Chapter 4, Problem 10E

Explanation of Solution

Validating the string based on Luhn’s formula:

• To reduce the likelihood of typographic errors, the digits comprising most credit card numbers are designed to satisfy the so called Luhn formula.
• It is standardized by ANSI in 1960s.
• The following grammar represents validating the sting based of Luhn’s formula.

$\begin{array}{l}\text{CCN }\to \text{DS}\Rightarrow \text{CCN}\text{.valid := (DS}\text{.sum 0 mod 10)}\\ \text{DS }\to \text{ digit}\Rightarrow \text{DS}\text{.even := false}\\ \Rightarrow \text{DS}\text{.sum := digit}\text{.value}\\ {\text{DS}}_1\to {\text{ digit DS}}_2\Rightarrow {\text{DS}}_1{\text{.even := not DS}}_2\text{.even}\end{array}$

        $\begin{array}{l}\Rightarrow {\text{DS}}_1{}_{}\end{array}$