Describe the set of strings defined by each of these sets of...

Describe the set of strings defined by each of these sets of productions in EBNF. $$ \begin{array}{ll}{\text { a) }} & {\text { string } : :=L+D ? L+} \\ {} & {L :=a|b| c} \\ {} & {D : :=0 | 1}\end{array} $$ $$ \begin{array}{l}{\text { b) string } : :=\operatorname{sign} D+| D+} \\ {\quad \operatorname{sign} : :=+|-}\end{array} $$ $$ D : :=0|1| 2|3| 4|5| 6|7| 8 | 9 $$ $$ \begin{array}{ll}{\text { c) }} & {\text { string } \therefore=L *(D+) ? L *} \\ {} & {L : :=x | y} \\ {} & {D : :=0 | 1}\end{array} $$

Describe the set of strings defined by each of these sets of productions in EBNF.
$$
\begin{array}{ll}{\text { a) }} & {\text { string } : :=L+D ? L+} \\ {} & {L :=a|b| c} \\ {} & {D : :=0 | 1}\end{array}
$$
$$
\begin{array}{l}{\text { b) string } : :=\operatorname{sign} D+| D+} \\ {\quad \operatorname{sign} : :=+|-}\end{array}
$$
$$
D : :=0|1| 2|3| 4|5| 6|7| 8 | 9
$$
$$
\begin{array}{ll}{\text { c) }} & {\text { string } \therefore=L *(D+) ? L *} \\ {} & {L : :=x | y} \\ {} & {D : :=0 | 1}\end{array}
$$

Key Concepts

EBNF Notation

EBNF (Extended Backus-Naur Form) is a formal way to describe the syntax of computer languages, defining production rules that specify how symbols in the language can be combined. It uses symbols, such as ':=', '|', '?', '+', and '*', to represent definitions, alternatives, optional elements, one or more repetitions, and zero or more repetitions respectively.

Terminal and Nonterminal Symbols

In formal grammars such as those defined in EBNF, terminal symbols are the actual characters or tokens that appear in the final strings, while nonterminal symbols serve as placeholders for patterns or groups of terminals. Understanding the distinction between these types of symbols is crucial for interpreting and constructing the language described by the grammar.

Optional and Repetition Operators

EBNF includes operators that simplify the description of repeated or potentially omitted parts of the language. The '?' operator indicates an optional element (meaning it may appear zero or one time), the '+' operator specifies that an element must appear one or more times, and the '*' operator indicates that an element can appear zero or more times. These operators allow for a concise representation of the structure of strings in a language.

Transcript

00:03 We are given sets of productions in ebnf, extended box in our form.

00:10 We're asked to describe the set of strings defined by each of these sets.

00:16 So, in part a, we're given the set of productions string colon -colon -colon equals l plus d question mark l plus l -colon -colon -equal equals a -line b -line c.

00:45 And d colon -colon equals 0, line 1.

00:54 So to describe the set of strings defined by this set, let's start from string.

01:03 So we see that from string, we can get l plus d question mark l plus.

01:33 Now l plus means that l occurs one or more times.

01:40 So this means that we can find some, positive integer n, such the l plus is l to the n.

01:50 Now, d question mark means that d occurs zero or one times.

01:58 So either d appears once, or d does not appear.

02:05 So we could have ln d, or we could have just ln to start the string.

02:12 And then we see another ln plus.

02:14 This means that we can find some integer m, such the l plus, is equal to l to the m, m is a non -negative integer.

02:24 And so we see that the two kinds of strings we can have are l -n -d, l to the m, l to the n, l to the m.

02:33 And l to the n, l to the m is really the same as l to the n plus m, which is the same as we could just say l to the p, where p is some non -negative integer.

02:59 Well, the problem is plus actually implies that l occurs one or more times.

03:03 So we have the n and m are both going to be greater than or equal to one, and therefore p is greater than or equal to two.

03:19 And we see that l represents a, b, and c, and d represents 0 or 1.

03:36 So we see that any string specified by these sets of productions is any sequence of letters, a, b, or c, containing at the least one letter, a, b or c of course, and then after this, optionally followed by a binary digit, so optionally followed by a bit, followed by any sequence of letters again, a, b, and c with at least one letter.

05:38 Now in part b, we're given the set of productions...

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