Functional Specification Software Specification Lecture 41

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Functional Specification Software
SpecificationLecture 41

• Prepared by
• Stephen M. Thebaut, Ph.D.
• University of Florida

2
Overview

• Any program behavior can be represented entirely
  by a mathematical function in its effect on data.
• The domain of a program function corresponds to
  an initial data state that is transformed into a
  final data state by the program.
• Functional specification scales well and is used
  in Cleanroom Software Development.

Software Engineering, 6th Edition. Chapter 19
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Cleanroom Software Development

• Developed in the 70s and 80s by Harlan Mills,
  et al.
• The name is derived from the Cleanroom process
  in semiconductor fabrication.
• The philosophy is defect avoidance rather than
  defect removal.
• Emphasizes precise, logical expression and a
  systematic process for developing correct
  programs.

Software Engineering, 6th Edition. Chapter 19
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Cleanroom Software Development (contd)

• A software development process based on
• Incremental development (if appropriate)
• Formal specification
• Static verification using correctness arguments
• Statistical testing to certify program
  reliability
• NO defect testing!

Software Engineering, 6th Edition. Chapter 19
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The Cleanroom Process
Software Engineering, 6th Edition. Chapter 19
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Cleanroom Process Teams

• Specification team responsible for developing
  and maintaining the system specification.
• Development team responsible for developing and
  verifying the software. The software is NOT
  executed or even compiled during this process.
• Certification team responsible for developing a
  set of statistical tests to measure reliability
  after development.

Software Engineering, 6th Edition. Chapter 19
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Cleanroom Process Evaluation

• Results at IBM and elsewhere have been very
  impressive with very few discovered faults in
  delivered systems.
• Independent assessment shows that the
  (steady-state) process is no more expensive than
  other approaches.

Software Engineering, 6th Edition. Chapter 19
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What is a Function? (A Brief Tutorial)

• Sets and Relations
• Functions
• Conditional Rules
• Recursive Functions
• Lists
• Assignment Functions

Software Engineering, 6th Edition. Chapter 19
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Sets and Relations

• A set is any well-defined collection of objects,
  called members or elements.
• The relation of membership between a member, m,
  and a set, S, is written
• m ? S
• If m is not a member of S, we write
• m ? S

Software Engineering, 6th Edition. Chapter 19
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Sets and Relations (contd)

• A relation, r, is a set whose members (if any)
  are all ordered pairs.
• The set composed of the first member of each pair
  is called the domain of r and is denoted D(r).
  Members of D(r) are called arguments of r.
• The set composed of the second member of each
  pair is called the range of r and is denoted
  R(r). Members of R(r) are called values of r.

Software Engineering, 6th Edition. Chapter 19
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Functions

• A function, f, is a relation such that for each x
  ? D(f), there exists a unique element
• (x, y) ? f.
• (We often express this as y f(x), where y is
  the unique value corresponding to x in the
  function f.)
• It is the uniqueness of y that distinguishes a
  function from other relations.

Software Engineering, 6th Edition. Chapter 19
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Functions (contd)

• It is often convenient to define a function by
  giving its domain and a rule for calculating the
  corresponding value for each argument in the
  domain. For example
• f (x, y) x?0,1, y x 3x 2

Software Engineering, 6th Edition. Chapter 19
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Conditional Rules

• Conditional rules are a sequence of (predicate ?
  rule) pairs separated by vertical bars and
  enclosed in parentheses
• ( p1 ? r1 p2 ? r2 ... pk ? rk )
• Its meaning is evaluate predicates p1, p2, ...,
  pk in order for the first predicate, pi, which
  evaluates to true, if any, use the rule ri if no
  predicate evaluates to true, the rule is
  undefined. (Note that ? ? ?.)

Software Engineering, 6th Edition. Chapter 19
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Conditional Rules (contd)

• The conditional rule above is read if p1 then
  use r1 else if p2 then use r2 ... else if pk
  then use rk. For example
• f ((x, y) (x divisible by 2 ? y x/2
• x divisible by 3 ? y
  x/3
• true
  ? y x)
• Note that true ? r has the effect of if all
  else fails, use r.

Software Engineering, 6th Edition. Chapter 19
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Recursive Functions

• A recursive function is a function that is
  defined by using the function itself in the rule
  that defines it. For example
• oddeven(x) (x?0,1 ? x
• xgt1 ?
  oddeven(x-2)
• xlt0 ?
  oddeven(x2))
• Exercise 1 define the factorial function
  recursively.

Software Engineering, 6th Edition. Chapter 19
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Lists

• A list is a sequence of items which are all
  members of a single set, called an alphabet.
• Any computing process must eventually be
  represented by, and be described in terms of
  operations on, a list.
• The empty list, denoted by ?, is a sequence of no
  items.
• The fundamental relationship in lists is between
  members of the alphabet and a list.

Software Engineering, 6th Edition. Chapter 19
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List Operations and Semantics

• The first item, say a, of a non-empty list, say
  L, is written
• a head(L), L ? ?
• A non-empty list L with its first member removed
  is written
• tail(L)
• Note that tail(L) may be the empty list, and that
  a ? (a).

Software Engineering, 6th Edition. Chapter 19
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List Operations and Semantics (contd)

• Two fundamental operations in lists are (1)
  adding a new item, a, to the head of a list L,
  written
• a L
• and (2) concatenating two lists L and M, written
• LM

Software Engineering, 6th Edition. Chapter 19
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Assignment Functions

• Initial and final state space conditions may be
  explicitly represented using assignment
  functions.
• For example, in a program with data space x, y,
  z, the assignment statement x ?? y corresponds to
  a set of ordered pairs of the form
• ((x, y, z), (y, y, z))
• The assignment function representing a program
  consisting of this statement is
• x, y, z ?? y, y, z

final variable values
initial variable values
Software Engineering, 6th Edition. Chapter 19
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Assignment Functions (contd)

• Likewise, the function
• f (x?0 y?0 ? x, y ?? xy, 0)
• specifies a program for which the final value of
  x is the sum of the initial values of x and y and
  the final value of y is 0 if x and y are both
  initially ? 0 otherwise the program does not
  terminate (since f is not defined in this case).
• Can you design a program, P, such that P f ?

Software Engineering, 6th Edition. Chapter 19
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Exercise 2

• For each of the following, give appropriate
  assignment functions for the program behavior
  described.
• Set variable MAX to the maximum value of two
  integers, A and B.
• Set variable MIN to the minimum value in the
  unsorted, non-empty array A1N.
• Set variable SUM to the sum of the elements in
  array A1N.

Software Engineering, 6th Edition. Chapter 19
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Exercise 2 (contd)

• Given three arrays A1N, B1N, and C1N,
  set each element of A equal to the sum of the
  corresponding elements of B and C.
• Set variable NPRIME to true if N is prime and to
  false otherwise.
• Set variable Y to the greatest common divisor of
  integers A and B.

Software Engineering, 6th Edition. Chapter 19
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Exercise 2 (contd)

• Set variable R to the remainder of dividing A by
  D.
• Set variable I to the index of the first instance
  of Y in the array A1N.
• Perform integer subtraction using the arithmetic
  primitive "subtract 1" and a while loop. Let M
  be the minuend, S be the subtrahend, and D be the
  difference. Assume that the subtrahend is
  nonnegative.

Software Engineering, 6th Edition. Chapter 19
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Sample Solutions

• Set variable MAX to the maximum value of two
  integers, A and B.
• (AgtB ? A, B, MAX ?? A, B, A
• true ? A, B, MAX ?? A, B,
  B)

Software Engineering, 6th Edition. Chapter 19
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Sample Solutions (contd)

• Set variable MIN to the minimum value in the
  unsorted, non-empty array A1N.
• (MIN, A ?? FIND_MIN(A), A)
• where
• FIND_MIN(A) (tail(A)? ? head(A)
• true ? LESS(head(A),
  FIND_MIN(tail(A))))
• where
• LESS(X,Y) (X?Y ? X true ? Y)

Software Engineering, 6th Edition. Chapter 19
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Functional Specification Software
SpecificationLecture 41

• Prepared by
• Stephen M. Thebaut, Ph.D.
• University of Florida