COMP20121 The Implementation and Power of Computer Languages Power - - PowerPoint PPT Presentation

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COMP20121 The Implementation and Power of Computer Languages Power - - PowerPoint PPT Presentation

Mar 27, 2023 137 likes •636 views

COMP20121 The Implementation and Power of Computer Languages Power Part http://www.cs.man.ac.uk/ petera/2121/index.html . Peter Aczel room: CS2.52, tel: 56155 petera@cs.man.ac.uk email: Department of Computer Science, University of

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SLIDE 1

COMP20121 The Implementation and Power of Computer Languages ‘Power’ Part

http://www.cs.man.ac.uk/∼petera/2121/index.html .

Peter Aczel room: CS2.52, tel: 56155 email:

petera@cs.man.ac.uk

Department of Computer Science, University of Manchester

COMP20121 - Section 0 – p.1/307

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SLIDE 2

COMP20121, ‘power’ part: section 2

lecture 5: Pushdown Automata

LECTURE SIX

From Grammars to PDAs

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SLIDE 3

From grammars to PDAs

Given a grammar, create a PDA:

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SLIDE 4

From grammars to PDAs

Given a grammar, create a PDA:

Three states, 0 (start state), 1 and

2 (accepting state), with transitions

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SLIDE 5

From grammars to PDAs

Given a grammar, create a PDA:

Three states, 0 (start state), 1 and

2 (accepting state), with transitions

( , )→push(S)

→ 1,

1

(EOF,EOS)→

→ 2,

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SLIDE 6

From grammars to PDAs

Given a grammar, create a PDA:

Three states, 0 (start state), 1 and

2 (accepting state), with transitions

( , )→push(S)

→ 1,

1

(EOF,EOS)→

→ 2,

1

( ,R)→pop;push(Xn)···push(X1)

→ 1 for every

production R → X1X2 · · · Xn.

COMP20121 - Section 2 – p.124/307

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SLIDE 7

From grammars to PDAs

Given a grammar, create a PDA:

Three states, 0 (start state), 1 and

2 (accepting state), with transitions

( , )→push(S)

→ 1,

1

(EOF,EOS)→

→ 2,

1

( ,R)→pop;push(Xn)···push(X1)

→ 1 for every

production R → X1X2 · · · Xn.

1

(x,x)→pop;advance

→ 1, for each x ∈ Σ.

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SLIDE 8

Example: From grammar to PDA

Consider the grammar S → aSb|ǫ;

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SLIDE 9

Example: From grammar to PDA

Consider the grammar S → aSb|ǫ; Build a PDA: Three states

1 2

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SLIDE 10

Example: From grammar to PDA

Consider the grammar S → aSb|ǫ; Build a PDA: Getting started.

1 2 ( , ) → push(S)

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SLIDE 11

Example: From grammar to PDA

Consider the grammar S → aSb|ǫ; Build a PDA: When finished.

1 (EOF, EOS) → 2 ( , ) → push(S)

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SLIDE 12

Example: From grammar to PDA

Consider the grammar S → aSb|ǫ; Build a PDA: S → aSb.

1 (EOF, EOS) → 2 ( , S) → pop; push(b); push(S); push(a) ( , ) → push(S)

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SLIDE 13

Example: From grammar to PDA

Consider the grammar S → aSb|ǫ; Build a PDA: S → ǫ.

1 (EOF, EOS) → 2 ( , S) → pop; push(b); push(S); push(a) ( , ) → push(S) ( , S) → pop

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SLIDE 14

Example: From grammar to PDA

Consider the grammar S → aSb|ǫ; Build a PDA: Recognize a.

1 (EOF, EOS) → 2 ( , S) → pop; push(b); push(S); push(a) ( , S) → pop (a, a) → pop; advance ( , ) → push(S)

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SLIDE 15

Example: From grammar to PDA

Consider the grammar S → aSb|ǫ; Build a PDA: Recognize b.

1 (EOF, EOS) → 2 ( , S) → pop; push(b); push(S); push(a) (b, b) → pop; advance ( , S) → pop (a, a) → pop; advance ( , ) → push(S)

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SLIDE 16

Example: From grammar to PDA

Consider the grammar S → aSb|ǫ; Finished PDA

1 (EOF, EOS) → 2 ( , S) → pop; push(b); push(S); push(a) (b, b) → pop; advance ( , S) → pop (a, a) → pop; advance ( , ) → push(S)

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SLIDE 17

Example: From grammar to PDA

Consider the grammar S → aSb|ǫ; which (with exception of the empty word) accepts the same language as the PDA

2 3 ( , S) → pop (a, a) → pop; advance ( , S) → ( , ) → push(S) (b, b) → pop; advance ( , S) → pop; push(b); push(S); push(a) ( , a) → ( , b) → (EOF, EOS) → 1

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SLIDE 18

From PDAs to grammars

Given a (non-deterministic) PDA, it is

possible to devise a context-free grammar such that the language generated by the grammar is the language accepted by the automaton.

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SLIDE 19

From PDAs to grammars

Given a (non-deterministic) PDA, it is

possible to devise a context-free grammar such that the language generated by the grammar is the language accepted by the automaton.

We will not prove this statement here.

See any of the recommended books.

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SLIDE 20

Context-free languages and PDAs

The following result connects context-free languages and PDAs. Theorem 2.1 A language is context-free if and only if there is a non-deterministic pushdown automaton that accepts it.

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SLIDE 21

Determinism versus non-determinism

Non-deterministic PDAs are more

powerful than deterministic PDAs.

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SLIDE 22

Determinism versus non-determinism

Non-deterministic PDAs are more

powerful than deterministic PDAs.

There are context-free grammars for

which there is no deterministic pushdown

automaton. See Exercise 19 for an

example.

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SLIDE 23

Determinism versus non-determinism

Non-deterministic PDAs are more

powerful than deterministic PDAs.

There are context-free grammars for

which there is no deterministic pushdown

automaton. See Exercise 19 for an

example. (This example is not examinable.)

COMP20121 - Section 2 – p.128/307

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SLIDE 24

Long words in context-free languages

If α is a ‘long enough’ word then in the parse tree for this word, at least one of the non-terminal symbols, say R, will appear twice.

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SLIDE 25

Long words in context-free languages

If α is a ‘long enough’ word then in the parse tree for this word, at least one of the non-terminal symbols, say R, will appear twice. It is not easy to define what ‘long enough’ means this time.

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SLIDE 26

Long words in context-free languages

Parse tree for α:

R S R

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SLIDE 27

Long words in context-free languages

Breaking up α:

R S R κ λ µ ν ξ

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SLIDE 28

Leaving out parts of α

Given a parse tree for α, with α broken up as before,

R S R κ λ µ ν ξ

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SLIDE 29

Leaving out parts of α

we can replace the big parse tree at R

R S R κ λ µ ν ξ

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SLIDE 30

Leaving out parts of α

we can replace the big parse tree at R

R S R κ λ µ ν ξ

by the small parse tree at R

R S R κ λ µ ν ξ

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SLIDE 31

Leaving out parts of α

we can replace the big parse tree at R

R S R κ λ µ ν ξ

by the small parse tree at R

R S R κ λ µ ν ξ

to get the parse tree for κµξ

S R κ µ ξ

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SLIDE 32

Repeating parts of α

Given a parse tree for α, with α broken up as before,

R S R κ λ µ ν ξ

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SLIDE 33

Repeating parts of α

we can replace the small parse tree at R

R S R κ λ µ ν ξ

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SLIDE 34

Repeating parts of α

we can replace the small parse tree at R

R S R κ λ µ ν ξ

by the big parse tree at R

R S R κ λ µ ν ξ

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SLIDE 35

Repeating parts of α

to get the parse tree for κλλµννξ

S R R R κ λ λ µ ν ν ξ

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SLIDE 36

The Pumping Lemma II

We can formalize this observation: Lemma 2.2 If L is a context-free language then there is a number n such that any word α of L that has length at least n can be divided into five pieces

COMP20121 - Section 2 – p.134/307

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SLIDE 37

The Pumping Lemma II

α = κλµνξ such that

the length of λν is greater than 0;
the length of λµν is at most n;
all words of the form κλkµνkξ, where

k ∈ N, belong to L.

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SLIDE 38

The Pumping Lemma II

α = κλµνξ such that

the length of λν is greater than 0;
the length of λµν is at most n;
all words of the form κλkµνkξ, where

k ∈ N, belong to L.

Note that if L is finite, it may not have any words of length n or longer!

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SLIDE 39

Applying Pumping: An example

The language L = {aibici | i ∈ N} is

not context-free.

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Applying Pumping: An example

The language L = {aibici | i ∈ N} is

not context-free.

To see this, suppose it were!

Choose n by the Pumping Lemma. Let α = anbncn. Write α = κλµνξ as in the lemma.

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SLIDE 41

Applying Pumping: An example

Then one of λ, ν is non-empty. Case 1: Each only contain one kind of character. Then κµξ ∈ L.

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SLIDE 42

Applying Pumping: An example

Then one of λ, ν is non-empty. Case 1: Each only contain one kind of character. Then κµξ ∈ L. Case 2: One of λ, ν contains more than

ne kind of character.

Then κλ2µν2ξ ∈ L.

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SLIDE 43

Section 2 summary

Context-free languages are generated

by context-free grammars.

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SLIDE 44

Section 2 summary

Context-free languages are generated

by context-free grammars.

Context-free languages are more

powerful than regular ones.

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Section 2 summary

Context-free languages are generated

by context-free grammars.

Context-free languages are more

powerful than regular ones.

Context-free languages correspond

precisely to those accepted by non-deterministic pushdown automata.

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Section 2 summary (ctd)

Deterministic PDAs are less powerful

than (non-deterministic) PDAs.

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SLIDE 47

Section 2 summary (ctd)

Deterministic PDAs are less powerful

than (non-deterministic) PDAs.

There is a Pumping Lemma for

context-free languages.

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SLIDE 48

Section 2 summary (ctd)

Deterministic PDAs are less powerful

than (non-deterministic) PDAs.

There is a Pumping Lemma for

context-free languages.

Context-free languages are sufficient

to parse programming languages but

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SLIDE 49

Section 2 summary (ctd)

Deterministic PDAs are less powerful

than (non-deterministic) PDAs.

There is a Pumping Lemma for

context-free languages.

Context-free languages are sufficient

to parse programming languages but

a compiler needs a proper

random-access memory.

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