Recursive Algorithm Interrogation Techniques

• Recursive algorithm interrogation techniques are methods for systematically analyzing and transforming recursive algorithms through explicit expansion, abstraction, and model-based queries.
• They utilize fold operators, static mode-systems, and execution modeling to demystify inherent recursive structures and expose hidden operational behaviors.
• Applications span bias auditing in search engines, zero-shot semantic extraction in LLMs, and safe compilation through rigorous recursive verification.

Recursive algorithm interrogation techniques constitute a body of methodologies and formal tools for systematically analyzing, auditing, or transforming recursive algorithms and algorithmic systems by interrogating their structure, operational behavior, and semantic properties through explicit recursive expansion, abstraction, and model-based queries. These methods span formal program transformation, static analysis, automated reasoning frameworks, prompt-based semantic extraction, and empirical black-box probing, unified by the core principle of recursively decomposing or operationalizing the algorithmic object to reveal properties inaccessible with shallow inspection.

1. Core Concepts and Formal Abstractions

Recursive algorithm interrogation is grounded in expressing recursive definitions or behaviors in terms of formally analyzable constructions. In functional program transformation, recursion abstraction replaces explicit self-reference by standard higher-order iterators—folds (catamorphisms) over algebraic datatypes—yielding equivalent, recursion-free forms that expose the structural essence of the algorithm (Widemann, 2016). For example, the Ackermann function, as a textbook case of double recursion, is re-expressed as a composition of two nested natural-number folds:

$$\mathit{ack}(m, n) = \Bigl(\mathit{foldn}\;(\lambda f. \mathit{foldn}\,f\,(f\,1))\;(+1)\;m\Bigr)\;n$$

Recursive interrogation extends beyond pure transformation: frameworks such as Retreet (Wang et al., 2019) introduce specification languages for recursive tree traversals, with formal configuration abstractions encoding the dynamic call stacks and parameter maps as stack-based records. These enable reasoning about reachable states, data dependences, and concurrency by encoding operational semantics into monadic second-order logic over trees (MSO), and then querying this logical model for properties of interest.

In black-box settings, recursive interrogation can be operationalized as the systematic construction and exploration of output spaces via recursive probing. For example, in auditing search query suggestion algorithms, a suggestion tree is constructed by recursively querying the system with each suggestion returned, treating each as a new root, and bounding the recursion by depth, custom pruning, or termination on empty expansion (Haak et al., 6 Jan 2026). Analogously, structured prompt interrogation in LLM-powered semantic extraction applies recursive prompt generation and answer parsing to construct deeply nested semantic instance trees conforming to arbitrary user schemas (Caufield et al., 2023).

2. Methodological Frameworks and Algorithms

The methodologies for recursive algorithm interrogation instantiate the formal abstractions above in a range of algorithmic frameworks:

• Recursion Abstraction via Fold Operators: The process identifies the primary recursive argument, transforms recursive equations into the canonical "successor in terms of predecessor" form, and rewrites the definition as a fold or nested folds over the datatype. This technique generalizes to primitive recursive representations for functions exhibiting complex, even double or triple, recursion patterns (e.g., Knuth's up-arrow and Conway's chained-arrow notations), facilitating equational reasoning and proof (Widemann, 2016).
• Static Mode-Systems for Recursive Definitions: In call-by-value languages, safe evaluation of mutually recursive bindings is checked by inferring "modes" of use—Ignore, Delay, Guard, Dereference—and computing, via a backwards fixpoint algorithm, whether any recursive definition is "vicious" (unsafe). The check is formalized as environment- and mode-based structural induction on the syntax tree, guaranteeing soundness against the operational semantics (no uninitialized read segfaults at runtime) (Reynaud et al., 2018).
• Recursive Interrogation via Execution Modeling: Retreet specifies programs in a dedicated recursive tree traversal language. Each run is abstracted as a sequence of stack "records," specifying the block, node, and parameter/return-value maps at each step. These traces are lifted into a logical model (MSO over trees), allowing automated verification of data-race freedom, correctness of transformation (fusion, parallelization), and dependency satisfaction, using WS2S decision procedures (Wang et al., 2019).
• Black-box Recursive Interrogation: The suggestion-tree method for search engines, as an example of black-box RAI, recursively applies the suggestion function:

$f : Q \to S^k, \quad f(q) = (s_1, \ldots, s_k)$

to each discovered suggestion, constructing a tree $T = (V, E)$, bounding with recursion depth $D$ and pruning irrelevant or non-root-containing branches. Pseudocode formalization specifies expand, prune, and stopping criteria (see (Haak et al., 6 Jan 2026), sections 1–2).

• Recursive Semantic Extraction (SPIRES): The SPIRES algorithm executes a mutually recursive pipeline: prompt generation for a schema-rooted class, LLM completion, answer parsing, and, for inlined class-typed attributes, recursive child calls, until the schema is fully populated. The maximum number of LLM calls is $O(b^d)$ given average inlined branching factor $b$ and depth $d$; final outputs are rooted, schema-conformant semantic trees with grounded identifiers (Caufield et al., 2023).

3. Applications and Empirical Case Studies

Recursive algorithm interrogation has been realized in distinct domains, as demonstrated by recent case studies:

• Bias Auditing in Query Suggestion Systems: Recursive interrogation through suggestion trees dramatically increases the depth and breadth of discovered suggestions, enabling robust estimation of topical diversity and bias. For the political domain, constructing trees to $D = 8$ for person-name queries yielded ≈8,000 unique suggestions (vs. ≤10 per root in classical audits) (Haak et al., 6 Jan 2026). Table 1 summarizes significant regression coefficients for group-level topical bias:

Meta-Attribute	Political (C1)	Locations (C2)	Personal (C3)
Female	+0.074*	–0.096*	n.s.
Current Minister	+0.106*	n.s.	–0.092*
Former Minister	n.s.	n.s.	+0.103*
CSU Party	–0.071*	n.s.	n.s.

(*significant at $p\le0.05$.)

• Zero-Shot Semantic Extraction: SPIRES operates in biomedical informatics and beyond, extracting arbitrarily nested mechanisms (e.g., multistep drug actions) from raw text by recursive LLM prompt/question pipelines, achieving F₁ ≈ 0.4380 for relation extraction without task-specific tuning (Caufield et al., 2023).
• Formal Verification of Recursive Transformations: Retreet has validated the correctness of traversal fusions and parallelization for CSS-AST minification, tree mutation, and complex mutual recursion, detecting genuine dependency violations and providing counterexample configurations via MSO model checking (Wang et al., 2019).
• Safe Compilation of Recursive Bindings: Incorporating the mode-system-based check into OCaml's compiler eliminated runtime failures from unsound recursive definitions, with robust handling of advanced features (float-array representation, exceptions, GADTs) (Reynaud et al., 2018).

4. Evaluation, Complexity, and Limitations

The practical performance and scaling of recursive algorithm interrogation is tightly linked to formal recursion depth, branching factor, and the efficiency of the underlying solvers or APIs:

• For tree-building RAI (suggestion trees), maximal resource use scales as $O(k^D)$, with $k$ suggestions per node and depth $D$; pruning and early stopping are essential to avoid combinatorial explosion (Haak et al., 6 Jan 2026).
• For SPIRES, worst-case LLM call count is $\sum_{\ell=0}^d b^\ell = O(b^d)$, linear in observed cases ($b, d \leq 5, 3$), but with cost and context-length limitations affecting recursive extraction in very deep or wide schemas (Caufield et al., 2023).
• In formal frameworks such as Retreet, all MSO-reducible queries are decidable, but with non-elementary worst-case complexity in quantifier depth; experiments confirm practical tractability for programs with dozens of blocks (Wang et al., 2019).

Limitations include hallucinations and omission in LLM-driven approaches, heuristic parsing risks, latency/cost from excessive recursion, reliance on schema and prompt design, pruning/splitting artifacts, and scale barriers in pure logic-based approaches given deep or wide structures (Caufield et al., 2023Wang et al., 2019Haak et al., 6 Jan 2026).

5. Generalization and Domain Transfer

RAI methods generalize across both transparent and opaque algorithmic systems. The underlying paradigm is recursive probing or transformation—unfolding either the program structure (via fold abstraction, stack-based modeling, or static analysis) or the system output space (by recursively querying APIs or LLMs, or by recursive prompt chains). Extension to new domains requires appropriate translation of the expansion function, domain-specific pruning or weighting criteria, and adjustment to input- or output-space discretization.

Table: Methodological Instantiations of Recursive Algorithm Interrogation

Method	Domain/Artifact	Recursion	Interrogation Target
Fold Abstraction (Widemann, 2016)	Functional programs	Program definition	Recursion in algebraic datatypes
Mode System (Reynaud et al., 2018)	Functional compiler	Mutual rec. bindings	Run-time safety
Retreet (Wang et al., 2019)	Tree traversals	Stack/configuration	Data dependencies, correctness
SPIRES (Caufield et al., 2023)	LLM prompt/semantic tree	Attribute schema	Nested semantic extraction
Suggestion Tree (Haak et al., 6 Jan 2026)	Black-box APIs	Output tree	Distributional, bias audits

A plausible implication is that any system where recursion generates otherwise inaccessible internal or derived outputs can, in principle, be interrogated recursively, provided the expansion step is computable or the output/discourse can be recursively decomposed.

6. Impact and Synthesis

Recursive algorithm interrogation techniques have produced advances in formal correctness proofs, safe compilation checks, robust statistical auditing, and scalable semantic extraction. The capacity to abstract arbitrarily complex recursive behaviors into analyzable forms—whether by fold-based rewrites, logical encodings, or recursive black-box probing—yields a unified toolset for understanding, diagnosing, and optimizing recursive computational processes (Widemann, 2016, Reynaud et al., 2018, Wang et al., 2019, Caufield et al., 2023, Haak et al., 6 Jan 2026). The generalization of these tools is limited primarily by domain-specific expansion/pruning bottlenecks, and the tractability of the underlying recursion in model or API. These techniques form a critical component of the analytic infrastructure for recursive algorithms, supporting rigorous investigation across both white-box and black-box systems.