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We want to recap the concepts of a grammar and a formal language in the following definition.

Definition: Formal Languages Generated From a Grammar

Let $\Sigma$ be an alphabet and $G=(V,T,R,S)$ be a grammar over this alphabet. We say a string $y\in\Sigma^*$ can be syntactically derived from another string $x$ using the grammar $G$ (denoted by $x\Rightarrow y$), if $$\exists u,w\in (V\cup T)^*, \exists P\to C\in R: (x=uPv\wedge y=uCv).$$

In other words, $x\Rightarrow y$ if and only the two strings have the form $x=uPw$ and $y=uCw$ (for some other strings $u,w$) and there is a rule $P\to C.$

Note that $y$ can be derived from $x$ by applying several rules, e.g. $x\Rightarrow z_1\Rightarrow\cdots\Rightarrow z_n\Rightarrow y$. It is, therefore, reasonable to require that the set $R$ of grammar rules is a transitive relation.

Denoting by $\Rightarrow^*$ the transitive hull of the grammar rule relation $R,$ we can now state that a language $L\subset\Sigma^*$ is said to be generated by the grammar $G$ if and only if

$$L=L(G)=\{y\in\Sigma^*\mid S\Rightarrow^* y\}.$$

In this case, we call $L$ a formal language.

Notes

| | | | | created: 2020-04-23 16:21:05 | modified: 2020-05-24 07:53:07 | by: bookofproofs | references: [1086], [7895]

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Bibliography (further reading)

[1086] Erk, Katrin; Priese, Lutz: “Theoretische Informatik”, Springer Verlag, 2000, 2. Auflage

[7895] Hoffmann, Dirk: “Theoretische Informatik, 3. Auflage”, Hanser, 2015