CIA2326: WEEK ??

1
CIA2326 WEEK ??

• LECTURE
• Introduction to Algebras
• SUPPORTING NOTES
• See chapters 8,9,10 of my online book on my
  homepage
• TUTORIAL hand out exercises
• Telescopes are to Astronomy what Computers are
  to Computer Science (Edgar Dijstra)

2
Algebras

• Q1. What is an algebra?
• Q2.What has it to do with computing?

3
Algebras

• A1. Roughly, an algebra is A SET OF VALUES the
  specification of some OPERATIONS on those values
• A2. Roughly, a data type is A SET OF VALUES the
  implementation of some OPERATIONS on those
  values.
• Hence we can give a computer-independent meaning
  to data types using algebras.

4
Abstraction in Programming

• Algebras inform us in both the THEORY and
  PRACTICE of programming
• Good modularity of software is ensured to a large
  degree by procedural and data abstractions
• Object technologies owe much of their success to
  the fact that their structure results in a high
  degree of procedural and data abstraction.
• Data abstraction or abstract data types
  abstract away the implementation of data types
  values and operations - just like ALGEBRAS!

5
Abstract Data Types

• A very abstract way of defining data types is as
  follows abstract away the implementation of data
  types values and operations in two ways
• define the operations in terms of each other
• dont represent values AT ALL except in terms of
  the operations that construct them
• This give us an abstract algebra.

6
Homogenous algebras Formal Definition

• (A,O) is a Homogenous algebra if
• A is a non-empty set which contains values of the
  algebra and is known as the carrier set.
• O is a set of closed, total operations defined
  over the carrier set which may include nullary
  operations (constants).
• EXAMPLE (Natural Numbers, ) ,
• EXAMPLE (True,False, and,not)
• COUNTER-EX (Natural Numbers, -)
• COUNTER-EX (Real Numbers, /)

7
heterogeneous algebras

• (A,O) is a heterogeneous algebra if
• A is a SET of carrier sets and
• O is a set of closed and total operations over
  these sets.
• The semantics of data types are often given by
  heterogeneous algebras as we shall see...

8
?Equational Specification of Algebras

• The most abstract form of Presentation (ie how
  to define them) is NOT TO GIVE ANY NAMES to their
  values.
• A very abstract way to specify a (family of)
  Algebras/Data Types is to give an Equational
  Specification.

9
Example - Equational Presentation of a Homogenous
Algebra

• SPEC Boolean
• SORT bool
• OPS
• true -gt bool
• false -gt bool
• not bool -gt bool
• and bool bool -gt bool
• AXIOMS FORALL b bool
• (1) not(true) false (2) not(false) true
  (3) and(true,b) b
• (4) and(b,true) b (5) and(false,b) false (6)
  and(b,false) false
• ENDSPEC

10
Syntax for Equational Specs of Algebras (ADTs)

• SPEC Name of Specification
• SORT Type of interest - algebra being
  defined
• OPS Signature of operations
• FORALL Universally defined variables
• AXIOMS Equations defining MEANING of
  operations
• ENDSPEC

11
Another Example

• SPEC Natural
• SORT nat
• OPS
• zero -gt nat
• succ nat -gt nat
• add nat nat -gt nat
• AXIOMS FORALL m, n nat
• (1) add(zero, n) n (2) add(succ(m), n)
  succ(add(m, n))
• ENDSPEC

12

• SPEC A_State_Machine
• SORT state
• OPS
• S1 -gt state S2 -gt state
• S3 -gt state
• a state -gt state b state -gt state
• c state -gt state
• AXIOMS FORALL m, n nat
• (1) a(S1) S3 (6) c(S2) S1
• (2) b(S1) S1 (7) a(S3) S3
• (3) c(S1) S2 (8) b(S3) S1
• (4) a(S2) S2 (9) c(S3) S3
• (5) b(S2) S2
• ENDSPEC

13
Another Example...

• SPEC Stack USING Natural Boolean Carriers
  are Stack, Natural
• SORT stack and Boolean
• OPS Type of Interest Stack
• init -gt stack Signature of each Operation
  
• push stack nat -gt stack
• pop stack -gt stack
• top stack -gt nat is-empty? stack -gt bool
• stack-error -gt stack
• nat-error -gt nat
• FORALL s stack, n nat universally
  quantified vars
• AXIOMS for is-empty?
• (1) is-empty?(init) true this part gives
  meaning'
• (2) is-empty?(push(s,n)) false to each
  operation AXIOMS for pop
• (3) pop(init) stack-error
• (4) pop(push(s,n)) s
• AXIOMS for top
• (5) top(init) nat-error
• (6) top(push(s,n)) n ENDSPEC

14
Conclusions

• ALGEBRA gives us an ABSTRACT way to specify DATA
  TYPES
• NEXT week we will examine how to REASON with
  algebraic expressions.