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Questions and Answers
A delay machine can be constructed that has ______ possible inputs, namely, 0 and 1.
A delay machine can be constructed that has ______ possible inputs, namely, 0 and 1.
two
The machine must have a start state ______.
The machine must have a start state ______.
s0
The state ______ remembers that the previous input was a 1.
The state ______ remembers that the previous input was a 1.
s1
An input of 1 takes s0 to s1, because now a 1, and not ______ consecutive 1s, has been read.
An input of 1 takes s0 to s1, because now a 1, and not ______ consecutive 1s, has been read.
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An input of 0 takes every state to ______, because this breaks up any string of consecutive 1s.
An input of 0 takes every state to ______, because this breaks up any string of consecutive 1s.
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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.
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