Final commit of 'initial version' of program for breeding surreal numbers. Program...

diff --git a/chapter-1-experiments/ch1-breeding-numbers.go b/chapter-1-experiments/ch1-breeding-numbers.go
index b990361..9e9e637 100644 (file)
--- a/chapter-1-experiments/ch1-breeding-numbers.go
+++ b/chapter-1-experiments/ch1-breeding-numbers.go
@@ -11,28 +11,36 @@ import (
 
 // =============================================================================
 
 
 // =============================================================================
 

-// TODO: Anywhere we do simple comparisons for number equality testing, we need
-// to upgrade to dual-LEQ comparisons.
+// This program recognizes four types of numbers:
+//   - Surreal symbols, like those generated by Axiom 1.
+//   - Surreal numbers, like those selected by Axiom 2.
+//   - Reduced form numbers, per Definition 7.
+//   - Named form numbers, which are the equivalence classes induced by
+//         Definition 6, represented by the first representative reduced
+//         form number encountered at runtime.
+//
+// Although the surrealNumber type can represent any of these four types of
+// numbers, the surrealUniverse contains only named form numbers.
+//
+// All equality tests (e.g. eq, geq, leq) in this program actually test for
+// similarity per Definition 6.

 
 // =============================================================================
 
 type surrealUniverse struct {
 
 // =============================================================================
 
 type surrealUniverse struct {

-    // TODO: Note that the elements of this array are maintained in order
-    // w.r.t. Axiom 2. Maybe I should just rename number -> numberLine?
+    // The elements in this slice are stored in order according to Axiom 3.
+    // Thus, numbers[0] < numbers[1] < ... < numbers[n].

     numbers []surrealNumber
     numbers []surrealNumber

-    nextUniqueID int
+    nextUniqueID uint

 }
 
 }
 

-func (u *surrealUniverse) getIndex(num surrealNumber) (exists bool, index int) {
-    // TODO: After I implement a LEQ comparison function, the fact that
-    // u.numbers[] is ordered by that comparison will allow me to greatly
-    // shorten the search.
+func (u *surrealUniverse) getIndex(num surrealNumber) int {

     for i := 0; i < len(u.numbers); i++ {
     for i := 0; i < len(u.numbers); i++ {

-        if num.number == u.numbers[i].number {
-            return true, i
+        if num.identifier == u.numbers[i].identifier {
+            return i

         }
     }
         }
     }

-    return false, 0
+    return -1 // This should be unreachable.

 }
 
 func (u *surrealUniverse) cardinality() int {
 }
 
 func (u *surrealUniverse) cardinality() int {


@@ -40,109 +48,139 @@ func (u *surrealUniverse) cardinality() int {
 }
 
 func (u *surrealUniverse) insert(num surrealNumber) {
 }
 
 func (u *surrealUniverse) insert(num surrealNumber) {

-    exists, _ := u.getIndex(num)
-    if !exists {
-        num.metadata.identifier = u.nextUniqueID
-        u.nextUniqueID++
-        // Find spot to insert based on ordering defined in Axiom 2.
-        // TODO: Insert some comments explaining the basic algorithm.
-        for i := 0; i < len(u.numbers); i++ {
-
-//            if num.number.leftSet != nil {
-//                // By assumption, everything is in reduced form by this point,
-//                // thus there is precisely one element at this time.
-//                _, leftIndex := u.getIndex(num.number.leftSet[0])
-//                _, rightIndex := u.getIndex(u.numbers[i])
-//                if leftIndex >= rightIndex {
-//                    // num is NOT LEQ to u.numbers[i]
-//                }
-//            }
-//            if u.numbers[i].rightSet != nil {
-//                // TODO: How do I check if the second number's rightSet is less
-//                // than or equal to the first number? After all, the first
-//                // number isn't on the numberline yet.
-//            }
-
-// =================================================================================================
-//// Algorithm
-//foreach u.numbers as oldNum, in increasing order:
-//    if newNum <= oldNum:
-//        if oldNum <= newNum:
-//            // the number are the same, just in a different reduced form.
-//        else:
-//            // Insert newNum immediately before oldNum
-//    else:
-//        if more oldNums exist:
-//            // loop and try the next oldNum
-//        else:
-//            // Found a new largest number?
-// =================================================================================================
-
-//            if !lessThanOrEqual(num, u.numbers[i]) {
-//                if i+1 < len(u.numbers) {
-//                    // There are still more entries to check.
-//                    continue
-//                } else {
-//                    // New largest number. Append to numberline.
-//                    u.numbers = append(u.numbers, num)
-//                }
-//            } else {
-//                // Found insertion spot between two existing numbers.
-//                u.numbers = append(u.numbers[:i], num, u.numbers[i:]...)
-//            }
-        }
+    // Before we can insert a number into the universe, we must first determine
+    // if it is newly discovered. We do this by looking at its ancestors, all
+    // guaranteed to be on the number line by construction, finding three
+    // cases:
+    //
+    //   1. There are zero ancestors.
+    //      We have rediscovered 0 and can abort the insertion unless creating
+    //      a brand new universe.
+    //
+    //   2. There is one ancestor.
+    //      If the number extends the number line, it is new, otherwise it will
+    //      be similar to a form with two ancestors and should be ignored since
+    //      the two ancestor form is a better visual representation.
+    //
+    //   3. There are two ancestors.
+    //      The number is only new if both ancestors are side-by-side on the
+    //      number line.
+
+    ancestorCount := num.leftSet.cardinality() + num.rightSet.cardinality()
+    switch ancestorCount {
+        case 0:
+            if u.cardinality() == 0 {
+                num.identifier = u.nextUniqueID
+                u.nextUniqueID++
+                u.numbers = append(u.numbers, num)
+            }
+        case 1:
+            if num.leftSet.cardinality() == 1{
+                index := u.getIndex(num.leftSet.members[0])
+                num.identifier = u.nextUniqueID
+                u.nextUniqueID++
+                if index+1 == u.cardinality() {
+                    u.numbers = append(u.numbers, num)
+                }
+            } else {
+                index := u.getIndex(num.rightSet.members[0])
+                num.identifier = u.nextUniqueID
+                u.nextUniqueID++
+                if index == 0 {
+                    u.numbers = append(u.numbers[:1], u.numbers...)
+                    u.numbers[0] = num
+                }
+            }
+        case 2:
+            leftIndex := u.getIndex(num.leftSet.members[0])
+            rightIndex := u.getIndex(num.rightSet.members[0])
+            if leftIndex+1 == rightIndex {
+                num.identifier = u.nextUniqueID
+                u.nextUniqueID++
+                u.numbers = append(u.numbers[:rightIndex+1], u.numbers[rightIndex:]...)
+                u.numbers[rightIndex] = num
+            }

     }
 }
 
     }
 }
 

-func (u *surrealUniverse) remove(num surrealNumber) {
-    for i := 0; i < len(u.numbers); i++ {
-        if num.number == u.numbers[i].number {
-            u.numbers = append(u.numbers[:i], u.numbers[i+1:]...)
+// =============================================================================
+
+// Note: The following comparison functions apply only to numbers already in
+// the universe, not to new numbers constructed from the universe.  To
+// determine where a (potentially) new number goes, first insert it into the
+// universe.
+
+// This function implements Axiom 3 and is the root of all other comparisons.
+func (u *surrealUniverse) isRelated(numA, numB surrealNumber) bool {
+    // By construction, no number in this program will ever have more than one
+    // element in its right or left sets, nor will those elements be of a form
+    // other than that 'registered' with the universe. That allows direct
+    // comparisons using the surrealNumber.identifier field via u.getIndex().
+
+    // ---------------------------------
+
+    // First we implement the check "no member of the first number's left set
+    // is greater than or equal to the second number".
+    if numA.leftSet.cardinality() == 1 {
+        if u.getIndex(numA.leftSet.members[0]) >= u.getIndex(numB) {
+            return false
+        }
+    }
+
+    // Now we implement the check "no member of the second number's right set
+    // is less than or equal to the first number".
+    if numB.rightSet.cardinality() == 1 {
+        if u.getIndex(numB.rightSet.members[0]) <= u.getIndex(numA) {
+            return false

         }
     }
         }
     }

+
+    return true

 }
 
 }
 

-// TODO: Note that this is only for numbers already in the universe, not for
-// new numbers constructed FROM the universe.  To determine where a new number
-// goes, insert it into the universe.
-func (u *surrealUniverse) lessThanOrEqual(left, right surrealNumber) bool {
-    _, leftIndex := u.getIndex(left)
-    _, rightIndex := u.getIndex(right)
-    if leftIndex <= rightIndex {
+func (u *surrealUniverse) isSimilar(numA, numB surrealNumber) bool {
+    if u.isRelated(numA, numB) && u.isRelated(numB, numA) {

         return true
         return true

-    } else {
-        return false

     }
     }

+    return false

 }
 
 }
 

-// =============================================================================
+func (u *surrealUniverse) equal(left, right surrealNumber) bool {
+    return u.isSimilar(left, right)
+}

 
 

-type surrealNumber struct {
-    number surrealValue
-    metadata surrealMetadata
+func (u *surrealUniverse) lessThanOrEqual(left, right surrealNumber) bool {
+    return u.isRelated(left, right)

 }
 
 }
 

-// TODO: Note that this is split from the metadata so we can do direct
-// comparisons when numbers are in reduced form.
-type surrealValue struct {
-    leftSet *surrealSet
-    rightSet *surrealSet
+func (u *surrealUniverse) lessThan(left, right surrealNumber) bool {
+    return u.lessThanOrEqual(left, right) && !u.equal(left, right)

 }
 
 }
 

-type surrealMetadata struct {
-    identifier int
-    generation int
+func (u *surrealUniverse) greaterThanOrEqual(left, right surrealNumber) bool {
+    return !u.lessThan(left, right)
+}
+
+func (u *surrealUniverse) greaterThan(left, right surrealNumber) bool {
+    return !u.isRelated(left, right)

 }
 
 // =============================================================================
 
 }
 
 // =============================================================================
 

+type surrealNumber struct {
+    leftSet surrealSet
+    rightSet surrealSet
+    identifier uint
+    generation int
+}
+

 type surrealSet struct {
     members []surrealNumber
 }
 
 type surrealSet struct {
     members []surrealNumber
 }
 

-func (s *surrealSet) isMember(num surrealNumber) bool {
+func (s *surrealSet) isMember(num surrealNumber, u surrealUniverse) bool {

     for i := 0; i < len(s.members); i++ {
     for i := 0; i < len(s.members); i++ {

-        if num.number == s.members[i].number {
+        if u.equal(num, s.members[i]) {

             return true
         }
     }
             return true
         }
     }


@@ -153,15 +191,15 @@ func (s *surrealSet) cardinality() int {
     return len(s.members)
 }
 
     return len(s.members)
 }
 

-func (s *surrealSet) insert(num surrealNumber) {
-    if !s.isMember(num) {
+func (s *surrealSet) insert(num surrealNumber, u surrealUniverse) {
+    if !s.isMember(num, u) {

         s.members = append(s.members, num)
     }
 }
 
         s.members = append(s.members, num)
     }
 }
 

-func (s *surrealSet) remove(num surrealNumber) {
+func (s *surrealSet) remove(num surrealNumber, u surrealUniverse) {

     for i := 0; i < len(s.members); i++ {
     for i := 0; i < len(s.members); i++ {

-        if num.number == s.members[i].number {
+        if u.equal(num, s.members[i]) {

             s.members[i] = s.members[len(s.members)-1]
             s.members = s.members[:len(s.members)-1]
         }
             s.members[i] = s.members[len(s.members)-1]
             s.members = s.members[:len(s.members)-1]
         }


@@ -172,27 +210,38 @@ func (s *surrealSet) remove(num surrealNumber) {
 
 func main() {
     // Obtain termination conditions from user.
 
 func main() {
     // Obtain termination conditions from user.

-    remainingGenerations := flag.Int("generations", 2, "Number of generations of surreal numbers to breed.")
+    totalGenerations := flag.Int("generations", 2, "Number of generations of surreal numbers to breed.")

     flag.Parse()
     flag.Parse()

-    if *remainingGenerations < 1 {
+    if *totalGenerations < 1 {

         log.Fatal("ERROR: The argument to '-generations' must be greater than zero.")
     }
         log.Fatal("ERROR: The argument to '-generations' must be greater than zero.")
     }

+    remainingGenerations := *totalGenerations - 1

 
     // Build a universe to contain all the surreal numbers we breed.
     // Seed it by hand with the number zero as generation-0.
     var universe surrealUniverse
 
     // Build a universe to contain all the surreal numbers we breed.
     // Seed it by hand with the number zero as generation-0.
     var universe surrealUniverse

-    universe.insert(surrealNumber{surrealValue{nil, nil}, surrealMetadata{0, 0}})
+    universe.nextUniqueID = 0
+    universe.insert(surrealNumber{surrealSet{}, surrealSet{}, 0, 0})

 
 

-    for generation := 1; generation <= *remainingGenerations; generation++ {
+    // Breed however many generations of numbers were requested by the user and
+    // add them all to the universe.
+    for generation := 1; generation <= remainingGenerations; generation++ {
+        // First generate all possible reduced form symbols per Axiom 1.

         potentialNumbers := permuteExistingNumbers(generation, universe)
         potentialNumbers := permuteExistingNumbers(generation, universe)

+        // Now prune out any symbols which are NOT valid numbers per Axiom 2.

         validNumbers := pruneInvalidNumbers(potentialNumbers, universe)
         validNumbers := pruneInvalidNumbers(potentialNumbers, universe)

+        // Attempt to add the new numbers to the universe. Any duplicates will
+        // be weeded out in the attempt.

         addNumbersToUniverse(validNumbers, &universe)
     }
 
         addNumbersToUniverse(validNumbers, &universe)
     }
 

-    // TODO: These are just for checking progress. How do I want to render the
-    // final generation vs magnitude chart?
-    fmt.Println("Total Numbers:", len(universe.numbers))
-    fmt.Println(universe.numbers)
+    // Print the number line with generation on the horizontal axis and
+    // magnitude on the vertical axis.
+    for i := 0; i < universe.cardinality(); i++ {
+        printlnNumber(universe.numbers[i])
+    }
+    fmt.Println("After", *totalGenerations, "generations, the universe contains", len(universe.numbers), "numbers.")
+    fmt.Println("If output looks poor, ensure tabstop is eight spaces and that output doesn't wrap.")

 }
 
 // =============================================================================
 }
 
 // =============================================================================


@@ -207,19 +256,19 @@ func permuteExistingNumbers(generation int, universe surrealUniverse) []surrealN
     for i := 0; i < universe.cardinality(); i++ {
         for j := 0; j < universe.cardinality(); j++ {
             var leftSet, rightSet surrealSet
     for i := 0; i < universe.cardinality(); i++ {
         for j := 0; j < universe.cardinality(); j++ {
             var leftSet, rightSet surrealSet

-            leftSet.insert(universe.numbers[i])
-            rightSet.insert(universe.numbers[j])
-            newSurrealNumber := surrealNumber{surrealValue{&leftSet,&rightSet},surrealMetadata{0,generation}}
+            leftSet.insert(universe.numbers[i], universe)
+            rightSet.insert(universe.numbers[j], universe)
+            newSurrealNumber := surrealNumber{leftSet,rightSet,0,generation}

             numbers = append(numbers, newSurrealNumber)
         }
     }
     // Now build permutations with one empty set and one 1-element set.
     for i := 0; i < universe.cardinality(); i++ {
         var tempSet surrealSet
             numbers = append(numbers, newSurrealNumber)
         }
     }
     // Now build permutations with one empty set and one 1-element set.
     for i := 0; i < universe.cardinality(); i++ {
         var tempSet surrealSet

-        tempSet.insert(universe.numbers[i])
-        newSurrealNumber := surrealNumber{surrealValue{&tempSet,nil},surrealMetadata{0,generation}}
+        tempSet.insert(universe.numbers[i], universe)
+        newSurrealNumber := surrealNumber{tempSet,surrealSet{},0,generation}

         numbers = append(numbers, newSurrealNumber)
         numbers = append(numbers, newSurrealNumber)

-        newSurrealNumber = surrealNumber{surrealValue{nil,&tempSet},surrealMetadata{0,generation}}
+        newSurrealNumber = surrealNumber{surrealSet{},tempSet,0,generation}

         numbers = append(numbers, newSurrealNumber)
     }
 
         numbers = append(numbers, newSurrealNumber)
     }
 


@@ -239,19 +288,20 @@ func pruneInvalidNumbers(candidates []surrealNumber, universe surrealUniverse) [
 // Although a non-reduced-form number is technically valid, for the purposes of
 // this program we're declaring it invalid.
 func isValidSurrealNumber(candidate surrealNumber, universe surrealUniverse) bool {
 // Although a non-reduced-form number is technically valid, for the purposes of
 // this program we're declaring it invalid.
 func isValidSurrealNumber(candidate surrealNumber, universe surrealUniverse) bool {

-    // Is the number in reduced form per Definition 3?
-    if candidate.number.leftSet.cardinality() > 1 || candidate.number.rightSet.cardinality() > 1 {
+    // Is the number in reduced form?
+    if candidate.leftSet.cardinality() > 1 || candidate.rightSet.cardinality() > 1 {

         return false
     }
 
         return false
     }
 

-    // Is the number valid per Axiom 1?
-    if candidate.number.leftSet != nil && candidate.number.rightSet != nil {
-        // TODO: This needs to be beefed up since <|> = <-1|1>.
-        if candidate.number.leftSet == candiate.number.rightSet {
-            return false
-        }
-        if !universe.lessThanOrEqual(candidate.number.leftSet[0], candidate.number.rightSet[0]) {
-            return false
+    // Is the number valid per Axiom 2?
+    if candidate.leftSet.cardinality() != 0 && candidate.rightSet.cardinality() != 0 {
+        for i := 0; i < candidate.leftSet.cardinality(); i++ {
+            for j := 0; j < candidate.leftSet.cardinality(); j++ {
+                // Since candidates are drawn from the universe, this comparison is allowed.
+                if universe.greaterThanOrEqual(candidate.leftSet.members[i], candidate.rightSet.members[j]) {
+                    return false
+                }
+            }

         }
     }
 
         }
     }
 


@@ -264,3 +314,48 @@ func addNumbersToUniverse(numbers []surrealNumber, universe *surrealUniverse) {
     }
 }
 
     }
 }
 

+// Pretty print a number, indenting to indicate generation.
+func printlnNumber(num surrealNumber) {
+    for i := 0; i < num.generation; i++ {
+        fmt.Printf(".\t\t")
+    }
+    fmt.Printf("%s = ", toBase26(num.identifier))
+    fmt.Printf("<")
+    if num.leftSet.cardinality() == 0 {
+        fmt.Printf("---")
+    } else {
+        fmt.Printf("%s", toBase26(num.leftSet.members[0].identifier))
+    }
+    fmt.Printf("|")
+    if num.rightSet.cardinality() == 0 {
+        fmt.Printf("---")
+    } else {
+        fmt.Printf("%s", toBase26(num.rightSet.members[0].identifier))
+    }
+    fmt.Printf(">")
+    fmt.Printf("\n")
+}
+
+// This function is a total hack, intended only to avoid printing identifiers
+// as actual numbers since we don't want to imply any particular association to
+// specific 'real' numbers yet. Instead, print them with [a-z] as base-26
+// integers.
+func toBase26(num uint) (base26String []byte) {
+    // For alignment reasons, we never print more than three digits. Thus, any
+    // number greater than (26^3)-1 is out of range.
+    if num >= 26*26*26 {
+        base26String = []byte("xxx")
+    } else {
+        var firstByte, secondByte, thirdByte byte
+        thirdByte = byte((num % 26) + uint('a'));
+        num = num / 26;
+        secondByte = byte((num % 26) + uint('a'));
+        num = num / 26;
+        firstByte = byte((num % 26) + uint('a'));
+        num = num / 26;
+        base26String = append(base26String, firstByte)
+        base26String = append(base26String, secondByte)
+        base26String = append(base26String, thirdByte)
+    }
+    return base26String
+}