Normative Hierarchy Graph

The graph must be a directed acyclic graph (DAG) to prevent circular precedence paradoxes. Each directed edge R1 → R2 asserts that Rule 1 strictly outranks Rule 2.

• No cycles allowed: A cycle (R1 > R2 > R1) represents a logical contradiction where no consistent resolution is possible
• Topological ordering: The DAG property guarantees at least one valid traversal order for conflict resolution
• Cycle detection: Algorithms like depth-first search (DFS) with back-edge detection validate graph integrity before reasoning begins

A cyclic graph indicates a drafting error or an unresolved normative antinomy that requires human intervention.

Edges in the graph are typed to encode the specific conflict resolution maxim being applied, enabling the system to justify its reasoning.

• lexSuperior: Edge from a constitutional rule to a statutory rule, encoding hierarchical authority
• lexSpecialis: Edge from a specific exception to a general rule, encoding specificity-based precedence
• lexPosterior: Edge from a later-enacted rule to an earlier one, encoding temporal precedence
• derogat: A generic override edge when the specific maxim is ambiguous or composite

Edge types are critical for explainability, allowing the system to output the legal basis for why one rule prevailed over another.

The graph's transitive closure determines the full set of rules a given norm can defeat, not just its immediate neighbors. If Rule A overrides Rule B, and Rule B overrides Rule C, then Rule A transitively overrides Rule C.

• Precomputation: Transitive closure is often computed offline using the Floyd-Warshall algorithm for small graphs or via graph database queries for large corpora
• Conflict path tracing: When resolving a conflict, the system finds the shortest directed path between the two conflicting nodes
• Path existence check: If no directed path exists between conflicting rules, the graph is incomplete and requires a tie-breaking heuristic

This property ensures that deeply nested legal hierarchies are correctly traversed.

Each node carries metadata attributes that influence conflict resolution beyond simple topological precedence.

• Effective date range: Temporal bounds defining when a rule is active, enabling the graph to model historical legal states
• Jurisdictional scope: Tags limiting a rule's applicability to specific geographic or subject-matter domains
• Authority score: A quantitative weight (e.g., 1.0 for constitutional, 0.3 for administrative guidance) used when graph topology alone is insufficient
• Deontic modality: The node's normative force — obligation, prohibition, or permission — which determines the conflict type

Rich metadata transforms the graph from a simple ranking into a computationally complete normative model.

The graph typically represents a partial order, not a total order. Many pairs of rules may be incomparable — no directed path exists in either direction.

• Incomparability is valid: Two rules from different jurisdictional branches or subject domains may never interact
• Conflict detection scope: Incomparable rules that nonetheless produce contradictory obligations represent a genuine antinomy requiring external resolution
• Tie-breaking policies: When incomparable rules conflict, the system falls back on meta-policies like "prefer the more specific rule" or "flag for human review"

This property reflects the reality that legal systems are not perfectly hierarchical but contain independent, parallel authority structures.

The graph is not static; it must support incremental updates as new statutes are enacted, old ones repealed, and judicial interpretations shift.

• Node insertion: Adding a new rule requires computing its edges to all existing nodes based on its authority level and specificity
• Node expiration: Rules past their effective end date are not deleted but marked as inactive, preserving the graph's ability to reason about historical legal states
• Edge reconfiguration: A landmark court decision may reweight or redirect edges, requiring efficient graph update algorithms
• Version snapshots: The full graph state at time t is preserved for auditing and temporal reasoning queries

This property ensures the graph functions as a living model of the legal system, not a one-time artifact.

A conflict resolution maxim dictating that a specific rule overrides a general rule on the same subject matter. Within the graph, this is modeled as a specificity edge that takes precedence over standard hierarchical links. For example, a rule governing 'commercial drone flights at night' will override a general 'aviation regulation' when both are active. This principle is the algorithmic basis for normative exception handling, allowing the system to carve out precise exceptions without invalidating the general norm.

The temporal precedence rule: a later-enacted statute takes precedence over an earlier one in irreconcilable conflict. The Normative Hierarchy Graph encodes this via temporal edges annotated with effective dates. During conflict resolution, the graph traversal algorithm compares the enactment_date metadata of conflicting nodes. This is critical for modeling dynamic legal systems where legislative updates must automatically supersede prior versions without manual rule-base reprogramming.

A mode of non-monotonic inference where a legally sound conclusion can be retracted in the face of new, superior evidence or rules. The Normative Hierarchy Graph is the structural backbone that enables defeasibility. When a reasoning engine encounters a conflict, it does not crash; it traverses the graph to find a defeater—a higher-priority or more specific rule. This allows the system to model real-world legal logic where conclusions are always prima facie and subject to override by a superior norm.

The algorithmic process of identifying contradictory modalities within a normative corpus before resolution is attempted. This is the triggering event for a Normative Hierarchy Graph traversal. The system scans for direct collisions, such as:

• Obligation vs. Prohibition: Rule A mandates filing Form X; Rule B forbids it.
• Obligation vs. Permission: Rule A requires a license; Rule B explicitly waives the requirement. Once detected, the graph's edges are queried to determine which norm prevails based on authority, specificity, and temporality.

A computational method for resolving global inconsistencies by identifying the largest subset of non-contradictory rules from an inconsistent rule base. The Normative Hierarchy Graph provides the preference ordering needed to select the correct MCS. Instead of a brute-force search, the graph's topology dictates which rules must be excluded. A rule lower in the hierarchy is pruned to preserve a conflicting superior rule, generating a conflict-free subgraph that can be used for coherent downstream reasoning.