Minimization of semilinear automata

We investigate finite deterministic automata in sets with non-homogeneous atoms: integers with successor. As there are uncount- ably many deterministic finite automata in this setting, we restrict our attention to automata with semilinear transition function. The main re- sults is a minimization procedure for semilinear automata. The proof is subtle and refers to decidability of existential Presburger arithmetic with divisibility predicates. Interestingly, the minimization is not obtained by the standard partition refinement procedure, and we demonstrate that this procedure does not necessarily terminate for semilinear automata.

💡 Research Summary

The paper investigates deterministic finite automata (DFA) whose underlying domain consists of the integers equipped with the successor function, a non‑homogeneous atom structure. In such a setting the class of all DFA is uncountably large, making algorithmic analysis infeasible. To obtain a tractable subclass the authors restrict attention to automata whose transition function is semilinear—that is, each transition relation can be expressed as a finite union of linear sets, the standard objects of Presburger arithmetic.

The central contribution is a minimization procedure for semilinear automata. The authors first show that the classic partition‑refinement algorithm (the Myhill‑Nerode based method that iteratively splits state blocks) does not necessarily terminate when applied to semilinear automata over the integer‑successor atom. They construct a concrete family of automata where each refinement step creates a strictly finer partition, leading to an infinite refinement chain. This demonstrates that the usual state‑equivalence computation cannot be relied upon in this non‑homogeneous context.

To overcome this obstacle the paper leverages the decidability of existential Presburger arithmetic with divisibility predicates (EPA). The key observation is that the behavior of a semilinear automaton can be encoded as a collection of EPA formulas: for any two states p and q, the statement “p and q accept exactly the same language” can be rewritten as an existential Presburger formula that quantifies over input words and uses linear constraints together with divisibility tests to capture the effect of the successor atom. Since EPA is decidable (though PSPACE‑complete), one can algorithmically test whether two states are equivalent.

The minimization algorithm proceeds as follows:

1. Initialize the candidate state set S with all states of the given automaton.
2. For every unordered pair (p, q) in S, construct an EPA formula Φ(p,q) that is true iff p and q are language‑equivalent.
3. Invoke an EPA decision procedure on Φ(p,q). If the formula is true, merge p and q into a single representative state.
4. Iterate the merging step until no further merges are possible. Because each merge reduces the cardinality of S and EPA is decidable, the process must terminate, yielding a minimal automaton that is language‑equivalent to the original.

The authors discuss the computational complexity of this approach. The number of state pairs is O(|S|²), and each EPA query lies in PSPACE, so the overall worst‑case complexity is at most exponential in the size of the input automaton. Nevertheless, this is a theoretically complete solution to a problem that was previously unsolvable for general automata over the integer‑successor atom.

Beyond the algorithmic result, the paper analyses the expressive power of semilinear transition functions. Because semilinear sets correspond precisely to those definable in Presburger arithmetic, semilinear automata can model a wide range of integer‑based behaviors, such as modular counters, periodic patterns, and constraints involving linear combinations of input positions. The authors point out that if the transition relation were non‑semilinear, the reduction to EPA would fail, and the presented minimization technique would not apply.

The paper concludes with several avenues for future work: improving the practical performance of EPA solvers for the specific formulas arising from automata minimization, extending the framework to richer atom structures (e.g., adding multiplication or other arithmetic operations), and applying the theory to verification problems where systems naturally involve integer counters and successor dynamics (such as protocol analysis or hardware verification).

In summary, the work provides a novel, logically grounded minimization method for a significant subclass of automata over a non‑homogeneous atom structure, demonstrates the limitations of traditional partition refinement in this context, and opens the door to further exploration of algorithmic techniques that exploit decidable fragments of arithmetic.