X-Git-Url: http://git.subgeniuskitty.com/surreal-numbers/.git/blobdiff_plain/24271b43a4bb149a3904ad2a1e4fdf439d654ad0..7757890d8346c126a73043f0ca3660996d35c02d:/chapter-1-experiments/ch1-breeding-numbers.go diff --git a/chapter-1-experiments/ch1-breeding-numbers.go b/chapter-1-experiments/ch1-breeding-numbers.go index 1e53fc3..c5baa14 100644 --- a/chapter-1-experiments/ch1-breeding-numbers.go +++ b/chapter-1-experiments/ch1-breeding-numbers.go @@ -4,9 +4,400 @@ package main import ( + "flag" "fmt" + "log" + "os" + "runtime/pprof" + "sort" ) +// ============================================================================= + +// 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 { + // The elements in this slice are stored in order according to Axiom 3. + // Thus, numbers[0] < numbers[1] < ... < numbers[n]. + numbers []surrealNumber + nextUniqueID uint +} + +func (u *surrealUniverse) getIndex(num surrealNumber) int { + // Since getIndex() is only ever called on 'num' which already exists in + // the universe, we return the result directly, without bothering to + // confirm an exact match. + // + // Also, we can get away with a direct comparison of these floating point + // numbers since there will be an exact match, guaranteed since 'num' is + // already in the universe. + return sort.Search(u.cardinality(), func(i int) bool {return u.numbers[i].value >= num.value}) +} + +func (u *surrealUniverse) cardinality() int { + return len(u.numbers) +} + +func (u *surrealUniverse) insert(num surrealNumber) { + // 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.value = 0.0 + 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]) + if index+1 == u.cardinality() { + num.value = u.numbers[u.cardinality()-1].value + 1.0 + num.identifier = u.nextUniqueID + u.nextUniqueID++ + u.numbers = append(u.numbers, num) + } + } else { + index := u.getIndex(num.rightSet.members[0]) + if index == 0 { + num.value = u.numbers[0].value - 1.0 + num.identifier = u.nextUniqueID + u.nextUniqueID++ + 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.value = (num.leftSet.members[0].value + num.rightSet.members[0].value) / 2 + num.identifier = u.nextUniqueID + u.nextUniqueID++ + u.numbers = append(u.numbers[:rightIndex+1], u.numbers[rightIndex:]...) + u.numbers[rightIndex] = num + } + } +} + +// ============================================================================= + +// 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 +} + +func (u *surrealUniverse) isSimilar(numA, numB surrealNumber) bool { + if u.isRelated(numA, numB) && u.isRelated(numB, numA) { + return true + } + return false +} + +func (u *surrealUniverse) equal(left, right surrealNumber) bool { + return u.isSimilar(left, right) +} + +func (u *surrealUniverse) lessThanOrEqual(left, right surrealNumber) bool { + return u.isRelated(left, right) +} + +func (u *surrealUniverse) lessThan(left, right surrealNumber) bool { + return u.lessThanOrEqual(left, right) && !u.equal(left, right) +} + +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 + value float64 + identifier uint + generation int +} + +type surrealSet struct { + members []surrealNumber +} + +func (s *surrealSet) isMember(num surrealNumber, u surrealUniverse) bool { + for i := 0; i < len(s.members); i++ { + if u.equal(num, s.members[i]) { + return true + } + } + return false +} + +func (s *surrealSet) cardinality() int { + return len(s.members) +} + +func (s *surrealSet) insert(num surrealNumber, u surrealUniverse) { + if !s.isMember(num, u) { + s.members = append(s.members, num) + } +} + +func (s *surrealSet) remove(num surrealNumber, u surrealUniverse) { + for i := 0; i < len(s.members); i++ { + if u.equal(num, s.members[i]) { + s.members[i] = s.members[len(s.members)-1] + s.members = s.members[:len(s.members)-1] + } + } +} + +// ============================================================================= + func main() { - fmt.Println("Hello, World!") + // Flags to enable various performance profiling options. + cpuProfile := flag.String("cpuprofile", "", "Filename for saving CPU profile output.") + memProfile := flag.String("memprofile", "", "Filename for saving memory profile output.") + suppressOutput := flag.Bool("silent", false, "Suppress printing of numberline at end of simulation. Useful when profiling.") + + // Obtain termination conditions from user. + totalGenerations := flag.Int("generations", 2, "Number of generations of surreal numbers to breed.") + flag.Parse() + if *totalGenerations < 1 { + log.Fatal("ERROR: The argument to '-generations' must be greater than zero.") + } + remainingGenerations := *totalGenerations - 1 + + // Setup any CPU performance profiling requested by the user. This will run + // throughout program execution. + if *cpuProfile != "" { + cpuProFile, err := os.Create(*cpuProfile) + if err != nil { + log.Fatal("ERROR: Unable to open CPU profiling output file:", err) + } + pprof.StartCPUProfile(cpuProFile) + defer pprof.StopCPUProfile() + } + + // 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.nextUniqueID = 0 + fmt.Println("Seeding generation 0 by hand.") + universe.insert(surrealNumber{surrealSet{}, surrealSet{}, 0.0, 0, 0}) + + // Breed however many generations of numbers were requested by the user and + // add them all to the universe. + fmt.Printf("Breeding Generation:") + for generation := 1; generation <= remainingGenerations; generation++ { + // Give the user some idea of overall progress during long jobs. + if generation != 1 { + fmt.Printf(",") + } + fmt.Printf(" %d", generation) + + // First generate all possible new numbers in this generation. + potentiallyNewNumbers := permuteExistingNumbers(generation, universe) + // Attempt to add the new numbers to the universe. Any duplicates will + // be weeded out in the attempt. + addNumbersToUniverse(potentiallyNewNumbers, &universe) + } + fmt.Printf(".\n") + + // Setup any memory profiling requested by the user. This will snapshot + // the heap only at this point, after all numbers were generated. + if *memProfile != "" { + memProFile, err := os.Create(*memProfile) + if err != nil { + log.Fatal("ERROR: Unable to open heap profile output file:", err) + } + pprof.WriteHeapProfile(memProFile) + memProFile.Close() + } + + // Print the number line with generation on the horizontal axis and + // magnitude on the vertical axis. + if !(*suppressOutput) { + 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.") +} + +// ============================================================================= + +// We will only build permutations with 0 or 1 elements in each of the left and +// right sets. This covers all the reduced form permutations. Any longer +// permutations will be similar to one of the reduced forms. +func permuteExistingNumbers(generation int, universe surrealUniverse) []surrealNumber { + // Profiling indicates that most calls to growSlice() and memMove() + // originate in this function. Allocating a sufficiently large slice + // upfront results in 2x speed improvement and 1/3 reduction in heap usage. + numberOfPermutations := ((universe.cardinality()+1) * (universe.cardinality()+1)) - 1 + numbers := make([]surrealNumber, 0, numberOfPermutations) + + // First build permutations with 1 element in each set. + for i := 0; i < universe.cardinality(); i++ { + for j := 0; j < universe.cardinality(); j++ { + var leftSet, rightSet surrealSet + leftSet.insert(universe.numbers[i], universe) + rightSet.insert(universe.numbers[j], universe) + newSurrealNumber := surrealNumber{leftSet,rightSet,0.0,0,generation} + if isValidSurrealNumber(newSurrealNumber, universe) { + 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], universe) + newSurrealNumber := surrealNumber{tempSet,surrealSet{},0.0,0,generation} + numbers = append(numbers, newSurrealNumber) + newSurrealNumber = surrealNumber{surrealSet{},tempSet,0.0,0,generation} + numbers = append(numbers, newSurrealNumber) + } + + return numbers +} + +// 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? + if candidate.leftSet.cardinality() > 1 || candidate.rightSet.cardinality() > 1 { + 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 + } + } + } + } + + return true +} + +func addNumbersToUniverse(numbers []surrealNumber, universe *surrealUniverse) { + for i := 0; i < len(numbers); i++ { + universe.insert(numbers[i]) + } +} + +// 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 }