TOC QB Unit 1 PDF

## **1. Explain DFA with suitable example** A Deterministic Finite Automaton (DFA) is denoted by 'D'. D is a subset of five entities. * D = {Q, Γ, δ, S, F} * Q is a set of finite elements called STATES. * Γ is a set of ALPHABETS. * δ is a Delta x S → Q state. * S is a starting state. * F is a fi...

## **1. Explain DFA with suitable example** A Deterministic Finite Automaton (DFA) is denoted by 'D'. D is a subset of five entities. * D = {Q, Γ, δ, S, F} * Q is a set of finite elements called STATES. * Γ is a set of ALPHABETS. * δ is a Delta x S → Q state. * S is a starting state. * F is a final state. * S ∈ Q and F ∈ Q. Both of them s and F belong to Q. **For example:** A Diagram of a DFA, containing a "starting" state labeled "So," and three "accepting" states labeled "S1," "S2," and "S3," are connected by three transitions: * So xa S1 * So xb S2 * So xc S3 * Q = {So, S1, S2} * Γ = {a, b, c} * S = So * F = S2 **Transition Table for DFA** | | a | b | c | |:----|:-----|:-----|:-----| | So | S1 | S2 | S3 | | S1 | S1 | S1 | S1 | | S2 | S2 | S2 | S2 | ## **2. Explain NDFA with suitable example.** A Non-Deterministic Finite Automaton (NDFA) is denoted by 'N'. N is represented by N. * N = {Q, Γ, δ, S, F} * Qis a set of finite elements called STATES * Γ is a set of ALPHABETS. * δ is a Mapping from Q X Γ U ε → {Q} * S is a starting state. * F ∈ Q. Both of them δ & F belongs to Q. **For example:** A Diagram of a NDFA, containing a "starting" state labeled "So," and two "accepting" states labeled "S1," "S2," are connected by four transitions: * So xe S2 * S1 xe So * So xa S1 * S1 xb S1 * S1 xc S2 * Q = {So, S1, S2} * Γ = {a, b, c, ε} * S = So * F=S2 **Transition Table for NDFA** | | a | b | c | ε | |:----|:-----|:-----|:-----|:-----| | So | S1 | - | - | S2 | | S1 | - | S1 | S2 | So | | S2 | - | - | - | - | ## **3. What is DFA and NDFA equivalence? Explain with an example.** * **DFA (Deterministic Finite Automaton)** and **NDFA (Non-Deterministic Finite Automaton)** are both types of machines used to recognize patterns in strings, but they work a little differently. * **DFA (Deterministic Finite Automaton):** In a DFA, for every state and each input, there's exactly one possible next state. It's like following a clear, one-way path. No guessing, no choices - just one option at every step. * **NDFA (Non-Deterministic Finite Automaton):** In an NDFA, for the same state and input, these could be multiple possible next states, or sometimes no next state at all. It's like being able to take multiple paths at a fork in the road - you can “choose” where to go next. * **DFA and NDFA Equivalence:** Even though NDFAs can “choose” multiple paths, any NDFA can be turned into a DFA that accepts the same strings. So, DFA and NDFA are equivalent in terms of what languages they can recognize, though the DFA might need more states than the NDFA. **Example:** Let's say we want a machine that accepts strings containing at least one "a." ### **NDFA Example:** * **States**: qo (Start), q1 (accepting state). * **Transition:** * **From qo:** * On input "a," go to q1 (because we've seen "a"). * On input "b," stay in qo (no “a” yet). * **From q1:** * On input "a," stay in q1. * On input "b," stay in q1 (once we've seen an "a,” we accept any input). In the NDFA, when you're in state qo and seen an "a," you move to q1. If you see a "b," you stay in qo. There's some flexibility, and you don't have to decide the next state immediately, because you might have multiple choices. ### **DFA Example:** * **States**: qo, q1 (same as NDFA, but here we'll treat them differently). * **Transition:** * **From qo (no "a" yet):** * On "a," move to q1 (now we've seen "a"). * On "b," stay in qo. * **From q1 (we've seen "a"):** * On "a,” stay in q1. * On "b," stay in q1. In the DFA, there's no choice - you always know exactly where to go. If you see "a," you move to q1, and if you see "b”, you either stay in qo or q1 depending on where you are. ## **4. Write a short note on Mealy machines.** **A Mealy machine** is a type of finite automaton that produces output based on both its current state and the input symbol. It's different from other automata (like a DFA) because it generates outputs as it processes inputs. ### **Key features of a Mealy Machine:** * **Input-Output Relationship:** Output is produced for each transition (input symbol and current state). * **Transitions:** Each transition is labeled with an input-output pair (e.g., a/1 means on input a, output 1 is produced). * **Faster Output:** Outputs are generated during transitions, not after the input string is fully processed. ### **Structure:** A Mealy machine consists of: * **States:** Represent different conditions. * **Inputs:** Symbols that cause transitions between states. * **Outputs:** Symbols produced based on inputs and current states. * **Transitions:** Define how the machine moves between states and what output is produced. **Example:** Imagine a machine that outputs 1 if the current input is "a" and outputs 0 for "b." * **States**: qo, q1 * **Input**: {a, b} * **Output**: {0, 1} * **Transitions:** * qo on all → stay in qo * qo on bl → move to q1 * q1 on alt → stay in q1 * q1 on blo → move to qo ### **Applications:** Mealy machines are used in: * Digital circuits * Text processing * Protocol design (e.g., detecting patterns in data structure streams). ## **5. Write a short note on Moore Machine.** **A Moore Machine** is a type of finite automaton that generates an output based only on its current state, not the input. The output is constant as long as the machine remains in the same state. ### **Key Points:** * **State-based output:** Each state has a fixed output, independent of the input. * **Outputs on states:** Outputs are associated with states, not transitions. * **Real-time behavior:** Outputs are updated when the machine enters a new state. ### **Structure:** A Moore machine consists of: * **States:** Each state has a fixed output * **Transitions:** Define how the machine moves between states based on inputs. **Example:** If a machine outputs 1 in qo and 0 in q1: * qo → Output 1 (for all inputs). * q1 → Output 0 (for all inputs). ### **Applications:** Moore machines are simple and predictable because the output depends only on the current state. * Used in digital circuits, control systems, and pattern recognition. ## **6. Define grammer and its components in easy words.** A grammar is a set of rules that defines how strings in a language are formed. It is used to generate valid sentences (or strings) in a specific language. ### **Components of a grammer** A grammar has four main components: 1. **Non-Terminals (Variables):** Symbols that represent intermediate states in the derivation process. * Example: S, A, B. 2. **Terminals:** They are placeholders that are replaced by other symbols or strings. * **Symbols** that appear in the final strings of the language (cannot be replaced further). * **Example**: Letters, numbers, or special characters like a, b, 0, 1. 3. **Start Symbol:** A special non-terminal where the derivation begins. * **Example**: S (usually the starting point). 4. **Production Rules:** Rules that define how non-terminals are replaced with other non-terminals or terminals. * Example: S→aB or B→b **Example**: A grammar to generate strings of a’s followed by b’s. * Non-Terminals: {S, Y} * Terminals: {a, b} * Start Symbol: S * Rules: * S→aS * S→ b * **Strings generated**: b, ab, aab, aaab, etc. ## **7. What is derivation? Explain with an example.** The process of applying production rules to replace non-terminals with terminals or other non-terminals to generate strings. ### **Derivations are classified into two types:** 1. **Leftmost Derivation** 2. **Rightmost Derivation.** * **a. Leftmost Derivation:** Always chooses the Leftmost non-terminal symbol for further expansion. It is applied for all the steps of derivation. * **For Ex:** * S → aBcD * B → b * D → d * **LM**: * S → aBcD * S → abcD (Substituting value ‘B’) * S → abcd (Substituting value ‘D’) * **b. Rightmost Derivation:** Chooses the rightmost non-terminal symbol for further expansion. It is applied for all the steps of derivation. * **For Ex:** * S → aBcD * B → b * D → d * **RM**: * S → aBcD * S → aBcd (Substituting value of ‘D’) * S → abcd (Substituting value of ‘B’) ## **8. Describe Chomsky classification of grammer.** The Chomsky hierarchy is a way to classify grammars based on their rules and the types of languages they generate. It divides grammars into four types, ranked by their expressive power (how complex the languages they can describe are). 1. **Type 0: Unrestricted Grammer:** * **Rules:** No restrictions on the production rules. * **Example:** aβ (where both a and β are any combinations of terminals and non-terminals, but β cannot be empty). * **Power:** Can describe any language that a Turing machine can recognise (most general and powerful). * **Language:** Includes recursively enumerable languages. * **Example Rule:** AB + BCA. 2. **Type 1: Context-sensitive Grammar:** * **Rules:** The length of the output (right-hand side) must be greater than or equal to the length of the input (left-hand side). * **Example:** xAB + xBB (where α and β are context, and A is replaced with B). * **Power:** Can describe context-sensitive languages (languages that depend on the surrounding "context"). * **Language:** Includes language like a^b^c^ (e.q.aaabbb.ed). * **Example Rule:** AB → Ab. 3. **Type 2: Context-Free grammer (CFG):** * **Rules:** The left-hand side of a rule must be a single non-terminal. * **Example:** A→α (where A is a non-terminal, and α is a combination of terminals and/or non-terminals) * **Power:** Can describe languages where context is non-needed, like balanced parentheses or mathematical expressions. * **Languages:** Includes most programming languages. * **Example Rule:** S → asb| ε. 4. **Type 3: Regular grammer:** * **Rules:** The most restricted form. The right-hand side must be a terminal, optionally followed by a single non-terminal. * **Power:** Can describe simple languages, such as patterns of repetition or strings with specific sequences. * **Languages:** Includes regular languages (used in finite automata). * **Example Rule:** S → aS | b ## **9. Define and explain recursive enumerable set.** A recursive enumerable (RE) set is a collection of elements (usually strings) that can be listed or enumerated by a computer program, but the program may not always tell you when something is not in the set. ### **Key Points:** 1. **What It Means to Be Enumerable:** If a set is recursive enumerable, there is a program (or Turing machine) that can: * **Generate** all elements in the set, one by one. * **Recognize** if a given element belongs to the set by eventually halting and accepting it. * However, the program may never stop if the element is not in the set. 2. **Difference from Recursive Set:** A recursive set is one where a program can tell you for sure if an element is in the set or not in the set (halting for both cases). 3. **Examples:** * A recursive enumerable set could include strings that satisfy a certain condition (e.g., strings accepted by a Turing machine). * Example: The set of all valid mathematical proofs is recursive enumerable because we can keep generating proofs to check if a statement belongs to the set. 4. **Non-Membership Challenge:** If an element does not belong to the set, the program may run forever without confirming it. This makes recursive enumerable sets semi-decidable (decidable only for elements in the set). * **Real-Life Analogy:** Imagine you're flipping through a library catalog to find books written by a specific author: * If a book exists by that author, you'll eventually find it. * But if no such book exists, you'll keep flipping pages forever without knowing for sure. ## **10. Write a short note on operations on language.** We can perform various operations on languages (set of strings) to create new languages. These operations help in understanding how languages combine and behave. ### **Common Operations on Languages** 1. **Union (L₁ ∪ L₂):** * Combines two languages into one, including all strings from both. * **Example:** * L₁ = {a, b}, L₂ = {b, c} * L₁ ∪ L₂ = {a, b, c} 2. **Concatenation (L₁ ⋅ L₂):** * Forms strings by joining one string from L₁ with one string from L₂. * **Example:** * L₁ = {a, b}, L₂ = {x, y} * L₁ ⋅ L₂ = {ax, ay, bx, by} 3. **Kleene Star (L*):** * Includes all possible combinations of zero or more strings from the language. * **Example:** * L = {a} * L* = {ε, a, aa, aaa, ….} 4. **Intersection (L₁ ∩ L₂):** * Includes only the strings that are common to both languages. * **Example:** * L₁ = {a, b}, L₂ = {b, c} * L₁ ∩ L₂ = {b} 5. **Complement (L̄):** * Includes all strings not in the language (over the same alphabet). * **Example:** * If L = {a, b} and the alphabet = {a, b, c}, then L̄ = {c, aa, ab, ….} ### **Importance:** These operators allow us to combine, modify, and analyze languages, helping in tasks like designing automata, building parsers, and studying computability.