# Decidable real arithmetic

I believe that what I'm about to describe has a name—I'm almost certain that I've seen this in model theory and term-rewriting systems, possibly having something to do with ‘signature’ or ‘carrier’?—so please feel free to tell me if I'm using terms in bad, or non-standard, ways.

Consider a set $C$ of real constants. (There's no reason that we couldn't consider constants from an arbitrary field, or ring, or … whatever; but let's say ‘real’ for definiteness.) Fix also, for each $n$, a (possibly empty) set $S_n$ of (partially defined) $n$-place, real-valued functions on $\mathbb R$.

Write $\Sigma = (C, (S_n)_n)$, and define the set of $\Sigma$-terms in the following (obvious) way. The collection of $\Sigma$-terms is the smallest subset $\mathcal T$ of the set of strings over the alphabet consisting of $C \sqcup \bigsqcup_n S_n$, together with 3 distinguished symbols (, ), and ,, satisfying:

1. Each element of $C$ is in $\mathcal T$.
2. If $\sigma \in S_n$, and $w_1, \ldots, w_n$ are $\Sigma$-terms, then $\sigma(w_1, \ldots, w_n) \in \mathcal T$.

Notice that there is an obvious (partially defined) map from $\Sigma$-terms to $\mathbb R$. Notice also that I am avoiding all questions of representability of elements of $C$; if this is worrisome, we can assume that $C$ is countable, fix an enumeration, and replace all elements of $C$ by their labels.

Now call $\Sigma$ decidable if the problem of whether a given $\Sigma$-term represents $0$ is decidable. (To avoid trivialities like $C = \mathbb R_{\ge0}$, with all operations preserving positivity, let's assume that subtraction lies in $S_2$.) Are there any results about what ensembles $\Sigma$ are decidable?

For example, it's obvious that, if $C = \mathbb Q$ and the operations allowed are the field operations, then $\Sigma$ is decidable; but what if we enlarge our operations to allow extraction of positive square roots, or enlarge $C$ to the algebraic closure of $\mathbb Q$?

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