## 3 Methods
### The Grammatical Analysis by the Bubble Parser  
The parsing method introduced in this paper can be categorized as \"global\" constraint grammar. As discussed in Section 2.2, constraint grammar aims to resolve ambiguities by disambiguating the multiple possible readings of words in a sentence. However, this process is typically applied to each word individually. In contrast, our parser operates within the product space of all possible readings for all words. Each point in this product space represents a unique interpretation, where every word has a fixed unambiguous reading. Each interpretation is parsed independently, and when the parser detects a contradiction within a particular interpretation, that interpretation is discarded. This approach eliminates incorrect word readings not in isolation, but in combination with the readings of other words. A specific reading of word \(A\) is only discarded if it conflicts with a particular reading of word \(B\). Consequently, a given reading of word \(A\) will only be eliminated in all interpretations if it leads to a contradiction in every possible combination with other word readings. The new linguistic model described by [PERSON] et al. (2023)[PERSON] et al. (2023) generates a bubble model for each interpretation, provided that the interpretation is free from ambiguity. Figure 2 illustrates a bubble model for one of the ten example sentences we use in our analysis (see Appendix B). In these models, clause boundaries are depicted by a double-lined gray border encircling the described scene, while nouns within the scene are enclosed by a single line. Additionally, the bubble model illustrates the types of clause connections, although these are not discussed in the current paper. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Paper** & **Theory** & **Parser** & **Contribution** \\ \hline “Semantic approach for solving the decision problem in natural language”[PERSON] et al. (2023) & Yes & Yes & First introduction to the bubble model and principles of the parser \\ \hline “A rule-based parser in comparison with a modern large language model in terms of grammar competence” (present paper) & No & Yes & Discusses the main steps in parsing \\ \hline Future work I & Yes & No & Monadic second-order propositions for every bubble model \\ \hline Future work II & No & Yes & Statistics over the entire vocabulary \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Relationships between previous, current, and future publications on this topic. Unlike dependency parsers (as discussed in Section 2.3), our parser has the capability to reject input sentences if any contradictions in grammar are present, which forms the basis for eliminating false interpretations. The construction of bubble models is similar to the well-known CYK algorithm [15], mentioned in Section 2.1. Similar to the CYK algorithm, our parser starts with the given sentence and merges several words into new objects. In [PERSON]'s grammar description of natural language [14], these new objects are represented by non-terminal symbols, which we call here \"linguistic objects\", namely noun phrases, main- and sub-clauses, and other elements with semantic and syntactic meaning. However, in contrast to the CYK algorithm, our bubble parser produces several interpretations, in which elements are differently merged into new objects by going through all possible ways of combining elements. The CYK algorithm does not address this multiple parsing of sentences. The bubble parser maintains an ongoing list of all interpretations that are possible up to the current step in the parsing process. Every subsequent parsing step takes this ongoing list of interpretations and processes every interpretation from this list. An interpretation can either be sorted out, stay the same, or split into several subsequent interpretations, which results in a new ongoing list of interpretations for the next parsing step. Figure 3 illustrates this procedure, which is applied to the grammatically ambiguous sentence \"Time flies like an arrow\".