The Machines · Runtime stack

seedalg

A reckoner seed: alg's symbolic algebra, staying at the text boundary — machines/seeds/seedalg.shoddy

Summary

seedalg is a reckoner seed: the file that bridges one machine's words into the reckoner, the calculator engine. It bridges alg: ALGSIMPLIFY, ALGEXPAND, ALGFACTOR, ALGSOLVE and ALGDESOLVE. Every word here works on text. An expression sits on the stack — the calculator's working pile of values — only as text that reads back as itself: "x^2 - 4", never as a tree only this machine could take apart. The reason is rule R4.16: there is no EXPRESSION cell at the keyboard. Inside alg, an expression (AlgEx) is a full symbolic tree of sums, products, powers and function calls. No dict — a table of named values — and no pair fits that shape without inventing a second grammar for it. So every word stays at the string boundary alg.shoddy already has: text in through AlgRead (the total half of AlgParse — total meaning it answers on every input instead of aborting), then the algebra, then AlgShow back out.

Why It's Useful

Every other seed converts a machine's record into a dict. This one cannot, and what it did instead is the whole design. AlgEx is a symbolic expression tree: sums, products, powers and function calls, nested to any depth. No dict or pair shape holds that without inventing a second grammar for expressions on top of the one alg already has. So the seed does not try. It stays at the string boundary alg itself already offers: text in through AlgRead, the algebra, AlgShow back out.

An expression on this stack is text. That is a real limitation, not only a convenience, so it is worth stating plainly. The dictionary cannot take the text apart. There is no walking a tree, no substituting a subexpression, no asking for the degree. Every word here composes by writing a string and reading it back — exactly the text AlgRead is total over. What this buys is that the answer is always something you can read, retype, save with TAPESAVE, and hand to any other alg word.

AlgRead being the total half of AlgParse is what makes this safe. A malformed expression gets a refusal naming the token it died at, not an aborted session.

User's Guide

The four things a computer algebra system is for, each one line:

> "(x+1)*(x-1)" ALGEXPAND
x: "-1 + x ^ 2"
> CLEAR "2*x + 3*x" ALGSIMPLIFY
x: "5 * x"
> CLEAR "x^2 - 4" "x" ALGFACTOR
x: "(-2 + x) * (2 + x)"
> CLEAR "y" "y" "x" ALGDESOLVE
x: "C * exp(x)"

The output is canonical, not pretty. Canonical means one fixed standard form. Terms come back in alg's own order — -1 + x ^ 2 rather than x^2 - 1. A canonical form is what lets two expressions be compared, and reordering it for the eye would cost that. The text reads back as itself, and that is the property that matters.

ALGSOLVE answers a LIST of strings. The roots are exact expressions, not evaluated numbers:

> CLEAR "x^2 - 4" "x" ALGSOLVE
x: { "(1 / 2) * 16 ^ (1 / 2)" "(-1 / 2) * 16 ^ (1 / 2)" }

Those are 2 and −2, written as the quadratic formula produced them and not simplified afterwards. That is the honest form: the exact root is what was computed. But when you want a number, you take the extra step of evaluating it yourself — ALGSOLVE will not have done so.

Three refusals, and each names a different kind of wrong:

> CLEAR "sin(x)" "x" ALGFACTOR
?: ALGFACTOR: NOT A POLYNOMIAL IN x
> CLEAR "x^2 + 1" "x" ALGFACTOR
?: ALGFACTOR: IRREDUCIBLE OVER THE RATIONALS
> CLEAR "((" ALGEXPAND
?: ALGEXPAND: UNEXPECTED END OF EXPRESSION AT TOKEN 3

The middle one is the interesting refusal. x^2 + 1 is a perfectly good polynomial. It simply has no factorisation over the rationals — the fractions. Saying so is different from saying the input was wrong.

Word Reference

Word	Description
ALGSIMPLIFY ( str -- str2 )	The expression, simplified — like terms combined, obvious identities applied.
ALGEXPAND ( str -- str2 )	The expression, multiplied out — (x+1)(x-1) becomes x^2 - 1.
ALGFACTOR ( str var -- str2 )	The expression factored, as one string — a polynomial in var only. Refuses on a non-polynomial, a constant, or one irreducible over the rationals.
ALGSOLVE ( str var -- list )	The roots, as a list of strings — degree <= 2 solved exactly, higher degrees bisected numerically. Refuses on a non-polynomial.
ALGDESOLVE ( rhs y x -- str2 )	The general solution of y' = rhs. Refuses when rhs fits neither case this covers.

Who Uses It

	User	How
	halifax	The calculator's symbolic-algebra words: ALGSIMPLIFY, ALGEXPAND, ALGFACTOR, ALGSOLVE and ALGDESOLVE.
	sparky	Sparky folds it too, so a model calling eval reaches the same words halifax puts at a prompt.

A mill claims this seed by folding RckSeedAlg over its reckoner state, which is all halifax does.

The Machines It Uses

	Machine	Why
	alg	The domain this seed bridges: AlgRead, AlgShow, AlgSimplify, AlgExpand, AlgFactor, AlgSolve, AlgDeSolve.
	cuttle	The Cell type every bridged word reads its arguments from and answers into.
	reckoner	RckReg and the text argument reader every registered word is built from.