Ochmanski's Conjecture and Persistent Permutability

• Ochmanski’s Conjecture is a hypothesis in Petri net theory asserting that finitary persistent equivalence (SPE) guarantees fair persistent equivalence (FPE) in pps Petri nets.
• The approach employs algebraic and combinatorial techniques to validate the conjecture for equal-conflict (EC) and pure dissymmetric choice (DC) nets, ensuring net persistence.
• A counterexample in asymmetric choice (AC) nets reveals the boundaries of the conjecture, highlighting its reliance on specific net properties and prompting further investigation.

Ochmański’s conjecture addresses the relationship between “short” (finite) and “fair” (infinite) persistent permutability properties in choice-restricted Petri nets. Its central claim is that, for plain, pure, safe (pps) Petri nets, the existence of persistent equivalent permutations for all finite runs (SPE) ensures the same for fair infinite runs (FPE). The conjecture has significant implications for the structural and behavioral analysis of Petri nets, establishing finite interleaving properties as sufficient for guaranteeing infinite fairness in broad net classes. The work of Best and Devillers rigorously validates this conjecture for two infinite families that generalize free-choice nets, while delineating the boundaries of its applicability (Best et al., 25 Jan 2026).

1. Mathematical Preliminaries

A labelled transition system (LTS) is a quadruple $(S,\to,T,s_0)$ where $S$ is the set of states, $T$ the set of transition labels, $\to\subseteq S\times T\times S$ the transition relation, and $s_0$ the initial state. The reachability set comprises all states reachable from $s_0$. Petri nets are described as $N=(P,T,F,M_0)$ with $P$ places, $T$ transitions, $F$ flow function, and $M_0$ initial marking. Each marking $M$ specifies the multiset of tokens; a transition $t$ is enabled if $M(p)\ge F(p,t)$ for all $p\in P$. Firing $t$ yields a new marking $M’=M-F(\cdot,t)+F(t,\cdot)$. The reachability graph of a Petri net is the LTS of all markings reachable via firing sequences from $M_0$.

2. Persistence and Permutation Equivalence

A persistent LTS requires that no transition firing can disable any concurrently enabled transition. Formally, $(S,\to,T,s_0)$ is persistent if for all reachable $s\in S$ and any distinct $t,u\in T$, if both $s\xrightarrow{t}s’$ and $s\xrightarrow{u}s’’$, then there exists $r$ such that $s’\xrightarrow{u}r$ and $s’’\xrightarrow{t}r$. Similarly, firing sequences are persistent if no transition enabled just before some step is disabled by that step—that is, enabledness “persists.” Persistence as a property of the net is equivalent to all firing sequences being persistent.

Permutation equivalence is defined via swaps of adjacent, independent transitions in run sequences: two finite firing sequences are in single-swap relation $(\equiv_0)$ if swapping two adjacent transitions leads to the same marking. The reflexive transitive closure $(\equiv)$ captures all sequences obtainable via sequences of such swaps. For infinite runs, equivalence demands sequences become arbitrarily close in prefixes after finitely many swaps. Parikh equivalence, crucially, is preserved by permutation equivalence.

3. Notions of Persistent Permutability

The principal notions are:

• SPE (Short Persistent Equivalent): Every finite firing sequence has a permutation-equivalent persistent sequence.
• S$\widetilde{P}$E (Short Parikh-Equivalent): Every finite sequence has a persistent sequence with the same Parikh vector.
• FPE (Fair Persistent Equivalent): Every fair (infinite) firing sequence admits a permutation-equivalent persistent sequence.

Fairness is defined by strong fairness: in an infinite run, each transition occurs infinitely often or is enabled only finitely often.

The conjecture’s significance arises from the S$\widetilde{P}$E (and more particularly SPE) being a finitary property, relatively tractable to check, in contrast to the infinitary demands of FPE.

4. Choice and Conflict Structures in Petri Nets

Structural properties of Petri nets are characterized by relations on input and output sets of transitions, affecting how concurrency, confusion, and choice are realized:

• Free-choice (FC): If two transitions share any input, they have exactly the same set of inputs.
• Equal-conflict (EC): If two transitions share an input, the incoming edge weights (columns of $F$) are identical.
• Dissymmetric choice (DC): If two transitions share an input, one’s input set strictly contains the other’s.
• Asymmetric choice (AC): If two transitions share an output, one’s output set strictly contains the other’s.

DC and AC exclude specific forms of confusion (“butterfly” and “M-shaped” patterns, respectively), which play a key role in (non-)persistence and permutation properties.

5. Ochmański’s Conjecture and Its Resolution

Ochmański’s conjecture asserts that, for pps Petri nets, every finite run’s ability to permute to persistent form (SPE) entails the same property for every infinite fair run (FPE). Formally,

$\text{SPE} \implies \text{FPE}.$

The conjecture is proven correct for both EC nets and for pure DC nets, regardless of boundedness (Best et al., 25 Jan 2026). The strategy is to show that SPE or S$\widetilde{P}$E implies persistence of the net, after which FPE follows trivially.

EC Nets

For EC nets, S$\widetilde{P}$E implies that any purported non-persistent marking would necessitate a persistent sequence containing an infinite alternation of two non-coexistent transitions, which is impossible in a finite sequence. Thus, the EC property precludes the existence of non-persistent markings in the presence of S$\widetilde{P}$E, leading directly to persistence and hence FPE.

Pure DC Nets

For pure DC nets, the proof is fundamentally combinatorial, exploiting two key properties:

• No 3/4-diamond lemma: In a pure net, partial diamond patterns must extend to full diamonds, preventing certain non-persistent behaviors.
• Unification via diamonds: Induction shows that persistent Parikh-equivalent runs ending in the same marking can be unified two steps before the end, given the diamond properties.

These results, together with pattern embedding arguments, eliminate the possibility of non-persistent markings under S$\widetilde{P}$E, yielding persistence and thus FPE.

Crucially, a counterexample in pure, plain, 2-bounded AC nets shows that the conjecture fails without the DC property, establishing its necessity.

6. Methodological Framework

The proof techniques in (Best et al., 25 Jan 2026) integrate algebraic and combinatorial methodologies:

• Reduction to persistence by showing that the existence of SPE or S$\widetilde{P}$E excludes non-persistent markings.
• For EC nets, algebraic exploitation of “equal enabling” to rule out persistent but non-persistent sequences.
• For DC nets, combinatorial analysis using forbidden LTS substructures (e.g., diamond patterns) and the induction-based unification of Parikh-equivalent persistent runs.
• Embedding techniques to certify violations of choice structure constraints, providing rigorous counterexamples and boundary cases.

The pattern-based approach and Parikh-equivalence unification are especially significant for the identification of minimal non-DC subnets, and serve as tools for further generalization.

7. Impact, Limitations, and Open Problems

The validation of Ochmański’s conjecture for EC and pure DC nets extends the classical field of free-choice Petri nets, consolidating the relationship between finite and infinite persistent permutability in broad classes. The necessity of the DC condition, marked by the AC counterexample, delineates the limits of the conjecture’s scope.

Open problems include:

1. Whether the “pure” assumption can be relaxed in the DC case,
2. Whether SPE implies FPE, or at least persistence, for all plain, pure, safe nets,
3. Full generalization to arbitrary pps nets.

The methodologies introduced—such as pattern-based embedding and diamond structure analysis—are poised for further application in the ongoing investigation of these questions, potentially enabling deeper structural characterization and decidability results for persistent permutability in Petri net theory (Best et al., 25 Jan 2026).