CSE 311 Lecture 23: Finite State Machines

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Topics

Finite state machines (FSMs): Definition and examples.
Finite state machines with output: Definition and examples.

Finite state machines (FSMs)

Definition and examples.

Finite state machines by example

An FSM recognizes (or accepts) a set of strings.: (1) Given a string $s$, begin at the start state, denoted by an incoming arrow with no source.; (2) If $s = aw$, take the edge labeled $a$ to get to the next state.; (3) Otherwise ($s$ is empty), stop.; (4) Let $s$ be $w$ and repeat (2)-(4) until the machine stops.; The FSM accepts $s$ iff it stops in a final state, denoted by a double circle.

Which strings are recognized by the example FSM?: $\varepsilon$ no; $0$ yes; $0110$ yes; $1011$ no; All binary strings that end in $0$.

Defining an FSM

Finite state machine (FSM) or deterministic finite automaton (DFA): A finite state machine (FSM) $M = (S, \Sigma_{\textrm{in}}, f, s_0, F)$ consists of; a finite set of states $S$, a finite input alphabet $\Sigma_{\textrm{in}}$,; a transition function $f$ that maps each state in $S$ and input in $\Sigma_{\textrm{in}}$ to a state in $S$,; a start state $s_0\in S$, and a set of final states $F\subseteq S$.

state	0	1
$s_0$	$s_0$	$s_1$
$s_1$	$s_0$	$s_2$
$s_2$	$s_0$	$s_3$
$s_3$	$s_3$	$s_3$

What language does this FSM accept?: The set of all binary strings that contain $111$ or end in $0$.

Applications of FSMs

Implementation of string matching.: For example, in grep.
Algorithms for communication and cache-coherence protocols.: Each agent runs its own FSM.
Specifications for controllers.: For example, device drivers.
Specifications for reactive systems.: Components are communicating FSMs.
Verification.: Is an unsafe state reachable?

Example: an FSM over binary strings

What language does this FSM recognize?: The set of all binary strings where the numbers of 1’s and 0’s are congruent mod 2. That is, both numbers are even or both are odd.

Example: FSMs over {0, 1, 2}

$M_1$: strings with an even number of 2’s
$M_2$: strings where the sum of digits mod 3 is 0

Example: a combined FSM over {0, 1, 2}

$M_1$ and $M_2$: strings with an even number of 2’s where digit sum mod 3 is 0

Example: another combined FSM over {0, 1, 2}

$M_1$ or $M_2$: strings with an even number of 2’s or digit sum mod 3 is 0

Finite state machines with output

Definition and examples.

Adding output to FSMs

FSMs can do more than just accept or reject strings.: So DFAs aren’t the only kind of FSM!
Another useful kind of FSM can also return output.: Whenever you enter specific states, the FSM produces an output.; These FSMs (Moore machines) are used as controllers.

Finite state machine (FSM) with output: A finite state machine (FSM) $M = (S, \Sigma_{\textrm{in}}, $ $\Sigma_{\textrm{out}}$$, f, $ $g$$, s_0)$ consists of; a finite set of states $S$, a finite input alphabet $\Sigma_{\textrm{in}}$,; a finite output alphabet $\Sigma_{\textrm{out}}$,; a transition function $f$ that maps each state in $S$ and input in $\Sigma_{\textrm{in}}$ to a state in $S$,; an output function $g$ that maps each state in $S$ to an output symbol in $\Sigma_{\textrm{out}}$,; and start state $s_0\in S$.

Example: vending machine

Consider a simple vending machine.: Enter 15 cents in dimes or nickels.; Press S (Snickers) or B (Butterfinger) for a candy bar.

Basic transitions on N (nickel), D (dime), B (Butterfinger), and S (Snickers).

Example: vending machine with output

Adding output to states: N (nickel), B (Butterfinger), and S (Snickers).

Example: complete vending machine with output

Add transitions to cover all symbols for each state.

Summary

A finite state machine (FSM) consists of states and transitions.: States include the start state and final states.; Transitions map states and input symbols to states.; Also known as deterministic finite automata (DFAs).
An FSM recognizes a set of strings (language).: These are all strings that reach a final state from the start.
FSMs have many applications.: Regular expression matching, cache coherence protocols.; Verification (e.g., of OS kernels and devices).; Controllers (FSMs with output).