A cost register automaton (CRA) over \((\bbZ, \min, +)\) is a DFA equipped with a finite number of registers that take values in \(\bbZ\). Each transition induces a transformation of the registers formulated with \(\min\) and \(+\); for instance \(x \leftarrow \min\{x, y+3\} + z\). Accepting states can then output one of the registers.

Consider the model in which:

- Each update uses only \(\min\)'s and addition
*with constants*, - For each transition, no register appears twice on the right-hand side of the updates.

This model, which we call $\bbZ$-CCRA (the extra C standing for
*copyless*), can express for instance:
\[f(a^{i_1}\#\ldots\#a^{i_n}) = \min \{i_1, \ldots, i_n\}\enspace,\]
but couldn't express \(g(b^i\#w) = i + f(w)\) —this would require either copying
\(i\), or adding two registers.

For weighted automata people, note that this model is equivalent to this, where \(\mathcal{W}\) is a deterministic WA over \((\mathbb{Z}, \min, +)\):

In [1], we showed that equivalence is undecidable for that model (even, and this is much more interesting, when restricted to \(\bbN\)). We also showed that it is undecidable, given a $\bbZ$-CCRA, to decide whether the function it expresses is always negative.

**Question:** Is it decidable, given a $\bbZ$-CCRA, whether the function it
expresses is upper-bounded? That is, whether \((\exists c)(\forall w)[f(w) \leq
c]\).

[1] S. Almagor, M. Cadilhac, F. Mazowiecki, G. A. Pérez. Weak Cost Register Automata are Still Powerful. DLT'18. https://arxiv.org/abs/1804.06336