Syntax diagrams as a formalism for representation of syntactic relations of formal languages

The new approach to representation of syntax of formal languages– a formalism of syntax diagrams is offered. Syntax diagrams look a convenient language for the description of syntactic relations in the languages having nonlinear representation of texts, for example, for representation of syntax lows of the language of structural chemical formulas. The formalism of neighbourhood grammar is used to describe the set of correct syntax constructs. The neighbourhood the grammar consists of a set of families of “neighbourhoods”– the diagrams defined for each symbol of the language’s alphabet. The syntax diagram is correct if each symbol is included into this diagram together with some neighbourhood. In other words, correct diagrams are needed to be covered by elements of the neighbourhood grammar. Thus, the grammar of formal language can be represented as system of the covers defined for each correct syntax diagram.

💡 Research Summary

The paper introduces a novel formalism called “syntax diagrams” for representing and validating the syntax of formal languages whose texts have a non‑linear structure. Traditional formal language theory models syntax as strings and uses context‑free or regular grammars to parse them. This approach becomes cumbersome when dealing with domains such as structural chemical formulas, circuit diagrams, or any representation where symbols are arranged on a plane or graph rather than a linear sequence. To address this limitation, the author proposes a graph‑based representation: a syntax diagram is a directed graph whose vertices are symbols from the language alphabet and whose edges encode the admissible relations between symbols.

The core of the formalism is the notion of a “neighbourhood grammar.” For each alphabet symbol a, a family of neighbourhoods is defined. A neighbourhood is a small sub‑diagram that specifies how the symbol may be locally connected to other symbols. The set of all neighbourhood families constitutes the neighbourhood grammar of the language. A syntax diagram is deemed correct if every vertex in the diagram belongs to at least one neighbourhood from the appropriate family; equivalently, the whole diagram must be covered by neighbourhoods. This covering condition replaces the traditional derivation tree: instead of generating a string step by step, one checks whether the graph can be tiled by locally admissible patterns.

The paper formalizes the covering problem mathematically and proposes an algorithmic solution. The diagram is first transformed into a tree‑like representation (e.g., by selecting a spanning tree). A depth‑first search then attempts to assign to each vertex a neighbourhood that matches its incident edges. If the assignment succeeds for all vertices, the diagram satisfies the grammar; otherwise a syntactic error is reported. The algorithm runs in time linear in the size of the diagram, because each vertex is examined only once and the neighbourhood families are finite.

A detailed example is given for the language of structural chemical formulas. Atoms (C, O, N, H, etc.) are vertices, bonds are edges, and each atom’s valence determines its neighbourhoods (e.g., carbon may have four single bonds, a double bond plus two singles, etc.). By enumerating these valence‑consistent neighbourhoods, the grammar captures all chemically valid structures. The covering test then automatically validates whether a given molecular graph respects valence rules, demonstrating the practical power of the approach.

Several theoretical advantages are highlighted. First, the neighbourhood grammar is modular: adding a new symbol or a new local rule only requires extending the neighbourhood family for that symbol, without rewriting the whole grammar. Second, because the grammar is expressed locally, there is no need to construct global parsing tables as in LR or LL parsers, which reduces memory consumption and simplifies implementation. Third, the graph‑oriented nature preserves spatial relationships that are lost when a non‑linear structure is linearized for string parsing.

The author also discusses integration with existing parsing technology. By converting a syntax diagram into a tree (or a set of trees) one can feed it to conventional parsers, enabling hybrid systems that handle both linear and non‑linear fragments of a language. This opens the door to mixed‑mode languages where, for example, a program may contain embedded circuit diagrams or chemical sub‑expressions.

In conclusion, the paper proposes syntax diagrams together with neighbourhood grammars as a robust, extensible formalism for describing syntactic relations in languages with non‑linear text representations. The covering‑based correctness criterion offers a clear, mathematically grounded method for validation, and the modular nature of neighbourhoods facilitates incremental language development. The approach is illustrated with chemical formula syntax but is applicable to any domain where symbols are arranged in a graph‑like fashion, such as electronic schematics, data‑flow diagrams, or visual programming languages.