Chapter 4, Problem 17E

Explanation of Solution

Context-free grammar for the polynomials in “x”:

• The context-free grammar (CFG) generates the context-free language (CFL).
• Production rules of CFG are simple replacement.
• Parsing is a method of finding a tree which has a given yield.
• Rules can also be applied in reverse to check whether a string is grammatically correct according to the grammar.

Assumes exponents in the input are all positive integers. The grammar is as follows:

$\begin{array}{c}\text{P }\to \text{ T more\_Ts}\\ ⊳\text{ more\_Ts}\text{.st := T}\text{.d }⊳\text{ P}\text{.d := more\_Ts}\text{.d}\\ \text{T }\to \text{ num T\_tail}\\ ⊳\text{ T\_tail}\text{.c := num}\text{.v }⊳\text{ T}\text{.d := T\_tail}\text{.d}\end{array}$

$\begin{array}{c}\text{T\_tail }\to \text{ x exp}\\ ⊳\text{ exp}\text{.c := T\_tail}\text{.c }⊳\text{ T\_tail}\text{.d := exp}\text{.d}\\ \text{T\_tail }\to \in \\ ⊳\text{ T\_tail}\text{.d := " " }\end{array}$

$\begin{array}{c}\text{exp }\to \text{ ** num}\\ ⊳\text{ exp}\end{array}$

Explanation of Solution

Replacing the semantic functions with action routines:

• Action routines are a semantic function that the user instructs the compiler to execute at a specific part in the parse.
• An ad hoc translation scheme which is inserted with parsing takes the form of a set of action routine.
• Many parser generators allow the user to identify action routines.

Assumes exponents in the input are all positive integers. The grammar is as follows:

$\begin{array}{c}\text{P }\to \text{ T {more\_Ts}\text{.st := T}\text{.d} more\_Ts {P}\text{.d := more\_Ts}\text{.d}}\\ \text{T }\to \text{ num {T\_tail}\text{.c := num}\text{.v} T\_tail {T}\text{.d := T\_tail}\text{.d}}\\ \text{T\_tail }\to \text{ x {exp}\text{.c := T\_tail}\text{.c} exp {T\_tail}\text{.d := exp}\text{.d}}\\ \to \in \text{ {T\_tail}\text{.d := "}\text{}\text{ "}}\end{array}$

$\begin{array}{c}\text{exp }\to \text{ ** num { exp}\end{array}$