# What is a “sentential form”?

I do know what a "left-most-" and a "right-most derivation" is, but I stumble across the term "sentential form" and its specific differences "right-sentential form" as well as "left-sentential form" and I just don't understand what is meant with this term.

Is the concept "sentential form" so different from the concept "derivation" that it is necessary to "have" it" ?

Can some of you please explain?

Yours sincerely

Assuming you know what leftmost and rightmost derivations are, let $S \Rightarrow^*W_1W_2\dots W_m$ be a derivation (a sequence of replacement using derivation rules), where $W_i$ is a terminal or nonterminal symbol. Then the string $W_1W_2\dots W_m$ is a sentential form of a grammar $G$. In addition, if $W_1W_2\dots W_m$ contains only terminal symbols then it is called a sentence.

If the derivation $S \Rightarrow^*W_1W_2\dots W_m$ is leftmost (rightmost) then the sentential form $W_1W_2\dots W_m$ is called left-sentential form (right-sentential form).

Is the concept "sentential form" so different from the concept "derivation"

Yes, these are different concepts. A derivation is a sequence of replacements of nonterminals using derivation rules given as a part of grammar, while a sentential form is a string over terminals and nonterminals. You generate/derive/obtain sentential form using derivation (process).

Example:

$S \rightarrow aSa \mid bSb \mid \epsilon$

Derivation: $S\Rightarrow aSa \Rightarrow abSba \Rightarrow abbSbba \Rightarrow abbbba$

Sentential form: $abbSbba$.
Sentence: $abbbba$ (since it has no nonterminal)

• Hello fade2black, thanks for your answer!! Is it enough to say that "sentential form" is the mere "product" and "derivation" is the "operation / chain of operations" by which this product is obtained? Is that the relationship between those concepts? Please excuse if I am still wrong. Best wishes – von spotz Oct 10 '17 at 17:56
• A sentential form is what you obtain (kind of product as you say), and derivation is a chain of operations if by a single operation you mean a replacement of a nonterminal with the right-side of a derivation rule. I strongly recommend you the "Dragon" book pg. 167 (The first edition). The relationship between them is that a sentential form is generated/obtained as a result of derivation. – fade2black Oct 10 '17 at 18:08
• Hello Is this not the same as the element of a language, that is, a "word", already ? A "sentence" at least. – von spotz Oct 11 '17 at 4:33
• A sentential form is not an element of a language, it may contain nonterminals and hence is clearly not an element of a language. – fade2black Oct 11 '17 at 4:37
• A sentential form is any string consisting of non-terminals and/or terminals that is derived from a start symbol. Therefore every sentence is a sentential form, but only sentential forms without non-terminals are called sentences. Only sentences are in the language. – Jochem Kuijpers Sep 27 '18 at 11:55