Let's carefully analyze the given grammar: **Grammar:** ``` S → A B A → a A | ε B → b B | C C → c C | d ``` --- ### 1. Compute First sets #### First(A) - A → a A | ε **First(A):** - From `a A`, the first terminal is `a` - From `ε`, include ε **Therefore:** **First(A) = { a, ε }** --- #### First(C) - C → c C | d **First(C):** - From `c C`, terminal `c` - From `d`, terminal `d` **Therefore:** **First(C) = { c, d }** --- #### First(B) - B → b B | C **First(B):** - From `b B`, terminal `b` - From `C`, First(C) = { c, d } (since C → c C | d) **Since B can derive `b B` or `C`, and `b B` starts with `b`, and `C` starts with `c` or `d`, then:** **First(B) = { b, c, d }** --- #### First(S) - S → A B **First(S):** - First of `A`, which is { a, ε } - Since `A` can be ε, then First(S) includes First(B) as well **If A → ε, then First(S) = First(B).** - First(B) = { b, c, d } - And if A starts with `a`, then the production S can start with `a` (via A), or with First(B). **Therefore:** **First(S) = (First(A) \ { ε }) ∪ (if ε in First(A), then First(B))** - First(A) = { a, ε } - Since ε ∈ First(A), include First(B): { b, c, d } - Also include terminals from A (excluding ε): `a` **Final First(S):** `First(S) = { a } ∪ { b, c, d } = { a, b, c, d }` --- ### **Summary of First sets:** | Non-terminal | First set | |----------------|--------------------------------| | **First(A)** | { a, ε } | | **First(C)** | { c, d } | | **First(B)** | { b, c, d } | | **First(S)** | { a, b, c, d } | --- ### 2. Compute Follow sets **Start symbol:** S #### Rules for Follow sets: - For a production `A → α B β`, everything in First(β) except ε is in Follow(B). - If β can derive ε, then Follow(A) also in Follow(B). - For the start symbol, Follow(S) includes `$` (end of input). --- #### Follow(S) - S is start symbol: **Follow(S) = { $ }** --- #### Follow(A) - S → A B Since `A` is followed by `B` in `S → A B`: - Add First(B) excluding ε to Follow(A): First(B) = { b, c, d } (none ε) So, **Follow(A) includes { b, c, d }** - Also, since `A` is directly before `B`, and `B` can derive ε? Let's check: `B → b B | C` - `b B` starts with `b`, no ε here - `C` can derive ε? Let's check: **Is C nullable?** - C → c C | d - Both productions start with terminals, so C is **not nullable** - Therefore, `B` is **not nullable**. Thus, no epsilon, so no additional follow from that. **Therefore:** **Follow(A) = { b, c, d }** --- #### Follow(B) - S → A B - `B` is at the end of the production. - So, Follow(B) includes Follow(S): **Follow(B) includes { $ }** - Any other productions? No. **Therefore:** **Follow(B) = { $ }** --- #### Follow(C) - B → C - C appears on the right side of `B → C`. - Since C is at the end of the production, Follow(C) includes Follow(B): **Follow(C) = Follow(B) = { $ }** --- ### **Summary of Follow sets:** | Non-terminal | Follow set | |----------------|----------------------| | **Follow(S)** | { $ } | | **Follow(A)** | { b, c, d } | | **Follow(B)** | { $ } | | **Follow(C)** | { $ } | --- ### 3. Predict sets for each production Predict sets are computed as: **Predict(Production) = First(α) if α does not derive ε; if α derives ε, then First(α) ∪ Follow(A)** --- #### For S → A B - First(A B): - First(A): - { a, ε } - Since First(A) includes ε, we look at First(B): - First(B) = { b, c, d } - Therefore: - First(A B) = (First(A) \ { ε }) ∪ (if ε in First(A), then First(B)) - { a } ∪ { b, c, d } = { a, b, c, d } - Since First(A) includes ε, and First(B) does not include ε, the first set is: **Predict(S → A B) = { a, b, c, d }** --- #### For A → a A - First(a A): starts with terminal `a` **Predict(A → a A) = { a }** --- #### For A → ε - When production is ε, Predict set is Follow(A): **Predict(A → ε) = Follow(A) = { b, c, d }** --- #### For B → b B - First(b B): starts with `b` **Predict(B → b B) = { b }** --- #### For B → C - First(C) = { c, d } **Predict(B → C) = { c, d }** --- #### For C → c C - First(c C): starts with `c` **Predict(C → c C) = { c }** --- #### For C → d - First(d): terminal `d` **Predict(C → d) = { d }** --- ### **Summary of Predict sets:** | Production | Predict set | |----------------------|-------------------------| | **S → A B** | { a, b, c, d } | | **A → a A** | { a } | | **A → ε** | { b, c, d } | | **B → b B** | { b } | | **B → C** | { c, d } | | **C → c C** | { c } | | **C → d** | { d } | --- ## **Summary:** ### First Sets: | Non-terminal | First set | |----------------|---------------------| | A | { a, ε } | | C | { c, d } | | B | { b, c, d } | | S | { a, b, c, d } | ### Follow Sets: | Non-terminal | Follow set | |----------------|-------------------| | S | { $ } | | A | { b, c, d } | | B | { $ } | | C | { $ } | ### Predict Sets: | Production | Predict set | |----------------------|-------------------------| | S → A B | { a, b, c, d } | | A → a A | { a } | | A → ε | { b, c, d } | | B → b B | { b } | | B → C | { c, d } | | C → c C | { c } | | C → d | { d } | --- If you need further explanation or assistance, feel free to ask!

**1.1 Introduction (Contextual Foundation)** In formal language theory, First and Follow sets are fundamental tools used in constructing parsers, especially for predictive parsing. First sets indicate the possible leading terminals of a string derivable from a non-terminal, while Follow sets specify the terminals that can appear immediately after a non-terminal in some derivation. These concepts are crucial for parsing table construction, particularly for LL(1) grammars. The current problem involves computing these sets for a given context-free grammar, which aids in understanding parsing predictability and conflict resolution. **Explanation Block:** Understanding First and Follow sets enables the determination of parser actions and conflict detection. First sets help identify the set of terminals that can begin strings derived from non-terminals, whereas Follow sets specify the terminals that can immediately follow a non-terminal in derivations. These are essential in constructing predictive parsing tables to ensure unambiguous, efficient parsing. --- **1.2 Relevant Formulas** - **First Set Calculation:** For any non-terminal \( A \): \[ \text{First}(A) = \bigcup_{A \to \alpha} \text{First}(\alpha) \] where \(\alpha\) is a production's right-hand side. - If \(\alpha\) begins with a terminal \( t \), then \( t \in \text{First}(\alpha) \). - If \(\alpha\) begins with a non-terminal \( B \), then include \( \text{First}(B) \setminus \{\varepsilon\} \). - If \( B \) is nullable (\(\varepsilon \in \text{First}(B)\)), then consider the next symbol. - If all symbols in \(\alpha\) are nullable, then \(\varepsilon \in \text{First}(\alpha)\). - **Follow Set Calculation:** For non-terminal \( A \): \[ \text{Follow}(A) = \bigcup_{A \text{ appears in } \beta \to \gamma A \delta} \left( \text{First}(\delta) \setminus \{\varepsilon\} \right) \cup \left\{ \text{Follow}(\beta) \text{ if } \varepsilon \in \text{First}(\delta) \text{ or } \delta \text{ is empty} \right\} \] - For the start symbol \( S \), \( \text{Follow}(S) \) always includes the end marker \( \$ \). - **Predict Set for a production \( A \to \alpha \):** \[ \text{Predict}(A \to \alpha) = \begin{cases} \text{First}(\alpha), & \text{if } \alpha \not\Rightarrow \varepsilon \\ \text{First}(\alpha) \cup \text{Follow}(A), & \text{if } \varepsilon \in \text{First}(\alpha) \end{cases} \] **Explanation Block:** These formulas systematically determine the potential starting terminals and subsequent terminals in derivations, which are pivotal for constructing parsing tables. First sets depend on the initial symbols of productions, considering nullable sequences, while Follow sets depend on the positions of non-terminals within productions and their nullable status. --- **1.3 Step-by-Step Solution** **Step 1: Compute First Sets** **Substep A: Calculate First(A)** - Production: \( A \to aA \ | \ \varepsilon \) - First(aA): - Starts with terminal \( a \), so \( a \in \text{First}(aA) \). - First(ε): - \( \varepsilon \in \text{First}(A) \). **Result:** \[ \boxed{ \text{First}(A) = \{ a, \varepsilon \} } \] **Substep B: Calculate First(C)** - Production: \( C \to c C \ | d \) - First(c C): - Starts with terminal \( c \), so \( c \in \text{First}(cC) \). - First(d): - Terminal \( d \), so \( d \in \text{First}(d) \). - Since neither production starts with a nullable sequence, First(C) is: \[ \boxed{ \text{First}(C) = \{ c, d \} } \] **Substep C: Calculate First(B)** - Production: \( B \to b B \ | C \) - First(b B): - Starts with terminal \( b \), so \( b \in \text{First}(

**1.1 Introduction (Contextual Foundation)** First and Follow sets are essential constructs in compiler design and formal language theory used to facilitate syntax analysis, particularly in predictive (LL) parsing. The First set identifies the set of terminals that can appear at the beginning of strings derived from a non-terminal, while the Follow set indicates the set of terminals that can immediately follow a non-terminal in some derivation. Accurate computation of these sets supports the construction of parsing tables and aids in detecting grammar conflicts. In this problem, these concepts are applied to a specific context-free grammar to systematically determine the First and Follow sets, which are foundational for parser design. **Explanation Block:** The First and Follow sets serve as predictive tools that inform parser actions during syntax analysis. By calculating these sets for each non-terminal, it becomes possible to construct parsing tables that guide the parser in choosing the correct production based on the next input symbol, thereby ensuring deterministic parsing and avoiding ambiguity. Their computation directly impacts the correctness and efficiency of the parsing process. --- **1.2 Relevant Formulas** - **First Set Formula:** For any non-terminal \(A\), the First set is computed as: \[ \text{First}(A) = \bigcup_{A \to \alpha} \text{First}(\alpha) \] where \(\alpha\) is a sequence of grammar symbols on the right-hand side of a production. - If \(\alpha\) begins with a terminal \(t\), then \( t \in \text{First}(\alpha) \). - If \(\alpha\) begins with a non-terminal \(B\), then include \(\text{First}(B) \setminus \{\varepsilon\}\). - If \(\text{First}(B)\) contains \(\varepsilon\), then consider the next symbol in \(\alpha\). - If all symbols in \(\alpha\) are nullable (contain \(\varepsilon\)), then \(\varepsilon \in \text{First}(\alpha)\). - **Follow Set Formula:** For a non-terminal \(A\): \[ \text{Follow}(A) = \bigcup_{X \to \alpha A \beta} \left( \text{First}(\beta) \setminus \{\varepsilon\} \right) \cup \left\{ \text{Follow}(X) \text{ if } \varepsilon \in \text{First}(\beta) \text{ or } \beta = \varepsilon \right\} \] Additionally, for the start symbol \(S\), \( \text{Follow}(S) \) always contains the end marker \(\$ \). - **Predict Set Formula:** For a production \(A \to \alpha\): \[ \text{Predict}(A \to \alpha) = \begin{cases} \text{First}(\alpha), & \text{if } \alpha \not\Rightarrow \varepsilon \\ \text{First}(\alpha) \cup \text{Follow}(A), & \text{if } \varepsilon \in \text{First}(\alpha) \end{cases} \] **Explanation Block:** These formulas provide a systematic approach to compute the First and Follow sets based on the grammar's productions. First sets are derived by analyzing the initial symbols of productions, considering nullable sequences. Follow sets are determined by the positions of non-terminals within productions and whether subsequent sequences can derive \(\varepsilon\). The Predict set combines First and Follow sets to identify the lookahead terminals necessary for predictive parsing, ensuring the parser can select the correct production without backtracking. --- **1.3 Step-by-Step Solution** **Step 1: Computing First Sets** **Substep A: Calculate First(A)** - Production: \(A \to aA \mid \varepsilon\) - For \(A \to aA\): - Starts with terminal \(a\), so \(a \in \text{First}(aA)\). - For \(A \to \varepsilon\): - \(\varepsilon \in \text{First}(A)\). **Result:** \[ \boxed{ \text{First}(A) = \{ a, \varepsilon \} } \] *Justification:* The production explicitly starts with terminal \(a\), and there is an ε-production, making both elements part of First(A). --- **Substep B: Calculate First(C)** - Production: \(C \to cC \mid d\) - For \(C \to cC\): - Starts with terminal \(c\), so \(c \in \text{First}(