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Incoming links: Representation invariant
The representation invariant isEvery reachable non-
NIL pointer designates an allocated linked list node of the appropriate extension type. The header is not a data element. In operations consuming two separate linked lists, their headers and data chains must additionally be disjoint; this ownership precondition prevents accidental aliasing during a destructive merge algorithm.Scan the polynomial coefficients from highest degree downward. Omit zero terms, omit coefficient magnitude one on nonconstant terms, print The representation invariant ensures a nonzero polynomial prints at least one term. Constant and retain their coefficients, whereas and omit magnitude one. Thus the requested example printsDifferent
- for negative terms, and print + only before a positive term after the first. A helper prints small integral magnitudes without a decimal suffix; other real numbers use Out.Real.PROCEDURE OutMagnitude(c: REAL);
VAR n: INTEGER;
BEGIN
IF c < MAX(INTEGER) THEN
n := ENTIER(c);
IF c = n THEN Out.Int(n, 0) ELSE Out.Real(c, 0) END
ELSE
Out.Real(c, 0)
END
END OutMagnitude;
PROCEDURE OutPoly(p: Poly);
VAR i: INTEGER; c, magnitude: REAL; first: BOOLEAN;
BEGIN
IF p.degree < 0 THEN Out.String("0"); RETURN END;
first := TRUE; i := p.degree;
WHILE i >= 0 DO
c := p.coeff[i];
IF c # 0.0 THEN
IF c < 0.0 THEN Out.Char("-")
ELSIF ~first THEN Out.Char("+")
END;
magnitude := ABS(c);
IF (i = 0) OR (magnitude # 1.0) THEN
OutMagnitude(magnitude)
END;
IF i > 0 THEN
Out.Char("x");
IF i > 1 THEN Out.Char("^"); Out.Int(i, 0) END
END;
first := FALSE
END;
DEC(i)
END
END OutPoly;Out.Real implementations may use decimal or scientific notation for nonintegral coefficients; either denotes the appropriate real number. The number of coefficient inspections is linear in the stored polynomial degree.Use the representation invariant that each input occurs at most once in
data. It is an immutable finite partial function, so extending or removing an input returns a new object. The source comment get(x) returns Some(x) contradicts both the type signature Option[U] and the input-output semantics: it must return Some(y) when .class PartialFn[T,U](val data: List[(T,U)]) {
require(data.map(_._1).distinct.size == data.size)
def get(x: T): Option[U] =
data.find(p => p._1 == x).map(_._2)
def add(x: T, y: U): PartialFn[T,U] =
new PartialFn[T,U]((x,y) :: data.filterNot(p => p._1 == x))
def remove(x: T): PartialFn[T,U] =
new PartialFn[T,U](data.filterNot(p => p._1 == x))
def compose[V](that: PartialFn[U,V]): PartialFn[T,V] = {
val pairs: List[(T,V)] = data.flatMap { case (x,y) =>
that.get(y).toList.map(z => (x,z))
}
new PartialFn[T,V](pairs)
}
}get returns an optional value rather than inventing a result outside the domain of a function. add removes any old pair for its input before inserting the replacement; remove retains exactly the other pairs. In compose, undefined intermediate outputs contribute an empty list; defined outputs contribute one pair. Hence uniqueness of input keys is preserved.The printed composition order is deliberately followed:This is the stated function composition, even though some libraries use the name
compose for the opposite order.A covariant type parameter may be produced but not consumed directly. Widen the output type of
add through a lower type bound. In compose, the first parameter of CoPartialFn is invariant, so replace its occurrence of U by a fresh type parameter W with lower type bound U.class CoPartialFn[T,+U](val data: List[(T,U)]) {
require(data.map(_._1).distinct.size == data.size)
def get(x: T): Option[U] =
data.find(p => p._1 == x).map(_._2)
def add[V >: U](x: T, y: V): CoPartialFn[T,V] = {
val old: List[(T,V)] = data.filterNot(p => p._1 == x)
new CoPartialFn[T,V]((x,y) :: old)
}
def remove(x: T): CoPartialFn[T,U] =
new CoPartialFn[T,U](data.filterNot(p => p._1 == x))
def compose[V, W >: U](that: CoPartialFn[W,V]): CoPartialFn[T,V] = {
val pairs: List[(T,V)] = data.flatMap { case (x,y) =>
that.get(y).toList.map(z => (x,z))
}
new CoPartialFn[T,V](pairs)
}
}List, Tuple2, and Option are covariant in their element/output parameters, so data, get, and remove are valid producers. The variance-position rule permits U in the lower type bounds; every value of U can be passed to that.get, since . No cast or mutation through a widened reference is needed.[ref]Scala 2.11 basic declarations and definitions specification. Variance Annotations: variance positions and lower-bound widening
The input-key representation invariant and the partial function semantics are unchanged.Use a history dynamic array and a cursor between elements. If the cursor is behind the history's end, The representation invariant is , with history exactly the prefix already consumed from the original source. The logical unread sequence is the history suffix starting at
next replays history; otherwise it pulls a new element from the source. prev moves the cursor left and returns the element crossed, so prev followed by next returns the same element twice and restores the cursor.def genBiIter[T](it: Iter[T]): BiIter[T] = new BiIter[T] {
private val history = scala.collection.mutable.ArrayBuffer.empty[T]
private var pos = 0
def hasPrev: Boolean = pos > 0
def hasNext: Boolean = pos < history.length || it.hasNext
def next(): T = {
if (pos < history.length) {
val v = history(pos); pos += 1; v
} else if (it.hasNext) {
val v = it.next()
history += v; pos += 1; v
} else {
throw new NoSuchElementException("BiIter.next")
}
}
def prev(): T = {
if (!hasPrev) throw new NoSuchElementException("BiIter.prev")
pos -= 1
history(pos)
}
}pos, followed by the source's remaining sequence. The logical backward history is the prefix before pos. Every operation preserves this model, including alternating backward and forward moves after source exhaustion.This bidirectional iterator uses storage after consuming distinct source elements. Replay and backward moves take constant time; appending has amortized time complexity under the usual dynamic array model. The input iterator must not be advanced separately.An Oberon pointer refers to a record containing an integer label and two subtree references:Use
TYPE
Tree = POINTER TO Node;
Node = RECORD
left, right: Tree;
label: INTEGER
END;NIL for the empty binary tree, including either absent child. The representation invariant requires a finite, acyclic binary tree with disjoint left and right subtrees. In particular, shared nodes would represent a different structure and invalidate destructive traversal arguments.Use an immutable list as private storage and cache its length. This makes the data refinement transparent without depending on library-specific dynamic array costs:The representation invariant is , with
class DocImpl extends Doc {
private var chars: List[Char] = Nil
private var length: Int = 0
def size: Int = length
def insert(x: Char, pos: Int): Unit = {
require(0 <= pos && pos <= length)
val parts = chars.splitAt(pos)
chars = parts._1 ::: (x :: parts._2)
length += 1
}
def delete(pos: Int): Char = {
require(0 <= pos && pos < length)
val parts = chars.splitAt(pos)
val old = parts._2.head
chars = parts._1 ::: parts._2.tail
length -= 1
old
}
def read(pos: Int): Char = {
require(0 <= pos && pos < length)
chars(pos)
}
}chars representing the abstract document sequence. splitAt(pos) isolates exactly the prefix before the selected array index-style position. Insertion retains that prefix and the old suffix, placing the new character between them. Deletion returns the old suffix head and retains its tail. Updating the cached length by one preserves the representation invariant. read and size preserve the abstract state.The preconditions permit insertion at the end, including position zero in an empty document, but forbid reading or deleting at the end.Maintain one frequency count for each lowercase alphabet character. Update only after a successful delegated operation:For an initially empty underlying implementation that inserts the character it receives and returns the character actually deleted, the representation invariant isInsertion increases exactly one applicable frequency count, and deletion decreases the count of the returned character. Uppercase and nonletter characters contribute nothing. Exceptions from failed underlying operations leave the counters unchanged. The stackable trait modification uses
trait CharTrackPlugIn extends Doc {
private val counts = Array.fill[Int](26)(0)
private def lower(x: Char): Boolean = 'a' <= x && x <= 'z'
abstract override def insert(x: Char, pos: Int): Unit = {
super.insert(x, pos)
if (lower(x)) counts(x - 'a') += 1
}
abstract override def delete(pos: Int): Char = {
val old = super.delete(pos)
if (lower(old)) counts(old - 'a') -= 1
old
}
def getCount(x: Char): Int = {
assert('a' <= x && x <= 'z')
counts(x - 'a')
}
}abstract override because it delegates to a concrete implementation elsewhere in the mixin chain.[ref]The counters depend on the character seen by this trait. A later trait that transforms characters can break the stated underlying-operation assumption; part (d) deliberately exposes that order dependence.For the implementations above, the output isIn
0
1A, the Scala trait linearization sends insert('a',0) first to CapitalisePlugIn, then to CharTrackPlugIn, then to DocImpl. The counter trait receives 'A', so it does not increment the lowercase 'a' count. The stored document is A.In
B, the counter trait receives 'a', delegates through CapitalisePlugIn, and increments its original argument's counter on return. The stored document is again A, but the counter records one lowercase insertion:This demonstrates a composition that violates the count-of-current-document representation invariant for B: an earlier counter cannot assume a later transforming trait inserts its argument unchanged. Subsequent deletion also receives uppercase 'A' and would leave that stale lowercase count. A different valid design could inspect read(pos) after insertion to count the actual stored character; that design would produce 0,0. The stated 0,1 follows the explicit implementation in part (b), whose correctness assumption has now failed in the second composition.If The first cell stores ; its lazily memoized successor stores ; the next stores , and so on. A cell's successor is evaluated at most once. The representation invariant is that each cursor points to the cell holding its next result.
f is a pure function on immutable values, a simple implementation stores the next value and copies it into a fresh FunIter. Part (d)'s suggested mutable programming closure, however, would be shared by such copies and violate independence. A memoized iterator snapshot avoids that contradiction: all cursors share already determined sequence values, but each has its own position.class FunIter[T](v: T, f: T => T) extends Iter[T] {
private class Cell(val head: T) {
lazy val tail: Cell = new Cell(f(head))
}
private class Cursor(private var cell: Cell) extends Iter[T] {
def hasNext: Boolean = true
def next(): T = {
val answer = cell.head
cell = cell.tail
answer
}
def copy(): Iter[T] = new Cursor(cell)
}
private val cursor = new Cursor(new Cell(v))
def hasNext: Boolean = true
def next(): T = cursor.next()
def copy(): Iter[T] = cursor.copy()
}next returns that result and advances only this cursor. copy shares the cell, whose future is memoized, rather than the mutable cursor. Advancing either copy therefore cannot skip entries in the other.Assume each demanded application of f terminates and generated values are immutable for purposes of observation. A stateful generator should be privately owned by this family of cursors; arbitrary external changes to it cannot be made independent by the supplied interface. Unlike the pure-function implementation, memoization can retain a long prefix while one copy lags far behind another. The question imposes no constant-storage requirement.An abstract data type specifies a set of abstract values and the operations on them, including their observable behavior, independently of their concrete representation. Its operation signatures say what can be requested; its laws or preconditions and postconditions say what those requests mean.
For example a stack can be specified by an empty value,
push, pop, and isEmpty, with pop(push(x,s)) returning x and leaving s. A linked list and an array can implement the same abstract data type if clients observe the same specified behavior. Encapsulation hides the representation, while a representation invariant describes which internal states are valid and an abstraction function explains which abstract value each valid internal state represents. An abstract class is a language mechanism; it is not by itself a full abstract data type specification.For a node tuple , write for its identifier. A predicate that is true exactly when identifiers repeat isThe inequality excludes choosing the same tuple twice. The set cannot contain two copies of an identical quadruple, but it can contain distinct quadruples with equal first component and different information or successors. The negation is the usual identifier-uniqueness representation invariant, namely .
A trie node needs a terminal flag as well as its children: a stored word can also be the prefix of a longer stored word.Index zero denotes
TYPE Trie = POINTER TO Node;
Node = RECORD
terminal: BOOLEAN;
child: ARRAY 26 OF Trie
END;a, index 25 denotes z; null pointers denote absent branches. The set represented by a nonnull node consists of the empty string if terminal is true, together with every child's words prefixed by that child's letter. Represent the empty set by NIL. The singleton containing the empty string instead has a terminal root and no children.The declarations alone do not force a unique representation. A nonterminal leaf, or a whole branch without terminal descendants, contributes no word. To obtain uniqueness up to pointer addresses, impose a representation invariant: the structure is a finite rooted tree, and each allocated node lies on a path to a terminal node. Then precisely the prefixes of stored words exist, all flags are determined, and the empty set has only the
NIL representation. The insertion procedures below preserve this invariant when starting with such a trie.Allocate nodes only as required, explicitly initializing every field:Here
PROCEDURE Make(VAR t: Trie);
VAR c: INTEGER;
BEGIN
NEW(t); t.terminal := FALSE; c := 0;
WHILE c < 26 DO t.child[c] := NIL; c := c+1 END
END Make;
PROCEDURE AddTail(VAR t: Trie; s: ARRAY OF CHAR; i: INTEGER);
BEGIN
IF t = NIL THEN Make(t) END;
IF s[i] = 0X THEN
t.terminal := TRUE
ELSE
AddTail(t.child[Index(s[i])], s, i+1)
END
END AddTail;
PROCEDURE Add(VAR t: Trie; s: ARRAY OF CHAR);
BEGIN
AddTail(t, s, 0)
END Add;0X is the zero character, and s must contain its terminator within its array bounds. Make establishes that the first c child fields are NIL as its loop invariant. Passing t by reference lets the base allocation replace a missing root or child pointer.For correctness, use induction on the number of characters before the terminator. At length zero, setting
terminal adds exactly the empty suffix. Otherwise the chosen child stores exactly words with the required next letter, and the induction hypothesis adds the remaining suffix while preserving all other words. Thus the resulting set is the old set union the supplied word, ignoring case. Repeated insertion changes no word membership. A newly allocated path reaches the newly terminal node, preserving the canonical representation invariant after the procedure finishes. The running time is for a word of length , using call stack space without optimization.class MSet[T](contents: List[T]) {
protected var data: List[T] = contents
def add(x: T): Unit = {
data = x :: data
}
def delete(): T = data match {
case x :: xs => data = xs; x
case Nil => throw new NoSuchElementException("MSet.delete")
}
}The representation invariant interprets the list by its multiset of occurrences; order is immaterial and duplicates remain distinct occurrences.
add increases one multiplicity, while delete returns and removes one occurrence. The field is protected so the minimum-search subclass in part (d) can inspect it. Both successful updates take list operations. Deleting from an empty bag raises an explicit exception.Use the ordinary fraction identities, returning fresh records rather than changing either argument:
PROCEDURE addRat(x,y: rational): rational;
BEGIN
RETURN mkRat(x.num*y.den + y.num*x.den, x.den*y.den)
END addRat;
PROCEDURE mulRat(x,y: rational): rational;
BEGIN
RETURN mkRat(x.num*y.num, x.den*y.den)
END mulRat;
PROCEDURE negRat(x: rational): rational;
BEGIN
RETURN mkRat(-x.num,x.den)
END negRat;Every result is reduced by
mkRat. This preserves the representation invariant even if an input was initially unreduced, provided its denominator is nonzero. Aliasing between the arguments causes no problem because their fields are only read.Assume The search loop invariant says that the structure is unchanged and every ordinary linked list node passed has index smaller than
vector already satisfies the full representation invariant in part (b). Check the local bounds and value preconditions, then find the first index not smaller than the requested one:def setElement(vector: Node, value: Double, index: Int): Unit = {
require(vector != null)
require(0 <= index && index < vector.index)
require(value != 0.0 && !java.lang.Double.isNaN(value) &&
!java.lang.Double.isInfinite(value))
var p = vector.next
while ((p ne vector) && p.index < index) p = p.next
if ((p ne vector) && p.index == index) p.value = value
else {
val left = p.prev
val fresh = new Node(value, index, left, p)
left.next = fresh
p.prev = fresh
}
}index; p is the next candidate. The number of unpassed ordinary entries is a loop variant. The short-circuit evaluation prevents testing the header index as an ordinary entry. If an equal index exists, changing only its nonzero value preserves the representation invariant. Otherwise left and p delimit the unique insertion position: initializing both new links before redirecting their neighbors preserves reciprocal links and strict index order. Exactly the requested coordinate changes. With stored entries, the search costs and the splice costs constant time; checking the whole input structure is a caller obligation, rather than an unmentioned scan inside each call.An already-zero entry needs no new linked list node:The full input representation invariant is a precondition. The search has the same loop invariant and loop variant as in part (d). At exit, either
def zeroElement(vector: Node, index: Int): Unit = {
require(vector != null)
require(0 <= index && index < vector.index)
var p = vector.next
while ((p ne vector) && p.index < index) p = p.next
if ((p ne vector) && p.index == index) {
val left = p.prev
val right = p.next
left.next = right
right.prev = left
}
}p is the only stored entry with that index or none exists. In the first case, bypassing it in both directions removes exactly that coordinate and preserves the order of all remaining entries. This also works when its two neighbors are the same list header, leaving a self-linked empty structure. The header itself is never removed. The requested coordinate is zero and all others are unchanged, in time complexity.Use the sorted sparse dot product merge, processing a coordinate only when it appears in both supports of a vector:Both structures must satisfy the representation invariant; equal dimension is checked explicitly. The loop invariant says that
def dotProduct(vec1: Node, vec2: Node): Double = {
require(vec1 != null && vec2 != null)
require(vec1.index == vec2.index)
var p = vec1.next
var q = vec2.next
var result = 0.0
while ((p ne vec1) && (q ne vec2)) {
if (p.index < q.index) p = p.next
else if (q.index < p.index) q = q.next
else {
result += p.value * q.value
p = p.next
q = q.next
}
}
result
}result contains exactly the products for all matching indices already passed, and that every skipped unequal index has zero in the other vector. The latter follows from sortedness: if p.index < q.index, that index cannot appear at q or later, and all earlier entries of the other vector have already been settled. Equality contributes its product exactly once. The total number of remaining ordinary entries is a loop variant; every iteration advances at least one cursor. Once either cursor reaches its header, all remaining entries pair with zeros. Thus the exact-arithmetic result is . With stored entries, time complexity is and auxiliary space is constant. Floating-point products and additions may round or overflow even when input coordinates are finite; no exact real-arithmetic accuracy claim is made for Double.A fixed-size array cannot implement unbounded successful insertion. Either specify a bounded double-ended queue with an extra not-full insertion precondition, or retain the unbounded interface and specify a possible full-capacity failure. Silently overwriting an endpoint would change the stated behavior. Here use the bounded contract, with a fixed positive capacity
MAX and abstract state . Both additions require ; the other postconditions remain unchanged. The class declaration in part (c) is interpreted as implementing this explicitly revised contract, rather than as a behavioral subtype of the unrestricted contract in part (a).The concrete state is an array
a of length MAX, a first-position index head, and an occupied-entry count size. Its representation invariant isThe abstraction function is for . Unoccupied slots are unconstrained. Storing size distinguishes empty and full states even though their two circular endpoint positions coincide. MAX=1 is valid. Using a circular buffer avoids shifting entries and makes every endpoint operation constant time.