1. Rosen 5th ed., ch. 11 Ref: Wikipedia
Modeling Computation
Rosen 5th ed., ch. 11
Ref: Wikipedia
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2. Finite State Machines (FSM)
Model machines (or components in a computer) using a particular structure
Classification
FSM with no output: determine whether the input is accepted (recognized) or not
FSM with output: generate output from the given input
Representations (p.752)
State diagram
State transition table
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3. FSM with Output Definition: M = (S, I, O, f, g, s0)
S: finite set of states
I, O: input/output alphabets
f: transition function, f: S  I  S
g: output function
s0: initial state
Mealy machine: g: SI O
Moore machine:g: S O
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4. Simple Example (p.753)
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5. Generating Output Input string: x = x1x2…xk
s1 = f(s0, x1), s2 = f(s1, x2), …, sk = f(sk-1, xk)
These state transition produces output y = y1y2…yk
y1 = g(s0, x1), y2 = g(s1, x2), …, yk = g(sk-1, xk)
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6. Another Example
start
Input:
Output:
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7. Ex: vending machines
[description]
State table
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8. Vending Machine: state diagram
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9. Ex: Unit Delay Machine Input: x1x2…xk; Output: 0x1x2…xk-1 f g 1 s0 s2
state f g 1 s0 s2 s1
Example & how the machine is designed (p.755)
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10. Ex: FSM for Addition Design: [example]
s0: to remember that the previous carry is 0
s1: to remember that the previous carry is 1
[example]
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11. FSM with No Output
Language recognition; design and implementation of compilers
A string is recognized IFF it takes the starting state to one of the final states
Def: Finite-state Automata: do not produce output, but have a set of final states
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12. Ex: automaton to recognize “nice”
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13. Finite-State Automata
M = (S, I, f, s0, F)
S: finite set of states
I: input alphabets
f: transition function, f: S  I  S
s0: initial state
F: final states, subset of S (indicated by double circles)
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14. Example: construct the state diagram
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15. Recognizing Language
A string x is said to be recognized by the machine M = (S, I, f, s0, F) if f(s0, x) is a state in F
The language recognized (accepted) by the machine M, denoted by L(M), is the set of all strings accepted by M.
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16. Deterministic finite automaton
Example
Deterministic finite automaton
The following example is of a DFA M, with a binary alphabet, which determines if the input contains an even number of 0s
Start
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17. Transformations from/to State Diagram
It is possible to draw a state diagram from the table.
Draw the circles to represent the states given.
For each of the states, scan across the corresponding row and draw an arrow to the destination state(s). There can be multiple arrows for an input character if the automaton is an NFA.
Designate a state as the start state. The start state is given in the formal definition of the automaton.
Designate one or more states as accept state. This is also given in the formal definition.
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18. Example
L(M1) = {1n | n = 0,1,2,…}
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19. Example (cont)
L(M2) = {1, 01}
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20. Example (cont) L(M3) = {0n,0n10x | n = 0,1,2,…, x is any string}
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