CORGI Bites Back at RETE's Exponential Explosion

The classic RETE pattern matching algorithm often suffers from catastrophic combinatorial explosions. The CORGI (Collection-Oriented Relational Graph Iteration) architecture is a robust replacement that guarantees $O(KN^2)$ worst-case time for finding a single match.

Introduction

If you're deploying a real-time system that relies on rule-based engines (think reactive AI control, complex event processing, or low-latency database querying), you've run into the same problem: unpredictable latency during pattern matching.

The long-standing champion, the RETE algorithm, achieves efficiency by memoizing intermediate results partial matches in its internal memory structures, called $\beta$ -memory. The issue is that when a rule has many underconstrained variables ( $K$ ), those intermediate matches multiply uncontrollably. The time and memory complexity can soar exponentially to $O(N^K)$ , where $N$ is the size of the working memory (the set of facts).

This isn't just a slowdown; it's a stability failure. When automated systems (like self-learning AI agents) generate rules from examples, they frequently produce exactly these unconstrained, worst-case patterns, which can overwhelm available memory and crash the system.

A new matching algorithm, CORGI (Collection-Oriented Relational Graph Iteration), is designed to solve this crisis. It introduces a fundamental redesign that guarantees quadratic time and space complexity for finding a single match.

How It Works

CORGI achieves its stability by eliminating the storage of combinatorially large partial match tuples and implementing a tight memory ceiling. It operates in two clean phases:

1. Forward Pass: Building the Constrained Relation Graph

The architecture avoids the core problem of RETE by not materializing multi-variable partial matches. Instead, it focuses solely on binary relationships:

No Unbounded Storage: The traditional

\beta

-memory is replaced by structures designed for binary relations.

Quadratic Mappings: Each node in the match network maintains a mapping between satisficing pairs of bindings for the two variables involved in that literal (e.g.,

V1.\text{num} < V2.\text{num}

). This "collection-oriented" design ensures that the total memory footprint for the relations is capped at quadratic

O(N^2)

regardless of the complexity of the full rule.

2. Backward Pass: Relational Graph Iteration (RGI)

This phase provides the efficiency gain crucial for real-time systems: matches are generated on demand using an iterator.

Iterative Generation: Instead of calculating and committing the entire conflict set (all possible matches), the algorithm works backward through the Relation Graph.

Binding Resolution: Starting from a binding at the end of the rule, the process traces back through the binary mappings to quickly resolve the bindings for all preceding variables in the required order.

Low-Latency Focus: Since execution cycles only require one match (the best one) to fire the next action, returning an iterator avoids the latency and memory cost of processing all

O(N^K)

possibilities, providing an

O(KN^2)

guarantee for finding the next match.

Comparison & Benchmarks

The difference between CORGI and standard algorithms is the shift from relying on hopeful average-case performance to enforcing a tight worst-case guarantee.

Feature	CORGI (RGI)	RETE (Standard)
Single Match Guarantee (Worst Case)	$O(KN^2)$	Exponential (up to $O(N^K)$ )
Memory Ceiling	Quadratic ( $O(N^2)$ )	Unbounded/Exponential
Operational Flow	Iterative, On-Demand Match Generation	Full Conflict Set Materialization
Stability	High, Predictable under all rule conditions	Low, Prone to Overflow on unconstrained rules

Performance evaluations using a challenging combinatorial task showed that standard implementations of the classic algorithm deteriorated catastrophically. As the complexity of the task increased, one standard implementation failed due to Memory Overflow, and another timed out after over 10 hours. The new algorithm, however, handled the identical task in under 2 milliseconds, demonstrating its ability to maintain a flat, predictable runtime that scales quadratically ( $O(N^2)$ ) with memory size.

Examples

The most direct benefit of this algorithm is in its ability to execute complex, multi-variable constraints instantly, regardless of the underlying data size, which is impossible with standard approaches. These examples show how a developer can write unconstrained rules without fear of catastrophic failure.

Real-Time Constraint Validation (High N, Low K)

A supply chain monitoring system needs to confirm if four entities exist that meet a set of unconstrained relations, where $N$ (the total number of entities) is high.

Find any two Products P1, P2 and two Warehouses W1, W2 such that:
P1, P2, W1, W2 = Var(Product, 'P1'), Var(Product, "P2"), Var(Warehouse, "W1"), Var(Warehouse, "W2")

conds = AND (
    P1.item\_id < P2.item\_id, # P1 is older than P2
    P1.supplier == W1.supplier, # P1 shares a supplier with W1
    W1.location != W2.location, # W1 and W2 are in different locations
    P2.status == "URGENT" # P2 is an urgent item
)

Expected RETE Behavior: Tracking partial match combinations (e.g., all pairs of P1 and W1 that share a supplier) can generate an intermediate memory structure of size O(N^4) that is too large to process quickly, leading to minutes of latency or failure.

Expected CORGI Behavior: Finds the first valid match instantly. The algorithm never calculates the full O(N^4) set. It only traverses the O(N^2) pair mappings backward until the first match is resolved, guaranteeing low, predictable latency.

Generative Agent Reasoning

In a cognitive agent that synthesizes new rules, a complex rule with shared variables might emerge naturally, which causes standard algorithms to struggle disproportionately.

Find three employees E1, E2, E3 such that E1 mentors E2 and E3, and E2 is in a different department than E3.
E1, E2, E3 = Var(Employee, 'E1'), Var(Employee, "E2"), Var(Employee, "E3")

conds = AND (
    is\_mentor\_of(E1, E2),
    is\_mentor\_of(E1, E3),
    E2.dept\_num != E3.dept\_num
)

Expected Output: True (Match found).

Practical Implication: This is a fan-out combinatorial pattern. With 1000 employees, the partial match set size can be huge, freezing a standard system. With CORGI, the query for the next action (which only needs one valid binding for E1, E2, E3) is served in milliseconds because the graph structure efficiently handles the shared variable E1 without combinatorial materialization.

Takeaways

Prioritize Predictability: For engineers building real-time systems, the quadratic time guarantee of this algorithm is more valuable than the theoretical average-case speed of classic algorithms. You gain the essential robustness needed for high-stakes applications where failure (a crash or timeout) is unacceptable.

Unblock AI Learning: If you're building an AI or machine learning system that uses induction to learn new rules or generalize code, you must assume the learned patterns will be unconstrained. This architecture is necessary to prevent the system from self-destructing when it generates a mathematically complex rule.

Modernize the Rule Engine: This isn't just a patch; it's a fundamental architectural improvement for forward-chaining systems. It enables cleaner rule writing, eliminates the need for manual, complexity-avoiding coding tricks (like reordering literals), and finally delivers the reliable performance that rule engines have always promised.

Conclusion

The CORGI algorithm resolves a decades-old crisis by replacing the combinatorial memory model with a tight, mathematically verifiable approach. By focusing on binary relations and on-demand iteration, it delivers the stability and predictable low latency required for modern, large-scale, and autonomously learning AI systems.

It's time to move past the bottlenecks of the 1980s and build rule engines that can confidently handle complexity at scale.