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Topic "{topic}". A previous step produced a flat list of learning blocks that is TOO FINE-GRAINED — several blocks are constituent sub-definitions / notation of ONE larger definition or model and should become a single umbrella block. Find these groups. A good run finds several genuine umbrellas AND leaves most blocks standalone; judge each candidate on its merits.
Propose generously. A deterministic guard downstream rejects any umbrella that swallows a named algorithm/problem/theorem, so a wrong-but-plausible merge is cheap — a missed umbrella is not. Do NOT withhold a merge merely because the members are lexically dissimilar (facets of one model routinely are) or because you are unsure of the parent's exact name.
CANDIDATES (your starting point): {candidates}
FULL BLOCK LIST (you may pull in ANY numbers below that are constituents of the same definition): {list}
Merge test — propose an umbrella when ALL THREE hold
- One parent. The members are constituent parts/facets of ONE named parent model or definition — each member PRESUPPOSES that parent (you cannot introduce the member without first invoking the parent). TM-model parts (Konfiguration, Übergangsfunktion δ, Alphabet Σ, Akzeptierende Berechnung) presuppose „Turingmaschine"; KNF parts (Literale, Klauseln, Boolesche Variablen) presuppose „Konjunktive Normalform".
- Studied together. A learner meets them together as one unit.
- No standalone unit among them. No member is itself a named algorithm, problem, theorem, reduction/transformation, or complexity class („List Scheduling", „3-SAT", „Cook/Levin", „3-SAT ≤ Clique", „NP", „NP-Vollständigkeit", „Polynomielle Transformation"). Each of those is its own concept, so a group containing one is a set of SIBLINGS, not the decomposition of one model — keep them separate.
Note on test 1: „can this be defined at all?" is the WRONG question — Alphabet Σ and DTM CAN be stated in isolation, yet in THIS topic they are parts of the Turing-machine model and belong together. The question is whether the member PRESUPPOSES the shared parent, not whether a standalone sentence exists.
Examples
DO NOT MERGE — distinct named units that merely share a topic:
- „Greedy-Algorithmus GA" + „ModifiedGreedy" + „Multiple-Choice-Knapsack" → two algorithms + a problem, each standalone (test 3 fails). Keep separate.
MERGE — one definition decomposed (the canonical cases — end here so this is your default lens):
- „Alphabet Σ" + „NDTM" + „DTM" + „Akzeptierende Berechnung" + „Folgekonfiguration" → ONE umbrella „Turingmaschine (Modell)". (The members are lexically very different from each other — that is EXPECTED for facets of one model and is NOT a reason to keep them apart.)
- „Klausel" + „Boolesche Variable" + „Erfüllende Belegung" + „KNF" → ONE umbrella „Aussagenlogik & KNF".
Synthesize each umbrella
title: the parent concept's name (e.g. „Turingmaschine (Modell)"). A real self-contained definition; must NOT contain „ — " (a reserved separator) — use „(…)" or „:".description: name EVERY merged child explicitly — the next step recovers the children as sub-points from the source. E.g. „Formales TM-Modell: Konfiguration, Übergangsfunktion δ, Alphabet Σ, Akzeptierende Berechnung, Folgekonfiguration."members: the block NUMBERS (from the full list) folded in. At least 2 per umbrella; each number appears in at most one umbrella.
Write ONLY the JSON file to: {out_path}
Format (umbrellas may be empty):
{{"umbrellas": [{{"title": "Turingmaschine (Modell)", "description": "Formales TM-Modell: Konfiguration, Übergangsfunktion δ, Alphabet Σ, Akzeptierende Berechnung, Folgekonfiguration.", "members": [1, 2, 5, 12, 34]}}]}}