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Formal Methods in Software Engineering

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Formal Methods in Software Engineering Machines can be required to maintain information about relationships on the information contained within them. Relation between sets If S and T are two sets, then their Cartesian product SxT is the set of all pairs (s, t) of elements from S and T respectively. This is defined as follows: s↦t pronounced s maps to t It exactly means the same as (s,t). it is a pair whose first element is s and whose second element is t. A relation R between sets S and T expresses a relationship between elements in S and elements in T. It is captured as a set of pairs (s, t) with s∈ S and t ∈ T representing those elements which are related. PHOTOGRAPHER={anna, bob, chris, dave, elizabeth, francis} and CAMERA={canon, kodak ,hasselblad ,minolta, olympus ,pentax} The relation owns detailing the ownership relation between photographers and their cameras can be defined as follows Owns={(anna, canon), (bob, canon), (bob, kodak), (chris, hasselblad), (chris, kodak), (chris, pentax), (dave, pentax), (elizabeth, pentax),(elizabeth, minolta)} Or using the notation map Owns={(anna ↦ canon), (bob ↦ canon), (bob ↦ kodak), (chris ↦ hasselblad), (chris ↦ kodak), (chris ↦ pentax), (dave ↦ pentax), (elizabeth ↦ pentax),(elizabeth ↦ minolta)} A relation between S and T is a subset of SxT or an element of P(SxT). The notation S⟷T denotes the set of relations between S and T. Hence S↦T is an abbreviation for the set of all substes of SxT S⟷T=P(SxT) To specify that R is a relation between S and T we write R∈S⟷T. In this case set S is said to be the source of R and the set T is said to be the target. For example owns ∈ PHOTOGRAPHER⟷CAMERA The statement that (dave, pentax) ∈ owns states that dave is related to pentax in the relation owns. dave↦pentax also states that dave is related to pentax. If owns records the cameras that are owned by the photographers, then this corresponds to the information that dave owns a pentax camera. When producing specifications, we are generally interested in extracting information from the relations our machines maintain, To answer particular questions such as: Does Bob own a camera? How many cameras does Elizabeth own? Who owns a Hasselblad camera? Which cameras are used by both Chris and Dave? How many cameras does Anna own? Is there anyone that does not have a camera? There are constructs for extracting the information required for questions to be expressed in relations. Extracting the domain and range of a relation Domain restriction: restricting the relation to a particular domain of interest Range restriction: restricting the relation to a particular range of interest Identifying the elements related to some item in a given set. Relational inverse the relation the opposite way round The domain of a relation R∈ S ↔T is the set of elements of S that R relates to something in T. Items in the source set that are related to something, it is written dom(R) Dom(owns)= {anna,bob, chris, dave, elizabeth} The range is the set of elements of T that are related to some element of S. it is written ran(R) For example Ran(owns)={canon, kodak, hasselblad, minolta, pentax} A relation R∈ S↔ T can be projected onto a particular domain U⊆S. Only the relationship for items in U is of interest. The result is those pairs in R whose first element is in U. It is written U⊲ R. This is domain restriction. U⊲ R={S ↦ t | s ↦ t ∈ R ∧ S ∈ U} For example To consider only the cameras that chris owns, the relation owns can be restricted to the set {chris}. A complementary restriction removes all pairs whose first element is in U. This is written U ⩤ R Domain anti restriction U ⩤ R= { S↦ t | t ∈R ∧ s ∉ U} The target set can also be restricted. The range restriction R⊳ V gives all pairs in R whose second element is in V⊆ T. For example, to consider only those photographers who have a kodak camera the relation owns can be range restricted to {kodak} Range anti restriction R⩥V gives all pairs apart from those whose second element is in V To identify all the elements in the target T that are related to some elements U of the source. If 푈 ⊆ 푆, then the set of elements in T related to U is called the relational image of U. It is written as R[U]. It is same as R[U]=ran(U⊲ R) R[U]={t | s↦t∈ 푅 ∧ 푠 ∈ 푈} The relational image of {chris, elizabeth} through owns; This is written owns{chris, elizabeth} R-1 it is the type T⟷S R-1={t↦s | s↦t ∈ 푅} The inverse of the owns relation could be called by ownedby It is concerned with the conjunction of relations. If s is related to t, and t is related to u, then it is often useful to put these two relations together and observe that s is related to u. If s↦t ∈ 푅0 푎푛푑 푡 ↦ 푢 ∈ 푅1 , 푡ℎ푒푛 푠 ↦u ∈ R0;R1 A relation might have a same set S and T Identity relation Written as id(S) and is of type S⟷ S A pair (s,t) will be in the identity relation if and only if s=t. thus Id(S)={(s,t)| s∈ 푆 ∧ 푡 ∈ 푆 ∧ 푠 = 푡} The identity relation is reflexive. Furthermore, it is the smallest reflexive relation on S: a relation is reflexive if and only if id(S)⊆R A relation R∈ 푆 ⟷ 푆 is symmetric, if it relates s1 to s0 and s0 to s1. The order of two elements does not matter. Example Relation id(S) is symmetric Relation R is symmetric if R=R-1 A relation R∈ 푆 ⟷ 푆 is anti-symmetric, if it relates two distinct elements in at most one way. For s0≠ 푠1. 푓 푠0 ↦ 푠1 ∈ 푅, 푡ℎ푒푛푠1 ↦ 푠0 ∉ 푅 It happens when R⋂R-1-id(S)={} If a relation relates two elements at most one way. This is equivalent to the property R⋂R-1={}. It follows that R cannot intersect at all with id(S): that R⋂id(S)={} Whenever s0↦s1 ∈ 푅 and s1 ↦s2 ∈ 푅 then s0 ↦s2 ∈ 푅 reflexive: For every element a in A, (a,a) is in R. symmetric: If an ordered pair (a,b) in R, (b,a) is in R. transitive: If ordered pairs (a,b) and (b,c) in R, (a,c) is in R. My name is ………………….. I remember ……………………… If a relation R: S⟷ S is reflexive, symmetric, and transitive, then it is called as equivalence relation. It means set S can be considered as the union of a collection of pairwise disjoint equivalence classes. Example If X is the set of all cars, and the equivalence relation is“ has the same color as", then one particular equivalence class consists of all green cars. For example, if A = {1, 2, 3}, we can define a relation R = {(1,2), (2,3), (3,1), (3,2)}. R = {(1,2), (2,3), (3,1), (3,2)}. For R to be reflexive it must contain (1,1), (2,2), and (3,3). Since R doesn't even contain (1,1), it is not reflexive. Here is an equivalence relation R2 on A: R2 = {(1,1), (1,3), (2,2), (3,1), (3,3)}. If a relation R: S⟷ S on elements of a set S can be composed with itself. The relation R;R is well defined. For example If parent∈ 푃퐸푅푆푂푁 ⟷ 푃퐸푅푆푂푁 then grandeparent= parent;parent grandgrandparent-=parent3 Using mathematical induction A transitive relation with itself can be written as If s0↦s1∈ 푅 and s1↦s2∈ R and s0 ↦ 푠2 ∈R. This is equivalent to the property R;R⊆R. For example the subset relation is transitive A reflexive relation R relates two elements of S, when there is some iterate of R including R0 which relates them. This is written as R* and it is defined as follows If R is symmetric then R* is an equivalence relation It excludes the 0th iteration of R and relates two elements of S, when there is iterate of R which relates them. Example succeeds∈ 푃퐸푅푆푂푁 ⟷ 푃퐸푅푆푂푁 The relation follows =succeeds+ holds between a person and any person anywhere behind him in the queue. in this example, a person is considered not to follow himself A relation R0 ∈ 푆 ⟷ 푇 푎푛푑 A relation 1R ∈ 푆 ⟷ 푇 Then the relational override R0 R1 Example Chris disposes off his camera collection and then buys an Olympus, the resulting relationship between photographers and their cameras will be given by This relation will be given by Functions are a particular kind of relations. For a relation to be a function, it must have one further property that it relates elements of S to no more than one element of T. If s is related to some element by the function f, then f(s) denotes the value of the function on s.(unique element that is related to s) It is defined only for those elements which are related to some value in T. Function from S to T is a relation which relates each element in S to at most one element in T. Some elements of S might not be related to anything, if they are related to some element then that element is unique. It generalizes the concept of a function ƒ: X → Y by not forcing f to map every element of X to an element of Y (only some subset X' of X The set of partial function is defined by S ⇸ T.
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