Cross Recursion Detection Module

• Cross Recursion Detection (CRD) Module is a tool that detects minimal cycles in graphs representing mutual, cross, or self recursion to enforce system consistency.
• It leverages algorithms such as Tarjan’s SCC and depth-first search to identify cycles in varied domains, including RDF data shapes, functional programming, and deep learning.
• The module supports applications in program verification, anomaly detection, and regression analysis by formalizing cycle detection and managing complexity in diverse computational settings.

A Cross Recursion Detection (CRD) module is a class of static or dynamic analysis tool designed to detect, characterize, or exploit recursion cycles, including “cross-shape,” “cross-function,” or mutual recursion, across various domains such as programming language semantics, program verification, intermediate representations, RDF data shape languages, and deep learning architectures. CRD modules formalize and implement algorithms for recognizing the presence of minimal cycles in call graphs or reference structures, providing foundational mechanisms for ensuring correctness, safety, equivalence, or anomaly detection depending on the application domain.

1. Formal Foundations: Recursion Cycle Detection

CRD fundamentally relies on cycle detection within directed graphs representing definitions, calls, or references. In the context of RDF data shapes, for example, each shape and its references can be modeled as a directed graph $G = (N, E)$ where $N$ is the set of named shapes and $(a, b) \in E$ iff shape $a$ refers to shape $b$. The recursion predicate is formalized as the existence of a cycle in $G$, i.e., $\exists n \in N$ such that $(n, n) \in E^+$ (where $E^+$ is the reflexive-transitive closure), or equivalently, the presence of a directed cycle, possibly self-referential (Ryman, 2015).

CRD modules compute all cross-shape recursion cycles, i.e., all minimal cycles $$c = (s_1 \rightarrow s_2 \rightarrow \dots \rightarrow s_k \rightarrow s_1)$$ in $(N, E)$, which generalizes to cross-function or cross-object in other settings. Algorithms used include Tarjan’s strongly connected components (SCC) (outputting all SCCs with $|SCC| > 1$ or singleton SCCs with self-loops) and depth-first search with a recursion stack (identifying cycles via back edges). The time and space complexity for cycle detection are linear in the total number of nodes and edges, i.e., $O(|N| + |E|)$ (Ryman, 2015).

2. Domain-Specific CRD Module Designs

CRD modules are instantiated differently depending on the language or framework:

RDF Data Shape Languages

In RDF shape languages, CRD is used to prohibit or handle recursive shapes by analyzing the neighbor/reference graph extracted from the shape specification. For each candidate recursion, the algorithm reports the minimal set of shapes involved in recursion, thereby supporting well-formedness checking and optimization (Ryman, 2015).

Call-by-Value Languages (Functional Programming)

In functional programming, particularly for OCaml and related languages, a CRD module is built atop a mode system that classifies each variable’s usage in a recursive block by an access mode lattice: Ignore $<$ Delay $<$ Guard $<$ Return $<$ Deref. Safety is enforced by statically rejecting recursively defined variables that are used in their own definition in mode Return or Deref, i.e., those that form a vicious recursive cycle likely to fail at runtime. The check is formulated via declarative inference rules and states that any let-rec group where a cycle maps to mode Return or Deref must be rejected. The algorithmic check reduces to solving a system of monotone equations in the lattice of environments representing variable-to-mode mappings (Reynaud et al., 2018).

Object-Oriented Intermediate Representations (EO)

Within the Elegant Objects (EO) paradigm, CRD modules address unanticipated mutual recursion in decorated-object hierarchies. The CRD algorithm constructs a resolved call graph for each object and attribute, identifying cycles of dynamic dispatches. This analysis is implemented as a bounded-depth DFS over (object, attribute) pairs, reporting chains of attribute dispatches that revisit the same (object, attribute)—i.e., recursion cycles potentially hidden by dynamic decoration patterns (Kudasov et al., 2022).

Program Equivalence and Regression Verification

In program equivalence checking, for instance in the Regression Verification Tool (RVT), CRD operates by detecting unbalanced recursive call structures across function versions. The module constructs harnesses that systematically unroll differently-structured recursion sites until the recursive argument sets “synchronize” across program versions. The algorithm encodes recursive function calls into verification conditions, checking via a SAT/SMT backend whether some unrolling produces matching call-site multisets (sync-unrolling), and thus enables or rules out inductive equivalence proofs for semantically but structurally distinct recursive functions (Sayedoff et al., 2022).

Anomaly Detection in Deep Learning

CRD modules have also been adapted to deep learning architectures, e.g., in the RcAE framework for unsupervised anomaly detection in images. Here, CRD encodes the dynamic across a sequence of progressively refined reconstructions as a 3D spatio-recursive volume. A 3D convolutional autoencoder learns patterns of recurring, stabilizing reconstructions; persistent “flickering” in certain regions across iterations is interpreted as an anomaly. CRD in this context outputs a pixel-wise anomaly map derived from learned inconsistency patterns across recursive steps (Wu et al., 12 Dec 2025).

3. Algorithmic Patterns and Implementations

The table below summarizes core algorithmic strategies in representative CRD modules:

Domain	Graph Structure	Cycle Detection Approach	Complexity
RDF/Shape Languages	Names + reference edges	Tarjan’s SCC/DFS-recursion stack	$O(|N|+|E|)$
Functional/OCaml	Variables in let-rec group	Mode-based fixpoint, env-lattice	$O(k^3V)$
EO IR/Object-Oriented	(Object, Attribute) pairs	Resolved call graph, bounded DFS	$O(n^2)$
RVT/Equivalence	Call multisets	Non-det harness, SAT/SMT search for cycles	NP, SAT-complete
RcAE/Deep Learning	Spatio-recursive tensor	3D ConvAE, inconsistency modeling	N/A (learned)

Each approach aligns detection of recursion cycles with the semantic structure under analysis. Standard graph algorithms suffice where explicit reference graphs exist (RDF, EO), while higher-order fixpoint computations or SAT-based searches appear in semantic analyses (OCaml, RVT).

4. Examples Illustrating Cross Recursion Detection

Illustrative cases clarify the practical operation of CRD:

• In RDF shape languages, mutual references $A \rightarrow B \rightarrow C \rightarrow A$ are detected as a cycle and reported as a single SCC (Ryman, 2015).
• In functional programming, a binding like $let\ rec\ x = x + 1$ yields a Deref-mode self-cycle and is rejected, while $let\ rec\ f = fun\ x \to f\ x$ is accepted due to mode Delay (Reynaud et al., 2018).
• In EO-based IR, cross-object call cycles such as $$(derived, n) \rightarrow (base, m) \rightarrow (derived, n)$$ are detected and listed using the resolved call graph (Kudasov et al., 2022).
• In equivalence verification, two versions of Fibonacci functions differentiated by their call-site parameterization are made comparable by unrolling until their recursive call sets are matched, at which point formal equivalence proof can proceed (Sayedoff et al., 2022).
• For RcAE, anomaly regions arise as pixels whose reconstruction values remain inconsistent across recursive steps, with 3D convolutions learning the temporal structure (Wu et al., 12 Dec 2025).

5. Integration, Computation, and Limitations

CRD modules are generally integrated as static analysis passes or components within a larger verification, linting, or training pipeline.

Complexity and scalability:

• Graph-based algorithms are linear to quadratic in program or model size.
• Mode-based recursive checks require solving fixpoints over environments; this is tractable for small mutually recursive blocks but can become cubic in the worst case for large groups.
• SAT/SMT-based approaches in program equivalence checking are NP-complete in unwinding depth.

Soundness and completeness:

• Some CRD instances, particularly in object-oriented and EO IR analysis, can report false positives due to over-approximation (i.e., cycles are detected even if unreachable in actual execution).
• Advanced features—such as GADTs, exceptions, row-polymorphism, or conditional branching—can introduce challenges, requiring careful semantic modeling or future language extensions (Reynaud et al., 2018, Kudasov et al., 2022).

Limitations:

• CRD modules may miss cycles masked by data-dependent control flow or recursion in dynamically generated structures.
• In deep learning, CRD performance relies on the breadth and fidelity of the temporal anomaly pattern the model can learn; manually specified losses and training regimes are necessary for effective anomaly localization (Wu et al., 12 Dec 2025).
• For program equivalence, cases where sync-unrolling is not possible or recursion indices are structurally incompatible can cause CRD-based strategies to fail (Sayedoff et al., 2022).

6. Impact and Extensions

CRD modules materially improve the robustness and expressiveness of their host frameworks:

• In RDF, precise recursion detection enables robust schema validation and optimization (Ryman, 2015).
• In programming languages, improved static checks have replaced earlier, syntactic-only criteria (e.g., OCaml’s prior letrec checks), closing subtle soundness gaps (Reynaud et al., 2018).
• EO IR’s CRD enables early detection of fragile base class patterns missed by standard object-oriented analysis (Kudasov et al., 2022).
• In program equivalence, CRD modules significantly enlarge the set of function pairs where regression verification is feasible, enabling automated proof of equivalence even across semantically equivalent but syntactically distinct recursion schemes (Sayedoff et al., 2022).
• In deep learning for anomaly detection, CRD modules leveraging multi-step reconstruction information outperform traditional one-shot reconstruction losses, closing the gap with larger diffusion-based approaches while maintaining low computational overhead (Wu et al., 12 Dec 2025).

Numerous extensions are possible, including integration with reachability analysis, use of additional type information, suppression mechanisms for known safe cycles, and adaptation to new language or architectural features as they emerge.