A powerful catalyst string representation must therefore be able to handle non-locality and an enormous variety of elemental compositions. A recently proposed approach to overcome these challenges is to use a coordinate-free representation based on crystallographic Wyckoff positions. By avoiding the definition of bonds between atoms altogether, this is potentially a viable route towards powerful string representations of catalysts. In the field of Zeolites and Metal-Organic Frameworks, the classification of network topologies offers similar advantages. 15,16 If a generally useful representation could be defined along these lines, this would give access to the wealth of techniques developed in natural language processing, both in terms of grammatical inference and machine learning (e.g. recurrent neural networks and transformers). Indeed, this type of interdisciplinary approach has recently led to significant advances in organic synthesis planning. Graph representations: An alternative route towards a catalyst grammar would be to use graphs instead of strings to represent the catalysts. Graphs have a long tradition for representing chemical structures in terms of atoms and their connectivity. Unlike strings, they can easily represent cycles, branches and other non-local features of arbitrary complexity. Moreover, graphs and nets (their periodic equivalent) are already used to characterize inorganic solids such as Zeolites, Metal-Organic-Frameworks and carbon allotropes. A further advantage of graphs over strings is that they can in principle directly encode the relative positions of atoms in three dimensional space and are thus overall more expressive. Furthermore, it is easier to define meaningful measures of similarity for graphs than for strings, which can be important in ML applications. An analogous concept to formal grammars also exists for graphs. Such graph grammars use production rules that define how subgraphs can be modified and replaced. As a downside, developing and using graph grammars is significantly more complicated, however, because they operate on a more complex type of object. This is particularly true for periodic graphs. We also note that defining bonds in inorganic solids is not always unambiguously possible, so that a straightforward graph representation based on valence rules is not necessarily equally well suited for all types of materials. However, it has been shown that graph neural networks are able to learn powerful graph representations in very diverse settings, without prior definition of chemical bonds. A combination of generative grammars with graph-based ML may therefore be a promising route. Validity: As noted above, the central advantage of using a grammar in the context of catalyst discovery is that it allows the exclusive generation of syntactically 'valid' candidates, thus avoiding the unnecessary consideration of 'invalid' ones. We have so far been fairly vague about what is meant by valid structures, however. Indeed, this is not clear and depends on the context. In the case of SMILES, validity simply means that the valence rules of organic chemistry are not violated. This implies that the corresponding molecules will also be reasonably stable in most (but not all) cases. Relying on valence rules alone is unlikely an adequate concept of validity for the full periodic table though, not least due to the ambiguous nature of chemical bonds. Similarly, the charge balance condition used in the toy example above is not sufficient to guarantee stability and only applies to ionic materials. Yet another type of validity criterion can be defined based on atomic or ionic radii, as e.g. used in the Goldschmidt tolerance factor for perovskites. In principle, a combination of these different validity measures could be encoded in a formal grammar, while in general and as noted above 'valid' could simply mean 'consistent' with available partial understanding.