Hereditary HHPB Equivalence in Concurrency

• Hereditary history-preserving bisimilarity (HHPB) is a true concurrency equivalence that rigorously preserves causality, conflict, and the entire downward-closed history of process computations.
• It refines interleaving bisimulation by distinguishing processes with identical traces but differing causal and conflict structures through label- and order-preserving bijections.
• Its operational, logical, and game-theoretic characterizations support robust compositional reasoning in reversible calculi, despite general undecidability in complex models.

Hereditary history-preserving bisimilarity (HHPB) is a behavioral equivalence arising in true-concurrency semantics, offering a fine-grained model for process equivalence that preserves causality, concurrency, and the full "downward-closure" of histories. Unlike interleaving bisimulation, which identifies processes based solely on sequences of actions, HHPB distinguishes structures with identical traces but differing causal dependencies and conflict relations. HHPB constitutes one of the finest equivalence relations on event- or configuration-based models of concurrency, subsuming history-preserving bisimilarity (HPB) and providing congruence with respect to truly concurrent process operators (Aubert et al., 2015, Wang, 2021). It is operationally characterized via reversible calculi with memory mechanisms (Aubert et al., 2020, Aubert et al., 2018), enjoys logical characterizations through modal logics with identifier and reverse modalities (Phillips et al., 2011, Baldan et al., 2011), and admits game-theoretic and denotational descriptions (Wang, 2019, Kahl, 2021). The relation is undecidable in general, though several decidable causal-reversible approximations for Petri nets exist (Gorrieri et al., 13 Jun 2025).

1. Formal Definition and Mathematical Framework

HHPB is defined on labeled configuration structures or event structures. A configuration structure is a triple $C = (E, \mathcal{C}, \ell)$ with $E$ a set of events, $\mathcal{C} \subseteq \mathcal{P}_{\text{fin}}(E)$ the set of finite, causally closed, conflict-free configurations, and $\ell: E \to L$ an action labeling. The causality order within a configuration $x$ is defined by $e_1 \le_x e_2$ iff every subconfiguration containing $e_2$ also contains $e_1$.

A hereditary history-preserving bisimulation is a symmetric relation: $$R \subseteq \mathcal{C}_1 \times \mathcal{C}_2 \times \{\text{label- and order-preserving bijections } f: x_1 \to x_2 \}$$ satisfying:

• $(\emptyset, \emptyset, \emptyset) \in R$
• For any $(x_1,x_2,f)\in R$:
  • If $x_1 \xrightarrow{e_1} y_1$ (forward step), there exist $e_2, y_2, f'$ such that $x_2 \xrightarrow{e_2} y_2$, $\ell_1(e_1)=\ell_2(e_2)$, $f'$ extends $f$ by $f'(e_1)=e_2$, and $(y_1, y_2, f') \in R$
  • Similarly for steps starting in $x_2$
  • For backward steps $x_1 \xleftarrow{e_1} y_1$, there is matching $x_2 \xleftarrow{e_2} y_2$, $f'$ restricts $f$ appropriately
  • (Heredity) For any restriction of $f$ to a subconfiguration, the corresponding triple is in $R$
  • This hereditary condition enforces downward closure: all subhistories (subsets of events and their causal pasts) are matched by suitable label- and order-preserving bijections (Aubert et al., 2015, Aubert et al., 2015, Wang, 2021).

For event structures with localities, HHPB requires bijections to respect both action and locality labels (Wang, 2021): $$\lambda_1(e_1) = \lambda_2(e_2), \quad \ell_1(e_1) = \ell_2(e_2)$$

2. Key Properties and Inclusion Relationships

HHPB refines standard bisimulations, yielding a strict hierarchy: $$\approx_{\mathrm{HHPB}} \subsetneq \approx_{\mathrm{HPB}} \subsetneq \approx_{\mathrm{pomset}} \subsetneq \approx_{\mathrm{step}} \subsetneq \approx_{\mathrm{interleaving}}$$ The hereditary condition distinguishes HHPB from HPB: while HPB requires isomorphisms between histories under forward/backward transitions, HHPB mandates that all restrictions to subconfigurations are preserved (i.e., the bisimulation is downward closed) (Wang, 2021, Wang, 2019).

HHPB is a congruence with respect to all truly concurrent process algebra operators, e.g., prefixing, parallel, choice, hiding, and locality labeling. If $P \hhpb P'$ and $Q \hhpb Q'$, then all contextually combined terms—via these operators—remain HHPB equivalent (Wang, 2021). This property enables compositional reasoning and equational reasoning in process algebras.

3. Operational Characterizations: Reversibility and Memory

The operational characterization of HHPB requires a reversible process calculus with explicit memory mechanisms—such as RCCS (Reversible CCS) (Aubert et al., 2018, Aubert et al., 2020). RCCS decorates processes with stacks of memory events. Each forward step augments the stack; each backward move pops from the stack, restoring prior state, with memory identifiers ensuring backward determinism. Under this encoding, forward and backward process transitions correspond exactly to the hereditary extension/restriction of bijections in HHPB.

For CCS processes without auto-concurrency, the back-and-forth bisimulation of RCCS coincides precisely with HHPB on configuration structures (Aubert et al., 2015, Aubert et al., 2020): $$P \approx_{\mathrm{RCCS-bf}} Q \iff \llbracket P \rrbracket \approx_{\mathrm{HHPB}} \llbracket Q \rrbracket$$ With auto-concurrency, one must refine configuration structures to carry unique identifiers for memory events, ensuring bijections can be tracked even when several concurrent events have the same label (Aubert et al., 2018).

4. Logical and Game-Theoretic Characterizations

HHPB is characterized by modal logics extending Hennessy-Milner logic with event identifiers and reverse modalities (Phillips et al., 2011, Baldan et al., 2011). Event Identifier Logic (EIL) and the logic $L$ for true concurrency allow formulas to refer to events via identifiers, match labeled transitions in both directions, and quantify over possible histories. HHPB coincides with logical equivalence in these logics: two processes (structures) are HHPB-equivalent iff they satisfy the same closed formulae.

For image-finite event structures, logical equivalence ($\equiv_L$) and HHPB coincide: $$\mathcal{E}_1 \approx_{\mathrm{hhp}} \mathcal{E}_2 \iff \mathcal{E}_1 \equiv_L \mathcal{E}_2$$ Game-theoretic characterizations of HHPB employ strong history-preserving bisimulation games, with positions tracking pairs of configurations and bijections. Duplicator’s winning strategy must itself be downward closed, matching Spoiler's moves and preserving subhistory conditions (Wang, 2019).

5. Examples, Applications, and Decidability

HHPB distinguishes concurrent and interleaved processes that standard interleaving bisimulation cannot. For example, $P = a \parallel b$ and $Q = a.b + b.a$ are equivalent under standard interleaving bisimulation, but not under HHPB:

• $P$ admits a configuration $\{a,b\}$ with concurrent $a, b$; $Q$'s configurations have $a$ and $b$ causally ordered.
• No label/order-preserving bijection relates these top configurations; hence, they are HHPB-distinct (Aubert et al., 2015, Aubert et al., 2015, Wang, 2021).

HHPB is undecidable for safe Petri nets (Gorrieri et al., 13 Jun 2025), but there exist reversible behavioral equivalences—causal-net bisimilarity and structure-preserving bisimilarity—that are decidable and coincide with hereditary causal-net bisimilarity (an HHPB refinement) in the finite net setting. Place bisimilarity provides an even finer, decidable equivalence.

For higher-dimensional automata (HDAs), HHPB equates ordinary HDAs with their symmetric counterparts, demonstrating coincidence in expressive power within concurrency models (Kahl, 2021).

6. Variants, Extensions, and Recent Characterizations

HHPB has been extended to richer contexts:

• Localities: Events may carry locality labels; HHPB remains unchanged except that bijections preserve locality in addition to action and causality (Wang, 2021).
• Reversibility: Bidirectional transitions enforce matching backward steps; HHPB naturally supports reversible process models.
• Probabilism: Probabilistic HHPB relates distributions over configurations, with matching probabilities for transitions and hereditary closure over support (Wang, 2021).
• Guards: Boolean guards on transitions require that bisimilar transitions have equivalent enabling conditions.

Recent work characterizes HHPB via backward ready multisets (brm), replacing event identifiers with multiset counts of enabled reverse actions (Bernardo et al., 7 Dec 2025). Such characterizations yield lighter-weight binary relations, valid in the absence of non-local conflicts, and connect classical HHPB (via event identifier logic) to forward-reverse bisimulation with brm equality. Modal logics with backward-ready multiset tests correspond precisely to these bisimulations, offering logic-based verification frameworks.

7. Significance and Role in the True-Concurrency Spectrum

HHPB provides the "gold standard" for true-concurrency equivalence, standing at the top of the spectrum by distinguishing structures with identical visible behavior but differing causal dependency or conflict. Its congruence and compositionality properties enable robust algebraic and logical reasoning. Operational characterizations via reversibility and memory close the gap between denotational and process-algebraic semantics, offering full abstraction theorems. Although undecidable in general, HHPB informs the design of practically checkable reversible equivalences and underpins the logical analysis of concurrent systems (Gorrieri et al., 13 Jun 2025). Its game-theoretic and modal logic correspondences unify logical and operational approaches to concurrency, and its algebraic axioms support sound and complete reasoning in truly concurrent process algebras (Wang, 2021).