I find myself floundering in terminology, with the result that I fear the article I am writing will only appeal to readers afflicted with masochistic tendencies. At the very least, there is the danger that by redefining such terms as "interpretation" and "representation," I risk being accused of muddying the waters. Can any soul brave and kind enough to endure the following torrent of words give this amateur mathematician advice on how he might reframe the associated concepts properly using already-established terminology? Even if you do not have the inclination to wade through the entire thing, I would greatly appreciate any assistance, no matter how modest.
I foolishly begin by proposing my own definition of a word already having an accepted definition in mathematics, because my deficiency of mathematical sophistication prevents me from understanding the nuances of the accepted definition and I fear growing senile before I can correct the deficiency:
Definition 2.1. Let $L$ be a language, and let $S$ be a set. If there exists some surjective function $f: L \rightarrow S$, then we will call the triplet $(L,S,f)$ an interpretation of L as S, specifically the interpretation of L as S according to f, and we may say any of the following:
- $L$ interpreted according to $f$ is $S$ (equivalently: $L$ is the underlying language in the interpretation $(L,S,f)$).
- $S$ is $L$ interpreted according to $f$ (equivalently: $S$ is the target set in the interpretation $(L,S,f)$).
- $f$ interprets $L$ as $S$ (equivalently: $f$ is the evaluation function in the interpretation $(L,S,f)$).
Remark. For example, let $B$ denote the set of nonempty strings over the alphabet $\{0,1\}$ and let ${f : B \rightarrow \mathbb{N}}$, where $f({b})$ is the nonnegative integer corresponding to $b$ such that the latter is regarded as a word in unsigned binary. Then ${(B,\mathbb{N}, f)}$ is the interpretation of $B$ as the set of natural numbers according to $f$. Now consider ${g : B \rightarrow \mathbb{Z}}$, where $g({b})$ is the integer corresponding to $b$ such that the latter is regarded as a word in 2s-complement binary. Then ${(B,\mathbb{Z}, g)}$ is the interpretation of $B$ as the set of integers according to $g$. Thus we have an illustration of a single language underlying multiple interpretations.
As if the preceding were not excruciating enough, I force my reader to endure another definition, this time pertaining to members of sets rather than the sets themselves:
The following definition allows us to speak of interpretations in terms of set members as well.
Definition 2.2. Let $(L,S,f)$ be an interpretation. For any $l \in L$, let $s \in S$ where $s = f(l)$. Then we may say any of the following:
- $s$ is $l$ interpreted according to $f$.
- $l$ interpreted according to $f$ is $s$.
- $f$ interprets $l$ as $s$.
Not satisfied with having intruded on the terminology of interpretations, I presumptuously redefine representations:
Sometimes the terminology of interpretations becomes awkward, resulting in a surfeit of passive participles ("l interpreted as..."; "l interpreted according to..."). We may remedy this to some extent by making use of the following definition.
Definition 2.3. Let $(L, S, f)$ be an interpretation, and let the ordered pair $(l,s)$ be in the graph of $f$. Then we say l represents s in $(L, S, f)$. If we do not wish to mention a particular interpretation (as for example when the interpretation would be clear from the context), we may simply say l represents s, implying some interpretation exists such that $l$ represents $s$ in that interpretation.
In addition to speaking of members of languages representing members of sets, we can speak of languages representing sets.
Definition 2.4. If $(L, S, f)$ is an interpretation, then we say L represents S in $(L, S, f)$, or, equivalently, $L$ is a representation of $S$ in $(L, S, f)$. In cases where we do not wish to mention a specific interpretation, we may simply say $L$ represents $S$, or, equivalently, $L$ is a representation of $S$.
A particularly sad aspect of all this is that none of the above definitions is what my paper is "about"; they are only devices meant to enable me to articulate what my paper is about.
Can someone help me out of this morass of my own making?