CSE 311 Lecture 24: FSM Minimization and NFAs

Emina Torlak and Sami Davies

Topics

FSM with output example
Review the vending machine example
FSM minimization
Algorithm and examples.
Nondeterministic finite automata (NFAs)
Definition and examples.

FSM with output example

Review the vending machine 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: complete vending machine with output

Add transitions to cover all symbols for each state.

FSM minimization

Algorithm and examples.

How would you compare two FSMs?

Suppose we are given two FSMs over the same input alphabet.
How can we tell if they are equivalent, i.e., accept the same language?

This is exactly the question answered by grinch! And many important practical applications beyond autograding, e.g., efficient implementation of grep :)

FSM minimization

Unique minimal FSM
Each finite state machine $M$ has a unique minimal equivalent machine (up to a renaming of states) $M_\min$ that accepts the same language as $M$.
The above definition means that …
No FSM that is equivalent to $M$ has fewer states than $M_\min$.
If an FSM is equivalent to $M$ and has the same number of states as $M_\min$, then it is equal to $M_\min$ up to a renaming of states.
So to check if two FSMs are equivalent:
(1) Minimize them, and
(2) Compare the minimal FSMs for equality (modulo renaming of states).

FSM minimization algorithm

  1. Given an FSM, partition its states into groups according to their outputs (for FSMs with output) or whether they are final states or not (for DFAs).
  2. Repeat the following until fixed point (no change happens):
    • If there is a symbol $a$ in a source group $G$ such that some states in $G$ disagree on which target group $a$ leads to, partition $G$ into smaller groups based on the target group for $a$.
  3. Convert groups to states and collapse edges with corresponding labels.

Example: minimizing a DFA

  1. Partition the states into groups according to whether they are final states or not.
  2. Every symbol causes the DFA to go from one group to the other so no partitioning needed.
  3. Convert groups to states and collapse edges with corresponding labels.

Example: minimizing an FSM with output

  1. Partition the states into groups according to their outputs.
  2. Repeat until fixed point:
    If there is a symbol $a$ in a source group $G$ such that some states in $G$ disagree on which target group $a$ leads to, partition $G$ into smaller groups based on the target group for $a$.
  3. Convert groups to states and collapse edges with corresponding labels.

Nondeterministic finite automata (NFAs)

Definition and examples.

Recall the definition of a DFA

Deterministic finite automaton (DFA)
A deterministic finite automaton (DFA) $M = (S, \Sigma_{\textrm{in}}, f, s_0, F)$ 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 $F\subseteq S$.
The label of a path in a DFA
The label of a path in a DFA is the concatenation of the labels of its edges in order.
Lemma: strings recognized by a DFA
A string $x$ is in the language recognized by a DFA if and only if $x$ labels a path from the start state to some final state.
What language does this DFA accept?
The set of all binary strings that contain $111$.

Defining an NFA

Nondeterministic finite automaton (NFA)
A nondeterministic finite automaton (NFA) $M = (S, \Sigma_{\textrm{in}}$$\cup$ {$\varepsilon$}$, f, s_0, F)$ consists of
a finite set of states $S$, a finite input alphabet $\Sigma_{\textrm{in}}$,
a transition function $f$ that maps each state in $S$ and input in $\Sigma_{\textrm{in}}$$\cup$ {$\varepsilon$} to a set of states in $S$,
a start state $s_0\in S$, and a set of final states $F\subseteq S$.
The label of a path in an NFA
The label of a path in an NFA is the concatenation of the labels of its edges in order.
Definition: strings recognized by an NFA
A string $x$ is in the language recognized by an NFA if and only if $x$ labels a path from the start state to some final state.
What language does this NFA accept?
The set of all binary strings that contain $111$.

Example NFAs

What language does this NFA accept?
$10(10)^* \cup 111(0\cup 1)^* $
What language does this NFA accept?
Strings over {0, 1, 2} with an even number of 2’s or with digits summing to 0 mod 3.

NFAs make it easy to union languages

A DFA and NFA for the same language.
Strings over {0, 1, 2} with an even number of 2’s or with digits summing to 0 mod 3.

NFAs can be much smaller than DFAs

What language does this NFA accept?
Binary strings with a 1 in the 3rd position from the end.

The smallest DFA that accepts the same language.

Three ways to understand NFAs

Outside observer
Is there a path labeled by $x$ from the start state to some final state?
Perfect guesser (oracle)
Given an input $x$, the NFA guesses the right edge to take (if one exists) whenever there is a choice to be made.
Parallel exploration
The NFA runs all possible computations on $x$ in parallel.

Summary

Every FSM has a unique minimal equivalent FSM (modulo state names).
We can compute the minimal FSM using the algorithm from this lecture.
We can use this to check if two FSMs are equivalent!
An NFA recognizes a set of strings (language).
An NFA differs from a DFA in that the transition function maps each state and input symbol to a set of states.
Determining if an NFA accepts string boils down to checking if there is a path from the start state to some final state, where the path consists of the edges labeled by the string’s characters.