. In fuzzy logic degree of truth of an elementary fuzzy statement takes a value from a closed interval 
, with 0 and 1 being the extreme values of the degree of truth and coinciding with the values “false” and “true”, respectively.
Fuzzy implication or implication of fuzzy statements A and B(reads “IF A, THEN B”) – is called a binary logical operation, the result of which is a fuzzy statement, the truth of which can take on a value, for example, determined by the formula proposed by E. Mamdani:
This form of fuzzy implication is also called fuzzy implication of Mamdani or fuzzy implication minimum correlation.
Modern science and technology cannot be imagined without the widespread use of mathematical modeling, since full-scale experiments cannot always be carried out, they are often too expensive and require considerable time, and in many cases they are associated with risk and large material or moral costs. The essence of mathematical modeling is to replace a real object with its “image” - a mathematical model - and further study of the model using computational and logical algorithms implemented on computers. The most important requirement for mathematical model, is the condition of its adequacy (correct correspondence) to the real object being studied relative to the selected system of its properties. This, first of all, means a correct quantitative description of the properties of the object under consideration. The construction of such quantitative models is possible for simple systems.
The situation is different with complex systems. To obtain significant conclusions about behavior complex systems it is necessary to abandon high accuracy and rigor when building a model and use approaches that are approximate in nature when constructing it. One of these approaches is associated with the introduction of linguistic variables that describe a person’s unclear reflection of the surrounding world. In order for a linguistic variable to become a full-fledged mathematical object, the concept of a fuzzy set was introduced.
In the theory of crisp sets, the characteristic function of a crisp set was considered 
in universal space 
,
equal to 1 if element 
satisfies the property 
and therefore belongs to the set 
, and equal to 0 otherwise. Thus, we were talking about a clear world (Boolean algebra), in which the presence or absence of a given property is determined by the values 0 or 1 (“no” or “yes”).
However, everything in the world cannot be divided only into white and black, truth and lies. So, even the Buddha saw a world filled with contradictions, things could be true to some extent and, to some extent, false at the same time. Plato laid the foundation for what would become fuzzy logic by pointing out that there was a third realm (beyond Truth and Falsehood) where these contradictions are relative.
University of California professor Zadeh published the paper “Fuzzy Sets” in 1965, in which he extended the two-valued estimate of 0 or 1 to an unlimited multi-valued estimate above 0 and below 1 in a closed interval and first introduced the concept of a “fuzzy set.” Instead of the term “characteristic function,” Zadeh used the term “membership function.” Fuzzy set 
(the same notation is retained as for a crisp set) in the universal space 
through the membership function 
(same notation as for the characteristic function) is defined as follows
(3.1)
The membership function is most often interpreted as follows: the value 
means subjective assessment degree of element membership 
fuzzy set 
, For example, 
means that 
80% owned 
. Therefore, there must be “my membership function”, “your membership function”, “specialist’s membership function”, etc. The graphical representation of a fuzzy set, a Venn diagram, is represented by concentric circles in Fig. 1. The membership function of a fuzzy set has a bell-shaped graph, in contrast to the rectangular characteristic function of a clear set, Fig. 1.
You should pay attention to the connection between the crisp and fuzzy sets. Two values (0,1) of the characteristic function belong to a closed interval of values of the membership function. Therefore, a crisp set is a special case of a fuzzy set, and the concept of a fuzzy set is an extended concept that also covers the concept of a crisp set. In other words, a crisp set is also a fuzzy set.
A fuzzy set is strictly defined using the membership function and does not contain any vagueness. The fact is that a fuzzy set is strictly defined using the estimated values of a closed interval, and this is the membership function. If the universal set 
consists of a discrete finite set of elements, then, based on practical considerations, indicate the value of the membership function and the corresponding element using the separation signs / and +. For example, let the universal set consist of integers less than 10, then the fuzzy set 
"small numbers" can be represented as
A=1/0 + 1/1 + 0.8/2 + 0.5/3 + 0.1/4
Here, for example, 0.8/2 means 
. The + sign denotes a union. When writing a fuzzy set in the above form, the elements of the universal set are omitted 
with membership function values equal to zero. Usually all elements of the universal set are written down with the corresponding values of the membership function. A fuzzy set notation is used, as in probability theory,
Definition. In general, a fuzzy subset 
universal set 
is defined as the set of ordered pairs
.
(3.2)
Lecture 4. Modeling and decision making in GIS.
1. Fuzzy sets
2. Optimization methods
Fuzzy sets
The most striking property human intelligence is the ability to accept right decisions in an environment of incomplete and unclear information. Building models of approximate human reasoning and using them in computer systems presents today one of important tasks development of GIS, especially in their application in various fields management.
Significant progress in this direction was made 30 years ago by the prophet of the University of California (Berkeley) Lotfi A. Zadeh. His work “Fuzzy Sets,” which appeared in 1965 in the journal Information and Control, No. 8, laid the foundations for modeling human intellectual activity and was the initial impetus for the development of a new mathematical theory.
What did Zadeh propose? Firstly, he expanded the classical Cantor concept of a set, admitting that the characteristic function (the function of membership of an element in a set) can take any values in the interval (0,1)), and not as in classical theory only values 0 or 1. Such sets were called fuzzy.
He also defined operations on fuzzy sets and generalizations of known methods of logical inference are proposed.
Let's consider some basic principles of the theory of fuzzy sets.
Let E be a universal set, X - element E, A TO- some property. Regular (crisp) subset A universal set E, whose elements satisfy the property R, is defined as the set of ordered pairs, where - characteristic function, taking the value 1 , If X satisfies the property R, And 0 - otherwise.
A fuzzy subset differs from a regular subset in that for elements X from E there is no clear answer "Not really" regarding property R. In this regard, the fuzzy subset A universal set E is defined as the set of ordered pairs, where - characteristic membership function(or simply a membership function) taking values in some well-ordered set M(for example, M = ). The membership function indicates the degree (or level) of membership of an element X subset A. A bunch of M called many accessories. If M = (0.1), then the fuzzy subset A can be considered as an ordinary or crisp set.
Let M = And A- fuzzy set with elements from the universal set E and many accessories M.
The quantity is called height fuzzy set A. Fuzzy set Is it okay, if its height is 1 , i.e. the upper bound of its membership function is equal to 1 ( =1 ). At< 1 нечеткое множество называется субнормальным.
Fuzzy set empty, If 
A non-empty subnormal set can be normalized by the formula 
The above examples used straight methods when the expert either simply sets the value for each or defines a compatibility function. Typically, direct methods for specifying the membership function are used for measurable concepts such as speed, time, distance, pressure, temperature, etc., or when polar values are extracted.
Indirect methods for determining the values of the membership function are used in cases where there are no elementary measurable properties through which the fuzzy set of interest to us is determined. Typically, these are pairwise comparison methods. If the values of the membership functions were known to us, for example, then pairwise comparisons can be represented by a matrix of relations , Where(division operation).
In practice, the expert himself forms the matrix A, it is assumed that diagonal elements are equal to 1, and for elements symmetrical about the diagonal, = 1/ , i.e. if one element is rated a times higher than another, then this latter should be 1/ times stronger. In the general case, the problem comes down to finding a vector that satisfies an equation of the form 
, where is the greatest eigenvalue matrices A.
The introduction of the concept of a linguistic variable, and the assumption that fuzzy sets act as its values (terms), actually makes it possible to create an apparatus for describing the processes of intellectual activity, including the fuzziness and uncertainty of expressions.
Since the matrix A positive definite by construction, the solution to this problem exists for accepted value() and is positive. C(T), where C(T) is the set of generated terms, is called the extended term set of a linguistic variable;
M is a semantic procedure that allows you to transform each new value of a linguistic variable generated by procedure C into a fuzzy variable, i.e., to form a corresponding fuzzy set.
By introducing the concept of a linguistic variable and assuming that its values (terms) are fuzzy sets, it actually makes it possible to create an apparatus for describing the processes of intellectual activity, including the fuzziness and uncertainty of expressions.
A fuzzy set is a set of pairs
So, for example, you can set a set for the number 7:
<0/1>,<0.4/3>,<1/7>This set says that 7 is 0% one, 40% three and 100% seven.
Fuzzy variable is defined as
A - variable name,
X=(x) - domain of definition of a variable, set of possible values x,
Ca=(
Example:<"Семь",{1,3,7},{<0/1>,<0.4/3>,<1/7>)>. With this entry, we determined the correspondence between the word and some numbers. Moreover, both in the name of the variable and in the x values, any records that carry any information could be used.
The linguistic variable is defined as
B - variable name.T is the set of its values (basic term set), consists of the names of fuzzy variables, the domain of definition of each of which is the set X.
G is a syntactic procedure (grammar) that allows you to operate with elements of the term set T, in particular, to generate new meaningful terms. T`=T U G(T) specifies an extended term set (U is a union sign).
M is a semantic procedure that allows you to assign fuzzy semantics to each new value of a linguistic variable by forming a new fuzzy set.
A fuzzy set (or a fuzzy number) describes some concepts in a functional form, i.e. concepts such as “approximately equal to 5”, “speed slightly more than 300 km/h”, etc., as you can see, these concepts cannot be represented by one in number, although in reality people use them very often.
A fuzzy variable is the same as a fuzzy number, only with the addition of a name that formalizes the concept described by this number.
A linguistic variable is a set of fuzzy variables, it is used to give verbal description some fuzzy number obtained as a result of some operations. That is, through some operations, the closest value from the linguistic variable is selected.
I want to give some advice for your program. It is better to store fuzzy numbers as a sorted set of pairs (sorted by media), due to this you can speed up the execution of all logical and mathematical operations. When you implement arithmetic operations, you need to take into account the calculation error, i.e. 2/4<>1/2 for a computer, when I encountered this, I had to make it a little more difficult to compare pairs, and I have to do a lot of comparisons. The carriers in fuzzy numbers must be multiples of some number, otherwise the results are arif. operations will be “ugly”, i.e. the result will be inaccurate, this is especially evident during multiplication.
By storing fuzzy numbers in sorted form, I ensured that arithmetic operations were performed according to an almost linear dependence (in time), i.e., as the amount of steam increased, the speed of calculations decreased linearly. I came up with and implemented the exact arif. operations in which the number and multiplicity of carriers does not matter, the result will always be accurate and “beautiful”, i.e. if the original numbers were similar to an inverted parabola, then the result will be similar, but with ordinary operations it turns out to be stepwise. I also introduced the concept of “inverse fuzzy numbers” (although I did not fully implement them), what are they for? As you know, when subtracting or dividing, the number from which the other is subtracted must be wider, and this is a big problem when solving complex equations, but “inverse fuzzy numbers” allow you to do this.
Basic operations on fuzzy sets.
UNION: a new set is created from elements of the original sets, and for identical elements membership is taken to be maximum.
A U B = (
top edge
(maximum membership value present in the set).
NORMALIZATION: a fuzzy set is normal if the supremum of the set is equal to one. To normalize, the affiliations of the elements are reread:
M"a(x) = Ma(x)/(Sup Ma(x)) ALPHA CUT: alpha level set - those elements of the original set whose membership is higher than or equal to a given threshold. The threshold equal to 1/2 is called the transition point . Aq = (x|x?X /\ Ma(x)>q) FUZZY INCLUSION: degree of inclusion of a fuzzy set V(A1,A2) = (Ma1(x0)->Ma2(x0))&(Ma1(x1) ->Ma2(x1))&.. According to Lukasiewicz: Ma1(x)->Ma2(x) = 1&(1-Ma1(x)+Ma2(x)) According to Zade: Ma1(x)->Ma2(x ) = (1-Ma1(x)) \/ Ma2(x) FUZZY EQUALITY: degree of fuzzy equality R(A1,A2) = V(A1,A2) & V(A2,A1)
Dictionary ADAPTATION - Any change in the structure or function of an organism that allows it to survive in its external environment. ALLELS -
Possible values genes. GA -
Genetic algorithm
. Intelligent exploration of random search. . Holland 1975 introduced.
ISLAND MODEL GA (IMGA) - A GA population is divided into several subpopulations, each of which is randomly initialized and performs an independent sequential GA on its own subpopulation. Sometimes, viable decision branches migrate between subpopulations. [For example. Levine 1994].
GENES - Variables on a chromosome.
GENETIC DRIFT - Members of a population converge to some point in the solution space outside the optimum due to the accumulation of stochastic errors. GENOTYPE - Actual structure. Encoded chromosome. GP -
Genetic programming
. Application programs using the principles of evolutionary adaptation to the design of procedural code.
DIPLOID - Each region of the chromosome contains a pair of genes. This allows long-term memory to be maintained.
KGA - Compact GA (CGA). In CGA, two or more gene assemblies constantly interact and mutually evolve.
CROSSINGOVER - Exchange of segments of parents' chromosomes. In the range from 75 to 95% the best individuals appear.
LOCUS - The position of a gene on a chromosome.
MUTATION - Arbitrary modification of a chromosome.
CONVERGENCE - Progression towards increasing homogeneity. A gene is considered to converge when 95% of the population has the same value.
UNN - Unified Neural Network.
FITNESS FUNCTION - A value that is the target functional value of a solution. It is also called evaluation function or objective function in optimization problems.
PHENOTYPE - Physical Expression structures. Decoded gene set.
CHROMOSOME - A constituent vector, string, or solution.
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