# HG changeset patch
# User Christian Urban <christian dot urban at kcl dot ac dot uk>
# Date 1384267246 0
# Node ID d35b2ee2e788b52523779d85890d950b056a8cda
# Parent 4e8482e505904c0d7ce36254a2f902a0dd62837b
add
diff -r 4e8482e50590 -r d35b2ee2e788 programs/prove2.scala
--- a/programs/prove2.scala Tue Nov 12 10:51:02 2013 +0000
+++ b/programs/prove2.scala Tue Nov 12 14:40:46 2013 +0000
@@ -1,5 +1,6 @@
import scala.language.implicitConversions
import scala.language.reflectiveCalls
+import scala.util._
abstract class Term
case class Var(s: String) extends Term
@@ -85,34 +86,32 @@
case _ => ()
}
-
-
// function that calls prove and returns immediately once a proof is found
def run (j : Judgement) : Unit = {
- try {
- def sc () = { println ("Yes!"); throw new Exception }
- prove(j, sc)
- }
- catch { case e: Exception => () }
+ def sc () = { println ("Yes!"); throw new Exception }
+ Try(prove(j, sc)) getOrElse ()
}
+run (goal)
+
run (Set[Form]() |- False -> Del)
run (Set[Form]() |- True -> Del)
run (Set[Form]() |- Del -> True)
+run (Set[Form]() |- Del -> Del)
+run (Set[Form]() |- Del -> Or(False, Del))
-run (goal)
val Gamma1 : Set[Form] =
Set( Admin says ((Bob says Del) -> Del),
Bob says Del )
val goal1 = Gamma1 |- Del // not provable
-
run (goal1)
-run (Set[Form]() |- (Del -> Del))
-run (Set[Form]() |- (Del -> Or(False, Del)))
+val f1 = Pred("F1", Nil)
+val f2 = Pred("F2", Nil)
+run (Set[Form](And(f1, f2)) |- And(f2, f1))
val Chr = "Christian"
diff -r 4e8482e50590 -r d35b2ee2e788 progs/prove.scala
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/progs/prove.scala Tue Nov 12 14:40:46 2013 +0000
@@ -0,0 +1,131 @@
+import scala.language.implicitConversions
+import scala.language.reflectiveCalls
+import scala.util._
+
+abstract class Term
+case class Var(s: String) extends Term
+case class Const(s: String) extends Term
+case class Fun(s: String, ts: List[Term]) extends Term
+
+abstract class Form
+case object True extends Form
+case object False extends Form
+case class Pred(s: String, ts: List[Term]) extends Form
+case class Imp(f1: Form, f2: Form) extends Form
+case class Says(p: String, f: Form) extends Form
+case class And(f1: Form, f2: Form) extends Form
+case class Or(f1: Form, f2: Form) extends Form
+
+case class Judgement(gamma: Set[Form], f: Form) {
+ def lhs = gamma
+ def rhs = f
+}
+
+// some syntactic sugar
+implicit def FormOps(f1: Form) = new {
+ def -> (f2: Form) = Imp(f1, f2)
+}
+implicit def StringOps(p: String) = new {
+ def says (f: Form) = Says(p, f)
+}
+implicit def SetFormOps(gamma: Set[Form]) = new {
+ def |- (f: Form) : Judgement = Judgement(gamma, f)
+}
+
+val Admin = "Admin"
+val Bob = "Bob"
+val Del = Pred("del_file", Nil)
+
+val Gamma: Set[Form] =
+ Set( (Admin says Del) -> Del,
+ Admin says ((Bob says Del) -> Del),
+ Bob says Del )
+
+val goal = Gamma |- Del // request: provable or not?
+
+def partitions[A](s: Set[A]): Set[(A, Set[A])] =
+ s.map (e => (e, s - e))
+
+
+def prove(j: Judgement, sc: () => Unit) : Unit = {
+ if (j.lhs.contains(j.rhs)) sc () // Axiom rule
+ else {
+ prove1(j, sc);
+ for ((f, lhs_rest) <- partitions(j.lhs)) prove2(f, lhs_rest, j.rhs, sc)
+ }
+}
+
+def prove1(j: Judgement, sc: () => Unit) : Unit =
+ j.rhs match {
+ case True => sc ()
+ case False => ()
+ case Imp(f1, f2) => prove(j.lhs + f1 |- f2, sc)
+ case Says(p, f1) => prove(j.lhs |- f1, sc)
+ case Or(f1, f2) =>
+ { prove(j.lhs |- f1, sc);
+ prove(j.lhs |- f2, sc) }
+ case And(f1, f2) =>
+ prove(j.lhs |- f1,
+ () => prove(j.lhs |- f2, sc))
+ case _ => ()
+ }
+
+def prove2(f: Form, lhs_rest: Set[Form], rhs: Form, sc: () => Unit) : Unit =
+ f match {
+ case True => prove(lhs_rest |- rhs, sc)
+ case False => sc ()
+ case And(f1, f2) =>
+ prove(lhs_rest + f1 + f2 |- rhs, sc)
+ case Imp(f1, f2) =>
+ prove(lhs_rest |- f1,
+ () => prove(lhs_rest + f2 |- rhs, sc))
+ case Or(f1, f2) =>
+ prove(lhs_rest + f1 |- rhs,
+ () => prove(lhs_rest + f2 |- rhs, sc))
+ case Says(p, Imp(f1, f2)) =>
+ prove(lhs_rest |- Says(p, f1),
+ () => prove(lhs_rest + Says(p, f2) |- rhs, sc))
+ case _ => ()
+ }
+
+// function that calls prove and returns immediately once a proof is found
+def run (j : Judgement) : Unit = {
+ def sc () = { println ("Yes!"); throw new Exception }
+ Try(prove(j, sc)) getOrElse ()
+}
+
+run (goal)
+
+run (Set[Form]() |- False -> Del)
+run (Set[Form]() |- True -> Del)
+run (Set[Form]() |- Del -> True)
+run (Set[Form]() |- Del -> Del)
+run (Set[Form]() |- Del -> Or(False, Del))
+
+
+val Gamma1 : Set[Form] =
+ Set( Admin says ((Bob says Del) -> Del),
+ Bob says Del )
+
+val goal1 = Gamma1 |- Del // not provable
+run (goal1)
+
+
+val f1 = "P" says Pred("F1", Nil)
+val f2 = "Q" says Pred("F2", Nil)
+run (Set[Form](And(f1, f2)) |- And(f2, f1))
+
+
+val Chr = "Christian"
+val HoD = "Peter"
+val Email = Pred("may_btain_email", List(Const(Chr)))
+val AtLib = Pred("is_at_library", List(Const(Chr)))
+val Chr_Staff = Pred("is_staff", List(Const(Chr)))
+
+val Policy_HoD = (HoD says Chr_Staff) -> Chr_Staff
+val Policy_Lib = And(Chr_Staff, AtLib) -> Email
+val HoD_says = HoD says Chr_Staff
+
+run (Set[Form](AtLib, Policy_HoD, Policy_Lib, HoD_says) |- Email)
+
+
diff -r 4e8482e50590 -r d35b2ee2e788 slides/slides06.pdf
Binary file slides/slides06.pdf has changed
diff -r 4e8482e50590 -r d35b2ee2e788 slides/slides06.tex
--- a/slides/slides06.tex Tue Nov 12 10:51:02 2013 +0000
+++ b/slides/slides06.tex Tue Nov 12 14:40:46 2013 +0000
@@ -606,7 +606,7 @@
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[c]
-\frametitle{Sudoku}
+\frametitle{BTW}
Are there sudokus that cannot be solved?\pause