{"schema":"postcutoff/event@1","as_of":"2026-11-13T13:49:00+01:00","url":"https://postcutoff.com/e/2026-11-12-test-free-pdl-no-craig-interpolation-claude-fable/","md":"https://postcutoff.com/e/2026-11-12-test-free-pdl-no-craig-interpolation-claude-fable/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"human editor","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-11-12-test-free-pdl-no-craig-interpolation-claude-fable","date":"2026-11-12","date_precision":"day","short_title":"Test-free PDL lacks interpolation; counterexample by Claude Fable in Lean","deck":null,"takeaway":"After Borzechowski et al. proved Craig interpolation for PDL (2026), the test-free fragment remained open. Malvin Gattinger asked Claude Fable to turn failed proof attempts into a counterexample search; it returned one with a Lean formalization in minutes. The paper, written by Gattinger, publishes all prompts.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":[{"n":1,"title":"arXiv 2611.16759: Propositional Dynamic Logic without tests does not have Craig Interpolation (Gattinger)","url":"https://arxiv.org/abs/2611.16759","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"Zenodo: prompts and LLM output summaries","url":"https://doi.org/10.5281/zenodo.23987961","type":"code","group":"primary","domain":"doi.org"},{"n":3,"title":"Lean formalization (malv.in)","url":"https://malv.in/2026/test-free-pdl-no-interpolation/","type":"code","group":"primary","domain":"malv.in"}],"official":3,"filed":"2026-11-13","updated":"2026-11-13","orgs":["Anthropic","Harmonic","University of Amsterdam"],"title":"Test-free Propositional Dynamic Logic has no Craig interpolation: Claude Fable found the counterexample 'within a few minutes', with a Lean proof, after failing to repair the positive proof","summary":"arXiv 2611.16759 (12 Nov 2026), Malvin Gattinger (ILLC Amsterdam): a test-free PDL validity with one atomic program and Kleene star that has no test-free interpolant, so test-free PDL lacks Craig interpolation (full PDL has it, by Borzechowski et al. 2026). While adapting the Lean formalization of the positive result, Gattinger prompted Aristotle and Claude Fable; Claude Fable, asked to use the failed proof ideas \"for the opposite goal\", found a counterexample \"within a few minutes, including a Lean formalization\". Prompts and output summaries are on Zenodo; the Lean proof is public.","key_facts":["Result: a valid test-free PDL implication with no test-free interpolant; it uses one atomic program and *, so it also re-proves Studer's (2009) result that K with reflexive transitive closure lacks interpolation","Method (paper, §7, verbatim): 'We then prompted both Aristotle and Claude Fable to explore alternative constructions … This did not lead anywhere, so we eventually also prompted Claude Fable to use the failed proof ideas as inspiration for the opposite goal, namely to find a counterexample. It did so within a few minutes, including a Lean formalization of the result.'","Human role: 'All text in this paper was written by its human author. The proof structure and lemma statements are strongly inspired by the Lean proof and its comments that were written by Claude.' The published example simplifies Claude's first one (fewer letters, smaller Kripke model)","Timeline: obtained in September 2026; paper 12 Nov. Prompts and model summaries: doi:10.5281/zenodo.23987961; Lean: malv.in/2026/test-free-pdl-no-interpolation/","Status: preprint with a Lean formalization of the counterexample"],"key_numbers":[],"tags":["math","logic","modal-logic","computer-science","claude-fable","aristotle","lean","counterexample","arxiv"],"science":{"field":"mathematics","subfield":"mathematical logic / modal logic","problem":"Does test-free Propositional Dynamic Logic have Craig interpolation?","result":"No: an explicit test-free validity with no test-free interpolant","open_since":"","ai_system":["Claude Fable","Aristotle"],"human_role":"Human set up and repaired the Lean development; Claude Fable found the counterexample and its Lean proof; human simplified it and wrote the paper","verification":{"key":"lean","label":"Lean-verified","text":"Formal proof in Lean (author's site); preprint"},"status":"confirmed","shock":""},"body_md":"## What happened\n\nMalvin Gattinger (University of Amsterdam), a co-author of the Lean formalization of interpolation for Propositional Dynamic Logic,\ntried to adapt that proof to PDL without the test operator. One construction could not be repaired, and attempts with Harmonic's\nAristotle and Claude Fable to find a different one failed. He then asked Claude Fable to look for a counterexample instead; it found\none in minutes, with a Lean proof. The 12 Nov paper presents a simplified version, written by Gattinger, and links all prompts.\n\n## Why it matters\n\nIt settles a question left open by the 2026 interpolation theorem for PDL, and documents a pattern seen elsewhere: a model asked to\nturn failed proof attempts around finds the counterexample. The counterexample is formally checked.","disputed":[],"related":[{"id":"2026-11-13-ai-assisted-results-november-13-listing","url":"https://postcutoff.com/e/2026-11-13-ai-assisted-results-november-13-listing/","date":"2026-11-13","date_precision":"day","short_title":"AI-assisted results, 13 Nov","deck":null,"takeaway":"The arXiv listings of 12–13 Nov carried about 190 maths papers with an AI statement. Most disclose editing or literature search; about forty credit models with proofs, constructions or counterexamples. Seven got their own entries; the rest are catalogued here with the disclosure quoted. All are unrefereed preprints.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":27,"official":27,"filed":"2026-11-13","updated":"2026-11-13","orgs":["OpenAI","Anthropic","Google DeepMind","Axiom Math","various researchers"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-11-13","type":"filed","text":"Created (science agent; arXiv PDF read)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-11-13","run":null,"sources_read":"Science agent; arXiv PDF read","updated":"2026-11-13","human_review":null,"version":null},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":196,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":135,"in_training_data":false},{"model_id":"gemini-4-hydrogen","name":"Gemini 4 Hydrogen","cutoff":"2026-03","days_after":226,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":165,"in_training_data":false}],"short_url":null}