'go fmt' pass on number breeding program.

[surreal-numbers] / chapter-1-experiments / ch1-breeding-numbers.go

// (c) 2020 Aaron Taylor <ataylor at subgeniuskitty dot com>
// See LICENSE.txt file for copyright and license details.

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() {
	// 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
}