Interesting languages

“Hardest” languages to test parsers. Most examples taken from Expressive power of LL(k) Boolean grammars ↗

TODO:

• add examples from The computational power of Parsing Expression Grammars ↗

LR(0)

$$\{a^n b^n | n \geq 1\}$$

on the other hand $a^*$ is regular, but not LR(0)

Linear context-free language

“Balanced brackets”:

$$\{a^n b^n | n \geq 0\}$$

And variations of the above:

$$\{a^n b^n c^l | n, l \geq 0\}$$ $$\{a^m b^m a^n b^n | m, n \geq 0\}$$

Classical context sensitive (LinConjLL)

Classical context sensitive example:

$$\{a^nb^nc^n | n \geq 0\}$$

S → A&C
A → aA | D
D → bDc | ε
C → aCc | B
B → bB | ε

Context-sensitive, Indexed

$$\{a^nb^nc^n \}$$ $$\{a^nb^{n^2}c^n 0\}$$ $$\{a^{2^n} | n \geq 1\}$$

S → ACaB
Ca → aaC
CB → DB|E
aD → Da
AD → AC
aE → Ea
AE → ε

“copy langauge”:

$$\{ ww | w \in \{a, b\}^* \}$$ $$\{ www | w \in \{a, b\}^* \}$$

Context-sensitive, not indexed

$$\{ a^{n!} \}$$

From: R.H. Gilman, A shrinking lemma for indexed languages. Theoret. Comput. Sci. 163 (1996) 277–281

$$\{ (ab^n)^n \}$$

PEG

Same example used by Ford ↗ to show that PEG can handle context sensitive grammar.

A ← aAb/ε
B ← bBc/ε
S ← &(A!b)a∗ B!.

Actually it can’t handle exactly this language, but it still can handle other context sensitive language. See https://github.com/SRI-CSL/PVSPackrat/issues/3 ↗

$$\{a^{2^n} | n \geq 1\}$$

Other context sensitive

$$\{a^{n} | n \text{ is prime}\}$$

TAG (Tree Adjoining Grammar)

$$\{ a^n b^m c^n d^m | n, m \geq 0 \}$$

MG (Minimalist Grammar)

$$\{ a^n b^n c^n e^n f^n | n \geq 0 \}$$

CFLL, but not in LinCFLL

$$\{a^n b^n cs | n \geq 0, s \in \{a, b\}\}$$

S → X &¬T
T → X &¬Aca&¬Acb
A → aAb | ε
X → aX | bX | cX | ε.

Strongly non-left-recursive Boolean P-complete language

$$a^{n−1−j_n} b a^{n−2−j_{n−1}} b \ldots a^{2−j_3} b a^{1−j_2} b$$

S → E&¬AbS&¬CS
A → aA | ε
C → aCAb | b
E → aE | bE | ε.

Strongly non-left-recursive conjunctive language

$$\{wcw | w \in \{a, b\}^*\}$$

S → C &D
C → XCX | c
X → a | b
D → aA&aD | bB&bD | cE
A → XAX | cEa
B → XBX | cEb
E → aE | bE | ε.

Jeż language

$$\{a^{4n} | n \geq 0\}$$

A1 → A1 A3 & A2 A2 | a
A2 → A1 A1 & A2 A6 | aa
A3 → A1 A2 & A6 A6 | aaa
A6 → A1 A2 & A3 A3.

LL(1) linear Boolean language

$$\{a^m b^n | 0 \leq m \leq n\}$$

S → aS&¬A | bB
A → aAb | ε
B → bB | ε

Boolean

$$\{ a^n b^{2^n} | n \geq 1\}$$

Context sensitive with backreferences

(a*)b\1b\1 ↗

$$a^nba^nba^n$$