Distilling Answer Set Programming Theories from Large Language Models

Writing Answer Set Programming (ASP) theories from scratch is a difficult and time-consuming task. We take a neurosymbolic approach to study whether a model can distill complete and correct theories, given a fixed agent harness with the solver in the loop. The protocol is dataset-agnostic: with a single prompt and an empty file as the starting point the model is given a 1-hour time limit to derive a complete theory. We chose VQA as the application domain, three benchmarks (CLEVR, GQA, CLEVRER), as these are publicly available and non-trivial. In order to study the model scale required for solving this task we nine different models: four frontier (Claude Sonnet 4.6, Claude Opus 4.7, GPT-5, DeepSeek V4 Pro), two mid-tier (DeepSeek V4 Flash, gpt-oss-120b), and three open-weights (qwen3.6-27b, gpt-oss-20b, qwen3.5-9b). Three of four frontier models reach 100% on CLEVR and 92.8%-98.8% on GQA; on CLEVRER, Sonnet, Opus, DeepSeek V4 Pro score 92.7%-95.3%. GPT-5 reaches 98.7% on CLEVR but drops to 41.8% on GQA and to 86.7% on CLEVRER. Adding handwritten reference theories from other datasets moves the other three frontier models by at most +/-3.4 pp but reduces GPT-5's accuracy by 3-19 pp. We release the code, prompts, and theories distilled.

A Verifiable Search Is Not a Learnable Chain-of-Thought

It is tempting to assume any task solvable by a short program can be taught to a model as its chain-of-thought: write the steps out, fine-tune, and the model follows. This paper shows the assumption fails for an identifiable class of procedures. The testbed is nine reasoning tasks, each from a deterministic generator; public and hidden splits share generators, so held-out data proxies test accuracy. I reverse-engineer the generators into Python solvers, render them as chain-of-thought, and distill into a rank-<= 32 LoRA over a 30B (3.5B-active) Nemotron model. Forward-computable tasks install readily: lookup/arithmetic and an 8-bit boolean task transfer (>= 0.99 and 0.68). Cryptarithm does not: distilling its backtracking search holds at 0.01-0.07 across eleven chain-of-thought designs, RL from verifiable rewards, and self-training, even though a search solver answers 71% of instances. This is not a capability gap. The model does the arithmetic on 97-100% of lines and ranks the correct cipher in its top eight on 71%; it cannot carry the search forward as a left-to-right derivation. Fine-tuning learns the shape of a verifiable elimination step while its verdicts become unconditional templates, correct only 16-57% of the time ("verdict-as-token"). The ceiling holds across backbones from 3B to 671B and across fine-tuning and prompting; a controlled intervention isolates the cause: revealing the cipher key, which turns the derivation forward, lifts the same instances from 0.03 to 0.57. When a procedure's only solution is search over information-free structure, no faithful forward chain-of-thought exists to imitate. The task becomes learnable only by removing the search, precomputing its combinatorial core into a catalog and reducing the trace to recall plus verification; the 1st-place solution reaches Private LB 0.92 this way. What distills is memorization and verification, not search.