discrete math. and logic ii. context-free grammars - sfwr ...se2fa3/notes/ln18.pdf · the grammar...

Discrete Math. and Logic II. Context-FreeGrammars

SFWR ENG 2FA3

Ryszard Janicki

Winter 2014

Acknowledgments: Material partially based on Automata and Computability by Dexter C. Kozen (Chapter 19).

Ryszard Janicki Discrete Math. and Logic II. Context-Free Grammars 1 / 16

Introduction

An example of a context-free grammar


Introduction

The objects 〈xxx〉 are called nonterminal symbols

Each nonterminal symbol generates a set of strings over a

�nite alphabet Σ in a systematic way

the nonterminal 〈arith-expr〉 in

〈assg-stmt〉 ::= 〈var〉 := 〈arith-expr〉

generates the set of syntactically correct arithmetic

expressions in this language

The strings corresponding to the nonterminal 〈xxx〉 aregenerated using rules with 〈xxx〉 on the left-hand side


Introduction

The alternatives on the righthand side, separated by vertical

bars |, describe di�erent ways strings corresponding to 〈xxx〉can be generated

These alternatives may involve other nonterminals 〈yyy〉,which must be further eliminated by applying rules with 〈yyy〉on the left-hand side

while x ≤ y do begin x := (x + 1); y := y − 1 end

is generated by the nonterminal 〈stmt〉


Introduction

We obtain the while statement from 〈stmt〉 through a

sequence of expressions called sentential forms

Each sentential form is derived from the previous by an

application of one of the rules

〈stmt〉〈while-stmt〉while 〈bool-expr〉 do 〈stmt〉while 〈arith-expr〉〈compare-op〉〈arith-expr〉 do 〈stmt〉while 〈var〉〈compare-op〉〈arith-expr〉 do 〈stmt〉while 〈var〉 ≤ 〈arith-expr〉 do 〈stmt〉while 〈var〉 ≤ 〈var〉 do 〈stmt〉while x ≤ 〈var〉 do 〈stmt〉while x ≤ y do 〈stmt〉while x ≤ y do 〈begin-stmt〉. . .


Introduction

Applying di�erent rules will yield di�erent results

begin if z = (x + 3) then y := z else y := x end

The set of all strings not containing any nonterminals

generated by the grammar is called the language generated by

the grammar

In general, this set of strings may be in�nite, even if the set of

rules is �nite

There may also be several di�erent derivations of the same

string

A grammar is said to be unambiguous if a string cannot have

more than one derivation


Introduction

The language (subset of Σ∗) generated by the context-free

grammar G is denoted L(G )

A subset of Σ∗ is called a context-free language (CFL) if it is

L(G ) for some CFG G

CFLs are good for describing in�nite sets of strings in a �nite

way

They are particularly useful in computer science for describingthe syntax of

programming languages,

well-formed arithmetic expressions,

well-nested begin-end blocks,

strings of balanced parentheses,

All regular sets are CFLs, but not necessarily vice versa.


Pushdown Automata (PDAs): A Preview

A pushdown automaton (PDA) is like a �nite automaton,

except it has a stack or pushdown store, which it can use to

record a potentially unbounded amount of information

Its input head is read-only and may only move right

The machine can store information on the stack in a

last-in-�rst-out (LIFO) fashion

It can push symbols onto the top of the stack or pop them o�

the top of the stack

It may not read down into the stack without popping the top

symbols o�


Pushdown Automata (PDAs): A Preview

a1a2 a3 a4 a5 a6 a7 a8 an

left to right, read only

push/pop

StackFiniteControl

QBA

CB


Formal De�nition of CFGs and CFL

De�nition

A context-free grammar (CFG) is a quadruple G = (N,Σ,P,S),where

N is a �nite set (the nonterminal symbols),

Σ is a �nite set (the terminal symbols) disjoint from N,

P is a �nite subset of N × (N ∪ Σ)∗ (the productions),

S ∈ N (the start symbol).



We use capital letters A,B,C , · · · for nonterminals

We use a, b, c , · · · for terminal symbols

Strings in (N ∪ Σ)∗ are denoted α, β, γ, · · ·

Instead of writing productions as (A, α), we write A −→ α

We often use the vertical bar | to abbreviate a set of

productions with the same left-hand side



Example

Instead of writing

A −→ α1, A −→ α2, A −→ α3,

we write

A −→ α1 | α2 | α3

De�nition

If α, β ∈ (N ∪ Σ)∗, we say that β is derivable from α in one step

and write

α1−→G

β

if β can be obtained from α by replacing some occurrence of a

nonterminal A in α with γ, where A −→ γ, is in P.



Example

We have A −→ γ a production in P

α = α1Aα2, where α1, α2 ∈ (N ∪ Σ)∗

β = α1γα2

Then, we have α1−→G

β



De�nition

Let∗−→G

be the re�exive transitive closure of the relation1−→G

; that

is, de�ne

α0−→G

α

αn+1−→G

β if there exists γ such that αn−→G

γ and γ1−→G

β

α∗−→G

β if ∃(n | n ≥ 0 : αn−→G

β )



Terminology:

A string in (N ∪ Σ)∗ derivable from the start symbol S is

called a sentential form

A sentential form is called a sentence if it consists only of

terminal symbols

The language generated by G , denoted L(G ), is the set of all

sentences:

L(G ) = {x ∈ Σ∗ | S ∗−→G

x}

A subset B ⊆ Σ∗ is a context-free language (CFL) if

B = L(G ) for some context-free grammar G



Example

The nonregular set {anbn | n ≥ 0} is a CFL

It is generated by the grammar G = (N,Σ,P, S), whereN = {S},Σ = {a, b},P = {S −→ aSb,S −→ ε}

Here is a derivation of a3b3 in G :

S1−→G

aSb1−→G

aaSbb1−→G

aaaSbbb1−→G

aaabbb

Sn+1−→G

anbn


discrete math. and logic ii. context-free grammars - sfwr ...se2fa3/notes/ln18.pdf · the grammar...

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