Abstract Consistency Property

Updated 24 November 2025

Abstract Consistency Property is a formal criterion that defines necessary and sufficient conditions for composing local or partial descriptions into a coherent global whole.
It is instantiated in databases, probabilistic models, and distributed systems via witness functions, partitions, and algebraic constraints that ensure global consistency.
The property underpins the design of algorithms in domains such as knowledge representation, machine learning, and blockchain by reconciling local constraints without requiring global arbitration.

The abstract consistency property is a formal, domain-independent criterion that ties together structural, algebraic, epistemic, and operational meanings of "consistency" across logic, distributed systems, databases, knowledge representation, and machine learning. Its instantiation specifies the minimal requirements under which local or partial descriptions (traces, relations, beliefs, attributions) can be realized as coherent global objects or explanations while satisfying various domain-specific constraints. The property captures both necessary and sufficient conditions for gluing, composing, or reconciling such local elements into a globally consistent whole.

1. Formal Definitions and Algebraic Frameworks

Abstract consistency properties are typically characterized by the existence of a global "witness" extension under algebraic or structural constraints that respect local or partial information:

Relational (database) setting: Given two $\mathbb{K}$ -relations $R(X)$ and $S(Y)$ over a positive commutative monoid $\mathbb{K}$ , consistency of $R$ and $S$ means there exists a $\mathbb{K}$ -relation $T$ on $X\cup Y$ such that $T[X]=R$ and $T[Y]=S$ . Any such $T$ is a consistency witness. The abstract consistency property of a binary operation $W$ is: whenever $R$ and $S$ are consistent, $W(R, S)$ is a witness (i.e., $W(R, S)[X]=R$ and $W(R,S)[Y]=S$ ) (Atserias et al., 23 Sep 2025).
Probabilistic (epistemic) setting: Given an ex ante prior $P$ $P$ , a family of ex post posteriors $\{P_s\}_{s \in S}$ ${P_{s}}_{s \in S}$ , and a possibility correspondence $I \colon S \to 2^S$ $I : S \to 2^{S}$ , the abstract consistency property is encoded by three axioms:
1. Invariance: $P(E) = \int_{S}P_s(E)P(ds)$ for all $E$ .
2. Entailment: $P_s(I(s)) = 1$ for all $s$ .
3. Self-Evidence: For all $s' \in I(s)$ , $P_{s'}(E)\geq P_s(E)$ (Fukuda, 2019).
Surrogate modeling (explainable ML): For high-level attributions $\alpha_j$ and low-level attributions $\beta_{j d}$ , consistency is specified by the constraint

$\alpha_j = \sum_{d=1}^{D_j} \beta_{j d}, \quad \forall j=1,\dots,J,$

enforcing that each high-level attribution is the sum of the attributions of its constituent low-level features (Yoshikawa et al., 23 May 2024).

The property is always predicated on strong algebraic or structural prerequisites: positivity, commutativity, additivity, monotonicity, or well-formedness, to ensure compositionality.

2. Semantic and Epistemic Instantiations

In epistemic and logical settings, abstract consistency is formulated as a lack of distributed knowledge of violation:

Epistemic logic: Consistency corresponds to the property that the group $G$ (e.g., threads, or threads plus observer) does not know that a state is incorrect,

$\neg D_G \neg \mathit{correct},$

where $D_G$ is distributed knowledge and $\mathit{correct}$ encodes the sequential or behavioral specification. All major consistency conditions in concurrency (sequential consistency, linearizability, eventual consistency) fit this pattern by appropriate choice of $G$ and predicate (Gleissenthall et al., 2013).

Partitioning in probability: The requirement that $P_s$ be the conditional measure on the cell $I(s)$ (which forms a partition up to null sets) enforces that local beliefs (the posteriors) globally partition the space, and that each belief is fully determined by this structure (Fukuda, 2019).

This epistemic view gives a unified schema for most classical and modern consistency criteria, allowing modular variations of "agents" or "correctness" to yield a wide spectrum of abstract consistency notions.

3. Structural, Distributed Systems, and Concurrency Models

The abstract consistency property operationalizes the manner in which local or partial orders must be extended or composed:

Distributed systems and databases: In the structural database setting, the existence of a consistency witness function generalizes the join operation and ensures that, for acyclic schemas (especially $\gamma$ -acyclic), all local constraints can be globally realized if and only if the underlying monoid has the transportation property (Atserias et al., 23 Sep 2025).
Distributed histories and visibility: Abstract consistency can be instantiated as closure or monotonicity properties of visibility relations, such as:
- Monotonic Visibility: $(a\,\mathrm{VIS}\,b) \wedge (b\,\mathrm{SO}\,c) \implies (a\,\mathrm{VIS}\,c)$ .
- Closed Past: If $a\,\mathrm{VIS}\,b$ and $c\,\mathrm{VIS}\,b$ for process $i$ , then $a\,S_i\,c$ (totally ordered in $i$ 's serialization).
- Arbitration: The existence of a global total order compatible with all process serializations (Almeida, 25 Nov 2024).

This suggests abstract consistency determines the boundaries of what global histories (or system behaviors) can coherently extend sets of local behaviors in distributed models.

4. Implementation Criteria and Necessary and Sufficient Conditions

Abstract consistency properties often yield sharp, checkable criteria for global solvability or implementability:

Witness function existence: Existence of a consistency witness function is equivalent to the transportation property of the value domain (monoid). When satisfied, every locally consistent set of relations (with matching marginals/projections) has a global extension. Non-uniqueness of such witnesses may arise, but existence is necessary and sufficient for many monotonicity theorems in database theory (Atserias et al., 23 Sep 2025).
Availability without arbitration: In distributed storage, arbitration-free consistency denotes that the consistency model does not require global arbitration order ( $\mathrm{ar}$ ); only local orders (session, causal, or visibility) are allowed. The AFC theorem asserts that a consistency model admits an available (partition-tolerant, non-blocking) implementation if and only if it is arbitration-free (Attiya et al., 24 Oct 2025). This generalizes the CAP theorem and is a necessary and sufficient condition for highly available systems over a wide object and model spectrum.
Bayesian updating and partition uniqueness: In epistemic models, the existence and uniqueness of a family of posteriors satisfying invariance, entailment, and self-evidence is equivalent to the cells $I(s)$ agreeing $P$ -almost surely with the partition induced by the posteriors, and each $P_s$ is conditional on its cell (Fukuda, 2019).

5. Illustrative Examples and Model Hierarchies

Domain-specific instantiations clarify the range and power of abstract consistency:

Domain	Consistency Property Instance	Sufficient Condition
Boolean relations	Join is witness iff schema is acyclic	Schema is $\gamma$ -acyclic and monoid is positive with transportation property (Atserias et al., 23 Sep 2025)
Probabilities	Posteriors are Bayes on partition	Regularity (additivity, invariance, entailment, self-evidence) (Fukuda, 2019)
ML attributions	HiFA sums to LoFAs per instance	Linear surrogate model fit, enforced as convex constraint (Yoshikawa et al., 23 May 2024)
Distributed systems	Available implementation iff arbitration-free	No $\mathrm{ar}$ constraint in consistency model (Attiya et al., 24 Oct 2025)

Specific examples include:

Graph transformations: A transformation is consistency-sustaining if it does not increase the count of constraint violations and consistency-improving if it strictly decreases this count—refining the classical preserving and guaranteeing concepts (Kosiol et al., 2020).
Blockchain ADTs: Abstract consistency criteria distinguish "strong" (prefix) consistency, requiring unique chain prefix, and "eventual" consistency, requiring only alignment in the limit. The corresponding oracles and selection functions express control over forks and the resultant global consistency (Anceaume et al., 2018).
Causal vs. sequential consistency for ADTs: The inclusion hierarchy for consistency properties situates weak causal, causal, and sequential consistency in strict containment, governed by the expressiveness and granularity of partial orderings (Perrin et al., 2016).

6. Role in Theory and Impossibility Results

Abstract consistency is instrumental in demarcating the boundary of achievable guarantees in concurrent, distributed, or logically compositional systems. Notable results include:

Monotonicity theorems: For (hyper)graph acyclic schemas and transportation monoids, every connected sequence of compositions (joins, witness functions) yields a globally defined, monotone expression if and only if the abstract consistency property holds for the value domain (Atserias et al., 23 Sep 2025).
Impossibility under availability: The CLAM theorem establishes that no wait-free implementation of nontrivial data types can simultaneously satisfy closed past, local visibility, global arbitration, and monotonic visibility, highlighting inevitable trade-offs in distributed settings (Almeida, 25 Nov 2024). Similarly, the AFC theorem exposes that any need for arbitration (total order) in a consistency model precludes availability (Attiya et al., 24 Oct 2025).
Uniqueness in epistemic models: Under the consistency properties, the information partition induced by posteriors is unique, and every compatible belief operator (qualitative or probabilistic) automatically satisfies the truth axiom almost surely (Fukuda, 2019).

7. Scope, Extensions, and Limitations

The abstract consistency property is a unifying abstraction that enables the systematic analysis and design of composition, reconciliation, and explanation protocols under both structural and behavioral constraints. However, its applicability depends crucially on the underlying algebraic properties (positivity, additivity), the possibility of partitioning or witnessing extensions, and tractable expressive power in the chosen logic or data model.

A plausible implication is that as new domains (e.g., deep learning explanation, blockchain consensus, knowledge representation) formulate increasingly complex local-global consistency requirements, the abstract consistency property will underpin formal characterizations of solvability, synthesizability, and explainability across these domains, provided their foundational algebraic and logical constraints can be precisely articulated and verified.