CS-3110: Formal Languages and AutomataDeterministic Finite Automata1 / 21

Example 1: DFA interpretion2 / 21

Example 1

Given the DFA M on the next slide, determine:

Then give an English description of the language the automaton accepts.

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Example 1

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Example 1

What does this automaton "do"?
Maybe we can find some sort of interpretation for the states?
How can we get to an accepting state?

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Example 1

We start in state $q_{00}$
We can add consecutive pairs of 0s and 1s to get back there
When we add the first 0, we get to state $q_{10}$
The only way to get back to our accepting state is by adding (at least) one more 0

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Example 1

What are the other options to get to $q_{10}$ ?
We can add a 1, then a 0, and then another 1
We can add a 1, then add three 0s, and then another 1
Suspicion: We always add an odd number of 0s

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Example 1

What are the other options to get to $q_{10}$ ?
We can add a 1, then a 0, and then another 1
We can add a 1, then add three 0s, and then another 1
Suspicion: We always add an odd number of 0s
... and an even number of 1s

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Example 1

Let's extend our suspicions to the other states
Each state is reachable after an even/odd number of 0s/1s has been added
There is one state for each combination
Our accepting state corresponds to words with an even number of 0s and an even number of 1s

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Example 1

This automaton recognizes the language over $\Sigma = \{0,1\}$ consisting of words with an even number of `0`s and an even number of `1`s

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Example 2: DFA11 / 21

Example 2Construct a Deterministic Finite Automaton that accepts words over the alphabet Σ={0,1} that represent binary numbers (without leading zeros) that are divisible by 312 / 21

The Horror

Remember from 2 weeks ago?

$L_4 = \{0\} \cup 1(01^\ast0)^\ast10^\ast (1(01^\ast0)^\ast10^\ast)^\ast$

As promised then, this is much nicer with an automaton!

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Basic Idea

How did we come up with this monstrosity?
We considered how adding digits to a number would change its divisibility
We did this by keeping track of the remainder, and how it changes when we add a digit
We can do this "keeping track" using states in a DFA, and only accept when the remainder (=state) is 0!

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Remainder vs. adding digits

	0	1
Remainder
0	0	1
1	2	0
2	1	2

This is basically our transition function!

We just need to add some extra pieces to handle the number "0"

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Automaton

Our automaton will consist of two parts

One part will handle the number "0": If we read a 0, we go to an accepting state, but if we read anything after that, we go to a "bad" state
The second part will handle numbers starting with 1: For each digit we add, we keep track what the remainder for a division by 3 would be. We only accept if we are in the state representing "remainder 0"
From the start state, if we read a 0 we go to the first part, and if we read a 1 we go to the second part

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Automaton: Recognizing 0

Only accept if you read a single 0. Any 1 or any additional 0 will lead to the state $q_2$ , which is not an accepting state.

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Automaton: Divisibility

	0	1
Remainder
0	0	1
1	2	0
2	1	2

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Full Automaton

Now we need to connect these two automata
Note: Our second automaton would allow leading 0s.
Here's what we do: Use the start state from the first automaton, if we read a 0, we stay there, but if we read a 1, we move to the second automaton (and set our "current remainder" to 1)
We accept in either of the accepting states of the two automata

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DFA

(0 decimal)

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Mojo

noam is the JavaScript library to simulate DFAs
The FSM Simulator served as inspiration
FSM Simulator Source
Behind the scenes, Viz.js does the graph visualization

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CS-3110: Formal Languages and Automata

Deterministic Finite Automata

Example 1: DFA interpretion

Example 1

Example 1

Example 1

Example 1

Example 1

Example 1

Example 1

Example 1

This automaton recognizes the language over Σ={0,1}\Sigma = \{0,1\} consisting of words with an even number of 0s and an even number of 1s

Example 2: DFA

Example 2

Construct a Deterministic Finite Automaton that accepts words over the alphabet Σ={0,1}\Sigma = \{0,1\} that represent binary numbers (without leading zeros) that are divisible by 3

The Horror

Basic Idea

Remainder vs. adding digits

Automaton

Automaton: Recognizing 0

Automaton: Divisibility

Full Automaton

DFA

Mojo

Example 1: DFA interpretion

Help

This automaton recognizes the language over $\Sigma = \{0,1\}$ consisting of words with an even number of `0`s and an even number of `1`s

Construct a Deterministic Finite Automaton that accepts words over the alphabet $\Sigma = \{0,1\}$ that represent binary numbers (without leading zeros) that are divisible by 3