State space search problem math equations and Die Hard movie

I have a state A.
I have operations
A -> AB
AB -> BB
B  -> AAA

Can I make symbol ABA from initial symbol A?


I have math expression 5x.
Given all math operations can I transform it to 10x?


All of these are state space search problems.

I have the code. Given the initial inputs
will it reach the state on infinite looping?

There is 3 and 5 gallons of jugs. How to get 4?

TLA+

----MODULE states----
EXTENDS Integers
VARIABLES s,b

typeOK == s \in 0..3 /\ b \in 0..5

Init == s = 0 /\ b = 0

FillBig == b' = 5 /\ s' = s
FillSmall == s' = 3 /\ b' = b 
EmptyBig == b' = 0 /\ s' = s
EmptySmall == s' = 0 /\ b' = b

SmallToBig == IF b + s >= 5 
              THEN s' = s + b - 5  /\ b' = 5
               ELSE s' = 0 /\ b' = b + s
               
BigToSmall == IF b+s >= 3
              THEN b' = b + s - 3 /\ s' = 3
               ELSE b' = 0 /\ s' = b + s
               

Next == FillBig \/ FillSmall \/ EmptyBig \/ EmptySmall \/ SmallToBig \/ BigToSmall
====

Check the model with invariant b /= 4 it will find the state sequence needed.