Modeling Computation

1
Modeling Computation

• Rosen 5th ed., ch. 11
• Ref Wikipedia

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

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 machineg S? O
4
Simple Example (p.753)

5
Generating Output

• Input string x x1x2xk
• s1 f(s0, x1), s2 f(s1, x2), , sk f(sk-1,
  xk)

• These state transition produces output y
  y1y2yk
• y1 g(s0, x1), y2 g(s1, x2), , yk g(sk-1,
  xk)

6
Another Example

start
Input 101011 Output 001000
7
Ex vending machines

• description
• State table

8
Vending Machine state diagram
9
Ex Unit Delay Machine

• Input x1x2xk Output 0x1x2xk-1

state f f g g
state 0 1 0 1
s0 s2 s1 0 0
s1 s2 s1 1 1
s2 s2 s1 0 0
Example how the machine is designed (p.755)
10
Ex FSM for Addition

• Design
• s0 to remember that the previous carry is 0
• s1 to remember that the previous carry is 1
• example

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

12
Ex automaton to recognize nice
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)

14
Example construct the state diagram

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.

16
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

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.

18
Example

• L(M1) 1n n 0,1,2,

19
Example (cont)

• L(M2) 1, 01

20
Example (cont)

• L(M3) 0n,0n10x n 0,1,2,, x is any string