Theory of Computation: Automata Theory

A finite state acceptor is a finite state machine that has no outputs and only cares about the final state after processing inputs.

The start state is represented by an arrow directed into the corresponding node, indicating that it is the starting point.

A finite automaton is a 5-tuple M = (Q, Σ, δ, q, F), consisting of a finite set of states Q, a finite alphabet Σ, a transition function δ, a start state q, and a set of accept states F.

<p>Nodes in finite-state machines correspond to different states of the machine and represent the current configuration.</p> Signup and view all the answers

<p>The transition function δ: Q × Σ → Q defines how the automaton moves between states based on the current state and input symbol.</p> Signup and view all the answers

<p>Double-lined nodes indicate accepting states, while single-lined nodes represent rejecting states in an acceptor.</p> Signup and view all the answers

<p>The length of a string is denoted by |S|, signifying the number of symbols present in the string.</p> Signup and view all the answers

<p>A language is characterized as a subset of Σ* for some alphabet Σ, which can be either finite or infinite.</p> Signup and view all the answers

<p>A finite state acceptor accepts a string if, after processing it, the machine ends in an accepting state.</p> Signup and view all the answers

<p>The regular expression is <code>ab(a + b)*</code>.</p> Signup and view all the answers

<p>An example of an alphabet is Σ = {a, b, c, d}, which includes the symbols 'a', 'b', 'c', and 'd'.</p> Signup and view all the answers

<p>A finite state acceptor does not produce any output; it only determines whether the input is accepted or rejected.</p> Signup and view all the answers

<p>An acceptor categorizes strings with exactly one edge by designating states that correspond to this condition as accepting states.</p> Signup and view all the answers

<p>A binary string is a sequence of symbols consisting only of '0's and '1's; the example language A specifically comprises binary strings that contain an odd number of '1's.</p> Signup and view all the answers

<p>The minimized DFA requires 4 states.</p> Signup and view all the answers

<p>An empty string indicates a string with |S| = 0 and is represented by λ or ε.</p> Signup and view all the answers

<p>The accepting states are <code>d</code> and <code>e</code>.</p> Signup and view all the answers

<p>The arrow in a state's transition indicates the state transition based on input, signifying a move from one state to another.</p> Signup and view all the answers

<p>An example of a language that can be accepted by a finite automaton is A = {w: w is a binary string containing an odd number of 1s}.</p> Signup and view all the answers

<p>The regular expression is <code>a(a + b)*</code>.</p> Signup and view all the answers

<p>The minimized DFA requires 3 states.</p> Signup and view all the answers

<p>An example string is <code>abab</code>.</p> Signup and view all the answers

<p>The initial substring defines the minimum number of states required in the DFA.</p> Signup and view all the answers

<p>The transition graph illustrates how the DFA moves between states upon reading inputs starting with 'a'.</p> Signup and view all the answers

<p>The primary purpose is to determine whether a binary number is divisible by a specific integer, in this case, 3.</p> Signup and view all the answers

<p>Three states are needed, representing the possible remainders: 0, 1, and 2.</p> Signup and view all the answers

<p>State q0 represents the situation where the current binary number is divisible by 3.</p> Signup and view all the answers

<p>If the DFA is in state q2, it means that the binary number leaves a remainder of 2 when divided by 3.</p> Signup and view all the answers

<p>When the input is '1' in state q1, the DFA transitions to state q0.</p> Signup and view all the answers

<p>Mealy and Moore machines are used to model finite state machines that perform transitions based on input symbols.</p> Signup and view all the answers

<p>Transition states in a DFA indicate how the machine processes input and determines the next state based on current input.</p> Signup and view all the answers

<p>The transition table for a ternary pattern with a divisor of 4 expands to include states q0, q1, q2, and q3 for inputs 0, 1, and 2.</p> Signup and view all the answers

<p>In a Mealy Machine, the output depends on both the present state and the current input, while in a Moore Machine, the output depends only on the present state.</p> Signup and view all the answers

<p>The components are Q (set of states), ∑ (input alphabet), O (output alphabet), δ (input transition function), X (output transition function), and q0 (initial state).</p> Signup and view all the answers

<p>In a Mealy Machine, X is defined as X: Q × ∑ → O, while in a Moore Machine, it is defined as X: Q → O, meaning the output in Moore depends only on the state, not the input.</p> Signup and view all the answers

<p>The initial state (q0) is the starting point from where any input is processed in both types of machines.</p> Signup and view all the answers

<p>The input transition function δ in a Mealy Machine is represented as δ: Q × ∑ → Q, defining how the machine transitions between states based on the current state and input symbol.</p> Signup and view all the answers

<p>The state table is organized by listing the present state, the next state based on input 0 and 1, and the corresponding outputs for each transition.</p> Signup and view all the answers

<p>A finite state machine (FSM) is a computational model with a limited number of states that can transition between them based on input, producing outputs accordingly.</p> Signup and view all the answers

<p>The dependence on both state and input means that a Mealy Machine can respond more dynamically to changing inputs, potentially leading to faster output changes compared to a Moore Machine.</p> Signup and view all the answers

<p>If all outputs of Qi are the same, then you copy state Qi. Otherwise, if there are n distinct outputs, you break Qi into n states.</p> Signup and view all the answers

<p>If the output of the initial state is 1, a new initial state should be added that produces an output of 0.</p> Signup and view all the answers

<p>States 'b' and 'c' deliver different outputs, with 'b' producing both 0 and 1 while 'c' produces both 0 and 1 as well but differs in its naming convention.</p> Signup and view all the answers

<p>When state 'b' receives input 'a=0', it transitions to state 'a' while providing an output of 1.</p> Signup and view all the answers

<p>For input 'a=0', state 'd' outputs 0 and transitions to state 'b0', whereas for 'a=1', it outputs 1 and transitions to state 'a'.</p> Signup and view all the answers

<p>Additional states like 'b0', 'b1', 'c0', and 'c1' are necessary to represent the distinct outputs from state 'b' and 'c' accurately.</p> Signup and view all the answers

<p>The key difference lies in how outputs are generated; Mealy machines produce outputs based on transitions, whereas Moore machines produce outputs based on states.</p> Signup and view all the answers

<p>State 'a' outputs 1 and state 'd' outputs 0, indicating a clear classification between their outputs.</p> Signup and view all the answers

A finite sequence of symbols taken from an alphabet.

A set of all possible strings that can be created using an alphabet. It can be finite or infinite.

A finite set of symbols used to create strings.

A finite set of states that represent different stages in a machine.

A function that determines the next state based on the current state and the input symbol.

A specific state where the machine starts its processing.

A subset of states where the machine accepts the input string as valid.

A mathematical model used to represent a system with a finite number of states and transitions.

A finite sequence of symbols taken from an alphabet. For example, 'hello' is a string over the alphabet {h, e, l, o}.

A set of all possible strings that can be created using an alphabet. It can be finite or infinite.

A finite set of symbols used to create strings. For example, the alphabet {a, b, c} can be used to create strings like 'abc', 'aba', and 'ccc'.

A finite set of states that represent different stages in a machine. The behavior of a machine depends on the current state.

A function that determines the next state based on the current state and the input symbol. It's like a rulebook for transitions.

A specific state where the machine starts its processing. The initial condition of the machine.

A subset of states where the machine accepts the input string as valid. These states indicate a successful operation.

A mathematical model used to represent a system with a finite number of states and transitions.

A sequence of symbols taken from an alphabet. It can be of any length, including zero (the empty string).

A set of all possible strings that can be created using a specific alphabet. It can be finite or infinite.

A finite set of symbols used to create strings. It can be any combination of characters, letters, or numbers.

Represents different stages in a state machine. The machine can be in only one state at a time.

A function that tells you how to move from one state to another based on the current state and the input symbol.

A specific state where the DFA starts processing the input string.

A set of states where the DFA accepts the input string as valid. It indicates the machine has successfully recognized a valid input.

A mathematical model that represents a system with a finite number of states and transitions. Used to recognize patterns in strings.

A type of finite automata where the output is determined only by the current state, not the input.

A type of finite automata where output is determined by both the current state and the input.

A specific state in a finite automata where the machine starts its processing.

A set of states in a finite automata where the machine accepts the input string as valid.

A function that maps a state and an input symbol to the next state. It dictates how the finite automata transitions between states.

A finite automata that has a minimum number of states while accepting the same language. It is the most efficient representation of the automaton.

In a Mealy machine, this function takes the current state and the input symbol, and returns the next state.

In a Mealy machine, this function takes the current state and the input symbol, and returns the output.

This is the first state of a Mealy machine, where processing begins.

The state table represents all possible combinations of current state, input, next state, and output in a Mealy machine.

The state diagram visually represents the transitions between states in a Mealy machine, with arrows showing potential moves based on input.

A set of symbols that the Mealy machine can receive as input.