Module:Limits

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This module depends on the following other modules: Module: ET Module: Rational

Todo: add documentation

---

local rat = require("Module:Rational")
local ET = require("Module:ET")
local p = {}

-- compute all positive ratios n/m with n and m <= q modulo powers of equave
-- previous: already computed ratios for q-1
function p.limit_modulo_equave(q, equave, previous)
	equave = equave or 2
	local ratios = {}
	if previous then
		for n = 1, q do
			local a = rat.new(n, q)
			a = rat.modulo_mul(a, equave)
			local a_key = rat.as_ratio(a)

			local b = rat.new(q, n)
			b = rat.modulo_mul(b, equave)
			local b_key = rat.as_ratio(b)

			if previous[a_key] == nil then
				ratios[a_key] = a
			end
			if previous[b_key] == nil then
				ratios[b_key] = b
			end
		end
	else
		for n = 1, q do
			for m = 1, q do
				local a = rat.new(n, m)
				a = rat.modulo_mul(a, equave)
				local key = rat.as_ratio(a)
				ratios[key] = a
			end
		end
	end
	return ratios
end

-- compute q-integer limit
-- if a function `norm` and a number `max_norm` are provided, the output will be additionally restricted
function p.integer_limit(q, norm, max_norm)
	local check_norm = type(norm) == "function" and type(max_norm) == "number"
	local ratios = {}
	for n = 1, q do
		for m = 1, q do
			local a = rat.new(n, m)
			if not check_norm or norm(a) <= max_norm then
				local key = rat.as_ratio(a)
				ratios[key] = a
			end
		end
	end
	return ratios
end

-- check additive consistency for a set of ratios (modulo powers of equave):
--   approx(a*b) = approx(a) + approx(b) forall a, b: a, b, ab in ratios
-- `distinct`: whether distinct ratios are required to be mapped to distinct approximations
-- `previous`: already computed ratios for the previous iteraton
function p.additively_consistent(et, ratios, distinct, previous)
	distinct = distinct or false
	previous = previous or {}
	if distinct then
		local approx_set = {}
		for a_key, a in pairs(previous) do
			local a_approx = ET.approximate(et, rat.as_float(a)) % et.size
			if approx_set[a_approx] then
				if not rat.eq(rat.modulo_mul(rat.div(a, approx_set[a_approx]), et.equave), 1) then
					mw.log(a_key .. " -> " .. a_approx .. ": conflict!")
					return false
				end
			end
			approx_set[a_approx] = a
			mw.log(a_key .. " -> " .. a_approx)
		end
		for a_key, a in pairs(ratios) do
			local a_approx = ET.approximate(et, rat.as_float(a)) % et.size
			if approx_set[a_approx] then
				if not rat.eq(rat.modulo_mul(rat.div(a, approx_set[a_approx]), et.equave), 1) then
					mw.log(a_key .. " -> " .. a_approx .. ": conflict!")
					return false
				end
			end
			approx_set[a_approx] = a
			mw.log(a_key .. " -> " .. a_approx)
		end
	end
	if type(distinct) == "number" then
		return true
	end
	local previous_ordered = {}
	for _, a in pairs(previous) do
		table.insert(previous_ordered, a)
	end
	local ratios_ordered = {}
	for _, a in pairs(ratios) do
		table.insert(ratios_ordered, a)
	end
	for i, a in ipairs(ratios_ordered) do
		local a_approx = ET.approximate(et, rat.as_float(a))
		for _, b in ipairs(previous_ordered) do
			local b_approx = ET.approximate(et, rat.as_float(b))

			local c = rat.mul(a, b)
			local c_approx = ET.approximate(et, rat.as_float(c))

			c = rat.modulo_mul(c, et.equave)
			local c_key = rat.as_ratio(c)
			if previous[c_key] or ratios[c_key] then
				if c_approx ~= a_approx + b_approx then
					mw.log("a = " .. rat.as_ratio(a) .. "; b = " .. rat.as_ratio(b) .. "; ab = " .. c_key)
					mw.log(a_approx .. " + " .. b_approx .. " != " .. c_approx)
					return false
				end
			end
		end
		for j, b in ipairs(ratios_ordered) do
			if i <= j then
				local b_approx = ET.approximate(et, rat.as_float(b))

				local c = rat.mul(a, b)
				local c_approx = ET.approximate(et, rat.as_float(c))

				c = rat.modulo_mul(c, et.equave)
				local c_key = rat.as_ratio(c)
				if previous[c_key] or ratios[c_key] then
					if c_approx ~= a_approx + b_approx then
						mw.log("a = " .. rat.as_ratio(a) .. "; b = " .. rat.as_ratio(b) .. "; ab = " .. c_key)
						mw.log(a_approx .. " + " .. b_approx .. " != " .. c_approx)
						return false
					end
				end
			end
		end
	end
	return true
end

-- find additive consistency limit
-- returns nil when at least `max_n`
-- `distinct`: whether distinct ratios are required to be mapped to distinct approximations
-- - if an integer, it is the regular consistency limit already known
function p.consistency_limit(et, distinct, max_n)
	if et.size == 0 then
		-- the answer is known already
		return "∞"
	end
	max_n = max_n or 1 / 0
	distinct = distinct or false
	local n = 1
	local last_n = 1
	local previous = {}
	while true do
		if type(distinct) == "number" and n > distinct then
			return last_n
		end
		local ratios = p.limit_modulo_equave(n, et.equave, previous)
		for key, _ in pairs(ratios) do
			mw.log("step " .. n .. ": " .. key)
		end
		if next(ratios) ~= nil then
			local consistent = p.additively_consistent(et, ratios, distinct, previous)
			if not consistent then
				mw.log("Not consistent at step " .. n .. ", returning " .. last_n)
				return last_n
			end
			for key, ratio in pairs(ratios) do
				previous[key] = ratio
			end
			last_n = n
		end
		n = n + 1
		if n > max_n then
			return nil
		end
	end
end

return p