Verbalized Sorting: Externalizing Comparisons

• Verbalized sorting is a paradigm that externalizes the atomic comparison operation using natural language queries, symbolic logic, and pattern-based rules.
• It incorporates fixed sorting networks, combinatorial methods, logic programming, and LLM or human-mediated oracles to ensure transparent and auditable decisions.
• These methods provide algorithmic guarantees, enhanced explainability, and robustness while addressing challenges like model alignment and efficiency in executing comparisons.

Verbalized sorting refers to a diverse set of paradigms where the process of sorting is reified, described, or executed through explicitly defined operations, symbolic logic, natural language queries, or constrained decision oracles—often with the aim of making the atomic comparison step accessible, explainable, or robust to modalities such as human or model-in-the-loop comparisons. The approaches to verbalized sorting span classical fixed sorting networks, combinatorial pattern-theoretic descriptions, abductive logic programming, benchmarking LLMs on sorting behaviors, and, most recently, the recasting of binary comparators in sorting algorithms as constrained LLM calls. While the term “verbalized sorting” is sometimes used loosely to refer to any explicable or natural-language-mediated ordering, its canonical technical instantiations revolve around the formalization and externalization of the comparison oracle and the stepwise communication of the sorting process or decision boundaries.

1. Paradigms for Verbalized Sorting

Verbalized sorting is instantiated across several research traditions, each specializing in a distinct formal or practical mechanism:

• Fixed sorting networks and (n, k)-schemes: Sorting can be expressed as the execution of a sequence of fully deterministic comparators or generalized (n, k)-comparators (processing k elements per step), with the comparison process “verbalized” as a predetermined list of operations devoid of runtime decision trees (Aslanyan, 2011).
• Pattern-based and combinatorial verbalization: Structural properties of input/output permutations under a sorting map (e.g., stack-sort, bubble-sort) are captured via forbidden or required mesh or decorated patterns, providing a combinatorial “verbalization” of when the operator sorts an input to a given pattern class (Claesson et al., 2012).
• Logic Program Transformation and Abductive Reasoning: Derivations of efficient sorting algorithms from naive permutation-generating baselines are facilitated by the “verbalized” justification of each recursive transformation and subgoal introduction, recorded explicitly via logic programming rules and abductive hypotheses (Hernández, 2018).
• LLM-mediated or Human Import Oracles: Sorting with unreliable comparators, such as human judgments or LLM-queried binary oracles, enables a discrete, explicitly “verbalized” comparison—turning the sorting process into a structured sequence of questions and answers, with uncertainty handled by Bayesian or Monte Carlo updating (Smith, 2016, Lall et al., 9 Sep 2025).
• Evaluation and benchmarking of LLMs: The ability of LLMs to faithfully and validly sort verbalized or semantically ambiguous data (e.g., numbers written as words) is itself a testbed for robustness in verbalized reasoning and a diagnostic for the degree to which sorting can be externalized and reliably followed in model-based settings (Herbold, 11 Apr 2025).

2. Explicit Externalization of Comparison

A unifying element in verbalized sorting is the externalization of the atomic comparison operation, either through fixed logic, pattern constraints, or by routing the comparison out to a human or model for explicit decision:

• Verbalization via patterns: In stack-sorting and similar operators, the transformation of an input permutation to a pattern class is fully characterized by a set of forbidden configurations, e.g., mesh patterns or decorated regions subject to additional local properties (Claesson et al., 2012). The corresponding verbalization amounts to describing, for each output pattern, the set of preimage patterns that—through stepwise operator actions—lead to that output.
• Binary oracle mediation: In recent work on verbalized algorithms, the comparison step $f(x, y)$ of a sorting algorithm is defined as a constrained prompt to an LLM (“Is $x$ better than $y$?”), yielding a yes/no response to be interpreted as a Boolean value (Lall et al., 9 Sep 2025). This electronic “verbalization” enables sorting algorithms to operate over arbitrarily complex domains, provided the binary oracle is reliable and repeatable.
• Human-in-the-loop Monte Carlo sort: For subjective or error-prone domains, the comparison results are “verbalized” through human input; an explicit probability model is maintained over the candidate orderings, updated at each verbal comparison. The algorithm actively selects the next pair for which the information gain is maximal, seeking to minimize the costly external (verbalized) queries (Smith, 2016).

3. Methodological Instantiations and Algorithmic Guarantees

Verbalized sorting methods are constructed to inherit the guarantees of their base algorithm, provided the comparison oracle behaves in a sufficiently reliable or controlled manner:

• Sorting Networks as Verbalized Algorithms: Using a fixed network such as the bitonic sorting network, each compare-and-exchange operation is replaced by a call to the (verbalized) binary oracle. The time complexity and correctness arise from the network structure: for $n$ elements, parallel time is $O((\log n)^2)$ and total comparisons are $O(n(\log n)^2)$ (Lall et al., 9 Sep 2025).
• Majority voting for robustness: To mitigate comparison errors (either due to human uncertainty or LLM stochasticity), each oracle call can be performed $K$ times, and the majority response is taken. The probability that the majority result reflects the correct comparison is bounded below via Hoeffding’s inequality (Lall et al., 9 Sep 2025).
• Constraint and scoring baselines: Direct output constraint approaches, where the LLM is asked to output a sorted list in a single response, are found to be less reliable—prone to introducing duplicates or missing elements—due to the lack of stepwise externalization and lack of a modular guarantee (Herbold, 11 Apr 2025, Lall et al., 9 Sep 2025).
• Pattern-based enumeration and characterization: In combinatorial settings, the precise enumeration and identification of sortable/preimage pattern classes is enabled by the complete list of mesh or decorated forbidden patterns; this enables both algorithmic enumeration and a verbal combinatorial understanding of sorting maps (Claesson et al., 2012, Defant et al., 2018).

4. Challenges and Empirical Findings

Several empirical and theoretical challenges are highlighted in the deployment and analysis of verbalized sorting techniques:

• Faithfulness and validity in LLMs: Evaluations on SortBench reveal that when presented with lists of number words, LLMs often confuse semantic meaning with syntactic order, sorting “three” by its numeric value rather than by string order. Models also drop or add elements, especially in long lists, demonstrating a lack of strict faithfulness and problems in preserving the set of input items (Herbold, 11 Apr 2025).
• Reasoning and overthinking: Models with test-time reasoning capabilities may enhance clarity but can also “overthink” and produce outputs with extra text or invalid formats, reducing both faithfulness and list validity (Herbold, 11 Apr 2025).
• Cost of verbalization: When each comparison is expensive—such as in human-in-the-loop or LLM-batched queries—the sorting algorithm must minimize the number of queries by active selection of the most informative comparison pair (Smith, 2016, Lall et al., 9 Sep 2025).

A summary table of core approaches and their main characteristics is as follows:

Paradigm	Comparison Oracle	Core Guarantee/Technique
Fixed Sorting Network	Logic/fixed logic	Data-independent, parallelizable
Pattern-based (mesh/decorated)	Symbolic pattern	Complete pattern-class characterization
Human/LLM as binary oracle	Human/LLM response	Flexibility, modularity, error-robust
Constraint Decoding (LLMs)	Direct output	Prone to validity/faithfulness error

5. Implications, Extensions, and Theoretical Impact

Verbalized sorting, as an explicit externalization of the comparison step, has notable implications for both theory and application:

• Robustness via modularity: Classical guarantees of sorting algorithms (e.g., O(n log n) runtime, correctness invariants) are preserved when the binary comparator is replaced by a reliable verbal oracle, enabling complex or semantically ambiguous objects to be sorted with well-understood behaviors (Lall et al., 9 Sep 2025).
• Transparency and explainability: By rendering each comparison explicit (either via a natural language query or a combinatorial constraint), the process becomes transparent and auditable—a property valuable for interactive systems, verification, and educational contexts (Hernández, 2018, Claesson et al., 2012).
• Combinatorial richness: The decorated and mesh pattern approaches “verbalize” sorting at the level of input structure, yielding avenues for new enumerative and structural results, including for multiset and word sorts (Defant et al., 2018, Claesson et al., 2012).
• Human/model alignment and benchmarking: Sorting tasks, especially with verbalized or ambiguous data types, expose systematic mismatches between LLM training biases and explicit symbolic requirements—driving new benchmarks and diagnostic tasks for contemporary NLU models (Herbold, 11 Apr 2025).

6. Future Directions and Open Problems

Research continues to probe the scalability, reliability, and applicability of verbalized sorting across new domains:

• Scalable hybrid architectures: Batching LLM calls in a network structure (e.g., bitonic network) may unlock practical large-scale verbalized sorting, though bounds on end-to-end runtime and the impact of oracle error rates require further clarification (Lall et al., 9 Sep 2025).
• Enriched oracle models: Modeling non-uniform error rates, adversarial settings, or domain-specific biases in the oracle is a natural extension, especially for human or LLM-oracle settings (Smith, 2016).
• Compositional “verbalized” algorithms: Extending the framework to clustering, ranking beyond sorting, and tasks such as code synthesis or semantic ordering, leveraging the same modular verbalization principle, is an emerging direction (Lall et al., 9 Sep 2025).
• Benchmarking and evaluation: Continued evolution of benchmarks such as SortBench is likely, with emphases on handling longer, more complex, or multilingual lists, and on explicit measurement of faithfulness, validity, and robustness under varying reasoning depths (Herbold, 11 Apr 2025).
• Abductive and logic-based verbalizations: Integration of abductive reasoning with externalized oracles in program synthesis and transformation for sorting typifies a cross-disciplinary advance, aligning explainable program generation with modular, auditable execution traces (Hernández, 2018).

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In sum, verbalized sorting encapsulates a multidimensional extension of classical sorting where the comparison operation is externalized, audited, or mapped into a constrained decision (human or model-based), combinatorial pattern, or logic program. Its formalizations provide new theoretical guarantees, modularity, and transparency, while also surfacing empirical challenges in model alignment, faithfulness, and efficient externalization of stepwise procedures.