--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/progs/automata/thompson.sc	Sat Jul 04 21:57:33 2020 +0100
@@ -0,0 +1,180 @@
+// Thompson Construction
+//=======================
+
+import $file.dfa, dfa._ 
+import $file.nfa, nfa._
+import $file.enfa, enfa._
+
+
+// states for Thompson construction
+case class TState(i: Int) extends State
+
+object TState {
+  var counter = 0
+  
+  def apply() : TState = {
+    counter += 1;
+    new TState(counter)
+  }
+}
+
+
+// some types abbreviations
+type NFAt = NFA[TState, Char]
+type NFAtrans = (TState, Char) :=> Set[TState]
+type eNFAtrans = (TState, Option[Char]) :=> Set[TState]
+
+
+// for composing an eNFA transition with an NFA transition
+// | is for set union
+implicit def nfaOps(f: eNFAtrans) = new {
+  def +++(g: NFAtrans) : eNFAtrans = 
+  { case (q, None) =>  applyOrElse(f, (q, None)) 
+    case (q, Some(c)) => applyOrElse(f, (q, Some(c))) | applyOrElse(g, (q, c))  }
+}
+
+ 
+// NFA that does not accept any string
+def NFA_ZERO(): NFAt = {
+  val Q = TState()
+  NFA(Set(Q), { case _ => Set() }, Set())
+}
+
+// NFA that accepts the empty string
+def NFA_ONE() : NFAt = {
+  val Q = TState()
+  NFA(Set(Q), { case _ => Set() }, Set(Q))
+}
+
+// NFA that accepts the string "c"
+def NFA_CHAR(c: Char) : NFAt = {
+  val Q1 = TState()
+  val Q2 = TState()
+  NFA(Set(Q1), { case (Q1, d) if (c == d) => Set(Q2) }, Set(Q2))
+}
+
+// sequence of two NFAs
+def NFA_SEQ(enfa1: NFAt, enfa2: NFAt) : NFAt = {
+  val new_delta : eNFAtrans = 
+    { case (q, None) if enfa1.fins(q) => enfa2.starts }
+  
+  eNFA(enfa1.starts, 
+       new_delta +++ enfa1.delta +++ enfa2.delta, 
+       enfa2.fins)
+}
+
+// alternative of two NFAs
+def NFA_ALT(enfa1: NFAt, enfa2: NFAt) : NFAt = {
+  val new_delta : NFAtrans = { 
+    case (q, c) =>  applyOrElse(enfa1.delta, (q, c)) | 
+                    applyOrElse(enfa2.delta, (q, c)) }
+  val new_fins = (q: TState) => enfa1.fins(q) || enfa2.fins(q)
+
+  NFA(enfa1.starts | enfa2.starts, new_delta, new_fins)
+}
+
+// star of a NFA
+def NFA_STAR(enfa: NFAt) : NFAt = {
+  val Q = TState()
+  val new_delta : eNFAtrans = 
+    { case (Q, None) => enfa.starts
+      case (q, None) if enfa.fins(q) => Set(Q) }
+
+  eNFA(Set(Q), new_delta +++ enfa.delta, Set(Q))
+}
+
+
+// We are now ready to translate regular expressions
+// into DFAs (via eNFAs and NFAs, and the subset construction)
+
+// regular expressions
+abstract class Rexp
+case object ZERO extends Rexp                    // matches nothing
+case object ONE extends Rexp                     // matches the empty string
+case class CHAR(c: Char) extends Rexp            // matches a character c
+case class ALT(r1: Rexp, r2: Rexp) extends Rexp  // alternative
+case class SEQ(r1: Rexp, r2: Rexp) extends Rexp  // sequence
+case class STAR(r: Rexp) extends Rexp            // star
+
+// thompson construction 
+def thompson (r: Rexp) : NFAt = r match {
+  case ZERO => NFA_ZERO()
+  case ONE => NFA_ONE()
+  case CHAR(c) => NFA_CHAR(c)  
+  case ALT(r1, r2) => NFA_ALT(thompson(r1), thompson(r2))
+  case SEQ(r1, r2) => NFA_SEQ(thompson(r1), thompson(r2))
+  case STAR(r1) => NFA_STAR(thompson(r1))
+}
+
+//optional regular expression (one or zero times)
+def OPT(r: Rexp) = ALT(r, ONE)
+
+//n-times regular expression (explicitly expanded)
+def NTIMES(r: Rexp, n: Int) : Rexp = n match {
+  case 0 => ONE
+  case 1 => r
+  case n => SEQ(r, NTIMES(r, n - 1))
+}
+
+
+def tmatches_nfa(r: Rexp, s: String) : Boolean =
+  thompson(r).accepts(s.toList)
+
+def tmatches_nfa2(r: Rexp, s: String) : Boolean =
+  thompson(r).accepts2(s.toList)
+
+// dfas via subset construction
+def tmatches_dfa(r: Rexp, s: String) : Boolean =
+  subset(thompson(r)).accepts(s.toList)
+
+// Test Cases
+//============
+
+// the evil regular expression  a?{n} a{n}
+def EVIL1(n: Int) : Rexp = SEQ(NTIMES(OPT(CHAR('a')), n), NTIMES(CHAR('a'), n))
+
+// the evil regular expression (a*)*b
+val EVIL2 : Rexp = SEQ(STAR(STAR(CHAR('a'))), CHAR('b'))
+
+//for measuring time
+def time_needed[T](i: Int, code: => T) = {
+  val start = System.nanoTime()
+  for (j <- 1 to i) code
+  val end = System.nanoTime()
+  (end - start)/(i * 1.0e9)
+}
+
+// the size of the NFA can be large, 
+// thus slowing down the breadth-first search
+
+for (i <- 1 to 13) {
+  println(i + ": " + "%.5f".format(time_needed(2, tmatches_nfa(EVIL1(i), "a" * i))))
+}
+
+for (i <- 1 to 100 by 5) {
+  println(i + " " + "%.5f".format(time_needed(2, tmatches_nfa(EVIL2, "a" * i))))
+}
+
+
+// the backtracking that is needed in depth-first 
+// search can be painfully slow
+
+for (i <- 1 to 8) {
+  println(i + " " + "%.5f".format(time_needed(2, tmatches_nfa2(EVIL2, "a" * i))))
+}
+
+
+
+// while my thompson->enfa->subset->partial-function-chain
+// is probably not the most effcient way to obtain a fast DFA 
+// (the test below should be much faster with a more direct 
+// construction), in general the DFAs can be slow because of 
+// the state explosion in the subset construction
+
+for (i <- 1 to 13) {
+  println(i + ": " + "%.5f".format(time_needed(2, tmatches_dfa(EVIL1(i), "a" * i))))
+}
+
+for (i <- 1 to 100 by 5) {
+  println(i + " " + "%.5f".format(time_needed(2, tmatches_dfa(EVIL2, "a" * i))))
+}