algebrite/sources/transform.coffee

Computer Algebra System in Coffeescript

###
Transform an expression using a pattern. The
pattern can come from the integrals table or
the user-defined patterns.

The expression and free variable are on the stack.

The argument s is a null terminated list of transform rules.

For example, see the itab (integrals table)

Internally, the following symbols are used:

	F	input expression

	X	free variable, i.e. F of X

	A	template expression

	B	result expression

	C	list of conditional expressions

Puts the final expression on top of stack
(whether it's transformed or not) and returns
true is successful, false if not.


###



# p1 and p2 are tmps

#define F p3
#define X p4
#define A p5
#define B p6
#define C p7

transform = (s, generalTransform) ->
	transform_h = 0

	save()

	p1 = null

	p4 = pop() # X i.e. free variable
	p3 = pop() # F i.e. input expression

	if DEBUG
		console.log "         !!!!!!!!!   transform on: " + p3


	saveMetaBindings()

	set_binding(symbol(METAX), p4)

	# put constants in F(X) on the stack

	transform_h = tos
	push_integer(1)
	push(p3)
	push(p4)
	polyform(); # collect coefficients of x, x^2, etc.
	push(p4)
	
	bookmarkTosToPrintDecomps = tos - 2
	decomp(generalTransform)
	numberOfDecomps = tos - bookmarkTosToPrintDecomps

	if DEBUG
		console.log "	" + numberOfDecomps + " decomposed elements ====== "
		for i in [0...numberOfDecomps]
			console.log "	decomposition element " + i + ": " + stack[tos-1-i]

	transformationSuccessful = false

	if generalTransform
		# "general tranform" mode is supposed to be more generic than
		# "integrals" mode.
		# In general transform mode we get only one transformation
		# in s

		# simple numbers can end up matching complicated templates,
		# which we don't want.
		# for example "1" ends up matching "inner(transpose(a_),a_)"
		# since "1" is decomposed to "1" and replacing "a_" with "1"
		# there is a match.
		# Although this match is OK at some fundamental level, we want to
		# avoid it because that's not what the spirit of this match
		# is: "1" does not have any structural resemblance with
		# "inner(transpose(a_),a_)". There are probably better ways
		# to so this, for example we might notice that "inner" is an
		# anchor since it "sits above" any meta variables, so we
		# might want to mandate it to be matched at the top
		# of the tree. For the time
		# being let's just skip matching on simple numbers.
		if !isnum(p3)

			theTransform = s
			if DEBUG then console.log "applying transform: " + theTransform
			if DEBUG then console.log "scanning table entry " + theTransform

			push theTransform

			# replacements of meta variables. Note that we don't
			# use scan_meta because the pattern is not a string
			# that we have to parse, it's a tree already.
			# replace a_ with METAA in the passed transformation
			push symbol(SYMBOL_A_UNDERSCORE)
			push symbol(METAA)
			subst()

			# replace b_ with METAB in the passed transformation
			push symbol(SYMBOL_B_UNDERSCORE)
			push symbol(METAB)
			subst()

			# replace x_ with METAX in the passed transformation
			push symbol(SYMBOL_X_UNDERSCORE)
			push symbol(METAX)
			subst()

			p1 = pop()

			p5 = car(p1)
			if DEBUG then console.log "template expression: " + p5
			p6 = cadr(p1)
			p7 = cddr(p1)

			###
			p5 = p1.tensor.elem[0]
			p6 = p1.tensor.elem[1]
			for i in [2..(p1.tensor.elem.length-1)]
				push p1.tensor.elem[i]
			list(p1.tensor.elem.length - 2)
			p7 = pop()
			###



			if (f_equals_a(transform_h, generalTransform))
				# successful transformation,
				# transformed result is in p6
				transformationSuccessful = true
			else
				# the match failed but perhaps we can match
				# something lower down in the tree, so
				# let's recurse the tree

				if DEBUG then console.log "p3 at this point: " + p3

				transformedTerms = []

				if DEBUG then console.log "car(p3): " + car(p3)
				restTerm = p3

				if iscons(restTerm)
					transformedTerms.push car(p3)
					restTerm = cdr(p3)

				while (iscons(restTerm))
					secondTerm = car(restTerm)
					restTerm = cdr(restTerm)

					if DEBUG then console.log "tos before recursive transform: " + tos
					
					push(secondTerm)
					push_symbol(NIL)
					if DEBUG then console.log "testing: " + secondTerm
					#if (secondTerm+"") == "eig(A x,transpose(A x))()"
					#	debugger
					if DEBUG then console.log "about to try to simplify other term: " + secondTerm
					success = transform(s, generalTransform)					
					transformationSuccessful = transformationSuccessful or success

					transformedTerms.push pop()

					if DEBUG then console.log "tried to simplify other term: " + secondTerm + " ...successful?: " + success + " ...transformed: " + transformedTerms[transformedTerms.length-1]


				# recreate the tree we were passed,
				# but with all the terms being transformed
				if transformedTerms.length != 0
					for i in transformedTerms
						push i
					list(transformedTerms.length)
					p6 = pop()



	else # "integrals" mode
		for eachTransformEntry in s
			if DEBUG
				console.log "scanning table entry " + eachTransformEntry
				if (eachTransformEntry + "").indexOf("f(sqrt(a+b*x),2/3*1/b*sqrt((a+b*x)^3))") != -1
					debugger
			if eachTransformEntry
				scan_meta(eachTransformEntry)
				p1 = pop()

				p5 = cadr(p1)
				p6 = caddr(p1)
				p7 = cdddr(p1)

				###
				p5 = p1.tensor.elem[0]
				p6 = p1.tensor.elem[1]
				for i in [2..(p1.tensor.elem.length-1)]
					push p1.tensor.elem[i]
				list(p1.tensor.elem.length - 2)
				p7 = pop()
				###


				if (f_equals_a(transform_h, generalTransform))
					# there is a successful transformation,
					# transformed result is in p6
					transformationSuccessful = true
					break




	moveTos transform_h


	if transformationSuccessful
		#console.log "transformation successful"
		# a transformation was successful
		push(p6)
		Eval()
		p1 = pop()
		#console.log "...into: " + p1
		transformationSuccessful = true
	else
		# transformations failed
		if generalTransform
			# result = original expression
			p1 = p3
		else
			p1 = symbol(NIL)

	restoreMetaBindings()

	push(p1)

	restore()
	return transformationSuccessful

saveMetaBindings = ->
	push(get_binding(symbol(METAA)))
	push(get_binding(symbol(METAB)))
	push(get_binding(symbol(METAX)))


restoreMetaBindings = ->
	set_binding(symbol(METAX), pop())
	set_binding(symbol(METAB), pop())
	set_binding(symbol(METAA), pop())

# search for a METAA and METAB such that F = A

f_equals_a = (h, generalTransform) ->
	fea_i = 0
	fea_j = 0
	for fea_i in [h...tos]

		set_binding(symbol(METAA), stack[fea_i])
		if DEBUG
			console.log "  binding METAA to " + get_binding(symbol(METAA))
		for fea_j in [h...tos]

			set_binding(symbol(METAB), stack[fea_j])
			if DEBUG
				console.log "  binding METAB to " + get_binding(symbol(METAB))

			# now test all the conditions (it's an and between them)
			p1 = p7
			while (iscons(p1))
				push(car(p1))
				Eval()
				p2 = pop()
				if (iszero(p2))
					break
				p1 = cdr(p1)

			if (iscons(p1))
				# conditions are not met,
				# skip to the next binding of metas
				continue
			push(p3);			# F = A?
			if DEBUG
				console.log "about to evaluate template expression: " + p5 +
					" binding METAA to " + get_binding(symbol(METAA)) +
					" and binding METAB to " + get_binding(symbol(METAB)) +
					" and binding METAX to " + get_binding(symbol(METAX))
			push(p5)
			if generalTransform
				originalexpanding = expanding
				expanding = false
			Eval()
			if generalTransform
				expanding = originalexpanding
			if DEBUG
				console.log "  comparing " + stack[tos-1] + " to: " + stack[tos-2]
			subtract()
			p1 = pop()
			if (iszero(p1))
				if DEBUG
					console.log "binding METAA to " + get_binding(symbol(METAA))
					console.log "binding METAB to " + get_binding(symbol(METAB))
					console.log "binding METAX to " + get_binding(symbol(METAX))
					console.log "comparing " + p3 + " to: " + p5
				return 1;		# yes
	return 0;					# no