Lecture Notes 13: More CFGs and Intro to Push-Down Automata

Outline

This class we'll discuss:

• More on CFGs
• Intro to Push-Down Automata

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Recap: CFGs

All REs can be created with CFGs

Approach 1: Recursive creation of CFG from definition of REs

For any RE \(R_1\) We can produce a CFG \(G_1\) by applying the following steps:

• Any base cases can be writen as generation of
  \(S \rightarrow_g \emptyset \vert \epsilon \vert a\) for all \(a \in \Sigma \quad from \ R_1's \ \Sigma\)
• Defining the more complex rules requires combining two rules. The recursive expression based on these two base rules can start with:
  \(S \rightarrow_g A \) (some expression)
  \(S \rightarrow_g B \) (another expression)
  and combined to generate more complex expressions:
  • Alternation:
    \(S \rightarrow_g S_1 \vert S_2 \) (equivalent to A + B for some A, B)
  
  
  • Concatenation:
    \(S \rightarrow_g S_1 S_2 \) (equivalent to A B for some A, B)
  
  
  • Kleene:
    \(S \rightarrow_g S_1 S \vert \epsilon \) (equivalent to \( A^* \) for some A)

Approach 2: RL's are a special case of CFLs

How do we prove there are languages that are NOT (beyond) CFLs?

How did we do this back when we did it for RLs?

• the middle part is not too big
• v and y (the repeating parts) are not both simultaneously empty
• repeating v and or y we will keep us in the language

Note that RLs are a special case of Context-Free-Languages (without the \(uv^i\)) part.



So, if we have a pumping lemma for CFGs, is there a "Machine" equivalent to the Finite Automatons as well?



We'll see those next class.

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Proving a language is NOT context-free

What does your untuition say? Is it a CFL?









Remember:

• Given a structure of \(w = uvxyz\), and \( \mid vy \mid \geq 1\)
• We want to find an \(i\) for which a word \(uv^ixy^iz\)
  does not have a prime length ( \( \mid uv^ixy^iz \mid \) is not prime ) after being "pumped" some number of times.
• Here, we can start with a word \(w\) with length \(p\geq N \) (\(N\) provided by the pumping Lemma)
• Now, the trick is to pump the pattern some number of times so that we can prove that the final length is NOT prime!
• Ideas?

(Wait; then Click)

Steps:

1. The length of a word \(\mid uv^ixy^iz \mid \)
   is the length of \( \mid w \mid \) plus any added repetitions of \(v\) and \(y\)
2. So, \( \mid uv^ixy^iz \mid \) is \( \mid w\mid + (i-1)\mid vy \mid \)
3. Since we said \(w\) is in PRIMEAL, then \( \mid w\mid \) is some prime number \(p\geq N \).
4. Then, \( \mid uv^ixy^iz \mid = \mid w\mid + (i-1)\mid vy \mid = p + (i-1)\mid vy \mid\)
5. Now, What possible choice of \(i\) could we choose to cause the overall length to be provably NOT prime?

(Wait; then Click)

If we choose \(i\) so that the \(i-1\) is equal to \(p\) in the following expression:
\[ \mid uv^ixy^iz \mid = \mid w\mid + (i-1)\mid vy \mid = p + (i-1)\mid vy \mid \]
Then substituting \(i-1\) for \(p\) ( by making \(i = p-1\)), we would get: \[ \mid uv^ixy^iz \mid = \mid w \mid + (i-1)\mid vy \mid = p + p\mid vy \mid \\ = p (1+\mid vy \mid) \] which means that, after pumping, the word is divisible by \(p\)! and therefore, not of prime length.

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Why are CFGs important?

check the article out: https://www.nature.com/articles/nature04675

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Push-Down Automata

Activity 1 [2 minutes]:

Notes

• we need a notation for READ, POP and PUSH... we'll use: \( READ, POP \rightarrow PUSH\)
• we need a symbol for "The stack is currently empty!"... we'll use \(z_0\)
• we need to be able to read a symbol without pushing ... we'll use \(\epsilon\) in place of PUSH
• we need to be able to read a symbol without popping ... we'll use \(\epsilon\) in place of POP
• we need to be able to push without reading a symbol ... we'll use \(\epsilon\) in place of READ
• we need to be able to pop without reading a symbol ... we'll use \(\epsilon\) in place of READ

So now, let's draw the PDA for HALF

An example for HALF:

And in a "compressed notation":







Note that we defined an Augmented NON-Deterministic Automaton















Tips:

• Think of phases:
  1. memorizing
  2. middle (even or odd)
  3. replicating in reverse
• write the PDA... and remember it is NON-Deterministic

Activity 2 [2 minutes]:

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Next class: Practice with CFGs and PDAs

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Homework

[Due for everyone]

problem Set 4 due October 13th 11:59 PM

[Optional]

TODO