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MATH.5:5.2 - Calculate the combined effect of a word of operations

Let a word contain commands I and D. Their mathematical effects on an integer are:

I(x)=x+1, D(x)=2*x.

Represent an affine transformation x -> s*x+t by the pair (s,t). Thus I receives (1,1) and D receives (2,0).

For (s,t) followed by (u,v), define:

(s,t) star (u,v)=(u*s,u*t+v).

Substitution derives this rule: u*(s*x+t)+v=(u*s)*x+(u*t+v). The identity is (1,0). For a third pair (w,z), either grouping gives (w*u*s,w*u*t+w*v+z), so the rule is associative.

Extend the two generator assignments to words. Then:

  • I;D receives (2,2), meaning x -> 2*x+2;
  • D;I receives (2,1), meaning x -> 2*x+1;
  • I;D;I receives (2,3), meaning x -> 2*x+3.

The third word sends 10 to 23. The pair describes its effect for every integer, allowing it to be composed with another affine operation without expanding the whole word again.

A different map can send each generator to cost 1 and concatenate by addition. It gives both I;D and D;I the value 2. This is a valid cost homomorphism but loses their different effects. Requiring the source equation I;D=D;I would preserve that cost map while preventing the stated effect map. The receiving question decides which distinction must remain.

If a command’s availability depends on intermediate state, the all-words construction is no longer the intended source. Use state-sensitive paths under MATH.1 and preserve their interfaces when constructing the interpretation.