Software as Math

Ryan Luna

Javier Garcia

(or why you should be paying attention in this class)

Overview

• What the heck is discrete math?

• Algebraic Specifications

• Theories and Models

• Specification Morphisms

• Homomorphisms and Isomorphisms

• Useful applications!!

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.

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.

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

Binary Search Tree ADT

• Sort: Any elementary or user defined data type.

BSTree <int> myTree;

BSTree <string> strTree;• Operators: The functions written for the tree.

void BSTree <ADT> insert (ADT element)

void BSTree <ADT> remove (ADT element)

void BSTree <ADT> retrieve (ADT element)

Heap ADT

• Sort: Any elementary or user defined data type.

Heap <int> myTbl;

Heap<string> strTbl;• Operators: Implementation of the Heap.

void insert (ADT element);

ADT removeMax ();

void clear();

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)

• E=MC2

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.

Homomorphisms

• A homomorphism exists between two specifications if every thing true in A

is true in B.• A ≠ B, but A B.

B

A

Homomorphism between A and B.A is more abstract than B.B is more general than A.

BC

A

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

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

A Good Example…

• Consider two data structures: a stack and a list.

• A homomorphism exists between the stack and the list.

• Stack ListList

Stack

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 ListList

Queue

Stack

No homomorphism exists between a stack and a queue!

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!

Questions?