leftmost derivation

Let $G=(\mathrm{\Sigma },N,P,\sigma )$ be a context-free grammar. Recall that a word $v$ is generated by $G$ if it is

• a word over the set $T:=\mathrm{\Sigma }-N$ of terminals, and

• derivable from the starting symbol $\sigma$.

The second condition simply says that there is a finite sequence of derivation steps, starting from $\sigma$, and ending at $v$:

$$\sigma =v_0\to v_1\to \mathrm{\cdots }\to v_n=v$$

For each derivation step $v_i\to v_{i+1}$, there is a production $X\to w$ in $P$ such that

$v_i$$=$$aXb,$

$v_{i+1}$$=$$awb.$

Note that $X$ is a non-terminal (an element of $N$), and $a,b,w$ are words over $\mathrm{\Sigma }$.

Definition. Using the notations above, if $X$ is the leftmost non-terminal occurring in $v_i$, then we say the derivation step $v_i\to v_{i+1}$ is leftmost. Dually, $v_i\to v_{i+1}$ is rightmost if $X$ is the rightmost non-terminal occurring in $v_i$.

Equivalently, $v_i\to v_{i+1}$ is leftmost (or rightmost) if $a$ (or $b$) is a word over $T$.

A derivation is said to be leftmost (or rightmost) if each of its derivation steps is leftmost (or rightmost).

Example. Let $G$ be the grammar consisting of $a,b$ as terminals, $\sigma ,X,Y,Z$ as non-terminals (with $\sigma$ as the starting symbol), and $\sigma \to XY$, $X\to a$, $Y\to b$, $\sigma \to ZY$, and $Z\to X\sigma$ as productions. $G$ is clearly context-free.

The word $a^2{b}^2=aabb$ can be generated by $G$ by the following three derivations:

1. $\sigma \to ZY\to X\sigma Y\to XXYY\to XaYY\to XaYb\to Xabb\to aabb$,

2. $\sigma \to ZY\to X\sigma Y\to a\sigma Y\to aXYY\to aaYY\to aabY\to aabb$,

3. $\sigma \to ZY\to Zb\to X\sigma b\to XXYb\to XXbb\to Xabb\to aabb$.

The second is leftmost, the third is rightmost, and the first is neither.

Note that in every derivation, the first and last derivation steps are always leftmost and rightmost.

Remarks.

• One of the main properties of a context-free grammar $G$ is that every word generated by $G$ is derivable by a leftmost (correspondingly rightmost) derivation, which can be used to show that for every context-free language $L$, there is a pushdown automaton accepting every word in $L$, and conversely that the set of words accepted by a pushdown automaton is context-free.

• Leftmost derivations may be defined for any arbitrary formal grammar $G$ satisfying the condition

  (*) no terminal symbols occur on the left side of any production.

  Given two words $u,v$, we define $u{\Rightarrow }_{L}v$ if there is a production $A\to B$ such that $u=xAy$ and $v=xBy$ such that $x$ is a terminal word. By taking the reflexive transitive closure of ${\Rightarrow }_L$, we have the leftmost derivation ${\Rightarrow }_L^{*}$. It can be shown that for any grammar $G$ satisfying condition (*), the language $L_L(G)$ consisting of terminal words generated by $G$ via leftmost derivations is always context-free.