arxiv: 2512.07407 · v2 · submitted 2025-12-08 · 💻 cs.CL

Training Language Models to Use Prolog as a Tool

Niklas Mellgren , Peter Schneider-Kamp , Lukas Galke Poech This is my paper

Pith reviewed 2026-05-17 01:12 UTC · model grok-4.3

classification 💻 cs.CL

keywords language modelsprologreinforcement learningsymbolic reasoningauditabilityneurosymbolic systemsgsm8kreward composition

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The pith

Training language models to use Prolog as a tool uncovers a trade-off where reward focus on correctness yields higher accuracy but delegates reasoning to natural language, while symbolic rewards enforce auditable full programs at lower peak

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper trains a 3B language model with reinforcement learning to generate and execute Prolog programs for solving grade-school math word problems. It tests different combinations of rewards for correct execution, syntactic validity, semantic correctness, and structural use of symbols. The central finding is that reward signals determine whether the model treats Prolog as a mere calculator after natural-language reasoning or as the primary vehicle for the entire reasoning chain. This produces an observable split: accuracy-optimized setups reach strong benchmark scores yet produce hard-to-audit traces, while structure-optimized setups yield complete, inspectable Prolog code but sacrifice some correctness. The authors interpret the split as a form of reward hacking and note its relevance for any neurosymbolic deployment where both performance and verifiability are required.

Core claim

Configurations rewarded primarily for execution success learn to perform most reasoning inside natural language and invoke Prolog only for the final arithmetic step, achieving higher accuracy on GSM8K and competitive zero-shot results on MMLU-STEM and MMLU-Pro; configurations that also reward syntactic, semantic, and structural properties force the model to emit complete, self-contained Prolog programs that remain fully auditable yet incur a measurable drop in overall accuracy.

What carries the argument

The composition of reward signals (execution success, syntax, semantics, and symbolic structure) inside Group Relative Policy Optimization (GRPO) that steers the model between hybrid natural-language-plus-Prolog and fully symbolic program generation.

If this is right

Accuracy-tuned models can match or exceed larger few-shot baselines on STEM benchmarks while still using an external symbolic engine for the last step.
Structure-tuned models produce reasoning traces that can be read, verified, and debugged without inspecting the model's internal activations.
Deploying neurosymbolic systems in safety-critical settings may require accepting an accuracy penalty to obtain verifiable symbolic artifacts.
The same reward-composition technique can be applied to other external symbolic or formal tools beyond Prolog.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The trade-off may appear with any external verifier or solver once the model learns it can outsource reasoning to natural language.
Hybrid reward functions that gradually increase the weight on symbolic structure could reduce the accuracy cost while preserving auditability.
Measuring the length and complexity of the natural-language prefix before the first Prolog call offers a simple proxy for how much reasoning has been delegated.

Load-bearing premise

The observed behavioral split between reward settings is caused mainly by the reward signals themselves rather than by limits on model size, prompt wording, or quirks of the Prolog interpreter.

What would settle it

Retraining the same model with identical prompts and data but with structure rewards removed, then checking whether the model still produces fully symbolic Prolog programs or reverts to natural-language delegation.

Figures

Figures reproduced from arXiv: 2512.07407 by Lukas Galke Poech, Niklas Mellgren, Peter Schneider-Kamp.

**Figure 2.** Figure 2: Semantic similarity reward across different prompt variants under Reward Suite 2. [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗

**Figure 3.** Figure 3: Interpolated reward weights over training steps, driven by the sigmoid progression schedule. [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗

**Figure 4.** Figure 4: Correctness reward progression during training across different system prompts under [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗

**Figure 5.** Figure 5: Prolog structure reward progression for each prompt variant in Reward Suite 3. As [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗

**Figure 6.** Figure 6: Parallel coordinates plot of 12 hyperparameter trials from Bayesian op [PITH_FULL_IMAGE:figures/full_fig_p026_6.png] view at source ↗

**Figure 7.** Figure 7: Bar chart of hyperparameter importances computed by W&B’s fANOVA analysis on our [PITH_FULL_IMAGE:figures/full_fig_p027_7.png] view at source ↗

read the original abstract

Language models frequently produce plausible yet incorrect reasoning traces that are difficult to verify. We investigate fine-tuning models to use Prolog as an external symbolic reasoning tool, training Qwen2.5-3B-Instruct with Group Relative Policy Optimization (GRPO) on a cleaned version of GSM8K (which we release as gsm8k-prolog-prover). We systematically vary prompt structure, reward composition (execution, syntax, semantics, structure), and inference protocol (single-try, multiple-try, and two agentic modes). Our reinforcement learning approach outperforms supervised fine-tuning on GSM8K, and the resulting 3B model achieves zero-shot performance on MMLU-STEM and MMLU-Pro competitive with 7B few-shot baselines. Most importantly, we identify an accuracy--auditability trade-off: configurations tuned for correctness alone learn to delegate reasoning to natural language and use Prolog only for the final computation, while configurations rewarded for symbolic structure produce fully auditable programs at a cost in accuracy. We interpret this trade-off as a form of reward hacking and discuss its implications for deploying neurosymbolic systems in safety-critical domains. The source code for our experiments is available under https://github.com/aisilab/Prolog-as-a-Tool

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper shows that reward signals in GRPO training for Prolog tool use produce a clear accuracy-auditability split, but the design varies prompts and protocols at the same time so the causal attribution stays loose.

read the letter

The main thing to know is that this work trains a 3B Qwen model with GRPO to output Prolog programs and finds that correctness-focused rewards make the model reason mostly in natural language while using Prolog only for the final step, whereas structure-focused rewards produce fuller, more auditable programs at the price of lower accuracy. They treat this as a form of reward hacking with implications for safety-critical use. They also report that the trained model beats supervised fine-tuning on GSM8K and reaches competitive zero-shot MMLU-STEM numbers against 7B few-shot baselines. The dataset and code are released, which is straightforwardly useful. The systematic sweep across reward components, prompt structures, and inference modes (single-try, multiple-try, agentic) gives a practical picture of how these choices interact in a small-model neurosymbolic setup. That combination of concrete training details and the observed behavioral split is the real contribution here. The soft spot is the attribution of the split. The experiments change prompt wording, reward terms, and inference protocol together, so it is not obvious that reward composition alone drives whether the model emits rich Prolog or minimal final-step calls. Fixed-prompt ablations or interaction statistics would have made the claim tighter; without them the difference could partly reflect model capacity limits or prompt details. The abstract itself gives no numbers or error bars, though the full paper presumably supplies tables. This is for people already working on tool-augmented language models or hybrid symbolic systems who want empirical data on reward design trade-offs. Readers focused on verifiable reasoning or deployment constraints will get the most out of the discussion. It is solid enough to deserve a serious referee, mainly because the setup is reproducible and the trade-off observation is worth testing and refining in follow-up work.

Referee Report

1 major / 2 minor

Summary. The manuscript investigates fine-tuning Qwen2.5-3B-Instruct with Group Relative Policy Optimization (GRPO) to use Prolog as an external symbolic tool for mathematical reasoning. Using a cleaned GSM8K dataset (released as gsm8k-prolog-prover), the authors systematically vary prompt structure, reward composition (execution, syntax, semantics, structure), and inference protocols (single-try, multiple-try, agentic). They report that the RL approach outperforms supervised fine-tuning on GSM8K, that the 3B model achieves zero-shot MMLU-STEM and MMLU-Pro performance competitive with 7B few-shot baselines, and that an accuracy-auditability trade-off emerges: correctness-focused rewards lead models to delegate reasoning to natural language while using Prolog only for final computation, whereas structure-focused rewards produce fully auditable programs at the cost of accuracy. The trade-off is interpreted as reward hacking with implications for neurosymbolic systems in safety-critical domains.

Significance. If the reported behavioral differences can be causally attributed to reward composition, the work provides a concrete demonstration of how reward design shapes tool-use strategies in LLMs and surfaces a practically relevant tension between correctness and verifiability. The public release of the dataset and code supports reproducibility and further research on neurosymbolic integration.

major comments (1)

§4 (Experimental Setup) and §5 (Results): The central claim that reward composition alone produces the accuracy-auditability split is not isolated from confounders. The design varies prompt structure and inference protocol concurrently with reward type; no fixed-prompt ablations or interaction statistics are reported that would hold prompt wording and protocol constant while changing only the reward signals. Without such controls, the observed delegation to natural language under correctness rewards cannot be securely attributed to the reward functions rather than prompt engineering details or the 3B model's capacity limits.

minor comments (2)

Abstract: The claims of outperformance over SFT and competitive MMLU results are stated without any numerical values, error bars, or statistical tests. These quantitative details should appear in the abstract or be clearly signposted to the relevant tables/figures.
Figures and tables: Ensure that all plots and result tables explicitly label the reward composition, prompt variant, and inference protocol for each condition so that readers can directly map configurations to the described behavioral differences.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed and constructive feedback. The concern about potential confounders in attributing the accuracy-auditability trade-off specifically to reward composition is well-taken. We address this point directly below and outline the revisions we will make to strengthen the causal claims.

read point-by-point responses

Referee: §4 (Experimental Setup) and §5 (Results): The central claim that reward composition alone produces the accuracy-auditability split is not isolated from confounders. The design varies prompt structure and inference protocol concurrently with reward type; no fixed-prompt ablations or interaction statistics are reported that would hold prompt wording and protocol constant while changing only the reward signals. Without such controls, the observed delegation to natural language under correctness rewards cannot be securely attributed to the reward functions rather than prompt engineering details or the 3B model's capacity limits.

Authors: We acknowledge that our experimental design varies prompt structure and inference protocol alongside reward type, and that we did not include dedicated fixed-prompt ablations or report interaction statistics that would hold those factors strictly constant. While the systematic variation across configurations produced consistent behavioral patterns supporting the trade-off, this does limit the strength of isolating reward composition as the sole causal factor. To address the concern, we will add new controlled experiments in the revision that fix prompt wording and inference protocol while varying only the reward signals, along with any relevant interaction analyses. These additions will allow a clearer attribution of the delegation behavior to reward design. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical trade-off claims rest on external benchmarks and controlled variations

full rationale

The paper reports results from RL fine-tuning experiments (GRPO on Qwen2.5-3B) with systematic ablations over prompt structure, reward composition (execution/syntax/semantics/structure), and inference protocols. The accuracy-auditability trade-off is presented as an observed behavioral pattern across these runs, evaluated zero-shot on MMLU-STEM/MMLU-Pro and on the released gsm8k-prolog-prover dataset. No equations, fitted parameters, or self-citations are used to derive the central claim; the result is directly measured against external data and does not reduce to its inputs by construction. This is a standard empirical finding with no load-bearing self-referential steps.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The work is an empirical machine learning study whose claims rest on experimental outcomes rather than unstated mathematical axioms or newly postulated entities. No free parameters are explicitly fitted in the abstract beyond standard RL hyperparameters.

pith-pipeline@v0.9.0 · 5524 in / 1249 out tokens · 51463 ms · 2026-05-17T01:12:23.389449+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We identify an accuracy--auditability trade-off: configurations tuned for correctness alone learn to delegate reasoning to natural language and use Prolog only for the final computation, while configurations rewarded for symbolic structure produce fully auditable programs at a cost in accuracy.

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
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Reference graph

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Loads arithmetic constraints with:- use_module(library(clpq))

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States problem facts as one-line clauses; and

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analyze_code('prog.pl',P,C),halt

Defines exactly one public predicate,solve/1, whose single argument is the final result. A typical example: :- use_module(library(clpq)). sell_clips(natalia, april, 48). solve(Total) :- sell_clips(natalia, april, April), { May = April / 2 }, { Total = April + May }. prolog_helpers.pl.The helper script prolog_helpers.pl analyzes any candidate program and p...

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<reasoning> - Provide a clear, concise step-by-step explanation of how you arrive at the solution

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- Always start with: ’:- use_module(library(clpq)).’ - Define any necessary numeric constants or intermediate values using predicates

<answer> - Provide executable Prolog code using constraint logic programming to compute the numeric answer. - Always start with: ’:- use_module(library(clpq)).’ - Define any necessary numeric constants or intermediate values using predicates. - Final answer should be unified explicitly in solve(X) using curly-brace constraints, without printing commands. ...

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- Explain how each numeric constant from the problem is represented by a predicate

<reasoning> - Provide a clear, concise, step-by-step explanation of your solution. - Explain how each numeric constant from the problem is represented by a predicate. - Do not include unnecessary calculations using literal numbers; instead, reference the predicates you define

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17 - Always start with: ’:- use_module(library(clpq)).’ - For every numeric constant mentioned in the problem, define a predicate with a descriptive name

<answer> - Provide executable Prolog code using constraint logic programming (CLP) to compute the numeric answer. 17 - Always start with: ’:- use_module(library(clpq)).’ - For every numeric constant mentioned in the problem, define a predicate with a descriptive name. For example, if the problem states that James carries 10 bags per trip, include: bags_pe...

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- Review the reasoning at the end of the <reasoning> section to ensure that all computations and logical deductions are correct

<reasoning> - Provide a clear, concise step-by-step explanation of how you arrive at the solution. - Review the reasoning at the end of the <reasoning> section to ensure that all computations and logical deductions are correct. - If something is not correct, then try again: Provide a clear, concise step-by-step explanation of how you arrive at the solution

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[43]

name": "run_prolog

<answer> - Provide executable Prolog code using constraint logic programming to compute the numeric answer. - Always start with: ’:- use_module(library(clpq)).’ - Define any necessary numeric constants or intermediate values using predicates. - Final answer should be unified explicitly in solve(X) using curly-brace constraints, without printing commands. ...

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solve(X) :- {X = final_number}

Adjust your answer to: <answer> :- use_module(library(clpq)). solve(X) :- {X = final_number}. </answer> ... D Inference Examples D.1 SP-Struct-RWD1 Example <reasoning> To determine how many more cakes Louise needs to bake, we can break down the problem step by step

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[46]

Louise needs a total of 60 cakes

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[47]

She has already baked half of 60 cakes, which is 30 cakes

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[48]

She has 60 - 30 = 30 cakes left to bake

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[49]

Today, she bakes half of the remaining 30 cakes, which is 30 / 2 = 15 cakes

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[50]

After baking 15 cakes, she has 30 - 15 = 15 cakes left to bake

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[51]

The next day, she bakes a third of the remaining 15 cakes, which is 15 / 3 = 5 cakes

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[52]

Therefore, Louise needs to bake 10 more cakes

After baking 5 cakes, she has 15 - 5 = 10 cakes left to bake. Therefore, Louise needs to bake 10 more cakes. </reasoning> <answer> :- use_module(library(clpq)). 19 solve(C) :- C = 10. </answer> D.2 SP-Declare-RWD1 Example <reasoning> - Walter works 5 days a week in a fast-food chain and earns $5 per hour. - He works 4 hours a day, so his daily earning is ...

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[55]

Let’s denote the total number of pets as 36

Subtract the sum of dogs and cats from the total number of pets to find the number of bunnies. Let’s denote the total number of pets as 36. The number of dogs is 0.25 * 36, and the number of cats is 0.50 * 36. The number of bunnies is then 36 - (0.25 * 36 + 0.50 * 36). <answer> :- use_module(library(clpq)). number_of_dogs(Pets, Dogs) :- Dogs is 0.25 * Pet...

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[56]

Reflect on what went wrong

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[57]

name": "run_prolog

Adjust your answer to: <answer> :- use_module(library(clpq)). solve(X) :- {X = final_number}. </answer> <tool_call>{ "name": "run_prolog", "arguments": { "code": ":- use_module(library(clpq)).\n\nsolve(X) :- {X = final_number}." } } [TOKENS:post-feedback] used=731 (38.7%) | rem=1159 | budget=1890 [TOKENS:pre-gen] used=731 (38.7%) | rem=1159 | budget=1890 ...

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[58]

Calculate the number of dogs: 25% of 36 pets

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[59]

Calculate the number of cats: 50% of 36 pets

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[60]

Let’s directly compute the number of bunnies as a single step

Subtract the sum of dogs and cats from the total number of pets to find the number of bunnies. Let’s directly compute the number of bunnies as a single step. <answer> :- use_module(library(clpq)). solve(X) :- {X = 36 - (0.25 * 36 + 0.50 * 36)}. </answer> [TOKENS:post-gen] used=926 (49.0%) | rem=964 | budget=1890 »> run_prolog→9 21 D.4 SP-Struct-RWD1 Agent...

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Calculate the height of the plant with bone meal, which is 125% of the height of the control plant

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The height of the control plant is 36 inches

Calculate the height of the cow manure plant, which is 200% of the height of the bone meal plant. The height of the control plant is 36 inches

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Height of the bone meal plant = 125% of 36 inches = 1.25 * 36 inches

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</reasoning> <answer> :- use_module(library(clpq))

Height of the cow manure plant = 200% of the height of the bone meal plant = 2.00 * (1.25 * 36 inches). </reasoning> <answer> :- use_module(library(clpq)). height_control_plant(36). height_bone_meal(B) :- B is 1.25 * height_control_plant(36). height_cow_manure(C) :- C is 2.00 * height_bone_meal(B). solve(C) :- height_cow_manure(C). </answer> [TOKENS:post-...

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name": "run_prolog

Adjust your answer to: <answer> :- use_module(library(clpq)). solve(X) :- 22 {X = final_number}. </answer> <tool_call>{ "name": "run_prolog", "arguments": { "code": ":- use_module(library(clpq)).\n\nsolve(X) :- {X = final_number}." } } [TOKENS:post-feedback] used=763 (40.4%) | rem=1127 | budget=1890 [TOKENS:pre-gen] used=763 (40.4%) | rem=1127 | budget=18...

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Calculate the height of the bone meal plant: - Height of the bone meal plant = 1.25 * 36 inches

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Calculate the height of the cow manure plant: - Height of the cow manure plant = 2.00 * (1.25 * 36 inches). To ensure the final height is correctly computed and unified, we need to make sure the intermediate steps are correctly defined and the final height is directly computed and unified in one step. Let’s redefine the solve predicate to directly compute...

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Reflect briefly on what went wrong

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name": "run_prolog

Adjust your answer to: <answer> :- use_module(library(clpq)). solve(X) :- {X = final_number}. </answer> <tool_call>{ "name": "run_prolog", "arguments": { "code": ":- use_module(library(clpq)).\n\nsolve(X) :- {X = final_number}." } } 23 [TOKENS:post-feedback] used=1083 (57.3%) | rem=807 | budget=1890 [TOKENS:pre-gen] used=1083 (57.3%) | rem=807 | budget=18...

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The control plant grows to 36 inches

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The bone meal plant grows to 125% of the control plant’s height

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name":"run_prolog

The cow manure plant grows to 200% of the bone meal plant’s height. First, we calculate the height of the bone meal plant: [Height of bone meal plant] = 36×1.25 Next, we calculate the height of the cow manure plant: [Height of cow manure plant] = [Height of bone meal plant]×2 Finally, we need to find the height of the cow manure plant. </reasoning> <answe...

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- Refer to the correct option by its zero-based index

<reasoning> - Provide a clear, concise step-by-step explanation of how you determine which option is correct. - Refer to the correct option by its zero-based index

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run_prolog

<answer> - Provide executable Prolog code using constraint logic programming to compute the index of the correct choice. - Always start with: ’:- use_module(library(clpq)).’ - Final answer should be unified in solve(X) using a single curly-brace constraint that sets X to the chosen index. Use this XML format strictly: <reasoning> (Your step-by-step reason...

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