Compound Data Structures

1
Compound Data Structures

• Structural decomposition into other values.
• Lists domain A
• Constructors NIL, CONS.
• Selectors HEAD, TAIL.
• Tuples domain A x B x C
• Constructor (, , )
• Selector ? i
• Problem imperative languages
• Variable forms of the objects exist.
• Objects subcomponents can be altered.
• Several versions of arrays variables.

2
Arrays

• Collection of homogeneous objects indexed by a
  set of scalar values.
• Homogeneity ? all components have same structure.
  
• Allocation of storage and typechecking easy.
• Both tasks can be performed by compiler.
• Selector operation indexing.
• Scalar index set primitive domain with
  relational and arithmetic operations.
• Restricted by lower and upper bounds.

3
Linear Vector of Values

• IDArray (Index ? Location) x Lowerbound x
  UpperBound
• Index is index set
• lessthan, greaterthan, equals.
• Lowerbound Upperbound Index.
• First component maps indices to the locations
  that contain the storable values.
• Second and third component denote the bounds
  allowed on array indices.

4
Multidimensional Arrays

• Array may contain other arrays.
• Threedimensional array is vector whose
  components are twodimensional vectors.
• Hierarchy of arrays defined as infinite sum
• 1DArray (Index ? Location) x Index x Index.
• (n1)DArray (Index ? nDArray) x Index x Index.
• 1DArray maps indices to locations, 2DArray maps
  indices to 1D arrays, . . . .

5
Multidimensional Arrays

• a ? MDArray
• inkDArray(map, lower, upper) for some k gt 1
• accessarray
• Index ? MDArray ?(Location MDArray Errvalue)
  
• accessarray ?i. ?r. cases r of
• is1DArray(a) ? index1 a i
• is2DArray(a) ? index2 a i
• . . .
• iskDArray(a) ? indexK a i
• . . .
• end

6

• index m ?(map, lower, upper). ? i.
• (i lessthan lower) or (i greaterthan upper) ?
• inErrvalue() MInject(map(i))
• 1Inject ? l.inLocation(l)
• . . .
• (n1)Inject ? a. inMDArray( innDArray(a))

7
Multidimensional Arrays

• accessarray is represented by an infinite
  function expression.
• By using pair representation of disjoint union
  elements, operation is convertible to finite,
  computable format.
• Operation performs onelevel indexing upon array
  a returning another array if a has more than one
  dimension.
• Still model is to clumsy to be used in practice.
• Real programming languages allow arrays of
  numbers, record structures, sets, . . . !

8
System of Type Declarations

• T ? Typestructure
• S ? Subscript
• T nat bool array N1... N2 of T record D
  end
• D D1D2var IT
• C IS E ...
• E ... IS ...
• S E E, S

9

• Denotablevalue
• (Natlocn Boollocn Array Record
  Errvalue)?
• l ? Natlocn Boollocn Location
• a ? Array (Nat ? Denotablevalue) x Nat x Nat
• r ? Record Environment Id ? Denotablevalue

10
Semantics of Type Declarations

• T Typestructure ? Store ? (Denotablevalue x
  Poststore)
• Tnat ? s
• let (l,p) (allocatelocn s)
• in (inNatlocn(l), p)
• Tbool ? s
• let (l,p) (allocatelocn s)
• in (inBoollocn(l), p)

11

• Tarray N1... N2of T ? s
• let n1 NN1 in let n2 NN2
• in n1 greaterthan n2 (inErrvalue(), (signalerr
  s))
• getstorage n1 (emptyarray n1 n2 ) s
• Trecord D end ? s let (e, p) (DD
  emptyenv s)
• in (inRecord(e), p)
• Type structure expressions are mapped to storage
  allocation
• actions!

12
Semantics of Type Declarations

• getstorage Nat ? Array ? Store ?
  (Denotablevalue x Poststore)
• getstorage ? n ? a ? s n greater n 2 ?
• (inArray(a), return s)
• let (d, p) TTs
• in (check(getstorage (n plus one)
• (augmentarray n d a)))(p)

13

• augmentarray Nat ? Denotablevalue ? Array ?
  Array
• augmentarray ? n ? d ?(map, lower, upper)
• (n ? d map, lower, upper)
• emptyarray Nat ? Nat ? Array
• emptyarray ? n1 ? n2 ((? ninErrvalue())
  n1 n2 )
• getstorage iterates from lower bound of array to
  upper bound allocating the proper amount of
  storage for a component.
• augmentarray inserts the component into the
  array.

14
Declarations

• D Declaration ? Environment ? Store ?
• (Environment x Poststore)
• DD1D2 ? e ? s
• let (e, p) (DD1e s) in (check (DD2
  e))(p)
• Dvar IT ? e ? s
• let (d, p) TTs in ((updateenv I d e),
  p)
• A declaration activates the storage allocation
  strategy specified by its type structure.

15
Array Indexing

• S Subscript ?Array ? Environment ? Store ?
  Storablevalue
• SE ? a?e ? s
• cases (EEe s) of . . .
• isNat(n) accessarray n a end
• SE, S ? a ? e ? s
• cases (EEe s) of
• isNat(n) ? (cases (accessarray n a) of
• isArray(a ) ? SSa e s ...end) ...
• end

16
Array Assignment

• CIS E ? e ? s
• cases (accessenv I e) of ...
• isArray(a) ? (cases (SSa e s) of ...
• isNatlocn(l) ?(cases (EEe s) of
• ... isNat(n) ? return(update l inNat(n) s)
  ...end)
• ...end)
• ...end
• Assignment is first order (location, not an
  array, is on lefthandside).