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//
// Chapter02.swift
// ElementsOfProgramming
//
import Darwin
import EOP
func absoluteValue(x: Int) -> Int {
return x < 0 ? -x : x
} // unary operation
func euclideanNorm(x: Double, y: Double) -> Double {
return sqrt(x * x + y * y)
} // binary operation
func euclideanNorm(x: Double, y: Double, z: Double) -> Double {
return sqrt(x * x + y * y + z * z)
} // ternary operation
// Exercise 2.1
func definitionSpacePredicateIntegerAddition<T: FixedWidthInteger>(
x: T,
y: T
) -> Bool {
// Precondition: T.min <= x <= T.max ∧ T.min <= y <= T.max
if x > 0 && y > 0 { return y <= T.max - x }
if x < 0 && y < 0 { return y >= T.min - x }
return true
}
func powerUnary<DomainF: Distance>(
_ x: DomainF,
power n: N,
transformation: Transformation<DomainF>
) -> DomainF {
var x = x, n = n
assert(n >= 0)
// Precondition: n ≥ 0 ∧ (∀i ∈ N), 0 < i ≤ n ⇒ f^i(x) is defined
while n != 0 {
n = n - 1
x = transformation(x)
}
return x
}
// See Distance protocol in EOP/TypeFunctions.swift
//func distance<DomainF: Distance>(
// x: DomainF,
// y: DomainF,
// f: Transformation<DomainF>
//) -> DistanceType {
// var x = x
// // Precondition: y is reachable from x under f
// var n = N(0)
// while x != y {
// x = f(x)
// n = n + 1
// }
// return n
//}
func collisionPoint<DomainFP: Distance>(
start x: DomainFP,
transformation f: Transformation<DomainFP>,
definitionSpace p: UnaryPredicate<DomainFP>
) -> DomainFP {
// Precondition: p(x) ⇔ f(x) is defined
guard p(x) else { return x }
var slow = x // slow = f^0(x)
var fast = f(x) // fast = f^1(x)
// n ← 0 (completed iterations)
#if !XCODE
var ft = [x, fast]
var st = [x]
#endif
while fast != slow { // slow = f^n(x) ∧ fast = f^{2n+1}(x)
slow = f(slow) // slow = f^{n+1}(x) ∧ fast = f^{2n+1}(x)
#if !XCODE
st.append(slow)
#endif
guard p(fast) else { return fast }
fast = f(fast) // slow = f^{n+1}(x) ∧ fast = f^{2n+2}(x)
#if !XCODE
ft.append(fast)
#endif
guard p(fast) else { return fast }
fast = f(fast) // slow = f^{n+1}(x) ∧ fast = f^{2n+3}(x)
#if !XCODE
ft.append(fast)
#endif
} // n ← n + 1
#if !XCODE
ft.map { $0 }
st.map { $0 }
#endif
return fast // slow = f^n(x) ∧ fast = f^{2n+1}(x)
// Postcondition: return value is terminal point or collision point
}
func terminating<DomainFP: Distance>(
start x: DomainFP,
transformation f: Transformation<DomainFP>,
definitionSpace p: UnaryPredicate<DomainFP>
) -> Bool{
// Precondition: p(x) ⇔ f(x) is defined
return !p(collisionPoint(start: x,
transformation: f,
definitionSpace: p))
}
func collisionPointNonterminatingOrbit<DomainF: Distance>(
start x: DomainF,
transformation f: Transformation<DomainF>
) -> DomainF {
var slow = x // slow = f^0(x)
var fast = f(x) // fast = f^1(x)
// n ← 0 (completed iterations)
#if !XCODE
var ft = [x, fast]
var st = [x]
#endif
while fast != slow { // slow = f^n(x) ∧ fast = f^{2 n + 1}(x)
slow = f(slow) // slow = f^{n+1}(x) ∧ fast = f^{2n+1}(x)
#if !XCODE
st.append(slow)
#endif
fast = f(fast) // slow = f^{n+1}(x) ∧ fast = f^{2n+2}(x)
#if !XCODE
ft.append(fast)
#endif
fast = f(fast) // slow = f^{n+1}(x) ∧ fast = f^{2n+3}(x)
#if !XCODE
ft.append(fast)
#endif
} // n ← n + 1
#if !XCODE
ft.map { $0 }
st.map { $0 }
#endif
return fast // slow = f^n(x) ∧ fast = f^{2n+1}(x)
// Postcondition: return value is collision point
}
func circularNonterminatingOrbit<DomainF: Distance>(
start x: DomainF,
transformation f: Transformation<DomainF>
) -> Bool {
let cp = collisionPointNonterminatingOrbit(start: x,
transformation: f)
return x == f(cp)
}
func circular<DomainFP: Distance>(
start x: DomainFP,
transformation f: Transformation<DomainFP>,
definitionSpace p: UnaryPredicate<DomainFP>
) -> Bool {
// Precondition: p(x) ⇔ f(x) is defined
let cp = collisionPoint(start: x,
transformation: f,
definitionSpace: p)
return p(cp) && x == f(cp)
}
func convergentPoint<DomainF: Distance>(
x0: DomainF,
x1: DomainF,
transformation f: Transformation<DomainF>
) -> DomainF {
var x0 = x0, x1 = x1
// Precondition: (∃n ∈ DistanceType(F)), n ≥ 0 ∧ f^n(x0) = f^n(x1)
while x0 != x1 {
x0 = f(x0)
x1 = f(x1)
}
return x0
}
func connectionPointNonterminatingOrbit<DomainF: Distance>(
start x: DomainF,
transformation f: Transformation<DomainF>
) -> DomainF {
let cp = collisionPointNonterminatingOrbit(start: x,
transformation: f)
return convergentPoint(x0: x,
x1: f(cp),
transformation: f)
}
func connectionPoint<DomainFP: Distance>(
start x: DomainFP,
transformation f: Transformation<DomainFP>,
definitionSpace p: UnaryPredicate<DomainFP>
) -> DomainFP {
// Precondition: p(x) ⇔ f(x) is defined
let cp = collisionPoint(start: x,
transformation: f,
definitionSpace: p)
guard p(cp) else { return cp }
return convergentPoint(x0: x,
x1: f(cp),
transformation: f)
}
/// Exercise 2.3
func convergentPointGuarded<DomainF: Distance>(
x0: DomainF,
x1: DomainF,
y: DomainF,
transformation f: Transformation<DomainF>
) -> DomainF {
var x0 = x0, x1 = x1
// Precondition: reachable(x0, y, f) ∧ reachable(x1, y, f)
let d0 = x0.distance(to: y, transformation: f)
let d1 = x1.distance(to: y, transformation: f)
if d0 < d1 {
x1 = powerUnary(x1, power: d1 - d0, transformation: f)
} else if d1 < d0 {
x0 = powerUnary(x0, power: d0 - d1, transformation: f)
}
return convergentPoint(x0: x0,
x1: x1,
transformation: f)
}
func orbitStructureNonterminatingOrbit<DomainF: Distance>(
start x: DomainF,
transformation f: Transformation<DomainF>
) -> Triple<DistanceType, DistanceType, DomainF> {
let y = connectionPointNonterminatingOrbit(start: x,
transformation: f)
return Triple(m0: x.distance(to: y, transformation: f),
m1: f(y).distance(to: y, transformation: f),
m2: y)
}
func orbitStructure<DomainFP: Distance>(
start x: DomainFP,
transformation f: Transformation<DomainFP>,
definitionSpace p: UnaryPredicate<DomainFP>
) -> Triple<DistanceType, DistanceType, DomainFP> {
// Precondition: p(x) ⇔ f(x) is defined
let y = connectionPoint(start: x,
transformation: f,
definitionSpace: p)
let m = x.distance(to: y, transformation: f)
var n = N(0)
if p(y) { n = f(y).distance(to: y, transformation: f) }
// Terminating: m = h - 1 ∧ n = 0
// Otherwise: m = h ∧ n = c - 1
return Triple(m0: m, m1: n, m2: y)
}
#if !XCODE
// MARK: Playground examples
func playgroundDSPAddition() {
// Exercise 2.1
definitionSpacePredicateIntegerAddition(x: Int32.max, y: Int32.min)
definitionSpacePredicateIntegerAddition(x: Int32.max - 1, y: Int32.max)
definitionSpacePredicateIntegerAddition(x: Int64.max, y: Int64.min)
definitionSpacePredicateIntegerAddition(x: Int64.max - 1, y: Int64.max)
definitionSpacePredicateIntegerAddition(x: UInt.max, y: UInt.min)
definitionSpacePredicateIntegerAddition(x: UInt.max - 1, y: UInt.max)
}
// playgroundDSPAddition()
func playgroundRhoShapedOrbit() {
let f: Transformation<UInt> = { ($0 % 113 + 2) * 2 }
let x: UInt = 0
orbitStructureNonterminatingOrbit(start: x, transformation: f)
}
// playgroundRhoShapedOrbit()
func playgroundTerminatingOrbit() {
let f: Transformation<UInt> = { $0 + 2 }
let p: UnaryPredicate<UInt> = { $0 < 208 }
let x: UInt = 0
orbitStructure(start: x, transformation: f, definitionSpace: p)
}
// playgroundTerminatingOrbit()
func playgroundCircularOrbit() {
let f: Transformation<UInt> = { ($0 % 113 + 2) * 2 }
let x: UInt = 4
orbitStructureNonterminatingOrbit(start: x, transformation: f)
}
// playgroundCircularOrbit()
#endif
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