# 19.1 Deciding upperboundedness of $$\bbZ$$-CCRA

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