DECOMAS: Deterministic Causal Structure

• DECOMAS Architecture is a formal framework for deterministic causal relations in distributed systems and control systems, using a unique global DAG to ensure auditability and consistency.
• It employs a directed-complete join-semilattice to guarantee convergence and policy-agnostic state consistency through immutable contributions and fixed causal links.
• The architecture sets itself apart from CRDTs and probabilistic models by explicitly encoding provenance and deterministic causal order, regardless of operational scheduling.

A DECOMAS (Deterministic Causal Structure) architecture provides a mathematically rigorous, policy-agnostic substrate for representing and reasoning about deterministic causal relations in distributed systems, control systems, and structured dynamical processes. Its foundations are especially relevant for asynchronous distributed computation, causal discovery in dynamical systems, and the structural analysis of equilibrated deterministic models. The DECOMAS approach delineates the boundary between correctness and the operational policies, ensuring that consistency, auditability, and causality are intrinsic to the structure, independent of the system's policy layer (Ren et al., 7 Oct 2025).

1. Mathematical Definition and Foundational Axioms

The DECOMAS paradigm formalizes the notion of deterministic causality over a set of “contributions” (events, updates, measurements) in a distributed multi-agent environment. The primary construction is the unique, global provenance DAG (Directed Acyclic Graph), assembled from individual contributions, each specified as a tuple

$\delta = (\rid,\, \parents,\, \payload,\, k)$

where

• $\rid$ is a globally unique identifier,
• $\parents$ is the set of immediate causal predecessors,
• $\payload$ is the (opaque) value,
• $k$ is a logical partition or key (Ren et al., 7 Oct 2025).

Local agent state for each key is the join (supremum) of all payloads received, computed within a directed-complete join-semilattice $(L_k, \sqsubseteq, \sqcup)$. This algebraic structure ensures that the partial order is closed under directed suprema, making state convergence well-defined and order-agnostic.

The minimal DECOMAS axioms are:

1. Localized Weak Fairness: Perpetually sent contributions for a key $k$ will eventually be seen by all relevant agents.
2. Directed-Complete Join-Semilattice: The value domain for each key forms a directed-complete poset with an associative, commutative, idempotent join.
3. Immutability of Contributions: Once created, rids and their payloads are fixed.
4. Immutability of Causal Links: The parents of each contribution never change.
5. Causal Well-Formedness: Each new contribution may only cite as parents previously observed contributions, precluding cycles.

These axioms guarantee that the resultant global graph $G^*=(\mathcal{R},\,E)$ is a unique DAG and that all converged agent states (across any valid delivery/execution policy) are invariant up to isomorphism (Ren et al., 7 Oct 2025).

2. Core Theorems: Invariance and Uniqueness

Four principal results anchor the DECOMAS architecture:

• Existence and Uniqueness: The global provenance DAG and converged agent states exist, are unique, and are constructible solely from the causal metadata, given the axioms.
• Policy-Agnostic Invariance: For any two admissible delivery or scheduling policies yielding the same multiset of contributions, the global DAG and local limit states are isomorphic; system behavior remains unchanged modulo isomorphism.
• Observational Equivalence: Applications making only ancestry/concurrency queries and homomorphic (join-based) aggregate operations cannot distinguish between two executions with isomorphic DAGs.
• Axiom Minimality: Dropping any one axiom permits counterexamples where existence, uniqueness, or convergence fails (Ren et al., 7 Oct 2025).

These results explicate why DECOMAS serves as a “correctness-as-a-chassis” for distributed computation, providing deterministic guarantees at the provenance level, independent of operational behavior such as routing, batching, or delivery order.

3. DECOMAS Versus Value-Centric and Probabilistic Models

DECOMAS strictly refines value-centric convergence approaches such as CRDTs (Conflict-Free Replicated Data Types). While CRDTs ensure value convergence via commutative, associative operations, they are oblivious to the shape of the global causal history. Minimal examples expose that two different sequences—concurrent versus causally dependent contributions—produce the same terminal state in CRDTs but yield distinct DAGs in DECOMAS. Thus, DECOMAS encodes provenance and partial order explicitly, enabling queries and invariants grounded in audit-ready, policy-agnostic causal structure (Ren et al., 7 Oct 2025).

In contrast with stochastic causal models, a DECOMAS is defined by deterministic transitions: stochasticity appears only in the operational policies, never in the core causal structure. Deterministic SCMs (as constructed from ODE equilibria or linear propagation models) can be viewed as special cases where the join-semilattice reduces to a functional graph of deterministic assignments (Mooij et al., 2013, Mooij et al., 2014, Yang et al., 2021).

4. Deterministic Causal Structure in Dynamical and Control Systems

The decomposition of correctness and policy in dynamical contexts parallels developments in causal discovery for deterministic and partially deterministic systems:

• In deterministic control systems, the causal structure links state and input variables via deterministic, possibly nonlinear relations, with the structure inferred by systematically designed interventions exploiting system controllability. Causal links are established by distinguishing the effect of perturbations via statistical divergences (e.g., MMD) under repeated, noise-attenuated experiments, a process that is robust in the deterministic regime (Baumann et al., 2020).
• For ODE-governed dynamical systems with unique equilibrium, the mapping from ODEs to deterministic SCMs ensures that perfect interventions (replacement of structural equations) commute with both the equilibrium computation and the causal update, preserving consistency. These deterministic SCMs capture the stationary causal logic of the underlying process (Mooij et al., 2013, Mooij et al., 2014).

DECOMAS formalism naturally encompasses these paradigms: the global DAG is the unique classical causal structure compatible with all possible agent actions and interventions, provided the integrity and convergence axioms hold.

5. Structural Comparison, Boundaries, and Practical Implications

DECOMAS establishes an asynchrony boundary principle: in asynchronous, crash-stop environments with no external coordination or strong clocks, the join-semilattice is the maximal expressive state model that supports deterministic, policy-free convergence. Any stronger consistency requirement (e.g., total order) invokes entanglements between policy and correctness, violating the core DECOMAS separation (Ren et al., 7 Oct 2025).

Illustrative scenarios confirm that vastly differing policies (e.g., FIFO vs. random batching) result in the same global DAG and agent states, confirming both theoretical invariance and practical robustness to operational evolutions.

Table: DECOMAS vs. Related Causal Formalisms

Aspect	DECOMAS Architecture	CRDTs	Deterministic SCMs (ODE mapping)
Causal Provenance Explicit	Yes	No	Yes (structural equations graph)
Value Convergence	By join-semilattice	Yes	Yes (unique solution per intervention)
Policy Independence	Guaranteed	Yes in value, not in history	Yes (interventions commute)
Asynchrony Boundary	Characterized	Implied	N/A (not distributed)

6. Generalizations and Open Directions

The DECOMAS framework generalizes to arbitrary agent sets, key spaces, and state domains admitting join-semilattice structure. It is compatible with nondeterministic delivery policies, arbitrary batching, and arbitrary schedulers, provided only the axioms are respected.

Emerging research directions include:

• Extending DECOMAS principles to nonlinear, hybrid, or quantum causal structures, where cause-effect relations may depart from classical DAG representations or involve noncommutative state evolution (Baumeler et al., 2024).
• Analyzing identifiability, algorithmic recovery, and robustness for DECOMAS-style models in noisy, partially observed, or adversarial settings, synthesizing techniques from deterministic and probabilistic causal inference (Daniusis et al., 2012, Yang et al., 2021).

7. Applications and Significance Across Domains

DECOMAS enables modular, evolvable system architectures where performance optimizations (e.g., new scheduling or communication patterns) do not threaten system correctness or auditable causal semantics. It provides foundational structure for distributed data systems, large-scale control infrastructures, epidemiological models with latent determinism, and multi-agent platforms requiring provenance guarantees. Its axiomatic clarity and policy separability make it a core architectural principle in the emerging landscape of causally robust intelligent systems (Ren et al., 7 Oct 2025, Baumann et al., 2020).