Foundations of Truly Concurrent Semantics

• Truly Concurrent Semantics is a computational model that natively represents concurrency, causality, and conflict using partial orders and pomsets.
• The framework employs algebraic constructs and automata-theoretic techniques, integrating operators like parallel and communication merge to enable compositional reasoning.
• Its robust foundation supports precise verification of distributed and mobile processes, enhancing analysis of synchronization, reversibility, and probabilistic behaviors.

A truly concurrent semantics foundation rigorously models computation in which concurrency, causality, and conflict among actions are treated as first-class entities, in contrast to classical interleaving-based frameworks where concurrency is reduced to nondeterministic shuffling of atomic steps. Modern theory achieves this by building on partial orders—pomsets, event structures, and their automata or algebraic analogues—and formalizes compositional operators, algebraic reasoning, and various behavioral equivalences that distinguish between concurrent and merely interleaved behavior. This foundation supports precise representation, analysis, and verification of concurrent, mobile, probabilistic, and effectful systems.

1. Mathematical Basis: Truly Concurrent Automata and Pomsets

At the core of a truly concurrent semantics lies the concept of partially ordered computations. The truly concurrent automaton (TCA) is the canonical automata-theoretic model. A TCA over action alphabet $\Sigma$ is a tuple $\mathcal A = (Q, q_0, F, \Sigma, \to, \leq, \#)$, where

• $Q$ is a finite set of control states,
• $q_0 \in Q$ the unique initial state,
• $F \subseteq Q$ the set of final states,
• $\to \subseteq Q \times \Sigma \times Q$ is the labelled transition relation,
• $\leq \subseteq \to \times \to$ is a partial order modeling causal dependence between transitions,
• $\# \subseteq \to \times \to$ is an irreflexive, symmetric incompatibility (conflict) relation.

Runs of a TCA yield pomsets: partially ordered multisets of events, recording both causal independence and conflict. Two transitions $e, e'$ are concurrent iff they are unrelated by either order or conflict: $$e \parallel e' \Longleftrightarrow \neg (e \leq e' \lor e' \leq e \lor e \# e')$$ (Wang, 2024). This encodes non-sequentializable concurrency at the semantic level.

2. Algebraic Constructions and Compositionality

Truly concurrent semantics supports an algebra of expressions over TCAs:

• $$E ::= 0 \mid 1 \mid a \mid E+E \mid E \cdot E \mid E^* \mid E \parallel E \mid E \mid E$$ with
• $+$ modeling nondeterministic choice,
• $\cdot$ sequential composition,
• $^*$ Kleene/star iteration,
• $\parallel$ parallel composition with independence,
• $\mid$ communicating parallel (communication merge).

Operators are interpreted denotationally in the automaton model. For instance, the communication merge is governed operationally by

$\infer {p \xrightarrow{a} p' \quad q \xrightarrow{b} q' \quad \gamma(a,b) = c} {p \mid q \xrightarrow{c} p' \mid q'}$

where $$\gamma : \Sigma \times \Sigma \rightharpoonup \Sigma \cup \{\tau\}$$ synchronizes actions (Wang, 2024). The algebra is closed under these operators and supports modular composition.

3. Semantic Equivalences and Bisimilarity Spectrum

Two main equivalence notions underpin the semantic foundation:

• Pomset-language equivalence ($E \equiv_{\Lang} F$): $E$ and $F$ recognize the same set of labeled pomsets.
• Pomset bisimilarity ($E \approx_{\mathrm{pb}} F$): processes are equivalent if there exists a relation matching each pomset transition in one with a pomset isomorphic transition in the other, inductively on reachable configurations.

These extend through a refinement spectrum: | Equivalence | Matching Criterion | Congruence Level | |----------------------|-----------------------|--------------------------------| | Step bisimulation | Steps (concurrent sets) | Coarser | | Pomset bisimulation | Labeled partial orders (pomsets) | Finer | | History-preserving | Partial order + event isomorphism | Yet finer | | Hereditary history-preserving | Downward closure on histories | Finest (Wang, 2019, Wang, 2020) |

Bisimilarity games provide fully abstract characterizations: Duplicator's existence of a winning strategy corresponds exactly to bisimilarity (Wang, 2019), and congruence closure is preserved by all TCA operators (Wang, 2017, Wang, 2020).

4. Proof Theory, Completeness, and Axiomatization

Axiomatic systems for the truly concurrent algebra (e.g., STC for $\pi_{tc}$, PA1/PA2 for APTC) provide sound and, where possible, complete equational characterizations for the equivalences above.

For parallel composition, the PA1 axioms suffice for pomset, step, and history-preserving bisimulation. For hereditary history-preserving bisimulation, finite axiomatization requires auxiliary operators: left parallel merge ($\leftmerge$) and communication merge ($\mid$). The uniform solution is

$x \parallel y = x\, \leftmerge\, y + y\, \leftmerge\, x + x \mid y$

with a finite, complete axiom system over $\{+, \cdot, \parallel, \leftmerge, \mid\}$ (Wang, 2020). Normal-form theorems and Arden-style fixpoint arguments establish decidability and completeness for series-communication terms (Wang, 2024).

5. Communication, Synchronization, and Extensions

Truly concurrent semantics is distinguished by its internalization of communication and interaction patterns:

• Communication functions $\gamma$ make send/receive (or read/write) synchronizations first-class, rather than exogenous as in interleaving models.
• The expansion law

$$a \between b = a \cdot b + b \cdot a + a \parallel b + (a \mid b)$$

exactly isolates the new behaviors enabled by true concurrency: both independent concurrency ($\parallel$) and direct synchronization ($\mid$) as orthogonally extending the interleaving sum (Wang, 2024).

Extensions such as reversibility, probabilism, and guards are accommodated smoothly. E.g., reversible steps and backward moves are modeled by paired (forward/backward) transitions and causal-consistency conditions (Wang, 2021, Wang, 2021).

6. Model-Theoretic and Operational Frameworks

Truly concurrent semantics supports foundational correspondence results:

• Pomset-labeled transition systems (PLTS) and pomset transition system specifications (PTSS) generalize classical SOS, allowing transitions labeled by pomsets/partial orders (Wang, 24 Jan 2026).
• Congruence formats for pomset/step/hp-bisimulations guarantee that all compositional operators are structurally well-behaved.
• Denotational models (e.g., pomset-based semantics for memory models (Kavanagh et al., 2018), interval-based refinement (Dongol et al., 2013), bicategory/pseudomonad frameworks (Paquet et al., 2023)) ensure existence of unambiguous semantics, robust fixed-point theory, and modularity.

The theory generalizes not only classical automata/Kleene algebra but also concurrent Kleene algebra, unifying the interleaving and truly concurrent perspectives in a single syntactic and semantic setting (Wang, 2023, Wang, 2024).

7. Significance and Impact Across Models

This foundation underpins:

• Precise modeling and verification of distributed and mobile processes, including those with dynamic reconfiguration and synchronization;
• Formal reasoning about program logics, effect systems, and refinement in the presence of real concurrency;
• Efficient symbolic and algorithmic methods for equivalence, model-checking, and verification, exploiting reductions to games, normalization, and compositionality (Wang, 2019, Horn et al., 2015, Narayanaswamy et al., 2016);
• Generalization to complex process calculi, probabilistic, and reversible systems, and categorical models capturing weak interchange laws for processes and effects (Paquet et al., 2023).

By directly expressing concurrency, causality, and conflict, the truly concurrent semantics foundation supplies the mathematical and operational infrastructure required for both theoretical investigation and practical analysis of complex concurrent systems (Wang, 2024).