Software Engineering

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Software Engineering

• Refining requirements using Formal methods

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Review of discrete math

• Sets
• Finite e.g. 1,2,3
• Infinite e.g. ?
• Bags
• Sequences e.g. lt 1, 3, 3, 1gt

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Review of discrete math

• Power set of a set, ?
• What is ?1,2,3?
• What is ??
• Consider the following sets, well use in later
  examples
• A 1, 2, 3
• B 3, 1, 4, 7

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Review of discrete math

• Set operations
• x ? A
• x ? A
• A ? B
• A ? B
• A ? B

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Review of discrete math

• Existential and universal quantification
• ?
• ?

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Review of discrete math

• Binary Relations
• A binary relation on two sets, A and B, is a
  subset of A x B
• Examples lt,?
• Properties
• Reflexive
• Symmetric
• Antisymmetric
• Transitive

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Why formal methods

• Our specifications (even with the diagrams) tend
  to be imprecise
• Particularly inputs and outputs are often
  ill-defined
• Mathematics offers the promise of a precision
  that doesnt exist in English

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Introducing the formal method, Z

• Definition A set of conventions for presenting
  mathematical text, chosen to make it convenient
  to use simple mathematics to describe computing
  systems. (Jacky)
• 3 parts of Z are
• Types
• Axiomatic descriptions
• Schemas

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Z types

• Definition A type is an expression of a
  restricted kind (Spivey 24)
• Zs only built-in types integers (?) and natural
  numbers (?)
• Some of the ways to declare new types
• New types 
• PLANETS mercury venus earth mars
  jupiter saturn uranus neptune pluto
• Using other types
• SOLAR_SYSTEM PLANETS ? ASTERIOD_BELT ? SUN 

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Types, continued

• What makes up a zip code?
• How can the ZIP type be created
• as a new type?
• using other types?

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Axiomatic Descriptions

• Definition An axiomatic description
  introducesglobal variableswhich may or may not
  specify a constraint on their values. (Spivey
  48)
• Two ways to declare axiomatic descriptions
• With the predicate (want to know exactly what is
  in there)

lower_letters, upper_letters ? CHAR

lower_letters a, b, c, d,
e, f, g, h, i, j, k, l, m, n,
o,p,q, r, s, t, u, v, w,
x, y, z upper _letters A, B, C,
D, E, F, G, H, I, J, K, L, M,
N, O, P,Q, R, S, T, U, V,
W, X, Y, Z
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• Two ways to declare axiomatic descriptions
• Without a predicate (give the type of everything
  and let the reader assume from there)

student_table ID ? NAME ? ADDRESS ? MAJOR ?
SCHEDULE
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Axiomatic descriptions

• takes a seq of any type and returns the nth
  element
• elementAt_ ? ? seq X ? X
• perm(A) B says that seq B is a permutation of
  set A, where both A and B are of type X
• perm_ ? X ? seq X
• the following allow me to find out how many xs
  are in the sequence modified Spivey 124 I took
  def for bags and made it for seq
• __ seq X ? X ? ?
• ? s seq X ? (? n ? ? 0 ? n lt s 1 ? dom(s)
  n ? elementAt(n) ran(s))
• ? s1 ? X s2 seq X ? perm(s1) s2 ? s1 s2
  ?
• (? x X ? x ? s1 ? x ? s2 ? s1x
  s2x)
• literally taken from Spivey 124
• ? x X S seq X ? Sx count Sx
• // count Aa simply counts how many as are in A

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Schemas

• Definition A schema isused to describe some
  part of the state or an operation on the state
  (Barden)
• Schemas structure
• ???????NameOfSchema?????????
• ? VariableDeclarations
• ? Predicate
  .
• VariableDeclarations introduce local variables
  and provides the type of them
• Predicate that which is true in order for this
  state to occur

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Schemas adding a new student to a table

• ????????NewStudent?????????????????????????
• ?// new students info
• ? newStudentID? ID
• ? newStudentName? NAME
• ? newStudentAddress? ADDRESS
• ? newStudentMajor? MAJOR
• ?  
• ? // local variables
• ? // used to go through student table
• ? temp ID ? NAME ? ADDRESS ? MAJOR ? SCHEDULE
• ? // make sure ID isnt already in the table
• ?(? temp temp ? student_table ? newStudentID? ?
  dom(temp))
• ?  
• ? // the ID is unique add student
• ? student_table student_table ? (newStudentID?
  ? newStudentName? ?
• ? newStudentAddres? ? newStudentMajor)
• ??????????????????????????????????????????????????
  ???????

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How to get the Z font?

• Go to
• http//www.cs.kent.ac.uk/people/staff/rej/Zedfont/
  latest/
• And follow the directions.