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Thank you.

Help with context free grammar excercise

So, I have an exercise in which I have to write a context free grammar for this language:

$ $ L = \{x \in L(a^∗b^∗c^∗) : |x|_a > |x|_c; |x|_b > 0; |x|_c ≥ 0\}$ $

meaning every string with any number of $ a$ ‘s, $ b$ ‘s and $ c$ ‘s in that order, with the amount of $ a$ ‘s greater than the amount of $ c$ ‘s and the amount of $ b$ ‘s greater than zero.

I am having trouble figuring out the rule that makes sure there are more $ a$ s than $ c$ s.

I have: $ $ \begin{align}S&\to aABC | ab\ A&\to aA | a\ B&\to bB | b\ C&\to cC | c\ \end{align}$ $ I know this is wrong because I should be adding an $ a$ every time I add a $ c$ , but I don’t know how to write that.

How to interpret this context free language?

$ S -> aAA$

$ A -> aS | bS | a$

Trivial thing:

starting and ending with a

Atleast 3 a’s are definitely present

(These are very layman observations…but seriously I am unable to figure out what exactly this language is all about. …) My attempt:

What I am able to generate

1)aaa

2)aAA

a bS A

a ba AA A

a ba ***

This suggests after b there should be atleast 1 a

*(Coz those *** have all

  1. a’s

or

  1. if bS used then again a ‘ba**’

Or

  1. if aS used length also increases by atleast 3.)*

I know this is not a good analysis..and so I am not expecting any answer but would definitely expect some comments about some intuition or idea..plz any help would be much regarded..

(Although answers are always welcome 😉 )

Are there context free grammars for all restricted Dyck paths?

A Dyck path is a finite list of $ 1$ ‘s and $ -1$ ‘s whose partial sums are nonnegative and whose total sum is $ 0$ . For example, [1, 1, -1, -1] is a Dyck path. Rather than use $ 1$ and $ -1$ , it is common to use "U" and "D" for $ 1$ and $ -1$ , respectively, so we might write UUDD instead. (These might be more familiar as Dyck words.)

It is well-known that Dyck paths have a standard "grammar." The "Dyck grammar" for a path $ P$ is

$ $ P = \epsilon \quad | \quad U P D P.$ $

This grammar is very useful because it lets us quickly compute the generating function which enumerates the number of Dyck paths of given lengths.

I am interested not in all Dyck paths, but restricted sets of Dyck paths. For example, consider the Dyck paths which avoid "peaks" (the bigram UD) at positive even heights; UUUDDD is such a path, while UUDD is not. If we could devise an unambiguous grammar for such paths, then we could write down an equation for the generating function which counts them. (There is such a grammar. It requires two production rules – one for the paths which avoid peaks at even heights, and one for paths which avoid peaks at odd heights.)

This idea leads to a natural question:

Given a (possibly infinite) set of positive integers $ A$ , does there exist a finite, unambiguous context-free grammar which generates precisely the Dyck paths whose peaks avoid $ A$ ?

The answer (I think) is "yes" when $ A$ is finite or an arithmetic progression. I suspect that the answer is "no" in every other case, but I have no idea how to begin proving this.

For example, here is a special case that I cannot decide:

Is the set of Dyck paths which avoid peaks at positive squares a context-free language?

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Thank You.

Can I use the haste action to break free of vines or similar things?

Haste states:

Choose a willing creature that you can see within range. Until the spell ends, […] it gains an additional action on each of its turns. That action can be used only to take the Attack (one weapon attack only), Dash, Disengage, Hide, or Use an Object action.

This obviously does not allow to use any other actions that you can usually use, like casting a spell, as the more specific rule beats the general rule of what actions can be used. But what if some other specific effect grants additional option to use an action for? For example say an Assassin Vine has entangled me, that grants me the additional action option of breaking free from the vines:

A creature restrained by the plants can use its action to make a DC 13 Strength (Athletics) check, freeing itself on a successful check.

Can I use my haste action to try to break free?

As far as I can tell, I have two specific rules contradicting each other. If there is no official ruling on this, does anyone know what be ruled in AL?

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