Finite State Machine

Description:

• Mathematical model of computation
• Can be in exactly one of a finite number of states at any given time.
• A finite-state automata $M=(S,I,f,s_0,F)$:
  • consists of
    • $S$: a finite set of states
    • $I$: a finite input alphabet
    • $f$: a transition function
      • assigns a next state to every pair of state and input
      • such that $f:S\times I\to S$
    • an initial or final state, $s_0$
    • a set of possible accepting state, $F$
      • also a subset of $S$
  • example question and my attempt solution

Types of FA

• FA without output:
  • Deterministic Finite-state Automata
  • Nondeterministic Finite-state Automata (NFA)
• FA with output:
  • Moore Machine
  • Mealy Machine

Transition:

• To change FA from one state to another in response to some inputs

Accept/reject string:

• A string is said to be recognized or accepted by the machine, $M=(S,I,f,s_0,F)$, if it takes the initial state $s_0$ to a final state, that is $f(s_0,x)$, is a state in $F$.
• example:
  • sample state machine
  • (question) 𝑥=01011, what is $f(s_0,x)$? Is it accepted?
    • (answer) Yes
    • $f(s_0,x)$ = S0 S1 S1 S0 S1, take from the initial to final state
  • (question) 𝑦=00110, what is $f(s_0,y)$? Is it accepted?
    • (answer) No
    • $f(s_0,x)$ = S0 S0 S1 S0 S0, doesnt take from init to final state

Language recognition by FSM

• The language recognized or accepted by the machine $M$, denoted by $L(M)$, is the set of all strings that are accepted by $M$
• Determine languages recognized by a machine M:
  • define final state(s) of M
  • per each final state, determine the string that take the initial to its final state
• example

Equivalent finite-state automata:

• Definition: Two finite-state automata are called equivalent if they recognize the same language
  • Meaning accept the same set of inputs
• example question & solution

State table:

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