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

arXiv:0802.3974v2 [cs.LO] 28 Feb 2008 SYNTAX DIAGRAMS AS A FORMALISM FOR REPRESENTATION OF SYNTACTIC RELATIONS OF FORMAL LANGUAGES VLADIMIR LAPSHIN Abstract. The new approach to representation of syntax of for- mal languages – a formalism of syntax diagrams is offered. Syntax diagrams look a convenient language for the description of syntac- tic 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 el- ements of the neighbourhood grammar. Thus, the grammar of formal language can be represented as system of the covers defined for each correct syntax diagram. 1. The work’s motivation The idea of representation of syntax relations of a formal language by means of the definition of families of language symbols’ neighbourhoods belongs to Soviet mathematician J. Shreider ([4]). The neighbourhood of a symbol here is understood as any chain of symbols containing this symbol. The chain is in the source language if, and only if each symbol belongs to this chain together with some it’s neighbourhood. Such the system of neighbourhoods has been named by Shreider as a neighbour- hood grammar. Let consider a concrete example. Let L be a formal language with the alphabet A = {a, b} and chains of language L are the sequences of alternating symbols a and b, where first and last sym- bols must be a. In other words, chains of language L are chains of a kind aba, ababa, abababa, etc. Let define the neighbourhood grammar for this language by enumerating a finite system of neighbourhoods for each symbol of the alphabet A. Let consider the symbol a and places in the language’s chains where it is occurred. This symbol necessarily appears in the beginning and the end of any chain of formal language L. To accent this fact, enter an additional pseudo-symbol # which will signal about the beginning and the end of a chain. Thus, there are two neighbourhoods of a symbol a: a neighbourhood #ab and a neighbourhood ba#. Except for the above-stated cases, the symbol 1 2 VLADIMIR LAPSHIN a can be between two symbols b. Add for this case a neighbourhood bab a symbol a. For a symbol b enough a unique neighbourhood – chains aba. So, any chain of language L becomes covered by the neigh- bourhoods specified above. It is easy to prove the contrary: any chain which becomes covered by the system of neighbourhoods defined above belongs to language L. Languages for which it is possible to define a neighbourhood grammar in sense of Shreider are named as Shreider’s ones. Shreider’s languages are simple enough in sense of expression of syntactic relations. The unique type of syntactic laws which can be expressed by neighbourhood grammars is the relation ”to be close to”. In Chomsky’s hierarchy Shreider’s languages represent own subset of linear languages. In other words, neighbourhood grammars as they be formulated by Shreider cannot be used to define the overwhelming majority of languages. The idea of neighbourhood grammar developed in works of the Soviet mathematicians of V.Borschev and M.Homyakov ([1], [2]). They have suggested to expand a traditional sight at formal language as on a set of chains defined on some alphabet. In Borschev and Homyakov’s works a neighbourhood grammar was used to define not chains of symbols, but wider concept of texts. Texts could represent everything: chemical formulas, graphs and etc. In particular, the neighbourhood interpreta- tion of context-free languages has been offered. As it is known, each chain belongs to context-free language has at least one derivation tree. Such tree has the top signed by an initial non-terminal symbol of the context-free grammar for the given language, internal units are signed by non-terminal symbols, and sheet units signed by terminals of the given grammar. The idea was to define the set of correctly constructed derivation trees of the given context-free language by a systems of neighbourhoods, defined for each symbol of the language (nonterminal and terminal one). The neighbourhood is understood here as some sub- tree containing the dedicated symbol – the center of the neighbourhood. Borschev and Homyakov found that such the neighbourhood grammar can be defined for each context-free language. The neighbourhoods of the grammar are either a bush consisting of the one level tree, where the center of the neighbourhood is the top nonterminal of a bush, or one knot tree consisting of a single terminal symbol. The grammar’s nonterminals have the first type of neighbourhoods (bushes) and single vert

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