## 3 Methods
### Numerical Comparison of the Five Parsing Methods  
We compare our bubble parser with two widely used concurrent parsing methods: dependency parsing, using the SpaCy dependency parser [10], and constituent parsing, using the Stanford constituent parser [11]. Both of these parsing approaches generate parse trees that adhere to the Robinson Axioms [12]. In a previous study [13], we demonstrated that our bubble parser produces parse structures (bubbles) that do not conform to the Robinson Axioms, yet it aligns with similar linguistic principles to the two concurrent parsing methods. In fact, the parse structures generated by our bubble parser are more linguistically informative than those produced by trees that follow the Robinson Axioms. While both concurrent parsers can identify word classes, they are not designed to distinguish between main- and sub-clauses or determine the cases of nouns. Although the constituent structure of the Stanford parser allows for the identification of clause boundaries, it does not provide information regarding whether the clauses are main or subordinate. In contrast, our bubble parser can identify both main- and sub-clauses as well as the cases of nouns since it draws bubbles around linguistic elements such as clauses. Within these identified clauses, the nouns can be cross-referenced to determine their respective cases. Given the limitations of the two concurrent parsing methods, we also utilize two large language models [9]--GPT-4 and GPT-3.5--for comparison. Although these LLMs are not explicitly designed for parsing, they are capable of performing a wide range of linguistic tasks. Since no existing parsing method can fully address the proposed grammatical queries, it is necessary to employ a more general statistical approach. Currently, LLMs are the exclusive tools capable of handling these tasks. We choose ten test sentences for our comparison of the five parsing methods which are listed in Appendix B (in the following, we also label the two LLMs as parsing methods). In selecting the test sentences, our objective is to cover the full range of difficulties inherent to the German language, including the complex case and gender system, ambiguities regarding word classes, and complex sentence structures involving main- and sub-clauses. The sentences differ in their complexity and grammatical characteristics. The four sentences, 1, 2, 3, and 5, are specific sentences containing complex main- and sub-clause structures and some less frequently used word classes. For example, sentence 1 contains the verb \"einen\", which is identical to a frequently used article in German. Sentence 2 has the word \"die\", which fits into a noun phrase as an article but is, in fact, a pronoun. Sentences 3 and 5 have a very nested clause structure. The remaining six sentences are random samples taken from German news articles. However, only those sentences are selected that contain grammatical structures that our parser can already process. We avoid sentences with too many proper names, because our parser has no additional information about the case of these nouns that leads to many additional interpretations. In the current development stage, our parser can process only two punctuation marks, namely full stop and comma, which cover all linguistic principles related to main- and sub-clauses. Therefore, we avoid sentences containing additional punctuation marks. We find that in the current development stage, our parser can recognize approximately 50% of the German sentences, which are collected in several corpora by the \"Wortschatz Leipzig\" [21]. We parse the ten example sentences using both the dependency and constituent parsers and derive answers to the grammatical questions based on the annotations provided by these parsers. The dependency parser is accessed through its API, specifically using the \"German-de-core-news-sm (v3.5.0)\" module to generate parse trees via SpaCy (version 3.8.3).