Automatic Reconciliation Algorithm

• Automatic reconciliation algorithms are procedures that merge divergent data replicas using algebraic models and topologically-sorted command sequences.
• They detect conflicts via commutativity checks and independence relations, ensuring only non-interfering operations are propagated.
• This approach underpins distributed synchronization, versioned filesystems, and collaborative editing by guaranteeing maximal, safe updates without semantic loss.

Automatic reconciliation algorithms are algorithmic procedures that compute, with no manual intervention, the maximal safe merging of divergent data replicas or data structure histories. These algorithms are central to distributed data synchronization, versioned filesystems, and collaborative editing systems, where independent update sequences must be merged without semantic loss or conflict. The term captures both early algebraic approaches to merging filesystem trees and more general frameworks for propagating non-interfering operations or edits in settings ranging from filesystems (Csirmaz, 2016) to replicated distributed systems (Csirmaz et al., 2021), and provides an underpinning for provably correct and complete conflict detection and resolution.

1. Algebraic Model of Filesystem State and Commands

The foundational algebraic approach formalizes a filesystem as a total function from a fixed rooted tree of nodes $N$ to a value set $V = [D] \cup [F] \cup \{\bot\}$, where $[D]$ represents directories, $[F]$ are file values, and $\bot$ denotes “empty.” A valid filesystem state $\sigma$ must satisfy the tree property: $$\forall n \in N,\ \sigma(n) \neq \bot \implies (\text{parent}(n)\ \text{is defined})\ \wedge\ [\sigma(\text{parent}(n))]=[D].$$ Commands are modeled as partial endofunctions on filesystems, each characterized by an input type, an output value, and a node target: $$x = ([x],\,y,\,n)\quad\left([x]\in\{[D],[F],[\bot]\},\;y\in V,\;n\in N\right),$$ with semantic application defined as

$$x(\sigma) = \begin{cases} \bot, &\text{if } [\sigma(n)] \neq [x] \text{ or tree‐property fails after}, \ \sigma[n \mapsto y], &\text{otherwise}. \end{cases}$$

Commands are categorized as Construction, Destruction, Replacement, or Assertion types, and can be composed sequentially. Command categories are defined by their input–output types; e.g., Construction includes $[\bot] \!\to [D]$, $[\bot] \!\to [F]$, $[D] \!\to [F]$.

2. Detection and Definition of Conflicts

Conflicts are defined algebraically using commutativity and the concept of command independence. Two commands $\alpha$ and $\beta$ are independent, written $\alpha \parallel \beta$, if and only if their applications commute on all filesystems and neither application is everywhere undefined: $$\alpha \parallel \beta \iff \alpha\beta \equiv \beta\alpha \text{ and neither side is identically } \bot.$$ A conflict between update-sets $A$ and $B$ is any pair $\alpha\in A,\ \beta\in B$ failing independence, i.e., $\neg(\alpha\parallel\beta)$. For non‐assertion commands, non-commutativity arises precisely when they act on comparable nodes without directory/empty assertion cases.

3. Update Detection Algorithm

Given a base state $\sigma_0$ and a modified replica $\sigma^*$, the update-detection algorithm constructs a canonical set of non-assertion commands $U$ with at most one command per node, such that $U(\sigma_0) = \sigma^*$. The construction is as follows:

1. For each $n \in N$ with $\sigma_0(n) \neq \sigma^*(n)$, add $([\sigma_0(n)], \sigma^*(n), n)$ to $U$.
2. Perform a topological sort of $U$ under the partial order $\prec$ (parent–child order for constructions, reverse for destructions), yielding a simple sequence $U$.

The key theorems establishing the correctness of this procedure include:

• Can‐Simplify (Theorem 4.3): Any command sequence can be syntactically rewritten into a simple, non-assertion, at-most-one-per-node sequence by swapping independent neighbors, collapsing same-node pairs, and dropping assertions, preserving semantics.
• Reorder‐Equiv (Lemma 4.2): Any permutation of a simple sequence respecting $\prec$ is semantically equivalent; equivalently, all simple $\prec$-respecting permutations are functionally identical.

Thus, the update detection procedure produces a unique, minimal, canonical command sequence effecting the transition from $\sigma_0$ to $\sigma^*$.

4. Operation-Based Reconciliation Algorithm

Automatic reconciliation proceeds by taking two simple command sequences $A$ and $B$ representing the local modifications from the base state for two replicas. The core of the algorithm is to propagate as many commands as safely possible from $A\setminus B$ to replica $B$ and symmetrically, without breaking the filesystem or overwriting conflicting changes. The key structure is:

$$A^\dagger = \{\alpha\in A\setminus B\,|\,\forall\beta \in B\setminus A: \alpha\parallel\beta\}$$

A propagable sequence $A_{\text{prop}}$ is any topological sort of $A^\dagger$.

Pseudocode for propagating $A$ into $\sigma_B$:

1. Initialize $C\gets \emptyset$.
2. For each $\alpha \in A\setminus B$, if $\alpha \parallel (B\setminus A)$, add $\alpha$ to $C$.
3. Return a $\prec$-respecting permutation of $C$ as $A_{\text{prop}}$.

Applying $A_{\text{prop}}$ to $B(\sigma_0)$ yields the maximally reconciled state feasible without semantic violations.

5. Adequacy and Completeness Theorems

The correctness and optimality of automatic reconciliation are established by the following results:

• Adequacy Theorem (5.3):

For simple command sets $A,B$, $B\,A_{\text{prop}}$ is defined (never breaks) on every filesystem where $B$ is defined, i.e.,

$\{A,B\}\;\sqsubseteq\;\{B,\,A_{\text{prop}}\}.$

• Completeness Theorem (5.4):

No further command in $A\setminus B$ with $\neg(\alpha\parallel B\setminus A)$ can be safely propagated to $B$; such attempts either break the tree property or overwrite a change made by $B$.

Thus, the algorithm is maximal, propagating exactly those commands that can commute with all opposing operations, and no more; any additional propagation is either semantically impossible or involves a genuine conflict.

6. Symmetry, Precision, and Optimality of the Algebraic Approach

Optimality arises from several technical properties:

• Maximality: The reconciler includes all non-interfering, commutative commands.
• Symmetric command set: Every operation specifies both its precondition (input type) and effect (output value), eliminating edge cases and ensuring a precise, symmetric definition of independence.
• Explicit semantic independence: The independence relation is precise and algebraically characterized, supporting completely automatic selection of propagable operations.
• Proof of impossibility: Completeness shows that any command not included in the propagable set must create an error or a true conflict, guaranteeing that the reconciliation is both correct and maximal.

In practice, this algebraic approach underpins strongly predictable, automatic merge procedures for versioned filesystems and similar tree-structured data, generalizing to a theoretical basis for error-free operation in a wide array of synchronization applications.

7. Applications, Limitations, and Extensions

Automatic reconciliation algorithms in this framework offer practical, exact, and automatically computable procedures for merging filesystem replicas, supporting applications in distributed filesystems, mobile data synchronization, and conflict detection engines within version control environments.

Trade-offs exist between completeness and granularity of conflict detection; the current framework precisely delineates, via the independence relation, all irreducibly semantically conflicting operations, avoiding both spurious conflicts and unsafe merges.

Extensions may include richer data models and algebras for more general data structures (Csirmaz et al., 2021), but the central principles—algebraic modeling, detection via commutativity and well-founded partial orders, and maximal safe propagation—remain central to all such further work.