Type Definitions

1
Type Definitions
2
Concrete Types

• Primitive types ( int, bool, char, string, etc )
• Type constructors ( -gt, list, etc )
• Real world concepts can be modeled better with
  concrete types rather than simulated using
  primitive types.
• datatype decision (yes,no,maybe)
• Readability (self-documenting)
• Reliability (automatic detection of errors)
• operations on values of a type are independent of
  the operations on their representations.

3
Language Aspects

• Naming the new type
• Constructing values of the type
• Inspecting values and testing their type for
  defining functions
• Canonical representation (for equality tests)
• In ML, one can use symbolic terms to construct
  values in a type, and use patterns to define
  functions on this type.

4
Introducing Type Names

• type pair int int
• type points int list
• Type Equivalence
• Structural Equivalence
• - val x (1,2) pair
• - (1,1) int int
• - x (1,1) ( val it false bool )

5
Enumerated types with constants

• datatype directions
• North South East West
• fun reverse North South
• reverse South North
• reverse East West
• reverse West East
• val reverse fn directions -gt directions
• fun isEast x (x East)

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• Constructors with Arguments
• datatype phy_quant
• Pressure of real
• Volume of real
• datatype position
• Coord of int int
• (Tagged values)
• (Cf. Ada Derived Types)

• Recursive Types
• datatype number
• Zero
• Succ of number
• datatype tree
• leaf of int
• node of
• (treeinttree)
• E.g.,
• node (leaf 25,10,leaf 20) tree

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Defining Functions

• fun add (Zero, n) n
• add (Succ(m),n) Succ(add(m,n))
• val add fn number number -gt number
• fun mul (Zero, n) Zero
• mul (Succ(m),n) add(n,mul(m,n))
• val mul fn number number -gt number
• mul ( Succ(Succ Zero), Succ Zero )
• Succ(Succ Zero)
• ( Expression evaluation - Normalization )

8
New Type Operators

• datatype e matrix
• Matrix of (e list list)
• datatype 'a matrix Matrix of 'a list list
• datatype a list nil
• of (a a list)
• datatype 'a list of 'a 'a list nil
• ( Using a instead of a generates
  errors. )
• ( Using instead of nil generates an error.
  )

9
Semantics of Concrete Types

• Algebra (Values, Operations)
• E.g, vector/matrix algebra, group theory, etc.
• Free Algebra Construction is a way of
  defining an algebra from a collection of
  CONSTRUCTORS using SIGNATURE information.
• E.g.,
• datatype t e
• u of t
• p of tt

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• datatype t e u of t p of tt
• Signatures
• e t
• u t -gt t
• p tt -gt t
• What are the values in the type?
• What are the operations on the type?

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Values

• e
• u(e), p(e,e)
• u(u(e)), u(p(e,e)),
• p(e,u(e)), p(e,p(e,e)), p(u(e),e), p(p(e,e),e),
• . . .
• ( Unique name hypothesis )
• Grammar
• V e u(V) p(V,V)

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Operations

• u (function from terms to terms)
• e gt u(e)
• u(e) gt u(u(e))
• p (function from termsterms to terms)
• e, e gt p(e,e)
• u(e),e gt p(u(e),e)
• p(e,e),e gt p(p(e,e),e)
• Semantics of t
• (e,u(e),p(e,e),, u,p)

13
Abstract Type

• Specification and Implementation through Examples

14
Integer Sets Algebraic specification

• empty intset
• insert intset -gt int -gt intset
• remove intset -gt int -gt intset
• member intset -gt int -gt bool
• for all s e intset, m,n e int
• member empty n false
• member (insert s m) n
• (nm) orelse (member s n)
• remove empty n empty
• remove (insert s m) n
• if (nm) then remove s n
• else insert (remove s n) m

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• abstype
• intset Empty Insert of intsetint
• with
• val empty Empty
• fun insert s n Insert(s,n)
• fun member Empty n false
• member (Insert(s,m)) n
• (nm) orelse (member s n)
• fun remove Empty n Empty
• remove (Insert(s,m)) n
• if (nm) then remove s n
• else Insert(remove s n, m)
• end

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• val s1 (insert empty 5)
• val s2 (insert s1 3)
• val s1 (insert s2 8)
• (member s1 8)
• (member s1 5)
• (member s1 1)
• val s3 (remove s1 5)
• (member s3 5)
• (member s1 5)

17

• abstype
• intset Set of int list
• with
• val empty Set
• fun insert (Set s) n Set(ns)
• fun member (Set s) n
• List.exists (fn i gt (in)) s
• fun remove (Set s) n
• Set (List.filter (fn i gt (in)) s)
• end
• ( member and remove are not primitives in
  structure List because they are defined only
  for equality types. )

18

• abstype
• intset Set of (int -gt bool)
  
• with
• val empty
• Set (fn n gt false)
• fun insert (Set s) n
• Set (fn m gt (m n) orelse (s m))
• fun member (Set s) n (s n)
• fun remove (Set s) n
• Set (fn m gt (not(m n)) andalso (s m))
• end