(* Content-type: application/mathematica *)

(*** Wolfram Notebook File ***)
(* http://www.wolfram.com/nb *)

(* CreatedBy='Mathematica 6.0' *)

(*CacheID: 234*)
(* Internal cache information:
NotebookFileLineBreakTest
NotebookFileLineBreakTest
NotebookDataPosition[       145,          7]
NotebookDataLength[      6757,        210]
NotebookOptionsPosition[      6100,        184]
NotebookOutlinePosition[      6439,        199]
CellTagsIndexPosition[      6396,        196]
WindowFrame->Normal
ContainsDynamic->False*)

(* Beginning of Notebook Content *)
Notebook[{
Cell[BoxData[
 RowBox[{"(*", "  ", 
  RowBox[{
   RowBox[{"An", " ", "Instance", " ", "of", " ", "Godel", " ", 
    RowBox[{"Numbering", ":", " ", 
     RowBox[{
      RowBox[{"S", "^"}], "*", " ", "--"}]}]}], "\[Rule]", " ", "N"}], "  ", 
  "*)"}]], "Input",
 CellChangeTimes->{{3.397246461202574*^9, 3.397246515573707*^9}}],

Cell[BoxData[
 RowBox[{"(*", "  ", 
  RowBox[{
   RowBox[{"For", " ", "example"}], ",", " ", 
   RowBox[{"S", "=", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{"1", ",", "2", ",", "3", ",", "4", ",", "5", ",", "6"}], "}"}], 
     "."}]}]}], "  ", "*)"}]], "Input",
 CellChangeTimes->{{3.3972465566210327`*^9, 3.397246568322157*^9}, {
  3.397246601764076*^9, 3.39724660964218*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"s", "=", "6"}], ";", " ", 
  RowBox[{
   RowBox[{"F", "[", "alpha_", "]"}], ":=", 
   RowBox[{"Sum", "[", 
    RowBox[{
     RowBox[{
      RowBox[{"alpha", "[", 
       RowBox[{"[", "j", "]"}], "]"}], "*", 
      RowBox[{"s", "^", 
       RowBox[{"(", 
        RowBox[{
         RowBox[{"Length", "[", "alpha", "]"}], "-", "j"}], ")"}]}]}], ",", 
     RowBox[{"{", 
      RowBox[{"j", ",", "1", ",", 
       RowBox[{"Length", "[", "alpha", "]"}]}], "}"}]}], "]"}]}], 
  ";"}]], "Input",
 CellChangeTimes->{{3.3972466426351357`*^9, 3.397246647869728*^9}, {
  3.397246692470194*^9, 3.397246704095315*^9}, {3.3972467606134167`*^9, 
  3.397246822468696*^9}, {3.3972474634530773`*^9, 3.397247536745469*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{
   RowBox[{"alpha", "=", 
    RowBox[{"{", 
     RowBox[{
     "3", ",", "3", ",", "4", ",", "2", ",", "3", ",", "6", ",", "5", ",", 
      "5", ",", "1", ",", "3"}], "}"}]}], ";"}], "  ", 
  RowBox[{"(*", 
   RowBox[{
    RowBox[{
     RowBox[{" ", 
      RowBox[{"\[Not]", 
       RowBox[{"\[Not]", "\[LongRightArrow]"}]}], ")"}], "\[Not]"}], "|", 
    RowBox[{"pp", 
     RowBox[{"(", "\[Not]", "  "}]}]}], "*)"}]}]], "Input",
 CellChangeTimes->{{3.397247549606336*^9, 3.397247569910615*^9}, {
  3.397328988274497*^9, 3.397328995835107*^9}, {3.3973291360312567`*^9, 
  3.397329256642324*^9}}],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"F", "[", "alpha", "]"}]], "Input",
 CellChangeTimes->{{3.3972475745797253`*^9, 3.39724757719049*^9}}],

Cell[BoxData["36517365"], "Output",
 CellChangeTimes->{3.397247579024734*^9, 3.39731590678948*^9, 
  3.397329443837317*^9}]
}, Open  ]],

Cell[BoxData[
 RowBox[{
  RowBox[{
   RowBox[{"G", "[", "xi_", "]"}], ":=", "\[IndentingNewLine]", 
   RowBox[{"{", 
    RowBox[{
     RowBox[{
      RowBox[{"Quotient", "[", 
       RowBox[{
        RowBox[{"xi", "[", 
         RowBox[{"[", "1", "]"}], "]"}], ",", "s"}], "]"}], "-", 
      RowBox[{"KroneckerDelta", "[", 
       RowBox[{
        RowBox[{"Mod", "[", 
         RowBox[{
          RowBox[{"xi", "[", 
           RowBox[{"[", "1", "]"}], "]"}], ",", "s"}], "]"}], ",", "0"}], 
       "]"}]}], ",", "\[IndentingNewLine]", 
     RowBox[{
      RowBox[{"Mod", "[", 
       RowBox[{
        RowBox[{"xi", "[", 
         RowBox[{"[", "1", "]"}], "]"}], ",", "s"}], "]"}], "+", 
      RowBox[{
       RowBox[{"KroneckerDelta", "[", 
        RowBox[{
         RowBox[{"Mod", "[", 
          RowBox[{
           RowBox[{"xi", "[", 
            RowBox[{"[", "1", "]"}], "]"}], ",", "s"}], "]"}], ",", "0"}], 
        "]"}], "*", "s"}]}]}], "}"}]}], ";"}]], "Input",
 CellChangeTimes->{{3.397316383068304*^9, 3.397316427656505*^9}, {
   3.3973208440623198`*^9, 3.3973208897735567`*^9}, {3.397321609864888*^9, 
   3.3973216124884443`*^9}, {3.397322117608465*^9, 3.3973221176945343`*^9}, {
   3.397322800316576*^9, 3.3973228469311657`*^9}, {3.397328874425671*^9, 
   3.3973288981487913`*^9}, 3.3973294226722803`*^9}],

Cell[BoxData[{
 RowBox[{
  RowBox[{
   RowBox[{"x", "=", "36517365"}], ";"}], " "}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{
   RowBox[{"gamma", "=", 
    RowBox[{"{", "}"}]}], ";", 
   RowBox[{"in", "=", 
    RowBox[{"{", 
     RowBox[{"x", ",", "0"}], "}"}]}], ";"}], " "}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"While", "[", 
   RowBox[{
    RowBox[{"0", "<", 
     RowBox[{"in", "[", 
      RowBox[{"[", "1", "]"}], "]"}]}], ",", 
    RowBox[{"{", 
     RowBox[{
      RowBox[{"out", "=", 
       RowBox[{"G", "[", "in", "]"}]}], ",", " ", 
      RowBox[{"gamma", "=", 
       RowBox[{"Prepend", "[", 
        RowBox[{"gamma", ",", 
         RowBox[{"out", "[", 
          RowBox[{"[", "2", "]"}], "]"}]}], "]"}]}], ",", 
      RowBox[{"in", "=", "out"}]}], "}"}]}], "]"}], ";"}]}], "Input",
 CellChangeTimes->{{3.397326145973015*^9, 3.397326200706417*^9}, 
   3.397327156398979*^9, {3.397327393367257*^9, 3.397327481975182*^9}, {
   3.3973275977654543`*^9, 3.397327665113275*^9}, {3.3973276993410397`*^9, 
   3.397327737000002*^9}, {3.39732797699874*^9, 3.397328030908935*^9}, {
   3.397328115461523*^9, 3.397328119173086*^9}, {3.397328165450429*^9, 
   3.397328167217477*^9}, {3.3973283257679977`*^9, 3.3973283873650846`*^9}, {
   3.3973288526945353`*^9, 3.397328855298751*^9}, {3.39732894101987*^9, 
   3.3973289434300203`*^9}}],

Cell[CellGroupData[{

Cell[BoxData["gamma"], "Input",
 CellChangeTimes->{{3.397328137890663*^9, 3.397328138217667*^9}, {
  3.39732839651513*^9, 3.397328398137191*^9}}],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
  "3", ",", "3", ",", "4", ",", "2", ",", "3", ",", "6", ",", "5", ",", "5", 
   ",", "1", ",", "3"}], "}"}]], "Output",
 CellChangeTimes->{3.397328139461472*^9, 3.397328175507208*^9, 
  3.3973283362814198`*^9, 3.3973283992029333`*^9, 3.3973294518680058`*^9}]
}, Open  ]]
},
WindowSize->{710, 849},
WindowMargins->{{4, Automatic}, {Automatic, 4}},
FrontEndVersion->"6.0 for Mac OS X PowerPC (32-bit) (April 20, 2007)",
StyleDefinitions->"Default.nb"
]
(* End of Notebook Content *)

(* Internal cache information *)
(*CellTagsOutline
CellTagsIndex->{}
*)
(*CellTagsIndex
CellTagsIndex->{}
*)
(*NotebookFileOutline
Notebook[{
Cell[568, 21, 325, 8, 27, "Input"],
Cell[896, 31, 382, 10, 27, "Input"],
Cell[1281, 43, 738, 20, 27, "Input"],
Cell[2022, 65, 630, 18, 27, "Input"],
Cell[CellGroupData[{
Cell[2677, 87, 125, 2, 27, "Input"],
Cell[2805, 91, 123, 2, 27, "Output"]
}, Open  ]],
Cell[2943, 96, 1319, 35, 58, "Input"],
Cell[4265, 133, 1347, 34, 58, "Input"],
Cell[CellGroupData[{
Cell[5637, 171, 145, 2, 27, "Input"],
Cell[5785, 175, 299, 6, 27, "Output"]
}, Open  ]]
}
]
*)

(* End of internal cache information *)