Theory of Computation Lecture 9

Questions and Answers

What is the primary purpose of the finite-state machine described for delaying an input string?

To filter out invalid inputs

To convert binary to decimal

To produce an output that is the input delayed by one unit of time (correct)

To recognize patterns in the input

Which output is produced during the initial transition from the start state of the delay machine?

0 (correct)

The same input bit

Undefined

What happens to the machine when an input of 0 is received in the error detection machine?

It transitions to state s1

It transitions to state s2

It remains in the current state

It transitions to the start state s0 (correct)

In the machine that detects transmission errors with three consecutive 1s, what does state s2 represent?

<p>The previous two inputs were 1s</p> Signup and view all the answers

How does the delay machine handle an input of 1 from state s1?

<p>It transitions to state s2</p> Signup and view all the answers

Study Notes

Finite-State Machines with Output

Finite-state machines (FSMs) are models that represent various technologies in computing, including components found in computers.
They consist of a finite set of states, an initial state, a designated input alphabet, and a transition function that maps state-input pairs to new states.
Common applications include spell checking, grammar checking, text indexing/searching, and speech recognition.

Formal Definition

An FSM with output is defined as M = (S, I, O, f, g, s0), where:
- S: Finite set of states
- I: Finite input alphabet
- O: Finite output alphabet
- f: Transition function defining state transitions
- g: Output function mapping state-input pairs to outputs
- s0: Initial state

Representation of Finite-State Machines

FSMs can be represented using state tables, showing states, inputs, and associated outputs.
State diagrams are graphical representations using circles for states and arrows for transitions, labeled with input-output pairs.

Input and Output Relations

The FSM processes an input string symbol by symbol, transitioning through states based on the transition function.
Each transition generates an output, leading to an output string that corresponds to the input string through a defined output function.

Examples of Finite-State Machines

Unit-Delay Machine:
- Delays the output string by one time unit through three states (s0, s1, s2).
- State s1 remembers if the previous input was 1; s2 remembers if it was 0.
- Outputs 0 for initial transition and reflects the delayed input thereafter.
Error Detection Machine:
- Detects three consecutive 1s to indicate a transmission error, reflecting output accordingly.
- Utilizes three states to track sequences of 1s:
  - State s0: previous input was not 1.
  - State s1: previous input was 1, but not two consecutive 1s.
  - State s2: two previous inputs were 1s.
- An input of 0 resets the machine to state s0.

Limitations of Finite-State Machines

FSMs have limited memory capabilities due to a finite number of states.
They cannot perform tasks requiring unbounded memory, hindering their effectiveness for certain applications.

Description

Explore the fundamentals of finite-state machines with outputs through this quiz. Understand the different types and applications of these models critical to computer science. Enhance your knowledge of machine modeling as discussed in the Theory of Computation class.