--- a/scala/recs.scala Tue Feb 26 17:39:47 2013 +0000
+++ b/scala/recs.scala Tue Feb 26 23:44:57 2013 +0000
@@ -5,6 +5,13 @@
abstract class Rec {
def eval(ns: List[Int]) : Int
def eval(ns: Int*) : Int = eval(ns.toList)
+
+ //syntactic convenience for composition
+ def o(r: Rec) = Cn(r.arity, this, List(r))
+ def o(r: Rec, f: Rec) = Cn(r.arity, this, List(r, f))
+ def o(r: Rec, f: Rec, g: Rec) = Cn(r.arity, this, List(r, f, g))
+
+ def arity : Int
}
case object Z extends Rec {
@@ -12,6 +19,7 @@
case n::Nil => 0
case _ => throw new IllegalArgumentException("Z args: " + ns)
}
+ override def arity = 1
}
case object S extends Rec {
@@ -19,18 +27,28 @@
case n::Nil => n + 1
case _ => throw new IllegalArgumentException("S args: " + ns)
}
+ override def arity = 1
}
case class Id(n: Int, m: Int) extends Rec {
override def eval(ns: List[Int]) =
if (ns.length == n && m < n) ns(m)
else throw new IllegalArgumentException("Id args: " + ns + "," + n + "," + m)
+
+ override def arity = n
}
case class Cn(n: Int, f: Rec, gs: List[Rec]) extends Rec {
override def eval(ns: List[Int]) =
- if (ns.length == n) f.eval(gs.map(_.eval(ns)))
- else throw new IllegalArgumentException("Cn: args")
+ if (ns.length == n && gs.forall(_.arity == n) && f.arity == gs.length) f.eval(gs.map(_.eval(ns)))
+ else throw new IllegalArgumentException("Cn args: " + ns + "," + n)
+
+ override def arity = n
+}
+
+// syntactic convenience
+object Cn {
+ def apply(n: Int, f: Rec, g: Rec) : Rec = new Cn(n, f, List(g))
}
case class Pr(n: Int, f: Rec, g: Rec) extends Rec {
@@ -43,6 +61,13 @@
}
}
else throw new IllegalArgumentException("Pr: args")
+
+ override def arity = n + 1
+}
+
+// syntactic convenience
+object Pr {
+ def apply(r: Rec, f: Rec) : Rec = Pr(r.arity, r, f)
}
case class Mn(n: Int, f: Rec) extends Rec {
@@ -52,41 +77,36 @@
override def eval(ns: List[Int]) =
if (ns.length == n) evaln(ns, 0)
else throw new IllegalArgumentException("Mn: args")
+
+ override def arity = n
}
// Recursive Function examples
-def arity(f: Rec) = f match {
- case Z => 1
- case S => 1
- case Id(n, _) => n
- case Cn(n, _, _) => n
- case Pr(n, _, _) => n + 1
- case Mn(n, _) => n
+def Const(n: Int) : Rec = n match {
+ case 0 => Z
+ case n => S o Const(n - 1)
}
-val Add = Pr(1, Id(1, 0), Cn(3, S, List(Id(3, 2))))
-val Mult = Pr(1, Z, Cn(3, Add, List(Id(3, 0), Id(3, 2))))
-val Twice = Cn(1, Mult, List(Id(1, 0), Const(2)))
-val Fourtimes = Cn(1, Mult, List(Id(1, 0), Const(4)))
-val Pred = Cn(1, Pr(1, Z, Id(3, 1)), List(Id(1, 0), Id(1, 0)))
-val Minus = Pr(1, Id(1, 0), Cn(3, Pred, List(Id(3, 2))))
-def Const(n: Int) : Rec = n match {
- case 0 => Z
- case n => Cn(1, S, List(Const(n - 1)))
-}
+val Add = Pr(Id(1, 0), S o Id(3, 2))
+val Mult = Pr(Z, Add o (Id(3, 0), Id(3, 2)))
+val Twice = Mult o (Id(1, 0), Const(2))
+val Fourtimes = Mult o (Id(1, 0), Const(4))
+val Pred = Pr(Z, Id(3, 1)) o (Id(1, 0), Id(1, 0))
+val Minus = Pr(Id(1, 0), Pred o Id(3, 2))
+val Power = Pr(Const(1), Mult o (Id(3, 0), Id(3, 2)))
+val Fact = Pr(Const(1), Mult o (Id(3, 2), S o Id(3, 1))) o (Id(1, 0), Id(1, 0))
-val Power = Pr(1, Const(1), Cn(3, Mult, List(Id(3, 0), Id(3, 2))))
-val Sign = Cn(1, Minus, List(Const(1), Cn(1, Minus, List(Const(1), Id(1, 0)))))
-val Less = Cn(2, Sign, List(Cn(2, Minus, List(Id(2, 1), Id(2, 0)))))
-val Not = Cn(1, Minus, List(Const(1), Id(1, 0)))
-val Eq = Cn(2, Minus, List(Cn(2, Const(1), List(Id(2, 0))),
- Cn(2, Add, List(Cn(2, Minus, List(Id(2, 0), Id(2, 1))),
- Cn(2, Minus, List(Id(2, 1), Id(2, 0)))))))
-val Noteq = Cn(2, Not, List(Cn(2, Eq, List(Id(2, 0), Id(2, 1)))))
-val Conj = Cn(2, Sign, List(Cn(2, Mult, List(Id(2, 0), Id(2, 1)))))
-val Disj = Cn(2, Sign, List(Cn(2, Add, List(Id(2, 0), Id(2, 1)))))
+val Sign = Minus o (Const(1), Minus o (Const(1), Id(1, 0)))
+val Less = Sign o (Minus o (Id(2, 1), Id(2, 0)))
+val Not = Minus o (Const(1), Id(1, 0))
+val Eq = Minus o (Const(1) o Id(2, 0),
+ Add o (Minus o (Id(2, 0), Id(2, 1)),
+ Minus o (Id(2, 1), Id(2, 0))))
+val Noteq = Not o (Eq o (Id(2, 0), Id(2, 1)))
+val Conj = Sign o (Mult o (Id(2, 0), Id(2, 1)))
+val Disj = Sign o (Add o (Id(2, 0), Id(2, 1)))
def Nargs(n: Int, m: Int) : List[Rec] = m match {
case 0 => Nil
@@ -94,112 +114,95 @@
}
def Sigma(f: Rec) = {
- val ar = arity(f)
- Pr(ar - 1, Cn(ar - 1, f, Nargs(ar - 1, ar - 1) :::
- List(Cn(ar - 1, Const(0), List(Id(ar - 1, 0))))),
- Cn(ar + 1, Add, List(Id(ar + 1, ar),
- Cn(ar + 1, f, Nargs(ar + 1, ar - 1) :::
- List(Cn(ar + 1, S, List(Id(ar + 1, ar - 1))))))))
+ val ar = f.arity
+ Pr(Cn(ar - 1, f, Nargs(ar - 1, ar - 1) ::: List(Const(0) o Id(ar - 1, 0))),
+ Add o (Id(ar + 1, ar),
+ Cn(ar + 1, f, Nargs(ar + 1, ar - 1) ::: List(S o (Id(ar + 1, ar - 1))))))
}
def Accum(f: Rec) = {
- val ar = arity(f)
- Pr(ar - 1, Cn(ar - 1, f, Nargs(ar - 1, ar - 1) :::
- List(Cn(ar - 1, Const(0), List(Id(ar - 1, 0))))),
- Cn(ar + 1, Mult, List(Id(ar + 1, ar),
- Cn(ar + 1, f, Nargs(ar + 1, ar - 1) :::
- List(Cn(ar + 1, S, List(Id(ar + 1, ar - 1))))))))
+ val ar = f.arity
+ Pr(Cn(ar - 1, f, Nargs(ar - 1, ar - 1) ::: List(Const(0) o Id(ar - 1, 0))),
+ Mult o (Id(ar + 1, ar),
+ Cn(ar + 1, f, Nargs(ar + 1, ar - 1) ::: List(S o Id(ar + 1, ar - 1)))))
}
def All(t: Rec, f: Rec) = {
- val ar = arity(f)
- Cn(ar - 1, Sign, List(Cn(ar - 1, Accum(f), Nargs(ar - 1, ar - 1) ::: List(t))))
+ val ar = f.arity
+ Sign o (Cn(ar - 1, Accum(f), Nargs(ar - 1, ar - 1) ::: List(t)))
}
def Ex(t: Rec, f: Rec) = {
- val ar = arity(f)
- Cn(ar - 1, Sign, List(Cn(ar - 1, Sigma(f), Nargs(ar - 1, ar - 1) ::: List(t))))
+ val ar = f.arity
+ Sign o (Cn(ar - 1, Sigma(f), Nargs(ar - 1, ar - 1) ::: List(t)))
}
//Definition on page 77 of Boolos's book.
def Minr(f: Rec) = {
- val ar = arity(f)
- val rq = All(Id(ar, ar - 1),
- Cn(ar + 1, Not, List(Cn(ar + 1, f, Nargs(ar + 1, ar - 1) ::: List(Id(ar + 1, ar))))))
- Sigma(rq)
+ val ar = f.arity
+ Sigma(All(Id(ar, ar - 1), Not o (Cn(ar + 1, f, Nargs(ar + 1, ar - 1) ::: List(Id(ar + 1, ar))))))
}
//Definition on page 77 of Boolos's book.
def Maxr(f: Rec) = {
- val ar = arity(f)
+ val ar = f.arity
val rt = Id(ar + 1, ar - 1)
- val rf1 = Cn(ar + 2, Less, List(Id(ar + 2, ar + 1), Id(ar + 2, ar)))
- val rf2 = Cn(ar + 2, Not, List(Cn (ar + 2, f, Nargs(ar + 2, ar - 1) ::: List(Id(ar + 2, ar + 1)))))
- val rf = Cn(ar + 2, Disj, List(rf1, rf2))
- val rq = All(rt, rf)
- val Qf = Cn(ar + 1, Not, List(rq))
+ val rf1 = Less o (Id(ar + 2, ar + 1), Id(ar + 2, ar))
+ val rf2 = Not o (Cn (ar + 2, f, Nargs(ar + 2, ar - 1) ::: List(Id(ar + 2, ar + 1))))
+ val Qf = Not o All(rt, Disj o (rf1, rf2))
Cn(ar, Sigma(Qf), Nargs(ar, ar) ::: List(Id(ar, ar - 1)))
}
-//Mutli-way branching statement on page 79 of Boolos's book
-def Branch(rs: List[(Rec, Rec)]) = {
- val ar = arity(rs.head._1)
-
- def Branch_aux(rs: List[(Rec, Rec)], l: Int) : Rec = rs match {
- case Nil => Cn(l, Z, List(Id(l, l - 1)))
- case (rg, rc)::recs => Cn(l, Add, List(Cn(l, Mult, List(rg, rc)), Branch_aux(recs, l)))
- }
-
- Branch_aux(rs, ar)
-}
-
-//Factorial
-val Fact = {
- val Fact_aux = Pr(1, Const(1), Cn(3, Mult, List(Id(3, 2), Cn(3, S, List(Id(3, 1))))))
- Cn(1, Fact_aux, List(Id(1, 0), Id(1, 0)))
-}
//Prime test
-val Prime = Cn(1, Conj, List(Cn(1, Less, List(Const(1), Id(1, 0))),
- All(Cn(1, Minus, List(Id(1, 0), Const(1))),
- All(Cn(2, Minus, List(Id(2, 0), Cn(2, Const(1), List(Id(2, 0))))),
- Cn(3, Noteq, List(Cn(3, Mult, List(Id(3, 1), Id(3, 2))), Id(3, 0)))))))
+val Prime = Conj o (Less o (Const(1), Id(1, 0)),
+ All(Minus o (Id(1, 0), Const(1)),
+ All(Minus o (Id(2, 0), Const(1) o Id(2, 0)),
+ Noteq o (Mult o (Id(3, 1), Id(3, 2)), Id(3, 0)))))
-//Returns the first prime number after n
+//Returns the first prime number after n (very slow for n > 4)
val NextPrime = {
- val R = Cn(2, Conj, List(Cn(2, Less, List(Id(2, 0), Id(2, 1))),
- Cn(2, Prime, List(Id(2, 1)))))
- Cn(1, Minr(R), List(Id(1, 0), Cn(1, S, List(Fact))))
+ val R = Conj o (Less o (Id(2, 0), Id(2, 1)), Prime o Id(2, 1))
+ Minr(R) o (Id(1, 0), S o Fact)
}
-val NthPrime = {
- val NthPrime_aux = Pr(1, Const(2), Cn(3, NextPrime, List(Id(3, 2))))
- Cn(1, NthPrime_aux, List(Id(1, 0), Id(1, 0)))
-}
+val NthPrime = Pr(Const(2), NextPrime o Id(3, 2)) o (Id(1, 0), Id(1, 0))
def Listsum(k: Int, m: Int) : Rec = m match {
- case 0 => Cn(k, Z, List(Id(k, 0)))
- case n => Cn(k, Add, List(Listsum(k, n - 1), Id(k, n - 1)))
+ case 0 => Z o Id(k, 0)
+ case n => Add o (Listsum(k, n - 1), Id(k, n - 1))
}
//strt-function on page 90 of Boolos, but our definition generalises
//the original one in order to deal with multiple input-arguments
def Strt(n: Int) = {
+
def Strt_aux(l: Int, k: Int) : Rec = k match {
- case 0 => Cn(l, Z, List(Id(l, 0)))
+ case 0 => Z o Id(l, 0)
case n => {
- val rec_dbound = Cn(l, Add, List(Listsum(l, n - 1), Cn(l, Const(n - 1), List(Id(l, 0)))))
- Cn(l, Add, List(Strt_aux(l, n - 1),
- Cn(l, Minus, List(Cn(l, Power, List(Cn(l, Const(2), List(Id(l, 0))),
- Cn(l, Add, List(Id(l, n - 1), rec_dbound)))),
- Cn(l, Power, List(Cn(l, Const(2), List(Id(l, 0))), rec_dbound))))))
+ val rec_dbound = Add o (Listsum(l, n - 1), Const(n - 1) o Id(l, 0))
+ Add o (Strt_aux(l, n - 1),
+ Minus o (Power o (Const(2) o Id(l, 0), Add o (Id(l, n - 1), rec_dbound)),
+ Power o (Const(2) o Id(l, 0), rec_dbound)))
}
}
- def Rmap(f: Rec, k: Int) = (0 until k).map{i => Cn(k, f, List(Id(k, i)))}.toList
+ def Rmap(f: Rec, k: Int) = (0 until k).map{i => f o Id(k, i)}.toList
Cn(n, Strt_aux(n, n), Rmap(S, n))
}
+
+//Mutli-way branching statement on page 79 of Boolos's book
+def Branch(rs: List[(Rec, Rec)]) = {
+
+ def Branch_aux(rs: List[(Rec, Rec)], l: Int) : Rec = rs match {
+ case Nil => Z o Id(l, l - 1)
+ case (rg, rc)::recs => Add o (Mult o (rg, rc), Branch_aux(recs, l))
+ }
+
+ Branch_aux(rs, rs.head._1.arity)
}
+
+}