Disclaimer: I'm a software engineer trying to use mathematics to solve a problem. I only have a shallow understanding of (abstract) algebra. My terminology will probably be wrong. Please be gentle.
I'm at the "Is this even possible stage?" of this idea/question.
Given a set of generators $\{a, b, c\}$, is there a non-Abelian algebra, such as quaternion multiplication, such that given a value $x$ that is the product of the generators, one can deduce something about the sequence of values that resulted in $x$?
There is an important constraint:
- $x$ is a prefix of a sequence that has a recursive grammar, using a quasi-regular expression syntax: $L = a(L|\epsilon)b(L|\epsilon)c$.
E.g. a full sequence is $a \cdot b \cdot a \cdot b \cdot c \cdot c$, and I'm interested in the prefix $a \cdot b \cdot a \cdot b \cdot c = x$.
I know what $x$ is. What I want to know from $x$ is whether it resulted from that sequence, or from say, $a \cdot a \cdot b \cdot c$.
Or to put it more specifically, can I deduce from $x$ and the knowledge of the structure/grammar of the equation, what the next value must be after any given $c$? Is it an $a$ or a $b$ next? It is determined by the grammar, right? I.e., if you had the prefix, you could tell what comes after every $c$. Can the prefix be 'encoded' into $x$?
I did some experimentation with quaternions letting $a = (1, 1, 0, 0), b = (1, 0, 1, 0), c = (1, 0, 0, 1)$, and hit a failure with two prefixes of different sequences. That is:
$ a \cdot b \cdot a \cdot b \cdot a \cdot b \cdot c = a \cdot a \cdot b \cdot a \cdot a \cdot b \cdot c = (-8, 0, 0, -8)$
but the left-hand prefix is followed by a $c$ whereas the right-hand prefix is followed by a $b$.
I was disappointed. So I tried letting $b = (-1, 0, 1, 0)$ and then each prefix evaluated to a different number. OK, that's nice, but now I need someone who really understands this stuff theoretically to tell me whether this is a fool's errand and it's not possible to encode an arbitrary amount of structure into a single number no matter how many dimensions it has, or tell me that it is possible and preferably how.
Any suggestions for improving this question (including the title) are welcome. Thank you for reading my amateur question!
PS. Even though this is probably the undecidable word problem, how much information can be extracted from such a value $x$? Presumably one can deduce the powers that the generators were raised to by simply dividing $x$ by a generator if the generators are prime, but is there anything else?