Existential boundedness for jumping automata quantitative semantics

Determine, for a given jumping automaton A and for each of the three quantitative semantics—absolute distance (A_abs), reversal (A_rev), and Hamming (A_ham)—whether there exists a finite bound k ∈ N such that for all words w in the jumping language J(A), the cost assigned by the corresponding semantics satisfies A_sem(w) ≤ k; equivalently, ascertain whether A_sem is k-bounded for some k.

Background

The paper introduces three natural quantitative semantics for jumping automata—absolute distance, reversal, and Hamming—and studies several boundedness problems: fixed-k boundedness, parameterized boundedness with k given as input, and their universal variants. For these variants, the authors establish decidability and provide complexity bounds, leveraging constructions of NFAs and connections to Parikh Automata.

However, the existential version—asking whether there exists some bound k for which the quantitative language induced by a given jumping automaton is k-bounded—remains unresolved. The authors explain that standard pumping-style arguments are hindered by the role of permutations in jumping runs, and that analyzing their constructions with a parametric k appears to lead to systems (e.g., labeled VASS in the Hamming case) for which the corresponding boundedness properties are generally undecidable. They note a decidable fragment via translation to weighted automata under restricted swaps, but emphasize that this does not settle the general existential boundedness question.