Software as Math

1
Software as Math
(or why you should be paying attention in this
class)

• Ryan Luna
• Javier Garcia

2
Overview

• What the heck is discrete math?
• Algebraic Specifications
• Theories and Models
• Specification Morphisms
• Homomorphisms and Isomorphisms
• Useful applications!!

3
Discrete Mathematics

• What is discrete mathematics?
• Discrete mathematics is the part of
  mathematics devoted to the study of discrete
  objects where discrete refers to consisting of
  distinct or unconnected elements.
• It could be said that these subjects define the
  mathematical basis for computing.
• A key reason for the growth in importance of
  discrete mathematics is that information is
  stored and manipulated by computing machines in a
  discrete fashion.

4
Algebraic Specifications

• An algebra consists of a set (sort) and a
  collection of operators defined over this set.
• A good example
• Natural Numbers 0,1,2,3,
• Operators - x
• Each operator is defined over two elements in
  the sort.
• An algebra is multi-sorted if it includes
  multiple sorts.

5
How does this relate to me?

• A good example of a multi-sorted algebra is the
  Abstract Data Type programming style.
• Remember
• When defining an ADT we define
• 1) Collection of types
• 2) Set of functions and procedures for
  those types

6
Binary Search Tree ADT

• Sort Any elementary or user defined data type.
• BSTree ltintgt myTree
• BSTree ltstringgt strTree
• Operators The functions written for the tree.
• void BSTree ltADTgt insert (ADT element)
• void BSTree ltADTgt remove (ADT element)
• void BSTree ltADTgt retrieve (ADT element)

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Heap ADT

• Sort Any elementary or user defined data type.
• Heap ltintgt myTbl
• Heapltstringgt strTbl
• Operators Implementation of the Heap.
• void insert (ADT element)
• ADT removeMax ()
• void clear()

8
Theories and Models

• Suppose a computer program could take an
  algebraic specification and from it generate all
  its implications this set of true statements
  would be the theory associated with the algebraic
  specification.
• A theory models a system if
• 1) Every observable behavior of the system is
  a theorem
• in the theory. (Completeness)
• 2) Every theorem in the theory corresponds to
  a behavior
• in the system. (Consistency)
• EMC2

9
Specification Morphisms

• Why do we care, really?
• Software development is all about changing
  models.
• Software Design Process
• Specification morphism represents the
  transformation of one specification into another.

10
Homomorphisms

• A homomorphism exists between two specifications
  if every thing true in A
• is true in B.
• A ? B, but A ? B.

Homomorphism between A and B.A is more abstract
than B.B is more general than A.
Homomorphism between C, A and BB is more general
than A or C.B represents both A and C.
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Isomorphisms

• An isomorphism exists between A and B when the
  two specifications represent the same theory.
• Specifications are structurally identical.
• Used less often in software engineering because
  they are too strong.
• Physical examples
• Big Ben and a wristwatch
• Two decks of cards, each a different color

12
A Good Example

• Consider two data structures a stack and a list.
• A homomorphism exists between the stack and the
  list.
• Stack ? List

13
Another Good Example

• Consider several data structures
• A queue, a stack, and a list.
• We can define a homomorphism between the queue
  and the list, as well as the stack and the list.
• Stack ? List and Queue ? List

No homomorphism exists between a stack and a
queue!
14
Finally

• Algebraic specifications relate well to ADT
  programming
• Knowing homomorphisms between two specifications
  can make life easier!
• Set Theory and Functions can have real
  applications!

15
Questions?