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Joined 2026-09-21 Message
Incoming links: Precondition
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.Take the abstract state to be a finite sequence of integers, initially empty. Let denote the entry state, the empty sequence, and concatenation. A Scala trait presents the interface; its comments give the mathematical preconditions and postconditions:The double-ended queue has no abstract capacity limit in this first specification. Both removals require nonemptiness; the query leaves the state unchanged, an instance of command-query separation.
trait DoubleEndedQueue {
// Pre: true. Post: result == s0.isEmpty and s == s0.
def isEmpty: Boolean
// Pre: true. Post: s == [x] ++ s0.
def addLeft(x: Int): Unit
// Pre: s0.nonEmpty. Post: result == s0.head, s == s0.tail.
def getLeft(): Int
// Pre: true. Post: s == s0 ++ [x].
def addRight(x: Int): Unit
// Pre: s0.nonEmpty. Post: result == s0.last, s == s0.init.
def getRight(): Int
}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.Place this object method inside the concrete class. Its precondition is the representation invariant; its postcondition is that the abstract data type state is unchanged and its left-to-right contents have been printed in bracket notation followed by a newline:Assume ordinary sequential execution and successful output. The loop invariant is , unchanged representation, and output consisting of
def display: Unit = {
print("[")
var i = 0
while (i < size) {
if (i > 0) print(", ")
print(a(slot(i)))
i += 1
}
println("]")
}[ followed by precisely the first abstract values, separated by commas and spaces. The loop variant decreases. At exit the closing bracket completes the required display, including [] for emptiness. This takes output operations and constant space.Use pattern matching and an optional value to distinguish absence from a present result:The representation precondition is that equal keys always have equal associated values, with a stable equality relation on keys. This permits harmless repeated identical pairs; it excludes conflicting outputs for one input.
class PartialFn[T, U](val data: List[(T, U)]) {
def get(x: T): Option[U] =
data.find { case (key, _) => key == x }.map(_._2)
def remove(x: T): PartialFn[T, U] =
new PartialFn[T, U](data.filterNot { case (key, _) => key == x })
def add(x: T, y: U): PartialFn[T, U] =
new PartialFn[T, U]((x, y) ::
data.filterNot { case (key, _) => key == x })
}get therefore returns Some(y) exactly when the represented partial function is defined at x. Removing filters out every occurrence of the key and leaves all other pairs unchanged. Adding first removes every old occurrence and prepends exactly the new pair, proving the partial function update postcondition even if the input contains duplicate identical pairs. The immutable linked list representation leaves the original mapping unchanged. Each operation has worst-case time complexity for pairs under constant-cost key equality.Compare values directly, avoiding subtraction that could overflow:Each branch returns an index whose value lies between the other two sampled values. Non-strict comparisons make ties harmless. The precondition is a nonempty valid segment, not a segment of at least three elements: for lengths one or two some sample indices coincide, but the code still returns a valid median of the sampled triple. This matters because selection can reach a two-element segment. With three distinct sample values, the pivot is not the minimum or maximum of the whole segment, but it need not be near the segment's true median. Median-of-three pivot selection is a heuristic, not a balance guarantee.
def bestIndex(l: Int, r: Int): Int = {
require(0 <= l && l < r && r <= a.length)
val first = l
val middle = l + (r - l) / 2
val last = r - 1
if (a(first) <= a(middle)) {
if (a(middle) <= a(last)) middle
else if (a(first) <= a(last)) last
else first
} else {
if (a(first) <= a(last)) first
else if (a(middle) <= a(last)) last
else middle
}
}Let denote the entry state. A conventional meaning for the Boolean returned by Initially . In particular, re-adding an existing element returns false and does not increase the size. All operations have the stated empty precondition at the abstract level. The comment that
add is whether the call changes the set:// add(elem):
// S' = S0 union {elem}
// result == !S0.contains(elem)
// contains(elem):
// S' = S0
// result == S0.contains(elem)
// size:
// S' = S0
// result == cardinality(S0)add adds an element does not itself completely specify its Boolean result; the explicit changed/not-changed convention above completes that contract. As with any Int-valued size operation, exact cardinalities are understood to be representable by that result type.Represent the characteristic values of a bounded universe by one Boolean slot per possible member, a Boolean-array form of a bitset:The abstraction function is . The representation invariant is that
class BitMapSet(val MAXVALUE: Int) extends IntSet {
require(MAXVALUE >= 0)
private val bits = Array.fill(MAXVALUE)(false)
private var count = 0
def contains(elem: Int): Boolean =
0 <= elem && elem < MAXVALUE && bits(elem)
def add(elem: Int): Boolean = {
require(0 <= elem && elem < MAXVALUE, "Outside bitmap universe")
if (bits(elem)) false
else {
bits(elem) = true
count += 1
true
}
}
def size: Int = count
}bits.length==MAXVALUE, its entries are Boolean, and count equals the number of true entries. Initialization gives the empty set; changing a false entry to true adds exactly its index and increments the count once. A repeated addition preserves both. All three operations have time complexity after initialization; storage and explicit array initialization are .The constructor requires , and
add requires . contains remains total by returning false outside the representable universe; short-circuit evaluation protects the index. This finite representation cannot implement unrestricted addition of every Int while preserving the abstract contract. Its strengthened add precondition therefore makes it a restricted implementation, not a behaviorally substitutable implementation of the unrestricted trait for arbitrary clients. The sequence client below proves that it stays within the restricted universe.For result entries the largest required index is , so the bound in part (a)(iii) permits universe size . Check that the chosen universe can itself be expressed as an array length:The loop invariant is that the result prefix contains exactly and the abstraction function of
def recamanBitmap(a: Array[Int]): Unit = {
val n = a.length
if (n == 0) return
val capacity = n.toLong * (n.toLong - 1L) / 2L + 1L
require(capacity <= Int.MaxValue.toLong, "Bitmap universe too large")
val seen = new BitMapSet(capacity.toInt)
a(0) = 0
seen.add(0)
var i = 1
while (i < n) {
val candidate = a(i - 1).toLong - i.toLong
val next =
if (candidate > 0L && !seen.contains(candidate.toInt)) candidate
else a(i - 1).toLong + i.toLong
a(i) = next.toInt
seen.add(a(i))
i += 1
}
}seen is their set of values. The universe bound ensures that each positive candidate being queried and each actual next value is less than capacity; values are nonnegative. Thus the conversions and the restricted add precondition are justified, rather than merely hoped to succeed. At a repeated addition, add returning false is harmless: the set of previously seen values is still updated correctly. The loop variant is . At return every result entry agrees with the original recurrence.Insert before the first entry that is at least the new value; equal values remain separate occurrences:In the scanning branch, the loop invariant says that
class ListPriorityQueue extends PriorityQueue[Int] {
private var first: Node = null
def clear(): Unit = { first = null }
def isEmpty(): Boolean = first == null
def insert(x: Int): Unit = {
if (first == null || x <= first.value) {
first = new Node(x, first)
} else {
var p = first
while (p.next != null && p.next.value < x) {
p = p.next
}
p.next = new Node(x, p.next)
}
}
def delMin(): Int = {
require(first != null, "Empty priority queue")
val result = first.value
first = first.next
result
}
}p is reachable and all values from first through p are strictly smaller than x. The rest of the chain is unchanged and remains sorted. The loop variant is the number of nodes after p. At exit either the next value is at least x or there is no next node, so inserting between p and p.next preserves the representation invariant and adds exactly one occurrence under the abstraction function. The head branch covers an empty chain, a new minimum and equality with the old minimum. delMin returns the head's minimum value and removes exactly that occurrence.Insertion is worst-case;
delMin, isEmpty and the head-resetting clear are . Reclaiming an abandoned chain later is garbage-collection work, not an explicit traversal performed by clear. Storage is , including separate nodes for duplicate values. No fixed insertion-capacity precondition has been added.Keep a deficient interval of possible values and only the array segment containing occurrences from that interval. This is missing-value search by deficient value intervals:The precondition is that every original entry lies in . The loop invariant istogether with: every active entry lies in , and every original occurrence with a value in that interval is in the active segment. The whole array remains a permutation of its initial contents. Initially these assertions follow from the precondition and .
def missing(): Int = {
val n = a.length
var lo = 0L
var hi = n.toLong + 1L
var l = 0
var r = n
while (hi - lo > 1L) {
val p = lo + (hi - lo) / 2L
val k = partition(l, r, p.toInt)
if ((k - l).toLong < p - lo) {
hi = p
r = k
} else {
lo = p
l = k
}
}
lo.toInt
}After array partitioning, the low subinterval has occurrences and possible values. If it is deficient, retaining it preserves the loop invariant. Otherwise , while the original active count is strictly less than ; subtraction givesThus the high subinterval is deficient and retaining it also preserves the loop invariant. Discarded occurrences lie outside the new candidate interval. The loop variant is the positive integer , which strictly decreases because the midpoint is strictly internal. At exit the interval has one possible value and fewer than one occurrence, so
lo is absent from the entire original array. For the loop is skipped and zero is returned. The Long value bounds avoid overflowing ; every internal pivot is at most , so its checked mathematical range fits Int.For the time complexity, let be the interval width and its active occurrence count before iteration . The loop invariant gives , and binary search gives . Each two-way array partition costs . Until the width reaches one, the widths are bounded by , so the total scan work is bounded by a geometric series plus loop overhead: overall, hence for positive . Partitioning the full array again at every iteration would instead lose this bound. Auxiliary space complexity is .
The abstract state is a finite partial function from strings to strings, initially empty. Names and numbers below are non-null mathematical strings. These preconditions and postconditions specify an abstract data type, independently of any linked list. In particular, a failed deletion leaves the abstract state unchanged, and recalling a missing name is outside the operation's specified domain.
store replaces an old binding if present; delete reports whether a binding existed.trait Book {
// Abstract state B: finite partial map String -> String; initially empty.
// Pre: name != null && number != null.
// Post: B' = B updated at name with number.
def store(name: String, number: String): Unit
// Pre: name != null && name is in dom(B).
// Post: result = B(name), and B' = B.
def recall(name: String): String
// Pre: name != null.
// Post: result iff name is in dom(B), and B' = B.
def isInBook(name: String): Boolean
// Pre: name != null.
// Post: result iff name was in dom(B); B' = B with name removed.
def delete(name: String): Boolean
}Search for the predecessor of the matching linked list node, or for the last linked list node if there is no match. This helper makes deletion uniform:The search loop invariant says that
class LinkedBook extends Book {
private val header = new Node("", "", null)
// Pre: name != null.
// Post: result is header or a real node.
// All real nodes through result have names different from name.
// result.next is null or has name equal to name.
// No state is changed.
private def findPrev(name: String): Node = {
require(name != null)
var p = header
while (p.next != null && p.next.name != name) {
p = p.next
}
p
}
def store(name: String, number: String): Unit = {
require(name != null && number != null)
val p = findPrev(name)
if (p.next == null)
p.next = new Node(name, number, null)
else
p.next.number = number
}
def recall(name: String): String = {
require(name != null)
val p = findPrev(name)
require(p.next != null, "Name is not in the book")
p.next.number
}
def isInBook(name: String): Boolean = {
require(name != null)
findPrev(name).next != null
}
def delete(name: String): Boolean = {
require(name != null)
val p = findPrev(name)
if (p.next == null) false
else {
p.next = p.next.next
true
}
}
}p is the header or a reachable real linked list node, and every real linked list node from the start through p has a name different from the requested name. The linked list has not been changed. Short-circuit evaluation protects p.next.name when the end is reached. The loop variant is the number of real linked list nodes strictly after p; finiteness and acyclicity make it a nonnegative decreasing integer. Consequently the helper's stated postcondition holds.For
store, a found name changes only its number; an absent name adds exactly one fresh binding at the end, maintaining uniqueness. For delete, unlinking the found linked list node removes exactly the requested binding; absence returns false without mutation. Both query operations preserve the abstraction function. Non-null preconditions are checked at entry; recall additionally checks membership using the one completed search, avoiding a redundant whole-list membership scan. The helper is the only additional operation and its complete behavioral contract is stated above.Let be the number of real contacts, and suppose the requested name is at position , counting from one. The implementation above makes exactly name-equality comparisons, including the successful comparison; null checks are not string comparisons. Thus the minimum is one, when the first real contact is requested, and the maximum is , when the last is requested. The precondition excludes unsuccessful recalls and recalls on an empty book. This counts calls to string equality, not individual character inspections within each call.
Represent an applied edit by a command pattern object with
undo and redo. Use zero-based positions. For move(from,to), define to as the slide's index in the resulting sequence; it is in for a presentation of length . Removing the slide first and then inserting at that index implements this convention in either direction.trait Change {
// Pre: this is the most recent applied change not yet undone.
def undo(): Unit
// Pre: this is the next change on the redo history.
def redo(): Unit
}
class SlideSuite(initial: Sequence[Slide]) {
private var slides = initial
private def insertRaw(i: Int, x: Slide): Unit = {
slides = slides.insert(i, x)
}
private def deleteRaw(i: Int): Unit = {
slides = slides.delete(i)
}
private def moveRaw(from: Int, to: Int): Unit = {
val x = slides.elementAt(from)
slides = slides.delete(from).insert(to, x)
}
def insert(i: Int, x: Slide): Change = {
insertRaw(i, x)
new Change {
def undo(): Unit = deleteRaw(i)
def redo(): Unit = insertRaw(i, x)
}
}
def delete(i: Int): Change = {
val x = slides.elementAt(i)
deleteRaw(i)
new Change {
def undo(): Unit = insertRaw(i, x)
def redo(): Unit = deleteRaw(i)
}
}
def move(from: Int, to: Int): Change = {
moveRaw(from, to)
new Change {
def undo(): Unit = moveRaw(to, from)
def redo(): Unit = moveRaw(from, to)
}
}
}Sequence and Slide are the supplied external types. Valid insertion positions are , deletion positions are , and both move positions are valid indices of the original length. These are the permitted unchecked argument preconditions. The private raw operations perform exactly their named state transformation and do not create history entries. A move to the same index is a legitimate no-op.The deleted slide must be saved because its index alone cannot reconstruct it. By contrast, insertion already receives the slide to retain. For a move, the inverse of moving the slide from
from to its final index to is to remove it at to and reinsert it at from; all other slides retain their relative order. Thus each returned object satisfies undo restores the state immediately before the edit, and redo restores the state immediately after it, under the stated history protocol.A controller can enforce the protocol with two stacks. This code records changes that have already been executed:For example,
class History {
private var done = List.empty[Change]
private var undone = List.empty[Change]
// Pre: c is a fresh, already-applied change of this history's suite.
def record(c: Change): Unit = {
done = c :: done
undone = Nil
}
def undo(): Unit = {
require(done.nonEmpty)
val c = done.head
c.undo()
done = done.tail
undone = c :: undone
}
def redo(): Unit = {
require(undone.nonEmpty)
val c = undone.head
c.redo()
undone = undone.tail
done = c :: done
}
}history.record(suite.move(i,j)) executes and records one move. The representation invariant says that done contains applied changes in reverse application order, while undone contains undone changes in the next-redo order, all belonging to the same suite. A fresh edit clears the redo branch. Undo and redo transfer one command between the two heads, so they neither skip intervening edits nor generate new history records. Unrecorded external edits and out-of-order use of old command objects are excluded by this protocol.Replace edit-specific inverse information by before/after snapshots implementing the Memento pattern. The same By-value evaluation computes
Change and History interfaces still work. A complete alternative suite is:class SnapshotSlideSuite(initial: Sequence[Slide]) {
private var slides = initial
private def changeTo(after: Sequence[Slide]): Change = {
val before = slides
slides = after
new Change {
def undo(): Unit = { slides = before }
def redo(): Unit = { slides = after }
}
}
def insert(i: Int, x: Slide): Change =
changeTo(slides.insert(i, x))
def delete(i: Int): Change =
changeTo(slides.delete(i))
def move(from: Int, to: Int): Change = {
val x = slides.elementAt(from)
changeTo(slides.delete(from).insert(to, x))
}
}after without changing slides, then changeTo captures the still-current before and installs the new value. The val bindings inside each call are stable snapshots, rather than later reads of a mutable variable. Under the same chronological history preconditions, undo and redo are constant-time reference assignments. This is the Memento pattern applied to an immutable sequence. Retaining a reference to an in-place-mutated sequence would not save its old state. The snapshot covers slide order and membership; it does not promise to undo unrelated mutations inside a Slide object.The inclusive interval is nonempty exactly when . Reject with a failed precondition, rather than assigning a nonexistent normalization to an empty interval. Compute its width in The loop invariant says that and
Long before taking a reciprocal:class Uniform(l: Int, u: Int) extends FiniteDist[Int] {
require(l <= u, "Empty uniform interval")
private val width = u.toLong - l.toLong + 1L
private val mass = 1.0 / width.toDouble
def getProb(x: Int): Double =
if (l <= x && x <= u) mass else 0.0
def getDom: List[Int] = {
var x = l.toLong
var rev = List.empty[Int]
while (x <= u.toLong) {
rev = x.toInt :: rev
x += 1L
}
rev.reverse
}
}rev contains exactly the integers in reverse order. The loop variant is while the guard is true. Long handles both the full interval width and the increment past Int.MaxValue. At exit, reversing yields each support value once. All support values have mass , so their exact sum is one; Double approximates that rational number. Very large valid domains require correspondingly large lists and sufficient memory; that resource issue does not make the mathematical endpoint values invalid.Implement the Bernoulli distribution, taking special care that a zero-mass outcome is excluded from the support of a probability distribution:The precondition excludes NaN and all values outside , including infinities. The two masses are nonnegative and sum mathematically to one. Endpoint domains are singleton lists, while an interior parameter lists both outcomes exactly once. Treating
class Flip(p: Double) extends FiniteDist[Boolean] {
require(!p.isNaN && 0.0 <= p && p <= 1.0)
def getDom: List[Boolean] =
if (p == 0.0) List(false)
else if (p == 1.0) List(true)
else List(true, false)
def getProb(x: Boolean): Double =
if (x) p else 1.0 - p
}getDom as always containing both Booleans would violate its positive-support contract at the endpoints.Inspect index zero for emptiness and the head; removing the head shifts the query in the other direction:The no-holes invariant makes the zero-index test sufficient for emptiness. Both
null' :: List a -> Bool
null' (List f) = case f 0 of
Nothing -> True
Just _ -> False
head' :: List a -> a
head' (List f) = case f 0 of
Nothing -> error "head of empty representation"
Just x -> x
tail' :: List a -> List a
tail' input@(List f)
| null' input = error "tail of empty representation"
| otherwise = List query
where
query i
| i < 0 || i == maxBound = Nothing
| otherwise = f (i+1)head' and tail' have the precondition that the representation is nonempty and signal an error when it is violated. On a nonempty representation, tail' removes exactly one position. The overflow guard is consistent with finite lengths at most maxBound.Maintain opposite signs at the endpoints, rather than assuming that the polynomial is monotone:
def floorRoot(X: Int): Int = {
require(X > 0 && eval(0.0) <= 0.0 && eval(X.toDouble) > 0.0)
var lo = 0
var hi = X
while (hi - lo > 1) {
val mid = lo + (hi - lo) / 2
if (eval(mid.toDouble) <= 0.0) lo = mid
else hi = mid
}
lo
}The loop invariant is and . Both branches preserve those signs, even when there are other roots of a polynomial or sign reversals elsewhere. When the interval width exceeds one, the midpoint is strictly between its endpoints, so the positive integer loop variant
hi - lo strictly decreases. On exit hi == lo + 1, giving the required postcondition . Continuity establishes a root of a polynomial between those adjacent integers, possibly at the left endpoint.The width is at most halved with rounding upwards, so there are at most midpoint evaluations, plus two endpoint evaluations in the precondition check. The midpoint expression avoids adding two large endpoints. The explicit positive upper-bound precondition supplies the interval order assumed in the question. For the literal computed-
eval contract, the same proof holds provided evaluated midpoints are not NaN; identifying its bracket with the real polynomial additionally requires trustworthy evaluation signs.