Counter-Minimisation for Tropical CRAs

• The paper establishes that register minimisation is undecidable for tropical CRAs with seven or more registers via reduction from the 0-halting problem of two-counter machines.
• The methodology exploits the expressive equivalence between k-register CRAs and width-k tropical WFAs to map register count to automata width.
• The results imply significant limits on optimization techniques in quantitative models and indicate unresolved cases for register counts k ≤ 6.

The counter-minimisation problem for tropical (min-plus) Cost Register Automata (CRAs) concerns the question of whether, for a given $k$-register CRA, one can construct an equivalent CRA with strictly fewer than $k$ registers. CRAs over the min-plus semiring $\mathbb{Z}_{\min}$ are expressively equivalent to min-plus (tropical) Weighted Finite Automata (WFAs), and minimizing the number of registers within CRAs is directly analogous to minimizing the width of these WFAs. The recent resolution of this problem establishes that register minimisation is undecidable for $k\geq7$ registers, whereas the problem remains open for smaller register counts (Almagor et al., 10 Dec 2025).

1. Min-Plus Semiring and Tropical Cost Register Automata

A tropical (min-plus) semiring is defined over the domain $\mathbb{Z}_{\min} = \mathbb{Z} \cup \{\infty\}$, with the operations $a \oplus b = \min(a, b)$ and $a \otimes b = a + b$. The key properties are $\infty + x = \infty$ and $\min(\infty, x) = x$. A $k$-register CRA over $k$0 is formalized as the tuple

$k$1

where:

• $k$2 is a finite set of states,
• $k$3 is the initial state,
• $k$4 is the set of accepting states,
• $k$5 is a finite set of registers over $k$6,
• $k$7 is the transition relation, with: $k$8 assigning to each register either a pair $k$9 for copy-update or $\mathbb{Z}_{\min}$0 for constant reset.

A run is a unique sequence of configurations and register assignments starting from all-zero registers. If the final state is not accepting, the output is $\mathbb{Z}_{\min}$1. Otherwise, the value in a designated output register gives $\mathbb{Z}_{\min}$2.

2. Formal Statement of the Counter-Minimisation Problem

The problem is specified as follows:

• Input: A $\mathbb{Z}_{\min}$3-register CRA $\mathbb{Z}_{\min}$4.
• Question: Does there exist an equivalent CRA $\mathbb{Z}_{\min}$5 with strictly fewer than $\mathbb{Z}_{\min}$6 registers (i.e., with $\mathbb{Z}_{\min}$7 registers)?

Equivalently, for fixed $\mathbb{Z}_{\min}$8, is it decidable if a $\mathbb{Z}_{\min}$9-register CRA admits a representation with fewer registers preserving its behavior over all input words? The set of such minimal CRAs is precisely those not equivalent to any CRA with one fewer register.

3. Central Undecidability Result

The core result establishes that for every $k\geq7$0, the following decision problem is undecidable:

$k\geq7$1

Thus, for $k\geq7$2 registers, counter-minimisation for tropical CRAs is undecidable (Almagor et al., 10 Dec 2025).

4. Reduction and Undecidability Proof Outline

The proof reduces from the classically undecidable “0-halting” problem for two-counter machines. The key technical construction involves:

• Encoding the behavior of a two-counter machine $k\geq7$3 in a tropical WFA $k\geq7$4 of width $k\geq7$5, linking the halting behavior to whether $k\geq7$6 is unbounded above.
• “Padding” $k\geq7$7 with a seventh register produces a CRA $k\geq7$8 with $k\geq7$9 registers. If $\mathbb{Z}_{\min} = \mathbb{Z} \cup \{\infty\}$0 is upper-bounded, the additional register is redundant and removable; if $\mathbb{Z}_{\min} = \mathbb{Z} \cup \{\infty\}$1 is unbounded, any equivalent CRA must maintain all $\mathbb{Z}_{\min} = \mathbb{Z} \cup \{\infty\}$2 registers to correctly describe the behavior.
• Thus, deciding whether $\mathbb{Z}_{\min} = \mathbb{Z} \cup \{\infty\}$3 can be realized with $\mathbb{Z}_{\min} = \mathbb{Z} \cup \{\infty\}$4 registers solves the undecidable question regarding boundedness of $\mathbb{Z}_{\min} = \mathbb{Z} \cup \{\infty\}$5.

This construction follows the earlier approach of Almagor–Chatterjee–Dima (2020) regarding undecidability in WFAs and exploits the expressive equivalence between $\mathbb{Z}_{\min} = \mathbb{Z} \cup \{\infty\}$6-register CRAs and width-$\mathbb{Z}_{\min} = \mathbb{Z} \cup \{\infty\}$7 WFAs.

5. Structural and Expressive Connections

There is an exact correspondence between $\mathbb{Z}_{\min} = \mathbb{Z} \cup \{\infty\}$8-register CRAs and width-$\mathbb{Z}_{\min} = \mathbb{Z} \cup \{\infty\}$9 min-plus WFAs, where width is the maximal number of simultaneous nondeterministic states reachable in the WFA. The conversion involves interpreting each register as a “dimension” in the WFA, and simulating nondeterministic branching by parallel register assignment.

A table comparing these concepts:

Model	Resource Parameter	Expressivity Determined By
CRA	$a \oplus b = \min(a, b)$0 registers	Number of independent cost tracks (registers)
Tropical WFA	Width $a \oplus b = \min(a, b)$1	Max number of concurrent states reachable

This equivalence underpins reductions between the two models and is crucial for the undecidability result.

6. Boundary Cases and Related Decidability Results

The status of the counter-minimisation problem for $a \oplus b = \min(a, b)$2 remains unresolved; no matching lower bounds exist for $a \oplus b = \min(a, b)$3 below $a \oplus b = \min(a, b)$4. In contrast, for CRAs over the rational field $a \oplus b = \min(a, b)$5, register minimisation is decidable, as established by Ben Alioua et al. 2024.

Further, related problems include:

• Determinisation of tropical WFAs (recently shown decidable).
• Unambiguisability (decidability of equivalence to an unambiguous WFA).

A plausible implication is that structural restrictions on the underlying semiring or register management could lead to decidability in other settings.

7. Key Constructions and Lemmas

Two pivotal techniques support the proof and its generalizations:

• Expressive Equivalence Lemma: $a \oplus b = \min(a, b)$6-register CRAs are expressively equivalent to width-$a \oplus b = \min(a, b)$7 min-plus WFAs by simulating register updates via WFA state space and by simulating WFA subset constructions through register allocation.
• Gadget for Width Jump: In the main reduction, adding a spare register allows mimicking the others only if the automaton is bounded; otherwise, the spare becomes essential and cannot be eliminated, forcing the need for all $a \oplus b = \min(a, b)$8 registers.

These technical constructions confirm the undecidability for $a \oplus b = \min(a, b)$9 and clarify the critical role of width and register tracking in capturing quantitative behaviors within min-plus models.

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For foundational and technical details see (Almagor et al., 10 Dec 2025), as well as [Alur, Moerkotte et al. (ICALP 2013)], [Almagor, Chatterjee, Dima (FSTTCS 2020)], and [Bell, Jecker, Seidl (LICS 2023)].