Formal Languages, Grammars and Automata Lecture 6 Helle Hvid Hansen - - PowerPoint PPT Presentation

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Formal Languages, Grammars and Automata Lecture 6

Helle Hvid Hansen

helle@cs.ru.nl http://www.cs.ru.nl/~helle/

Foundations Group – Intelligent Systems Section Institute for Computing and Information Sciences Radboud University Nijmegen

13 June 2014

Helle Hvid Hansen 13 June 2014 FLGA 1 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Overview

Last lecture:

• Context-free grammars and context-free languages.
• Regular grammars.
• CYK-algorithm (parsing: w ∈ L?)

Helle Hvid Hansen 13 June 2014 FLGA 2 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Overview

Last lecture:

• Context-free grammars and context-free languages.
• Regular grammars.
• CYK-algorithm (parsing: w ∈ L?)

Today:

• Pushdown automata (NL: stapelautomaten)
• Closure properties of context-free languages
• Beyond context-free languages

Helle Hvid Hansen 13 June 2014 FLGA 2 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Automata for Context-Free Languages

Language class Syntax/Grammar Automata Context-free context-free grammar ? Regular regular expressions, DFA, NFA, NFA-λ regular grammar

Helle Hvid Hansen 13 June 2014 FLGA 3 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Automata for Context-Free Languages

Language class Syntax/Grammar Automata Context-free context-free grammar ? Regular regular expressions, DFA, NFA, NFA-λ regular grammar

• DFA, NFA, NFA-λ: finite memory (states can encode only

finite amount of information), for example (for Σ = {a, b}),

• even or odd number of a’s read: two states even, odd
• the last 2 letters read: four states for aa, ab, ba, bb.

Helle Hvid Hansen 13 June 2014 FLGA 3 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Automata for Context-Free Languages

Language class Syntax/Grammar Automata Context-free context-free grammar ? Regular regular expressions, DFA, NFA, NFA-λ regular grammar

• DFA, NFA, NFA-λ: finite memory (states can encode only

finite amount of information), for example (for Σ = {a, b}),

• even or odd number of a’s read: two states even, odd
• the last 2 letters read: four states for aa, ab, ba, bb.
• For example: {anbn | n ≥ 0} not regular. We would need to

remember how many a’s we have seen. A DFA with k states can only “count to k”.

Helle Hvid Hansen 13 June 2014 FLGA 3 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Automata for Context-Free Languages

Language class Syntax/Grammar Automata Context-free context-free grammar ? Regular regular expressions, DFA, NFA, NFA-λ regular grammar

• DFA, NFA, NFA-λ: finite memory (states can encode only

finite amount of information), for example (for Σ = {a, b}),

• even or odd number of a’s read: two states even, odd
• the last 2 letters read: four states for aa, ab, ba, bb.
• For example: {anbn | n ≥ 0} not regular. We would need to

remember how many a’s we have seen. A DFA with k states can only “count to k”. We need some form of unbounded memory.

Helle Hvid Hansen 13 June 2014 FLGA 3 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Stack Memory

Stack of data elements (“stapel”) Note:

• nly top of stack is visible.

Helle Hvid Hansen 13 June 2014 FLGA 4 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Stack Memory

Stack of data elements (“stapel”) Note:

• nly top of stack is visible.

Stack operations: push a new element

• nto stack

pop the top element from stack Stack is a Last-In-First-Out (LIFO) data structure.

Helle Hvid Hansen 13 June 2014 FLGA 4 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Stack Memory: Formal Notation

• Stack content described as word over a stack alphabet

Γ = {A, B, C, ..., X, Y , ...}. Last-in = Left-most: A B C = ABC

• The empty stack is λ ∈ Γ∗.

Helle Hvid Hansen 13 June 2014 FLGA 5 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Stack Memory: Formal Notation

• Stack content described as word over a stack alphabet

Γ = {A, B, C, ..., X, Y , ...}. Last-in = Left-most: A B C = ABC

• The empty stack is λ ∈ Γ∗.
• Stack operations:

push(X, A B C ) = X A B C pop( A B C ) = B C push(X, ABC) = XABC pop(ABC) = BC

• Note: The empty stack has no top.

Helle Hvid Hansen 13 June 2014 FLGA 5 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Pushdown Automaton

A pushdown automaton is an NFA-λ equipped with a stack.

Helle Hvid Hansen 13 June 2014 FLGA 6 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Pushdown Automaton

A pushdown automaton is an NFA-λ equipped with a stack.

• Def. A pushdown automaton (PDA) M = (Q, Σ, Γ, δ, q0, F)

consists of

• a finite set of (control) states Q
• an input alphabet Σ
• an initial state q0 ∈ Q
• a set of final states F ⊆ Q

Helle Hvid Hansen 13 June 2014 FLGA 6 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Pushdown Automaton

A pushdown automaton is an NFA-λ equipped with a stack.

• Def. A pushdown automaton (PDA) M = (Q, Σ, Γ, δ, q0, F)

consists of

• a finite set of (control) states Q
• an input alphabet Σ
• an initial state q0 ∈ Q
• a set of final states F ⊆ Q
• a stack alphabet Γ
• a transition function δ: Q × Σλ × Γλ → P(Q × Γλ)

where Σλ = Σ ∪ {λ} and Γλ = Γ ∪ {λ}. That is, for p ∈ Q, a ∈ Σλ, A ∈ Γλ, δ(p, a, A) is a set of pairs (q, B) where q ∈ Q, B ∈ Γλ (nondeterministic).

Helle Hvid Hansen 13 June 2014 FLGA 6 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Pushdown Automaton Computation

A computation of a PDA M on input word w:

Helle Hvid Hansen 13 June 2014 FLGA 7 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Pushdown Automaton Computation

A computation of a PDA M on input word w:

• Start configuration (q0, w, λ)

(start in initial state with empty stack)

Helle Hvid Hansen 13 June 2014 FLGA 7 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Pushdown Automaton Computation

A computation of a PDA M on input word w:

• Start configuration (q0, w, λ)

(start in initial state with empty stack)

• Transitions are taken (nondeterministically) depending on

– the next input symbol (as in DFA, NFA) and – the stack top symbol. (Details next slide.) Changes configuration (p, u, α) → (q, v, β) according to transition function (where p, q ∈ Q, ; u, v ∈ Σ∗, α, β ∈ Γ∗).

Helle Hvid Hansen 13 June 2014 FLGA 7 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Pushdown Automaton Computation

A computation of a PDA M on input word w:

• Start configuration (q0, w, λ)

(start in initial state with empty stack)

• Transitions are taken (nondeterministically) depending on

– the next input symbol (as in DFA, NFA) and – the stack top symbol. (Details next slide.) Changes configuration (p, u, α) → (q, v, β) according to transition function (where p, q ∈ Q, ; u, v ∈ Σ∗, α, β ∈ Γ∗).

• Computation is successful if it ends in a configuration

(q, λ, λ) where q ∈ F. (Acceptance by final state and empty stack)

Helle Hvid Hansen 13 June 2014 FLGA 7 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Pushdown Automaton Transitions

(q, B) ∈ δ(p, a, A) i.e. p

a, A/B q :

• can be taken if: next input symbol is a, stack top is A
• actions: read/consume a, pop A from stack, push B.

Helle Hvid Hansen 13 June 2014 FLGA 8 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Pushdown Automaton Transitions

(q, B) ∈ δ(p, a, A) i.e. p

a, A/B q :

• can be taken if: next input symbol is a, stack top is A
• actions: read/consume a, pop A from stack, push B.

(q, B) ∈ δ(p, λ, A) i.e. p

λ, A/B q

(input λ-transition):

• can be taken if: stack top is A (regardless of next input)
• actions: pop A from stack, push B.

Helle Hvid Hansen 13 June 2014 FLGA 8 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Pushdown Automaton Transitions

(q, B) ∈ δ(p, a, A) i.e. p

a, A/B q :

• can be taken if: next input symbol is a, stack top is A
• actions: read/consume a, pop A from stack, push B.

(q, B) ∈ δ(p, λ, A) i.e. p

λ, A/B q

(input λ-transition):

• can be taken if: stack top is A (regardless of next input)
• actions: pop A from stack, push B.

(q, B) ∈ δ(p, a, λ) i.e. p

a, λ/B q :

• can be taken if: next input is a (regardless of stack top)
• actions: read/consume a, push B.

Helle Hvid Hansen 13 June 2014 FLGA 8 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Pushdown Automaton Transitions

(q, B) ∈ δ(p, a, A) i.e. p

a, A/B q :

• can be taken if: next input symbol is a, stack top is A
• actions: read/consume a, pop A from stack, push B.

(q, B) ∈ δ(p, λ, A) i.e. p

λ, A/B q

(input λ-transition):

• can be taken if: stack top is A (regardless of next input)
• actions: pop A from stack, push B.

(q, B) ∈ δ(p, a, λ) i.e. p

a, λ/B q :

• can be taken if: next input is a (regardless of stack top)
• actions: read/consume a, push B.

(q, B) ∈ δ(p, λ, λ) i.e. p

λ, λ/B q :

• can be taken: always (regardless of next input and stack top)
• actions: push B.

Helle Hvid Hansen 13 June 2014 FLGA 8 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Pushdown Automaton Transitions II

Transitions of the form (q, λ) ∈ δ(p, −, −) i.e. p

−, −/λ q :

• can be taken if: as before.
• actions: as before, except nothing is pushed onto stack.

Helle Hvid Hansen 13 June 2014 FLGA 9 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Pushdown Automaton Transitions II

Transitions of the form (q, λ) ∈ δ(p, −, −) i.e. p

−, −/λ q :

• can be taken if: as before.
• actions: as before, except nothing is pushed onto stack.

For example: (q, λ) ∈ δ(p, a, λ) i.e. p

a, λ/λ q

(DFA-transition)

• can be taken if: next input symbol is a
• actions: read/consume a.

(q, λ) ∈ δ(p, λ, λ) i.e. p

λ, λ/λ q :

• Can be taken: always
• Actions: nothing.

Helle Hvid Hansen 13 June 2014 FLGA 9 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Remarks on PDA Transitions

• Note:

p

a, λ/B q

does NOT mean that the stack must be empty to take the transition.

Helle Hvid Hansen 13 June 2014 FLGA 10 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Remarks on PDA Transitions

• Note:

p

a, λ/B q

does NOT mean that the stack must be empty to take the transition.

• Note:

p

a, A/λ q

does NOT mean that the stack is empty after the transition.

Helle Hvid Hansen 13 June 2014 FLGA 10 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Remarks on PDA Transitions

• Note:

p

a, λ/B q

does NOT mean that the stack must be empty to take the transition.

• Note:

p

a, A/λ q

does NOT mean that the stack is empty after the transition.

• Shorthand notation (multiple push):

p

a,A/BC q

= p

a, A/C p′ λ, λ/B q

Helle Hvid Hansen 13 June 2014 FLGA 10 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Language Accepted by a PDA

Let M = (Q, Σ, Γ, δ, q0, F) be a PDA, and let p, q ∈ Q, u, v ∈ Σ∗, α, β ∈ Γ∗. Notation: We write: (p, u, α) ⇒ (q, v, β) if there exists a sequence of transitions that change (p, u, α) into (q, v, β).

Helle Hvid Hansen 13 June 2014 FLGA 11 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Language Accepted by a PDA

Let M = (Q, Σ, Γ, δ, q0, F) be a PDA, and let p, q ∈ Q, u, v ∈ Σ∗, α, β ∈ Γ∗. Notation: We write: (p, u, α) ⇒ (q, v, β) if there exists a sequence of transitions that change (p, u, α) into (q, v, β).

• Def. The language accepted by M is:

L(M) = {w ∈ Σ∗ | (q0, w, λ) ⇒ (q, λ, λ), q ∈ F}.

Helle Hvid Hansen 13 June 2014 FLGA 11 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

PDA Example 1

L = {anbn | n ≥ 0} ∋ λ, ab, aabb, . . .

Helle Hvid Hansen 13 June 2014 FLGA 12 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

PDA Example 1

L = {anbn | n ≥ 0} ∋ λ, ab, aabb, . . . Idea: count the a’s.

Helle Hvid Hansen 13 June 2014 FLGA 12 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

PDA Example 1

L = {anbn | n ≥ 0} ∋ λ, ab, aabb, . . . Idea: count the a’s. Take Σ = {a, b}, Γ = {A} and q0

a, λ/A

• b, A/λ q1

b, A/λ

• Example computations:

Helle Hvid Hansen 13 June 2014 FLGA 12 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

PDA Example 1

L = {anbn | n ≥ 0} ∋ λ, ab, aabb, . . . Idea: count the a’s. Take Σ = {a, b}, Γ = {A} and q0

a, λ/A

• b, A/λ q1

b, A/λ

• Example computations:
• (q0, λ, λ)

Helle Hvid Hansen 13 June 2014 FLGA 12 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

PDA Example 1

L = {anbn | n ≥ 0} ∋ λ, ab, aabb, . . . Idea: count the a’s. Take Σ = {a, b}, Γ = {A} and q0

a, λ/A

• b, A/λ q1

b, A/λ

• Example computations:
• (q0, λ, λ) (SUCCESS)

Helle Hvid Hansen 13 June 2014 FLGA 12 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

PDA Example 1

L = {anbn | n ≥ 0} ∋ λ, ab, aabb, . . . Idea: count the a’s. Take Σ = {a, b}, Γ = {A} and q0

a, λ/A

• b, A/λ q1

b, A/λ

• Example computations:
• (q0, λ, λ) (SUCCESS)
• (q0, aabb, λ) →

Helle Hvid Hansen 13 June 2014 FLGA 12 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

PDA Example 1

L = {anbn | n ≥ 0} ∋ λ, ab, aabb, . . . Idea: count the a’s. Take Σ = {a, b}, Γ = {A} and q0

a, λ/A

• b, A/λ q1

b, A/λ

• Example computations:
• (q0, λ, λ) (SUCCESS)
• (q0, aabb, λ) → (q0, aabb, A) →

Helle Hvid Hansen 13 June 2014 FLGA 12 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

PDA Example 1

L = {anbn | n ≥ 0} ∋ λ, ab, aabb, . . . Idea: count the a’s. Take Σ = {a, b}, Γ = {A} and q0

a, λ/A

• b, A/λ q1

b, A/λ

• Example computations:
• (q0, λ, λ) (SUCCESS)
• (q0, aabb, λ) → (q0, aabb, A) → (q0, aabb, AA) →

Helle Hvid Hansen 13 June 2014 FLGA 12 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

PDA Example 1

L = {anbn | n ≥ 0} ∋ λ, ab, aabb, . . . Idea: count the a’s. Take Σ = {a, b}, Γ = {A} and q0

a, λ/A

• b, A/λ q1

b, A/λ

• Example computations:
• (q0, λ, λ) (SUCCESS)
• (q0, aabb, λ) → (q0, aabb, A) → (q0, aabb, AA) →

(q1, aabb, A) →

Helle Hvid Hansen 13 June 2014 FLGA 12 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

PDA Example 1

L = {anbn | n ≥ 0} ∋ λ, ab, aabb, . . . Idea: count the a’s. Take Σ = {a, b}, Γ = {A} and q0

a, λ/A

• b, A/λ q1

b, A/λ

• Example computations:
• (q0, λ, λ) (SUCCESS)
• (q0, aabb, λ) → (q0, aabb, A) → (q0, aabb, AA) →

(q1, aabb, A) → (q1, aabbλ, λ)

Helle Hvid Hansen 13 June 2014 FLGA 12 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

PDA Example 1

L = {anbn | n ≥ 0} ∋ λ, ab, aabb, . . . Idea: count the a’s. Take Σ = {a, b}, Γ = {A} and q0

a, λ/A

• b, A/λ q1

b, A/λ

• Example computations:
• (q0, λ, λ) (SUCCESS)
• (q0, aabb, λ) → (q0, aabb, A) → (q0, aabb, AA) →

(q1, aabb, A) → (q1, aabbλ, λ) (SUCCESS)

Helle Hvid Hansen 13 June 2014 FLGA 12 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

PDA Example 1

L = {anbn | n ≥ 0} ∋ λ, ab, aabb, . . . Idea: count the a’s. Take Σ = {a, b}, Γ = {A} and q0

a, λ/A

• b, A/λ q1

b, A/λ

• Example computations:
• (q0, λ, λ) (SUCCESS)
• (q0, aabb, λ) → (q0, aabb, A) → (q0, aabb, AA) →

(q1, aabb, A) → (q1, aabbλ, λ) (SUCCESS)

• (q0, aba, λ) →

Helle Hvid Hansen 13 June 2014 FLGA 12 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

PDA Example 1

L = {anbn | n ≥ 0} ∋ λ, ab, aabb, . . . Idea: count the a’s. Take Σ = {a, b}, Γ = {A} and q0

a, λ/A

• b, A/λ q1

b, A/λ

• Example computations:
• (q0, λ, λ) (SUCCESS)
• (q0, aabb, λ) → (q0, aabb, A) → (q0, aabb, AA) →

(q1, aabb, A) → (q1, aabbλ, λ) (SUCCESS)

• (q0, aba, λ) → (q0, aba, A) →

Helle Hvid Hansen 13 June 2014 FLGA 12 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

PDA Example 1

L = {anbn | n ≥ 0} ∋ λ, ab, aabb, . . . Idea: count the a’s. Take Σ = {a, b}, Γ = {A} and q0

a, λ/A

• b, A/λ q1

b, A/λ

• Example computations:
• (q0, λ, λ) (SUCCESS)
• (q0, aabb, λ) → (q0, aabb, A) → (q0, aabb, AA) →

(q1, aabb, A) → (q1, aabbλ, λ) (SUCCESS)

• (q0, aba, λ) → (q0, aba, A) → (q1, aba, λ)

Helle Hvid Hansen 13 June 2014 FLGA 12 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

PDA Example 1

L = {anbn | n ≥ 0} ∋ λ, ab, aabb, . . . Idea: count the a’s. Take Σ = {a, b}, Γ = {A} and q0

a, λ/A

• b, A/λ q1

b, A/λ

• Example computations:
• (q0, λ, λ) (SUCCESS)
• (q0, aabb, λ) → (q0, aabb, A) → (q0, aabb, AA) →

(q1, aabb, A) → (q1, aabbλ, λ) (SUCCESS)

• (q0, aba, λ) → (q0, aba, A) → (q1, aba, λ) (STUCK)

Helle Hvid Hansen 13 June 2014 FLGA 12 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

PDA Example 1

L = {anbn | n ≥ 0} ∋ λ, ab, aabb, . . . Idea: count the a’s. Take Σ = {a, b}, Γ = {A} and q0

a, λ/A

• b, A/λ q1

b, A/λ

• Example computations:
• (q0, λ, λ) (SUCCESS)
• (q0, aabb, λ) → (q0, aabb, A) → (q0, aabb, AA) →

(q1, aabb, A) → (q1, aabbλ, λ) (SUCCESS)

• (q0, aba, λ) → (q0, aba, A) → (q1, aba, λ) (STUCK)
• (q0, abb, λ) →

Helle Hvid Hansen 13 June 2014 FLGA 12 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

PDA Example 1

L = {anbn | n ≥ 0} ∋ λ, ab, aabb, . . . Idea: count the a’s. Take Σ = {a, b}, Γ = {A} and q0

a, λ/A

• b, A/λ q1

b, A/λ

• Example computations:
• (q0, λ, λ) (SUCCESS)
• (q0, aabb, λ) → (q0, aabb, A) → (q0, aabb, AA) →

(q1, aabb, A) → (q1, aabbλ, λ) (SUCCESS)

• (q0, aba, λ) → (q0, aba, A) → (q1, aba, λ) (STUCK)
• (q0, abb, λ) → (q0, abb, A) →

Helle Hvid Hansen 13 June 2014 FLGA 12 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

PDA Example 1

L = {anbn | n ≥ 0} ∋ λ, ab, aabb, . . . Idea: count the a’s. Take Σ = {a, b}, Γ = {A} and q0

a, λ/A

• b, A/λ q1

b, A/λ

• Example computations:
• (q0, λ, λ) (SUCCESS)
• (q0, aabb, λ) → (q0, aabb, A) → (q0, aabb, AA) →

(q1, aabb, A) → (q1, aabbλ, λ) (SUCCESS)

• (q0, aba, λ) → (q0, aba, A) → (q1, aba, λ) (STUCK)
• (q0, abb, λ) → (q0, abb, A) → (q1, abb, λ)

Helle Hvid Hansen 13 June 2014 FLGA 12 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

PDA Example 1

L = {anbn | n ≥ 0} ∋ λ, ab, aabb, . . . Idea: count the a’s. Take Σ = {a, b}, Γ = {A} and q0

a, λ/A

• b, A/λ q1

b, A/λ

• Example computations:
• (q0, λ, λ) (SUCCESS)
• (q0, aabb, λ) → (q0, aabb, A) → (q0, aabb, AA) →

(q1, aabb, A) → (q1, aabbλ, λ) (SUCCESS)

• (q0, aba, λ) → (q0, aba, A) → (q1, aba, λ) (STUCK)
• (q0, abb, λ) → (q0, abb, A) → (q1, abb, λ) (STUCK)

Helle Hvid Hansen 13 June 2014 FLGA 12 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

PDA Example 1

L = {anbn | n ≥ 0} ∋ λ, ab, aabb, . . . Idea: count the a’s. Take Σ = {a, b}, Γ = {A} and q0

a, λ/A

• b, A/λ q1

b, A/λ

• Example computations:
• (q0, λ, λ) (SUCCESS)
• (q0, aabb, λ) → (q0, aabb, A) → (q0, aabb, AA) →

(q1, aabb, A) → (q1, aabbλ, λ) (SUCCESS)

• (q0, aba, λ) → (q0, aba, A) → (q1, aba, λ) (STUCK)
• (q0, abb, λ) → (q0, abb, A) → (q1, abb, λ) (STUCK)

Helle Hvid Hansen 13 June 2014 FLGA 12 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

PDA Example 2

L = {wcwR ∈ {a, b, c}∗ | w ∈ {a, b}∗} ∋ c, abcba, bbcbb. Can we find a PDA that accepts L?

Helle Hvid Hansen 13 June 2014 FLGA 13 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

PDA Example 2

L = {wcwR ∈ {a, b, c}∗ | w ∈ {a, b}∗} ∋ c, abcba, bbcbb. Can we find a PDA that accepts L? Idea: Memorise w-part using the stack, and use it to check for wR.

Helle Hvid Hansen 13 June 2014 FLGA 13 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

PDA Example 2

L = {wcwR ∈ {a, b, c}∗ | w ∈ {a, b}∗} ∋ c, abcba, bbcbb. Can we find a PDA that accepts L? Idea: Memorise w-part using the stack, and use it to check for wR. Take: Σ = {a, b, c} and Γ = {A, B}, and Q, δ, F as follows: q0

b,λ/B

• a,λ/A
• c,λ/λ q1

b,B/λ

• a,A/λ
• Helle Hvid Hansen

13 June 2014 FLGA 13 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

PDA Example 2

L = {wcwR ∈ {a, b, c}∗ | w ∈ {a, b}∗} ∋ c, abcba, bbcbb. Can we find a PDA that accepts L? Idea: Memorise w-part using the stack, and use it to check for wR. Take: Σ = {a, b, c} and Γ = {A, B}, and Q, δ, F as follows: q0

b,λ/B

• a,λ/A
• c,λ/λ q1

b,B/λ

• a,A/λ
• Example computations:
• (q0, abcba, λ) →

Helle Hvid Hansen 13 June 2014 FLGA 13 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

PDA Example 2

L = {wcwR ∈ {a, b, c}∗ | w ∈ {a, b}∗} ∋ c, abcba, bbcbb. Can we find a PDA that accepts L? Idea: Memorise w-part using the stack, and use it to check for wR. Take: Σ = {a, b, c} and Γ = {A, B}, and Q, δ, F as follows: q0

b,λ/B

• a,λ/A
• c,λ/λ q1

b,B/λ

• a,A/λ
• Example computations:
• (q0, abcba, λ) → (q0, abcba, A) →

Helle Hvid Hansen 13 June 2014 FLGA 13 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

PDA Example 2

L = {wcwR ∈ {a, b, c}∗ | w ∈ {a, b}∗} ∋ c, abcba, bbcbb. Can we find a PDA that accepts L? Idea: Memorise w-part using the stack, and use it to check for wR. Take: Σ = {a, b, c} and Γ = {A, B}, and Q, δ, F as follows: q0

b,λ/B

• a,λ/A
• c,λ/λ q1

b,B/λ

• a,A/λ
• Example computations:
• (q0, abcba, λ) → (q0, abcba, A) → (q0, abcba, BA) →

Helle Hvid Hansen 13 June 2014 FLGA 13 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

PDA Example 2

L = {wcwR ∈ {a, b, c}∗ | w ∈ {a, b}∗} ∋ c, abcba, bbcbb. Can we find a PDA that accepts L? Idea: Memorise w-part using the stack, and use it to check for wR. Take: Σ = {a, b, c} and Γ = {A, B}, and Q, δ, F as follows: q0

b,λ/B

• a,λ/A
• c,λ/λ q1

b,B/λ

• a,A/λ
• Example computations:
• (q0, abcba, λ) → (q0, abcba, A) → (q0, abcba, BA) →

(q1, abcba, BA) →

Helle Hvid Hansen 13 June 2014 FLGA 13 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

PDA Example 2

L = {wcwR ∈ {a, b, c}∗ | w ∈ {a, b}∗} ∋ c, abcba, bbcbb. Can we find a PDA that accepts L? Idea: Memorise w-part using the stack, and use it to check for wR. Take: Σ = {a, b, c} and Γ = {A, B}, and Q, δ, F as follows: q0

b,λ/B

• a,λ/A
• c,λ/λ q1

b,B/λ

• a,A/λ
• Example computations:
• (q0, abcba, λ) → (q0, abcba, A) → (q0, abcba, BA) →

(q1, abcba, BA) → (q1, abcba, A) →

Helle Hvid Hansen 13 June 2014 FLGA 13 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

PDA Example 2

L = {wcwR ∈ {a, b, c}∗ | w ∈ {a, b}∗} ∋ c, abcba, bbcbb. Can we find a PDA that accepts L? Idea: Memorise w-part using the stack, and use it to check for wR. Take: Σ = {a, b, c} and Γ = {A, B}, and Q, δ, F as follows: q0

b,λ/B

• a,λ/A
• c,λ/λ q1

b,B/λ

• a,A/λ
• Example computations:
• (q0, abcba, λ) → (q0, abcba, A) → (q0, abcba, BA) →

(q1, abcba, BA) → (q1, abcba, A) → (q1, abcbaλ, λ) (SUCCESS)

Helle Hvid Hansen 13 June 2014 FLGA 13 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

PDA Example 2

L = {wcwR ∈ {a, b, c}∗ | w ∈ {a, b}∗} ∋ c, abcba, bbcbb. Can we find a PDA that accepts L? Idea: Memorise w-part using the stack, and use it to check for wR. Take: Σ = {a, b, c} and Γ = {A, B}, and Q, δ, F as follows: q0

b,λ/B

• a,λ/A
• c,λ/λ q1

b,B/λ

• a,A/λ
• Example computations:
• (q0, abcba, λ) → (q0, abcba, A) → (q0, abcba, BA) →

(q1, abcba, BA) → (q1, abcba, A) → (q1, abcbaλ, λ) (SUCCESS)

• (q0, acb, λ) →

Helle Hvid Hansen 13 June 2014 FLGA 13 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

PDA Example 2

L = {wcwR ∈ {a, b, c}∗ | w ∈ {a, b}∗} ∋ c, abcba, bbcbb. Can we find a PDA that accepts L? Idea: Memorise w-part using the stack, and use it to check for wR. Take: Σ = {a, b, c} and Γ = {A, B}, and Q, δ, F as follows: q0

b,λ/B

• a,λ/A
• c,λ/λ q1

b,B/λ

• a,A/λ
• Example computations:
• (q0, abcba, λ) → (q0, abcba, A) → (q0, abcba, BA) →

(q1, abcba, BA) → (q1, abcba, A) → (q1, abcbaλ, λ) (SUCCESS)

• (q0, acb, λ) → (q0, acb, A) →

Helle Hvid Hansen 13 June 2014 FLGA 13 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

PDA Example 2

L = {wcwR ∈ {a, b, c}∗ | w ∈ {a, b}∗} ∋ c, abcba, bbcbb. Can we find a PDA that accepts L? Idea: Memorise w-part using the stack, and use it to check for wR. Take: Σ = {a, b, c} and Γ = {A, B}, and Q, δ, F as follows: q0

b,λ/B

• a,λ/A
• c,λ/λ q1

b,B/λ

• a,A/λ
• Example computations:
• (q0, abcba, λ) → (q0, abcba, A) → (q0, abcba, BA) →

(q1, abcba, BA) → (q1, abcba, A) → (q1, abcbaλ, λ) (SUCCESS)

• (q0, acb, λ) → (q0, acb, A) → (q1, acb, A)

(STUCK)

Helle Hvid Hansen 13 June 2014 FLGA 13 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Context-Free Languages and PDAs

Helle Hvid Hansen 13 June 2014 FLGA 14 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Context-Free Languages and PDAs

Last lecture: From regular grammar G to NFA-λ MG. Now: From context-free grammar G to PDA MG. Requires G to be in Greibach normal form.

Helle Hvid Hansen 13 June 2014 FLGA 14 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Context-Free Languages and PDAs

Last lecture: From regular grammar G to NFA-λ MG. Now: From context-free grammar G to PDA MG. Requires G to be in Greibach normal form.

• Def. A CFG G is in Greibach normal form if all its production

rules are of one of the following forms: A → aA1 · · · An A → a S → λ In leftmost derivation, the prefix of terminals increases in each step. where a ∈ Σ, Ai ∈ V , Ai = S, i = 1, . . . , n.

Helle Hvid Hansen 13 June 2014 FLGA 14 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Context-Free Languages and PDAs

Last lecture: From regular grammar G to NFA-λ MG. Now: From context-free grammar G to PDA MG. Requires G to be in Greibach normal form.

• Def. A CFG G is in Greibach normal form if all its production

rules are of one of the following forms: A → aA1 · · · An A → a S → λ In leftmost derivation, the prefix of terminals increases in each step. where a ∈ Σ, Ai ∈ V , Ai = S, i = 1, . . . , n. Lemma: Every CFG G can be transformed into an equivalent CFG in Greibach normal form (see e.g. Sudkamp 4.7-4.8).

Helle Hvid Hansen 13 June 2014 FLGA 14 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

From Greibach CFG to two-state PDA

Let G = (V , Σ, S, P) be a Greibach CFG. Define M = (Q, Σ, Γ, δ, q0, F0) where Q = {q0, q1}, Γ = V − {S}, F = {q1} and transitions in δ are defined from P by:

Helle Hvid Hansen 13 June 2014 FLGA 15 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

From Greibach CFG to two-state PDA

Let G = (V , Σ, S, P) be a Greibach CFG. Define M = (Q, Σ, Γ, δ, q0, F0) where Q = {q0, q1}, Γ = V − {S}, F = {q1} and transitions in δ are defined from P by:

• S → aw ∈ P

⇒ q0

a, λ/w

q1

• S → λ ∈ P

⇒ q0

λ, λ/λ

q1

• A → aw ∈ P

⇒ q1

a, A/w

q1

where a ∈ Σ, A ∈ Γ, w ∈ Γ∗. Example: on blackboard.

Helle Hvid Hansen 13 June 2014 FLGA 15 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Context-Free Languages and Pushdown Automata

Theorem: L is context-free ⇐ ⇒ L is accepted by some PDA.

• Proof. (⇒): Use previous construction.

(⇐): Ingenious, but more involved, see Sudkamp 7.3.

Helle Hvid Hansen 13 June 2014 FLGA 16 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Context-Free Languages and Pushdown Automata

Theorem: L is context-free ⇐ ⇒ L is accepted by some PDA.

• Proof. (⇒): Use previous construction.

(⇐): Ingenious, but more involved, see Sudkamp 7.3. Language class Syntax/Grammar Automata Context-free context-free grammar PDA Regular regular expressions, DFA, NFA, NFA-λ regular grammar

Helle Hvid Hansen 13 June 2014 FLGA 16 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Variations on PDAs

• Acceptance criteria:
• Acceptance by final state and empty stack (our def.)
• Acceptance by final state (only)
• Acceptance by empty stack (only)

All are equivalent (accept same class of languages).

Helle Hvid Hansen 13 June 2014 FLGA 17 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Variations on PDAs

• Acceptance criteria:
• Acceptance by final state and empty stack (our def.)
• Acceptance by final state (only)
• Acceptance by empty stack (only)

All are equivalent (accept same class of languages).

• A PDA is deterministic if for any combination of state, input

symbol and stack top, there is at most one transition possible.

Helle Hvid Hansen 13 June 2014 FLGA 17 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Variations on PDAs

• Acceptance criteria:
• Acceptance by final state and empty stack (our def.)
• Acceptance by final state (only)
• Acceptance by empty stack (only)

All are equivalent (accept same class of languages).

• A PDA is deterministic if for any combination of state, input

symbol and stack top, there is at most one transition possible. q0

a, λ/A

• b, A/λ q1

b, A/λ

• q0

b, λ/B

• a, λ/A
• λ, λ/λ q1

b, B/λ

• a, A/λ
• deterministic

nondeterministic (guess middle of string)

Helle Hvid Hansen 13 June 2014 FLGA 17 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Deterministic Context-Free Languages

• Def. A language is called deterministic context-free if there exists

a deterministic PDA M with L = L(M). Examples of deterministic CFLs:

• {anbn | n ≥ 0}
• {wcwR | w ∈ {a, b}∗}

Helle Hvid Hansen 13 June 2014 FLGA 18 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Deterministic Context-Free Languages

• Def. A language is called deterministic context-free if there exists

a deterministic PDA M with L = L(M). Examples of deterministic CFLs:

• {anbn | n ≥ 0}
• {wcwR | w ∈ {a, b}∗}

Examples of CFLS that are not deterministic:

• {wwR | w ∈ {a, b}∗}
• {w ∈ {a, b}∗ | w = wr, |w|} (palindromes)

Helle Hvid Hansen 13 June 2014 FLGA 18 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Deterministic Context-Free Languages

• Def. A language is called deterministic context-free if there exists

a deterministic PDA M with L = L(M). Examples of deterministic CFLs:

• {anbn | n ≥ 0}
• {wcwR | w ∈ {a, b}∗}

Examples of CFLS that are not deterministic:

• {wwR | w ∈ {a, b}∗}
• {w ∈ {a, b}∗ | w = wr, |w|} (palindromes)

Note: L is deterministic CFL ⇒ L is unambiguous, but there are unambiguous CFLs that are not deterministic (e.g., palindromes).

Helle Hvid Hansen 13 June 2014 FLGA 18 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Beyond Context-Free Languages

Examples of languages that are not context-free:

• {anbncn | n ≥ 0}
• {anbmanbm | n, m ≥ 0}
• {w ∈ {a, b}∗ | |w| is prime number}

(Prove using pumping lemma for CFLs, Sudkamp 7.4)

Helle Hvid Hansen 13 June 2014 FLGA 19 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Closure Properties of Context-Free Languages

Helle Hvid Hansen 13 June 2014 FLGA 20 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Closure Properties of Context-Free Languages

If L1 and L2 are context-free languages, then so are:

• L1 ∪ L2 (union),
• L1L2 (concatenation),
• L∗

1 (star),

• LR

1 (reversal)

Helle Hvid Hansen 13 June 2014 FLGA 20 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Closure Properties of Context-Free Languages

If L1 and L2 are context-free languages, then so are:

• L1 ∪ L2 (union),
• L1L2 (concatenation),
• L∗

1 (star),

• LR

1 (reversal)

but, in general, NOT

• L1 (complement) or
• L1 ∩ L2 (intersection).

Helle Hvid Hansen 13 June 2014 FLGA 20 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Closure Properties of Context-Free Languages

If L1 and L2 are context-free languages, then so are:

• L1 ∪ L2 (union),
• L1L2 (concatenation),
• L∗

1 (star),

• LR

1 (reversal)

but, in general, NOT

• L1 (complement) or
• L1 ∩ L2 (intersection).

Let S1 and S2 be start symbols in grammars G1 and G2. Make new grammar with start symbol S:

Helle Hvid Hansen 13 June 2014 FLGA 20 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Closure Properties of Context-Free Languages

If L1 and L2 are context-free languages, then so are:

• L1 ∪ L2 (union),
• L1L2 (concatenation),
• L∗

1 (star),

• LR

1 (reversal)

but, in general, NOT

• L1 (complement) or
• L1 ∩ L2 (intersection).

Let S1 and S2 be start symbols in grammars G1 and G2. Make new grammar with start symbol S: union: S → S1 | S2,

Helle Hvid Hansen 13 June 2014 FLGA 20 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Closure Properties of Context-Free Languages

If L1 and L2 are context-free languages, then so are:

• L1 ∪ L2 (union),
• L1L2 (concatenation),
• L∗

1 (star),

• LR

1 (reversal)

but, in general, NOT

• L1 (complement) or
• L1 ∩ L2 (intersection).

Let S1 and S2 be start symbols in grammars G1 and G2. Make new grammar with start symbol S: union: S → S1 | S2, concatenation: S → S1S2

Helle Hvid Hansen 13 June 2014 FLGA 20 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Closure Properties of Context-Free Languages

If L1 and L2 are context-free languages, then so are:

• L1 ∪ L2 (union),
• L1L2 (concatenation),
• L∗

1 (star),

• LR

1 (reversal)

but, in general, NOT

• L1 (complement) or
• L1 ∩ L2 (intersection).

Let S1 and S2 be start symbols in grammars G1 and G2. Make new grammar with start symbol S: union: S → S1 | S2, concatenation: S → S1S2 star: S → S1S | λ,

Helle Hvid Hansen 13 June 2014 FLGA 20 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Closure Properties of Context-Free Languages

If L1 and L2 are context-free languages, then so are:

• L1 ∪ L2 (union),
• L1L2 (concatenation),
• L∗

1 (star),

• LR

1 (reversal)

but, in general, NOT

• L1 (complement) or
• L1 ∩ L2 (intersection).

Let S1 and S2 be start symbols in grammars G1 and G2. Make new grammar with start symbol S: union: S → S1 | S2, concatenation: S → S1S2 star: S → S1S | λ, reversal: (exercise)

Helle Hvid Hansen 13 June 2014 FLGA 20 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Closure Properties of Context-Free Languages

If L1 and L2 are context-free languages, then so are:

• L1 ∪ L2 (union),
• L1L2 (concatenation),
• L∗

1 (star),

• LR

1 (reversal)

but, in general, NOT

• L1 (complement) or
• L1 ∩ L2 (intersection).

Let S1 and S2 be start symbols in grammars G1 and G2. Make new grammar with start symbol S: union: S → S1 | S2, concatenation: S → S1S2 star: S → S1S | λ, reversal: (exercise)

• Determ. CFLs are closed under complement, but NOT under union.

Helle Hvid Hansen 13 June 2014 FLGA 20 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Questions about Context-Free Languages

Decidable (general algorithm exists)

• Given any CFG G and w ∈ Σ∗, is w ∈ L(G)?

(CYK-algorithm or build PDA).

• Given PDA M, is L(M) = ∅?
• Given PDA M, is L(M) finite?
• Given any CFG G, is L(G) regular?

Undecidable (no general algorithm exists)

• Given any CFG G, is L(G) = Σ∗?
• Given any CFGs G1 and G2, is L(G1) = L(G2)?
• Given any CFG G, is L(G) deterministic?
• Given any CFG G, is it ambiguous?

Helle Hvid Hansen 13 June 2014 FLGA 21 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Chomsky Hierarchy

Languages Grammars Automata Recursively phrase-structure, Turing machines enumerable unrestricted Context-sensitive context-sensitive Linear bounded (natural lang.) grammar Turing machine Context-free context-free (nondet.) PDA (program.lang.) grammar Regular regular grammar DFA, NFA, NFA-λ

Helle Hvid Hansen 13 June 2014 FLGA 22 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Chomsky Hierarchy

Languages Grammars Automata Recursively phrase-structure, Turing machines enumerable unrestricted Context-sensitive context-sensitive Linear bounded (natural lang.) grammar Turing machine Context-free context-free (nondet.) PDA (program.lang.) grammar Regular regular grammar DFA, NFA, NFA-λ – Noam Chomsky, born 1928, American linguist.

Helle Hvid Hansen 13 June 2014 FLGA 22 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Chomsky Hierarchy

Languages Grammars Automata Recursively phrase-structure, Turing machines enumerable unrestricted Context-sensitive context-sensitive Linear bounded (natural lang.) grammar Turing machine Context-free context-free (nondet.) PDA (program.lang.) grammar Regular regular grammar DFA, NFA, NFA-λ – Noam Chomsky, born 1928, American linguist. – Sheila Adele Greibach, born 1939, American computer scientist.

Helle Hvid Hansen 13 June 2014 FLGA 22 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Chomsky Hierarchy

Languages Grammars Automata Recursively phrase-structure, Turing machines enumerable unrestricted Context-sensitive context-sensitive Linear bounded (natural lang.) grammar Turing machine Context-free context-free (nondet.) PDA (program.lang.) grammar Regular regular grammar DFA, NFA, NFA-λ – Noam Chomsky, born 1928, American linguist. – Sheila Adele Greibach, born 1939, American computer scientist. Learning more: “Berekenbaarheid” (NWI-IBC003) (Informatica BSc vak, 3ec)

Helle Hvid Hansen 13 June 2014 FLGA 22 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Summary

Formal Languages Grammars (generators) Automata (acceptors)

Helle Hvid Hansen 13 June 2014 FLGA 23 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Summary

Formal Languages Grammars (generators) Automata (acceptors)

• Context-free grammars generate context-free languages.
• All regular languages are context-free, but not vice versa.
• Context-free languages are accepted by PDAs.
• PDAs cannot be determinised (no subset construction)

Helle Hvid Hansen 13 June 2014 FLGA 23 / 24

Introduction Pushdown Automata CFLs and PDAs Closure Properties and Concluding Remarks

Radboud University Nijmegen

Next (last) lecture

• Wrapping up the course.
• Solutions to the midterm test (online later today).
• Examp preparation.
• Question hour.

Remember to register for the exam in Osiris!

Helle Hvid Hansen 13 June 2014 FLGA 24 / 24