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comp2022/2922 A2 (90 marks) – CFLs and non-regularity s2 2023
• This assignment is due in Week 8 on Sunday 24 September, 11:59pm.
• Submission is on GradeScope.
• All work must be done individually without consulting anyone else’s solutions in accordance with the University’s “Academic Dishonesty and Plagiarism” policies.
• For clarifications and more details on all aspects of this assignment (e.g., level of justification expected, late penalties, repeated submissions, what to do if you are stuck, etc.) you
are expected to regularly monitor the Ed Forum post “Assignment FAQ”.
Problem 1. (10 marks) Fix Σ = {a, b}. For each of the following grammars,
provide five strings that are in the language and five strings that are not (you
score max(0, 5 − num errors) points per part).
1.
S → aSbb | ϵ
2.
S → aA
A → aA | bB
B → bB | ϵ
Problem 2. (10 marks) Consider the following context-free grammar G over
Σ = {a, b}.
S → AA
A → AAA | bA | Ab | a
1. Show the grammar is ambiguous by providing two leftmost derivations of
the same string.
2. Provide a grammar in Chomsky Normal Form equivalent to G.
Problem 3. (20 marks) Fix Σ = {a, b}. Design context-free grammars for the
following languages:
1. The set of strings of length 6 that have the same number of a and b.
2. The set of strings of the form uvw with u, v, w ∈ {a
n
b
n
|n ∈ Z
+}.
3. The set of palindromes whose length is a multiple of 20. Use at most 100
rules.
4. The set of strings not in the set {a
n
b
2n+3
|n ∈ Z
+}.
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comp2022/2922 A2 (90 marks) – CFLs and non-regularity s2 2023
Problem 4. (20 marks) Fix Σ = {a, b}. A string is extreme if its amount of a is
more than twice its amount of b plus one, or the other way around. For example,
aaaab and babbabbbbba are both extreme, while bbbbbaa and ababab are not.
1. Prove that the set E of extreme strings is not regular (10 marks).
2. Prove that the set P of strings that have an extreme string as a prefix is not
regular. Recall that a string x is a prefix of a string y if y = xw for some
string w. Thus aaaabbbb is in P (since it has aaaab as a prefix), while ababab
is not in P (since no prefix of it is extreme). (10 marks)
Problem 5. (10 marks) Fix Σ = {a, b}. Write a program that takes in a contextfree grammar G in CNF and outputs a context-free grammar for the language of
strings that are in L(G) and whose number of as is a multiple of 5.
For full marks, your solution should be able to take an arbitrary CNF grammar
with 10 variables and output a grammar with at most 100 variables.
Problem 6. (20 marks) You’re honing your skills for the world championship
of popular card game Magic: the Gathering. Your goal is to draw through all
the cards in your deck, and to this end, you’ve included lots of cards that help
you draw more cards. However, if you get unlucky, with some deck orderings
you’ve found that this is impossible. You’re interested in determining which
deck orderings can be won, and which can’t.
At the start of the game, you draw seven cards from your deck into your hand.
The cards in your deck are of the following types:
• Ancestral Recall. When played, you draw three cards.
• Wheel of Fortune. When played, you discard your hand, and then draw seven
cards.
• Cruel Bargain. When played for the first time in a game, you draw four
cards. It cannot be played more than once in a game.
• Other, useless cards.
Aside: Unlike in the actual game, you can play any number of cards from your
hand in any order, without needing to pay mana.
Your goal is to draw all the cards in your deck, at which point you win. Note that
”overdraw” (drawing more cards than exist in your deck) meets this condition,
and so is also a win. We encode a possible ordering of your deck as a string
over the alphabet Σ = {a, w, c, x} where a, w, c correspond to the cards starting
with those letters, and x corresponds to a useless card. We are interested in the
language of strings for which it is possible for you to win.
For example, the following strings are in the language:
aax11
, wx13
, cawx18
, awx5wx13
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comp2022/2922 A2 (90 marks) – CFLs and non-regularity s2 2023
while the following strings are not in the language:
aax12
, wx14
, cawx19
, awx5wx14
Design a context-free grammar for the language described.
• 4 marks will be awarded for cases with strings containing only a, x.
• 4 marks will be awarded for cases with strings containing only a, c, x.
• 4 marks will be awarded for cases with strings containing only w, x.
• 4 marks will be awarded for cases with strings containing only a, w, x.
• The final 4 marks will be awarded for the general case, i.e., for strings over
the alphabet a, w, c, x.
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