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Sequences for system modelling
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At the end of this lecture you should be able to: provide a definition of a VDM sequence; identify situations in which a sequence is an appropriate data type; utilize and interpret sequence notation; make appropriate use of the VDM sequence operators; define a sequence by comprehension; write VDM specifications using the sequence type.
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A sequence is an ordered collection of objects in which repetitions are significant. A queue of jobs waiting for a printer
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A sequence is an ordered collection of objects in which repetitions are significant. A group of planes circling an airport
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Declaring sequences in VDM-SL
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To declare a variable to be of type sequence we place an asterisk after the name of the type contained within the sequence. seq : * convoy : SpaceCraft *
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Sequence Notation queue = [ michael, varinder, elizabeth, winston, judith ] s = [ a, d, f, a, d, d, c ] [ a, d, f ] [ a, f, d ] [ ]
![Sequence Notation queue = [ michael, varinder, elizabeth, winston, judith ] s = [ a, d, f, a, d, d, c ] [ a, d, f ] [ a, f, d ] [ ]](http://images.slideplayer.com./20/6225753/slides/slide_7.jpg "Sequence Notation queue = [ michael, varinder, elizabeth, winston, judith ] s = [ a, d, f, a, d, d, c ] [ a, d, f ] [ a, f, d ] [ ]")
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Retrieving items from the sequence s = [ a, d, f, a, d, d, c ] queue = [ michael, varinder, elizabeth, winston, judith ] s(3) = queue(4) = s(10) = f winston undefined
![Retrieving items from the sequence s = [ a, d, f, a, d, d, c ] queue = [ michael, varinder, elizabeth, winston, judith ] s(3) = queue(4) = s(10) = f winston undefined](http://images.slideplayer.com./20/6225753/slides/slide_8.jpg "Retrieving items from the sequence s = [ a, d, f, a, d, d, c ] queue = [ michael, varinder, elizabeth, winston, judith ] s(3) = queue(4) = s(10) = f winston undefined")
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Sequence operators len operator:Returns the length of a sequence s = [ a, d, f, a, d, d, c ] queue = [ michael, varinder, elizabeth, winston, judith ] len s = len queue = 7 5
![Sequence operators len operator:Returns the length of a sequence s = [ a, d, f, a, d, d, c ] queue = [ michael, varinder, elizabeth, winston, judith ] len s = len queue = 7 5](http://images.slideplayer.com./20/6225753/slides/slide_9.jpg "Sequence operators len operator:Returns the length of a sequence s = [ a, d, f, a, d, d, c ] queue = [ michael, varinder, elizabeth, winston, judith ] len s = len queue = 7 5")
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Sequence operators s = [ a, d, f, a, d, d, c ] queue = [ michael, varinder, elizabeth, winston, judith ] elems operator:Returns a set that contains all the members of the sequence elems s = elems queue = { a, d, f, c } {michael, varinder, elizabeth, winston, judith}
![Sequence operators s = [ a, d, f, a, d, d, c ] queue = [ michael, varinder, elizabeth, winston, judith ] elems operator:Returns a set that contains all the members of the sequence elems s = elems queue = { a, d, f, c } {michael, varinder, elizabeth, winston, judith}](http://images.slideplayer.com./20/6225753/slides/slide_10.jpg "Sequence operators s = [ a, d, f, a, d, d, c ] queue = [ michael, varinder, elizabeth, winston, judith ] elems operator:Returns a set that contains all the members of the sequence elems s = elems queue = { a, d, f, c } {michael, varinder, elizabeth, winston, judith}")
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Sequence operators inds operator :Returns a set of all the indices of the sequence s = [ a, d, f, a, d, d, c ] queue = [ michael, varinder, elizabeth, winston, judith ] inds s = inds queue = {1, 2, 3, 4, 5, 6, 7 } {1, 2, 3, 4, 5} inds [] ={ }
![Sequence operators inds operator :Returns a set of all the indices of the sequence s = [ a, d, f, a, d, d, c ] queue = [ michael, varinder, elizabeth, winston, judith ] inds s = inds queue = {1, 2, 3, 4, 5, 6, 7 } {1, 2, 3, 4, 5} inds [] ={ }](http://images.slideplayer.com./20/6225753/slides/slide_11.jpg "Sequence operators inds operator :Returns a set of all the indices of the sequence s = [ a, d, f, a, d, d, c ] queue = [ michael, varinder, elizabeth, winston, judith ] inds s = inds queue = {1, 2, 3, 4, 5, 6, 7 } {1, 2, 3, 4, 5} inds [] ={ }")
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Sequence operators head (hd) operator :Returns the first element in the sequence s = [ a, d, f, a, d, d, c ] queue = [ michael, varinder, elizabeth, winston, judith ] hd s = hd queue = a michael hd [] =undefined
![Sequence operators head (hd) operator :Returns the first element in the sequence s = [ a, d, f, a, d, d, c ] queue = [ michael, varinder, elizabeth, winston, judith ] hd s = hd queue = a michael hd [] =undefined](http://images.slideplayer.com./20/6225753/slides/slide_12.jpg "Sequence operators head (hd) operator :Returns the first element in the sequence s = [ a, d, f, a, d, d, c ] queue = [ michael, varinder, elizabeth, winston, judith ] hd s = hd queue = a michael hd [] =undefined")
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Sequence operators tail (tl) operator :Returns a sequence containing all but the first element s = [ a, d, f, a, d, d, c ] queue = [ michael, varinder, elizabeth, winston, judith ] tl s = tl queue = [d, f, a, d, d, c ] [varinder, elizabeth, winston, judith ] tl [] =undefinedtl [a] =[ ]
![Sequence operators tail (tl) operator :Returns a sequence containing all but the first element s = [ a, d, f, a, d, d, c ] queue = [ michael, varinder, elizabeth, winston, judith ] tl s = tl queue = [d, f, a, d, d, c ] [varinder, elizabeth, winston, judith ] tl [] =undefinedtl [a] =[ ]](http://images.slideplayer.com./20/6225753/slides/slide_13.jpg "Sequence operators tail (tl) operator :Returns a sequence containing all but the first element s = [ a, d, f, a, d, d, c ] queue = [ michael, varinder, elizabeth, winston, judith ] tl s = tl queue = [d, f, a, d, d, c ] [varinder, elizabeth, winston, judith ] tl [] =undefinedtl [a] =[ ]")
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Sequence operators concatenation operator ( ^ ) operator: operates on two sequences, and returns a sequence that consists of the two sequences joined together first = [ w, e, r, w ]second = [ t, w, q ] first ^ second =[ w, e, r, w, t, w, q ] second ^ first =[t, w, q, w, e, r, w ] first ^ [ ] =[ w, e, r, w ]
![Sequence operators concatenation operator ( ^ ) operator: operates on two sequences, and returns a sequence that consists of the two sequences joined together first = [ w, e, r, w ]second = [ t, w, q ] first ^ second =[ w, e, r, w, t, w, q ] second ^ first =[t, w, q, w, e, r, w ] first ^ [ ] =[ w, e, r, w ]](http://images.slideplayer.com./20/6225753/slides/slide_14.jpg "Sequence operators concatenation operator ( ^ ) operator: operates on two sequences, and returns a sequence that consists of the two sequences joined together first = [ w, e, r, w ]second = [ t, w, q ] first ^ second =[ w, e, r, w, t, w, q ] second ^ first =[t, w, q, w, e, r, w ] first ^ [ ] =[ w, e, r, w ]")
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Sequence operators the override operator (†) Takes a sequence and gives us a new sequence with a particular element of the old sequence overridden by a new element [a, c, d, e] † {1 z} =[z, c, d, e] [a, c, d, e] † {2 x, 4 y} =[a, x, d, y] [a, c, d, e] † {7 g} =undefined
![Sequence operators the override operator (†) Takes a sequence and gives us a new sequence with a particular element of the old sequence overridden by a new element [a, c, d, e] † {1 z} =[z, c, d, e] [a, c, d, e] † {2 x, 4 y} =[a, x, d, y] [a, c, d, e] † {7 g} =undefined](http://images.slideplayer.com./20/6225753/slides/slide_15.jpg "Sequence operators the override operator (†) Takes a sequence and gives us a new sequence with a particular element of the old sequence overridden by a new element [a, c, d, e] † {1 z} =[z, c, d, e] [a, c, d, e] † {2 x, 4 y} =[a, x, d, y] [a, c, d, e] † {7 g} =undefined")
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Sequence operators subsequence operator allow us to extract a part of a sequence between two indices s = [ a, d, f, a, d, d, c ] s(2,..., 5) =[d, f, a, d] s(1,...,0) = [ ]s(8,..., 7) = [ ] s(2,..., 2) =[d] s(2,..., 13) =undefined
![Sequence operators subsequence operator allow us to extract a part of a sequence between two indices s = [ a, d, f, a, d, d, c ] s(2,..., 5) =[d, f, a, d] s(1,...,0) = [ ]s(8,..., 7) = [ ] s(2,..., 2) =[d] s(2,..., 13) =undefined](http://images.slideplayer.com./20/6225753/slides/slide_16.jpg "Sequence operators subsequence operator allow us to extract a part of a sequence between two indices s = [ a, d, f, a, d, d, c ] s(2,..., 5) =[d, f, a, d] s(1,...,0) = [ ]s(8,..., 7) = [ ] s(2,..., 2) =[d] s(2,..., 13) =undefined")
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Sequence comprehension [ expression(a) | a SomeSet test (a) ]
![Sequence comprehension [ expression(a) | a SomeSet test (a) ]](http://images.slideplayer.com./20/6225753/slides/slide_17.jpg "Sequence comprehension [ expression(a) | a SomeSet test (a) ]")
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[ expression(a) | a SomeSet test (a) ]
![[ expression(a) | a SomeSet test (a) ]](http://images.slideplayer.com./20/6225753/slides/slide_18.jpg "[ expression(a) | a SomeSet test (a) ]")
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Sequence comprehension [ expression(a) | a SomeSet test (a) ] [ a | a {1,…,10} is-odd(a)]
![Sequence comprehension [ expression(a) | a SomeSet test (a) ] [ a | a {1,…,10} is-odd(a)]](http://images.slideplayer.com./20/6225753/slides/slide_19.jpg "Sequence comprehension [ expression(a) | a SomeSet test (a) ] [ a | a {1,…,10} is-odd(a)]")
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Sequence comprehension [ expression(a) | a SomeSet test (a) ] [ a | a {1,…,10} is-odd(a)] [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
![Sequence comprehension [ expression(a) | a SomeSet test (a) ] [ a | a {1,…,10} is-odd(a)] [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]](http://images.slideplayer.com./20/6225753/slides/slide_20.jpg "Sequence comprehension [ expression(a) | a SomeSet test (a) ] [ a | a {1,…,10} is-odd(a)] [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]")
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Sequence comprehension [ expression(a) | a SomeSet test (a) ] [ a | a {1,…,10} is-odd(a)] [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
![Sequence comprehension [ expression(a) | a SomeSet test (a) ] [ a | a {1,…,10} is-odd(a)] [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]](http://images.slideplayer.com./20/6225753/slides/slide_21.jpg "Sequence comprehension [ expression(a) | a SomeSet test (a) ] [ a | a {1,…,10} is-odd(a)] [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]")
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Sequence comprehension [ expression(a) | a SomeSet test (a) ] [ a | a {1,…,10} is-odd(a)] [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
![Sequence comprehension [ expression(a) | a SomeSet test (a) ] [ a | a {1,…,10} is-odd(a)] [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]](http://images.slideplayer.com./20/6225753/slides/slide_22.jpg "Sequence comprehension [ expression(a) | a SomeSet test (a) ] [ a | a {1,…,10} is-odd(a)] [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]")
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Sequence comprehension [ expression(a) | a SomeSet test (a) ] s1 = [2, 3, 4, 7, 9, 39, 6, 7, 8, 11, 45, 39, 3] Let’s filter this sequence so that we only have values greater than 10
![Sequence comprehension [ expression(a) | a SomeSet test (a) ] s1 = [2, 3, 4, 7, 9, 39, 6, 7, 8, 11, 45, 39, 3] Let’s filter this sequence so that we only have values greater than 10](http://images.slideplayer.com./20/6225753/slides/slide_23.jpg "Sequence comprehension [ expression(a) | a SomeSet test (a) ] s1 = [2, 3, 4, 7, 9, 39, 6, 7, 8, 11, 45, 39, 3] Let’s filter this sequence so that we only have values greater than 10")
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Sequence comprehension [ expression(a) | a SomeSet test (a) ] s1 = [2, 3, 4, 7, 9, 39, 6, 7, 8, 11, 45, 39, 3] [39, 11, 45, 39]
![Sequence comprehension [ expression(a) | a SomeSet test (a) ] s1 = [2, 3, 4, 7, 9, 39, 6, 7, 8, 11, 45, 39, 3] [39, 11, 45, 39]](http://images.slideplayer.com./20/6225753/slides/slide_24.jpg "Sequence comprehension [ expression(a) | a SomeSet test (a) ] s1 = [2, 3, 4, 7, 9, 39, 6, 7, 8, 11, 45, 39, 3] [39, 11, 45, 39]")
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Sequence comprehension [ expression(a) | a SomeSet test (a) ] s1 = [2, 3, 4, 7, 9, 39, 6, 7, 8, 11, 45, 39, 3] s2 = [ s1(i) | i inds s1 s1(i) > 10 ]
![Sequence comprehension [ expression(a) | a SomeSet test (a) ] s1 = [2, 3, 4, 7, 9, 39, 6, 7, 8, 11, 45, 39, 3] s2 = [ s1(i) | i inds s1 s1(i) > 10 ]](http://images.slideplayer.com./20/6225753/slides/slide_25.jpg "Sequence comprehension [ expression(a) | a SomeSet test (a) ] s1 = [2, 3, 4, 7, 9, 39, 6, 7, 8, 11, 45, 39, 3] s2 = [ s1(i) | i inds s1 s1(i) > 10 ]")
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Sequence comprehension [ expression(a) | a SomeSet test (a) ] s1 = [2, 3, 4, 7, 9, 39, 6, 7, 8, 11, 45, 39, 3] s2 = [ | i elems s1 ] [ 2, 3, 4, 6, 7, 8, 9, 11, 39, 45]
![Sequence comprehension [ expression(a) | a SomeSet test (a) ] s1 = [2, 3, 4, 7, 9, 39, 6, 7, 8, 11, 45, 39, 3] s2 = [ | i elems s1 ] [ 2, 3, 4, 6, 7, 8, 9, 11, 39, 45]](http://images.slideplayer.com./20/6225753/slides/slide_26.jpg "Sequence comprehension [ expression(a) | a SomeSet test (a) ] s1 = [2, 3, 4, 7, 9, 39, 6, 7, 8, 11, 45, 39, 3] s2 = [ | i elems s1 ] [ 2, 3, 4, 6, 7, 8, 9, 11, 39, 45]")
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Sequence comprehension [ expression(a) | a SomeSet test (a) ] s1 = [2, 3, 4, 7, 9, 39, 6, 7, 8, 11, 45, 39, 3] s2 = [ | i inds s1 ] [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]
![Sequence comprehension [ expression(a) | a SomeSet test (a) ] s1 = [2, 3, 4, 7, 9, 39, 6, 7, 8, 11, 45, 39, 3] s2 = [ | i inds s1 ] [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]](http://images.slideplayer.com./20/6225753/slides/slide_27.jpg "Sequence comprehension [ expression(a) | a SomeSet test (a) ] s1 = [2, 3, 4, 7, 9, 39, 6, 7, 8, 11, 45, 39, 3] s2 = [ | i inds s1 ] [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]")
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Sequence comprehension [ expression(a) | a SomeSet test (a) ] s1 = [2, 3, 4, 7, 9, 39, 6, 7, 8, 11, 45, 39, 3] s2 = [ | i inds s1 s1(i) > 10 ] [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]
![Sequence comprehension [ expression(a) | a SomeSet test (a) ] s1 = [2, 3, 4, 7, 9, 39, 6, 7, 8, 11, 45, 39, 3] s2 = [ | i inds s1 s1(i) > 10 ] [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]](http://images.slideplayer.com./20/6225753/slides/slide_28.jpg "Sequence comprehension [ expression(a) | a SomeSet test (a) ] s1 = [2, 3, 4, 7, 9, 39, 6, 7, 8, 11, 45, 39, 3] s2 = [ | i inds s1 s1(i) > 10 ] [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]")
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Sequence comprehension [ expression(a) | a SomeSet test (a) ] s1 = [2, 3, 4, 7, 9, 39, 6, 7, 8, 11, 45, 39, 3] s2 = [ s1(i) | i inds s1 s1(i) > 10 ] [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14]
![Sequence comprehension [ expression(a) | a SomeSet test (a) ] s1 = [2, 3, 4, 7, 9, 39, 6, 7, 8, 11, 45, 39, 3] s2 = [ s1(i) | i inds s1 s1(i) > 10 ] [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14]](http://images.slideplayer.com./20/6225753/slides/slide_29.jpg "Sequence comprehension [ expression(a) | a SomeSet test (a) ] s1 = [2, 3, 4, 7, 9, 39, 6, 7, 8, 11, 45, 39, 3] s2 = [ s1(i) | i inds s1 s1(i) > 10 ] [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14]")
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Sequence comprehension [ expression(a) | a SomeSet test (a) ] s1 = [2, 3, 4, 7, 9, 39, 6, 7, 8, 11, 45, 39, 3] [1, 2, 3, 4, 5, 39, 7, 8, 9, 10, 11, 12, 14] s2 = [ s1(i) | i inds s1 s1(i) > 10 ]
![Sequence comprehension [ expression(a) | a SomeSet test (a) ] s1 = [2, 3, 4, 7, 9, 39, 6, 7, 8, 11, 45, 39, 3] [1, 2, 3, 4, 5, 39, 7, 8, 9, 10, 11, 12, 14] s2 = [ s1(i) | i inds s1 s1(i) > 10 ]](http://images.slideplayer.com./20/6225753/slides/slide_30.jpg "Sequence comprehension [ expression(a) | a SomeSet test (a) ] s1 = [2, 3, 4, 7, 9, 39, 6, 7, 8, 11, 45, 39, 3] [1, 2, 3, 4, 5, 39, 7, 8, 9, 10, 11, 12, 14] s2 = [ s1(i) | i inds s1 s1(i) > 10 ]")
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Sequence comprehension [ expression(a) | a SomeSet test (a) ] s1 = [2, 3, 4, 7, 9, 39, 6, 7, 8, 11, 45, 39, 3] [1, 2, 3, 4, 5, 39, 7, 8, 9, 11, 11, 12, 14] s2 = [ s1(i) | i inds s1 s1(i) > 10 ]
![Sequence comprehension [ expression(a) | a SomeSet test (a) ] s1 = [2, 3, 4, 7, 9, 39, 6, 7, 8, 11, 45, 39, 3] [1, 2, 3, 4, 5, 39, 7, 8, 9, 11, 11, 12, 14] s2 = [ s1(i) | i inds s1 s1(i) > 10 ]](http://images.slideplayer.com./20/6225753/slides/slide_31.jpg "Sequence comprehension [ expression(a) | a SomeSet test (a) ] s1 = [2, 3, 4, 7, 9, 39, 6, 7, 8, 11, 45, 39, 3] [1, 2, 3, 4, 5, 39, 7, 8, 9, 11, 11, 12, 14] s2 = [ s1(i) | i inds s1 s1(i) > 10 ]")
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Sequence comprehension [ expression(a) | a SomeSet test (a) ] s1 = [2, 3, 4, 7, 9, 39, 6, 7, 8, 11, 45, 39, 3] [1, 2, 3, 4, 5, 39, 7, 8, 9, 11, 45, 12, 14] s2 = [ s1(i) | i inds s1 s1(i) > 10 ]
![Sequence comprehension [ expression(a) | a SomeSet test (a) ] s1 = [2, 3, 4, 7, 9, 39, 6, 7, 8, 11, 45, 39, 3] [1, 2, 3, 4, 5, 39, 7, 8, 9, 11, 45, 12, 14] s2 = [ s1(i) | i inds s1 s1(i) > 10 ]](http://images.slideplayer.com./20/6225753/slides/slide_32.jpg "Sequence comprehension [ expression(a) | a SomeSet test (a) ] s1 = [2, 3, 4, 7, 9, 39, 6, 7, 8, 11, 45, 39, 3] [1, 2, 3, 4, 5, 39, 7, 8, 9, 11, 45, 12, 14] s2 = [ s1(i) | i inds s1 s1(i) > 10 ]")
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Sequence comprehension [ expression(a) | a SomeSet test (a) ] s1 = [2, 3, 4, 7, 9, 39, 6, 7, 8, 11, 45, 39, 3] [1, 2, 3, 4, 5, 39, 7, 8, 9, 11, 45, 39, 14] s2 = [ s1(i) | i inds s1 s1(i) > 10 ]
![Sequence comprehension [ expression(a) | a SomeSet test (a) ] s1 = [2, 3, 4, 7, 9, 39, 6, 7, 8, 11, 45, 39, 3] [1, 2, 3, 4, 5, 39, 7, 8, 9, 11, 45, 39, 14] s2 = [ s1(i) | i inds s1 s1(i) > 10 ]](http://images.slideplayer.com./20/6225753/slides/slide_33.jpg "Sequence comprehension [ expression(a) | a SomeSet test (a) ] s1 = [2, 3, 4, 7, 9, 39, 6, 7, 8, 11, 45, 39, 3] [1, 2, 3, 4, 5, 39, 7, 8, 9, 11, 45, 39, 14] s2 = [ s1(i) | i inds s1 s1(i) > 10 ]")
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Specifying a stack
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Stack stack : Element [*] push(Element) pop() : Element isEmpty(): Boolean
![Stack stack : Element [*] push(Element) pop() : Element isEmpty(): Boolean](http://images.slideplayer.com./20/6225753/slides/slide_35.jpg "Stack stack : Element [*] push(Element) pop() : Element isEmpty(): Boolean")
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Stack stack : Element [*] push(Element) pop() : Element isEmpty(): Boolean
![Stack stack : Element [*] push(Element) pop() : Element isEmpty(): Boolean](http://images.slideplayer.com./20/6225753/slides/slide_36.jpg "Stack stack : Element [*] push(Element) pop() : Element isEmpty(): Boolean")
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types Element = TOKEN state Stack of stack : init mk-Stack(s) end Element * s = [ ]
![types Element = TOKEN state Stack of stack : init mk-Stack(s) end Element * s = [ ]](http://images.slideplayer.com./20/6225753/slides/slide_37.jpg "types Element = TOKEN state Stack of stack : init mk-Stack(s) end Element * s = [ ]")
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Stack stack : Element [*] push(Element) pop() : Element isEmpty(): Boolean
![Stack stack : Element [*] push(Element) pop() : Element isEmpty(): Boolean](http://images.slideplayer.com./20/6225753/slides/slide_38.jpg "Stack stack : Element [*] push(Element) pop() : Element isEmpty(): Boolean")
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push( ) ext pre post itemIn : Element stack : Element* wr stack = [itemIn] ^ stack TRUE
![push( ) ext pre post itemIn : Element stack : Element* wr stack = [itemIn] ^ stack TRUE](http://images.slideplayer.com./20/6225753/slides/slide_39.jpg "push( ) ext pre post itemIn : Element stack : Element* wr stack = [itemIn] ^ stack TRUE")
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Stack stack : Element [*] push(Element) pop() : Element isEmpty(): Boolean
![Stack stack : Element [*] push(Element) pop() : Element isEmpty(): Boolean](http://images.slideplayer.com./20/6225753/slides/slide_40.jpg "Stack stack : Element [*] push(Element) pop() : Element isEmpty(): Boolean")
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pop( ) ext pre post itemRemoved : Element stack : Element* wr stack = tl stack stack [ ] itemRemoved = hd stack
![pop( ) ext pre post itemRemoved : Element stack : Element* wr stack = tl stack stack [ ] itemRemoved = hd stack ](http://images.slideplayer.com./20/6225753/slides/slide_41.jpg "pop( ) ext pre post itemRemoved : Element stack : Element* wr stack = tl stack stack [ ] itemRemoved = hd stack ")
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Stack stack : Element [*] push(Element) pop() : Element isEmpty(): Boolean
![Stack stack : Element [*] push(Element) pop() : Element isEmpty(): Boolean](http://images.slideplayer.com./20/6225753/slides/slide_42.jpg "Stack stack : Element [*] push(Element) pop() : Element isEmpty(): Boolean")
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isEmpty( ) ext pre post query : stack : Element* rd TRUE querystack = [ ]
![isEmpty( ) ext pre post query : stack : Element* rd TRUE querystack = [ ]](http://images.slideplayer.com./20/6225753/slides/slide_43.jpg "isEmpty( ) ext pre post query : stack : Element* rd TRUE querystack = [ ]")
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Re-thinking the Airport system
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Airport2 permission: Aircraft [*] landed: Aircraft [*] circling: Aircraft [*] givePermission(Aircraft) recordLanding( ) recordTakeOff(Aircraft) getPermission( ): Aircraft [*] getLanded( ): Aircraft [*] numberWaiting(): Integer getCircling( ): Aircraft [*] allowToCircle (Aircraft)
![Airport2 permission: Aircraft [*] landed: Aircraft [*] circling: Aircraft [*] givePermission(Aircraft) recordLanding( ) recordTakeOff(Aircraft) getPermission( ): Aircraft [*] getLanded( ): Aircraft [*] numberWaiting(): Integer getCircling( ): Aircraft [*] allowToCircle (Aircraft)](http://images.slideplayer.com./20/6225753/slides/slide_45.jpg "Airport2 permission: Aircraft [*] landed: Aircraft [*] circling: Aircraft [*] givePermission(Aircraft) recordLanding( ) recordTakeOff(Aircraft) getPermission( ): Aircraft [*] getLanded( ): Aircraft [*] numberWaiting(): Integer getCircling( ): Aircraft [*] allowToCircle (Aircraft)")
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types state Airport2 of init mk-Airport2 ( ) end Aircraft = TOKEN permission: Aircraft-set landed: Aircraft-set circling: Aircraft* p, l, cp = { } l = { } c = [ ] inv mk-Airport2(p,l,c) ?
![types state Airport2 of init mk-Airport2 ( ) end Aircraft = TOKEN permission: Aircraft-set landed: Aircraft-set circling: Aircraft* p, l, cp = { } l = { } c = [ ] inv mk-Airport2(p,l,c) ](http://images.slideplayer.com./20/6225753/slides/slide_46.jpg "types state Airport2 of init mk-Airport2 ( ) end Aircraft = TOKEN permission: Aircraft-set landed: Aircraft-set circling: Aircraft* p, l, cp = { } l = { } c = [ ] inv mk-Airport2(p,l,c) ")
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The new invariant inv mk-Airport2(p,l,c) 1.Landed planes must have permission 2.Circling planes must have permission 3.Circling planes can not be landed 4.All circling planes are unique l p elems c p elems c l = { } isUnique(c) isUnique(seqIn : Aircraft*) query : pre true post query len seqIn = card elems seqIn
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The new invariant inv mk-Airport2(p,l,c) 1.Landed planes must have permission 2.Circling planes must have permission 3.Circling planes can not be landed 4.All circling planes are unique l p elems c p elems c l = { } isUnique(c) isUnique(seqIn : Aircraft*) query : pre true post query i1,i2 inds seqIn i1 i2 seqIn(i1) seqIn(i2)
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i1,i2 inds seqIn i1 i2 seqIn(i1) seqIn(i2) 12345
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Airport2 permission: Aircraft [*] landed: Aircraft [*] circling: Aircraft [*] givePermission(Aircraft) recordLanding( ) recordTakeOff(Aircraft) getPermission( ): Aircraft [*] getLanded( ): Aircraft [*] numberWaiting(): Integer getCircling( ): Aircraft [*] allowToCircle (Aircraft)
![Airport2 permission: Aircraft [*] landed: Aircraft [*] circling: Aircraft [*] givePermission(Aircraft) recordLanding( ) recordTakeOff(Aircraft) getPermission( ): Aircraft [*] getLanded( ): Aircraft [*] numberWaiting(): Integer getCircling( ): Aircraft [*] allowToCircle (Aircraft)](http://images.slideplayer.com./20/6225753/slides/slide_50.jpg "Airport2 permission: Aircraft [*] landed: Aircraft [*] circling: Aircraft [*] givePermission(Aircraft) recordLanding( ) recordTakeOff(Aircraft) getPermission( ): Aircraft [*] getLanded( ): Aircraft [*] numberWaiting(): Integer getCircling( ): Aircraft [*] allowToCircle (Aircraft)")
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allowToCircle ( ) ext pre post craftIn : Aircraft circling : Aircraft*wr permission : Aircraft-set rd landed : Aircraft-set rd circling = circling ^ [craftIn] craftIn permission craftIn elems circling craftIn landed
![allowToCircle ( ) ext pre post craftIn : Aircraft circling : Aircraft*wr permission : Aircraft-set rd landed : Aircraft-set rd circling = circling ^ [craftIn] craftIn permission craftIn elems circling craftIn landed](http://images.slideplayer.com./20/6225753/slides/slide_51.jpg "allowToCircle ( ) ext pre post craftIn : Aircraft circling : Aircraft*wr permission : Aircraft-set rd landed : Aircraft-set rd circling = circling ^ [craftIn] craftIn permission craftIn elems circling craftIn landed")
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Airport2 permission: Aircraft [*] landed: Aircraft [*] circling: Aircraft [*] givePermission(Aircraft) recordLanding( ) recordTakeOff(Aircraft) getPermission( ): Aircraft [*] getLanded( ): Aircraft [*] numberWaiting(): Integer getCircling( ): Aircraft [*] allowToCircle (Aircraft)
![Airport2 permission: Aircraft [*] landed: Aircraft [*] circling: Aircraft [*] givePermission(Aircraft) recordLanding( ) recordTakeOff(Aircraft) getPermission( ): Aircraft [*] getLanded( ): Aircraft [*] numberWaiting(): Integer getCircling( ): Aircraft [*] allowToCircle (Aircraft)](http://images.slideplayer.com./20/6225753/slides/slide_52.jpg "Airport2 permission: Aircraft [*] landed: Aircraft [*] circling: Aircraft [*] givePermission(Aircraft) recordLanding( ) recordTakeOff(Aircraft) getPermission( ): Aircraft [*] getLanded( ): Aircraft [*] numberWaiting(): Integer getCircling( ): Aircraft [*] allowToCircle (Aircraft)")
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recordLanding( ) ext pre post circling : Aircraft* landed : Aircraft-set wr circling = tl circling landed = landed {hd circling } circling [ ]
![recordLanding( ) ext pre post circling : Aircraft* landed : Aircraft-set wr circling = tl circling landed = landed {hd circling } circling [ ]](http://images.slideplayer.com./20/6225753/slides/slide_53.jpg "recordLanding( ) ext pre post circling : Aircraft* landed : Aircraft-set wr circling = tl circling landed = landed {hd circling } circling [ ]")
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See you next week!
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