Theory of Computer Science C5. Context-free Languages: Normal Form and PDA Gabriele R¨oger

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Theory of Computer Science

C5. Context-free Languages: Normal Form and PDA

Gabriele R¨oger

University of Basel

April 1, 2019

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Overview

Automata &

Formal Languages

Languages

& Grammars

Regular Languages

Context-free Languages

ε-rules

Chomsky Normal Form

PDAs

Pumping Lemma Closure Properties

Decidability Context-sensitive &

Type-0 Languages

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Context-free Grammars andε-Rules Chomsky Normal Form Push-Down Automata Summary

Context-free Grammars and ε-Rules

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Repetition: Context-free Grammars

Definition (Context-free Grammar)

Acontext-free grammar is a 4-tuplehΣ,V,P,Siwith

1 Σ finite alphabet of terminal symbols,

2 V finite set of variables (with V ∩Σ =∅),

3 P ⊆(V ×(V ∪Σ)⁺)∪ {hS, εi}finite set of rules,

4 IfS →ε∈P, then all other rules inV ×((V \ {S})∪Σ)⁺.

5 S ∈V start variable.

andS never occurs on a right-hand side.

With regular grammars, this restriction could be lifted. How about context-free grammars?

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Repetition: Context-free Grammars

RuleX →εis only allowed if X =S andS never occurs on a right-hand side.

With regular grammars, this restriction could be lifted. How about context-free grammars?

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Repetition: Context-free Grammars

RuleX →εis only allowed if X =S andS never occurs on a right-hand side.

With regular grammars, this restriction could be lifted.

How about context-free grammars?

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Overview

Automata &

Formal Languages

Languages

& Grammars

Regular Languages

ε-rules

Chomsky Normal Form

PDAs

Type-0 Languages

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ε-Rules

Theorem

For every grammar G with rules P⊆V ×(V ∪Σ)^∗ there is a context-free grammar G⁰ with L(G) =L(G⁰).

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ε-Rules

Theorem

Proof.

LetG =hΣ,V,P,Sibe a grammar withP ⊆V ×(V ∪Σ)^∗. LetVε={A∈V |A⇒^∗ ε}. We can find this set Vε by first collecting all variablesAwith rule A→ε∈P and then successively adding additional variablesB if there is a rule B→A1A2. . .Ak ∈P and the variablesAi are already in the set

for all 1≤i ≤k. . . .

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ε-Rules

Theorem

Proof (continued).

LetP⁰ be the rule set that is constructed from P by adding rules that obviate the need for A→εrules:

for every existing ruleB →w with B ∈V,w ∈(V ∪Σ)⁺, let Iε be the set of positions where w contains a variable A∈V_ε. For every non-empty setI⁰ ⊆I_ε, add a new rule B →w⁰, wherew⁰ is constructed fromw by removing the variables at all positions in I⁰.

removing all rules of the form A→ε(after the previous step).

. . .

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ε-Rules

Theorem

Proof (continued).

ThenL(G)\ {ε}=L(hΣ,V,P⁰,Si) andP⁰ contains no rule A→ε. If the start variableS of G is not in Vε, we are done.

Otherwise, letS⁰ be a new variable and construct P⁰⁰ fromP⁰ by

1 replacing all occurrences ofS on the right-hand side of rules with S⁰,

2 adding the rule S⁰ →w for every rule S →w, and

3 adding the rule S →ε.

ThenL(G) =L(hΣ,V ∪ {S⁰},P⁰⁰,Si).

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ε-Rules

Theorem

Proof (continued).

ThenL(G)\ {ε}=L(hΣ,V,P⁰,Si) andP⁰ contains no rule A→ε. If the start variableS of G is not in Vε, we are done.

Otherwise, letS⁰ be a new variable and construct P⁰⁰ fromP⁰ by

1 replacing all occurrences ofS on the right-hand side of rules with S⁰,

2 adding the rule S⁰ →w for every rule S →w, and

3 adding the rule S →ε.

ThenL(G) =L(hΣ,V ∪ {S⁰},P⁰⁰,Si).

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Questions

Questions?

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Chomsky Normal Form

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Overview

Automata &

Formal Languages

Languages

& Grammars

Regular Languages

ε-rules

Chomsky Normal Form

PDAs

Type-0 Languages

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Chomsky Normal Form: Motivation

As in logical formulas (and other kinds of structured objects), normal formsfor grammars are useful:

they show which aspects are critical for defining grammars and which ones are just syntactic sugar

they allow proofs and algorithms to be restricted

to a limited set of grammars (inputs): those in normal form Hence we now consider anormal form for context-free grammars.

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Chomsky Normal Form: Definition

Definition (Chomsky Normal Form)

A context-free grammarG is inChomsky normal form (CNF) if all rules have one of the following three forms:

A→BC with variables A,B,C, or

A→awith variableA, terminal symbol a, or S →εwith start variable S.

German: Chomsky-Normalform

in short: rule setP ⊆(V ×(VV ∪Σ))∪ {hS, εi}

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Chomsky Normal Form: Theorem

Theorem

For every context-free grammar G there is a context-free grammar G⁰ in Chomsky normal form with L(G) =L(G⁰).

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Chomsky Normal Form: Theorem

Theorem

Proof.

The following algorithm converts the rule set ofG into CNF:

Step 1: Eliminate rules of the form A→B with variablesA,B. If there are sets of variables{B₁, . . . ,B_k} with rules

B₁ →B₂,B₂ →B₃, . . . ,Bk−1 →B_k,B_k →B₁, then replace these variables by a new variableB.

Define a strict total order<on the variables such thatA→B∈P implies thatA<B. Iterate from the largest to the smallest variableAand eliminate all rules of the form A→B while adding rulesA→w for every ruleB →w withw ∈(V ∪Σ)⁺. . . .

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Chomsky Normal Form: Theorem

Theorem

Proof (continued).

Step 2: Eliminate rules with terminal symbols on the right-hand side that do not have the formA→a.

For every terminal symbola∈Σ add a new variable Aa

and the ruleA_a →a.

Replace all terminal symbols in all rules that do not have

the formA→awith the corresponding newly added variables. . . .

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Chomsky Normal Form: Theorem

Theorem

Proof (continued).

Step 3: Eliminate rules of the form A→B1B2. . .Bk with k >2 For every rule of the formA→B₁B₂. . .B_k with k >2, add new variablesC2, . . . ,Ck−1 and replace the rule with

A→B₁C₂ C₂→B₂C₃

...

C_k−1→B_k−1B_k

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Chomsky Normal Form: Length of Derivations

Observation

LetG be a grammar in Chomsky normal form,

and letw ∈ L(G) be a non-empty word generated byG.

Then all derivations ofw have exactly 2|w| −1 derivation steps.

Proof.

Exercises

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Push-Down Automata

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Overview

Automata &

Formal Languages

Languages

& Grammars

Regular Languages

ε-rules

Chomsky Normal Form

PDAs

Type-0 Languages

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Limitations of Finite Automata

q0 0 q1 q2

0,1

0

Language Lis regular.

⇐⇒ There is a finite automaton that acceptsL.

What information can a finite automaton “store”

about the already read part of the word?

Infinite memory would be required for

L={x₁x₂. . .x_nx_n. . .x₂x₁|n>0,x_i ∈ {a,b}}.

therefore: extension of the automata model with memory

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Limitations of Finite Automata

q0 0 q1 q2

0,1

0

L={x₁x₂. . .x_nx_n. . .x₂x₁|n>0,x_i ∈ {a,b}}.

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Limitations of Finite Automata

q0 0 q1 q2

0,1

0

L={x₁x₂. . .x_nx_n. . .x₂x₁|n>0,x_i ∈ {a,b}}.

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Stack

Astack is a data structure following thelast-in-first-out (LIFO) principle supporting the following operations:

push: puts an object on top of the stack

pop: removes the object at the top of the stack

peek: returns the top object without removing it

Pop Push

German: Keller, Stapel

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Push-down Automata: Visually

Input tape

I n p u t

Read head

Push-down automaton

Stack access

Stack

German: Kellerautomat, Eingabeband, Lesekopf, Kellerzugriff

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Push-down Automata: Definition

Definition (Push-down Automaton)

Apush-down automaton (PDA) is a 6-tupleM =hQ,Σ,Γ, δ,q₀,#i with

Q finite set of states Σ the input alphabet Γ the stack alphabet

δ :Q×(Σ∪ {ε})×Γ→ P_f(Q×Γ^∗) the transition function (where P_f is the set of all finitesubsets)

q0∈Q the start state

#∈Γ the bottommost stack symbol

German: Kellerautomat, Eingabealphabet, Kelleralphabet, German: Uberf¨¨ uhrungsfunktion

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Push-down Automata: Transition Function

LetM =hQ,Σ,Γ, δ,q0,#i be a push-down automaton.

What is the Intuitive Meaning of the Transition Functionδ?

hq⁰,B₁. . .B_ki ∈δ(q,a,A): If M is in stateq, reads symbol a and has Aas the topmost stack symbol,

then M cantransition toq⁰ in the next step while replacing A with B₁. . .B_k (afterwards B₁ is the topmost stack symbol)

q a,A→B1. . .Bk q⁰

special casea=εis allowed (spontaneous transition)

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Push-down Automata: Example

q q⁰

a,A→AA a,B→AB a,#→A#

b,A→BA b,B→BB b,#→B#

a,A→ε b,B→ε

a,A→ε b,B→ε ε,#→ε

M =h{q,q⁰},{a,b},{A,B,#}, δ,q,#i with

δ(q,a,A) ={hq,AAi,hq⁰, εi}δ(q,b,A) ={hq,BAi} δ(q, ε,A) =∅ δ(q,a,B) ={hq,ABi} δ(q,b,B) ={hq,BBi,hq⁰, εi}δ(q, ε,B) =∅ δ(q,a,#) ={hq,A#i} δ(q,b,#) ={hq,B#i} δ(q, ε,#) =∅ δ(q⁰,a,A) ={hq⁰, εi} δ(q⁰,b,A) =∅ δ(q⁰, ε,A) =∅ δ(q⁰,a,B) =∅ δ(q⁰,b,B) ={hq⁰, εi} δ(q⁰, ε,B) =∅

δ(q⁰,a,#) =∅ δ(q⁰,b,#) =∅ δ(q⁰, ε,#) ={hq⁰, εi}

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Push-down Automata: Configuration

Definition (Configuration of a Push-down Automaton)

Aconfiguration of a push-down automatonM =hQ,Σ,Γ, δ,q₀,#i is given by a triplec ∈Q×Σ^∗×Γ^∗.

German: Konfiguration Example

I n p u t

q

Configuration hq,ut,BAC#i.

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Push-down Automata: Steps

Definition (Transition/Step of a Push-down Automaton)

We writec `_M c⁰ if a push-down automatonM =hQ,Σ,Γ, δ,q₀,#i can transition from configurationc to configuration c⁰ in one step.

Exactly the following transitions are possible:

hq,a₁. . .a_n,A₁. . .A_mi `_M











hq⁰,a2. . .an,B1. . .BkA2. . .Ami ifhq⁰,B₁. . .B_ki ∈δ(q,a₁,A₁) hq⁰,a1a2. . .an,B1. . .B_kA2. . .Ami

ifhq⁰,B₁. . .B_ki ∈δ(q, ε,A₁)

German: Ubergang¨

IfM is clear from context, we only writec `c⁰.

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Push-down Automata: Reachability of Configurations

Definition (Reachable Configuration)

Configurationc⁰ isreachable from configuration c in PDAM (c `^∗_M c⁰) if there are configurations c₀, . . . ,c_n (n ≥0) where

c0=c,

ci `_M ci+1 for all i ∈ {0, . . . ,n−1}, and cn=c⁰.

German: c⁰ ist inM vonc erreichbar

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Push-down Automata: Recognized Words

Definition (Recognized Word of a Push-down Automaton) PDAM =hQ,Σ,Γ, δ,q0,#i recognizes the wordw =a1. . .an

iff the configurationhq, ε, εi(word processedandstack empty) for someq ∈Q is reachable from thestart configurationhq₀,w,#i.

M recognizesw iff hq₀,w,#i `^∗_M hq, ε, εi for someq ∈Q.

German: M erkenntw, Startkonfiguration

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Push-down Automata: Recognized Word Example

q q⁰

a,A→AA a,B→AB a,#→A#

b,A→BA b,B→BB b,#→B#

a,A→ε b,B→ε

a,A→ε b,B→ε ε,#→ε

example: this PDA recognizesbbabbabb blackboard

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Push-down Automata: Accepted Language

Definition (Accepted Language of a Push-down Automaton) LetM be a push-down automaton with input alphabet Σ.

Thelanguage accepted by M is defined as

L(M) ={w ∈Σ^∗ |M recognizesw}.

example: blackboard

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PDAs Accept Exactly the Context-free Languages

Theorem

A language L is context-free if and only if L is accepted by a push-down automaton.

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Questions

Questions?

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Summary

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Summary

Every context-free language has a grammar in Chomsky normal form. All rules have form

A→BC with variables A,B,C, or

A→awith variable A, terminal symbola, or S →εwith start variableS.

Push-down automata (PDAs) extend NFAs with memory.

PDAs acceptnot with end states but with an empty stack.

The languages accepted by PDAsare exactly thecontext-free languages.