Context Free Grammars

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Context Free Languages (CFL)

• The pumping lemma showed there are languages that are not regular– There are many classes “larger” than that of regular languages– One of these classes are called “Context Free” languages

• Described by Context-Free Grammars (CFG)– Why named context-free?– Property that we can substitute strings for variables regardless of

context (implies context sensitive languages exist)• CFG’s are useful in many applications

– Describing syntax of programming languages– Parsing– Structure of documents, e.g.XML

• Analogy of the day:– DFA:Regular Expression as Pushdown Automata : CFG

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Regular Languages

}0:{ nba nn }{ Rww

**ba *)( ba

Context-Free Languages

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Example: CFG for { 0n1n | n > 1}

• Productions:S -> 01

S -> 0S1

• Basis: 01 is in the language.

• Induction: if w is in the language, then so is 0w1.

CFG Example• Language of palindromes

– We can easily show using the pumping lemma that the language L = { w | w = wR } is not regular.

– However, we can describe this language by the following context-free grammar over the alphabet {0,1}:

P P 0P 1P 0P0P 1P1

Inductive definition

More compactly: P | 0 | 1 | 0P0 | 1P1 5

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Grammars• Grammars express languages

• Example: the English language grammar

verbpredicate

nounarticlephrasenoun

predicatephrasenounsentence

_

_

Fall 2006 Costas Busch - RPI 7

•

sleepsverb

runsverb

dognoun

catnoun

thearticle

aarticle

Fall 2006 Costas Busch - RPI 8

• Derivation of string “the dog walks”:

sleepsdogthe

verbdogthe

verbnounthe

verbnounarticle

verbphrasenoun

predicatephrasenounsentence

_

_

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• Derivation of string “a cat runs”:

runscata

verbcata

verbnouna

verbnounarticle

verbphrasenoun

predicatephrasenounsentence

_

_

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• Language of the grammar:

L = { “a cat runs”, “a cat sleeps”, “the cat runs”, “the cat sleeps”, “a dog runs”, “a dog sleeps”, “the dog runs”, “the dog sleeps” }

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catnoun

Variables

Sequence of Terminals (symbols)

Productions

predicatephrasenounsentence _

Sequence of Variables

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CFG Formalism

• Terminals = symbols of the alphabet of the language being defined.

• Variables = nonterminals = a finite set of other symbols, each of which represents a language.

• Start symbol = the variable whose language is the one being defined.

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Productions

• A production has the form variable (head) -> string of variables and terminals (body).

• Convention:– A, B, C,… and also S are variables.– a, b, c,… are terminals.– …, X, Y, Z are either terminals or variables.– …, w, x, y, z are strings of terminals only. , , ,… are strings of terminals and/or variables.

CFG Notation

• A CFG G may then be represented by these four components, denoted G=(V,T,R,S) – V is the set of variables– T is the set of terminals– R is the set of production rules– S is the start symbol.

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Derivations – Intuition

• We derive strings in the language of a CFG by starting with the start symbol, and repeatedly replacing some variable A by the body of one of its productions.– That is, the “productions for A” are those that

have head A.

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Derivations – Formalism

• We say A => if A -> is a production.

• Example: S -> 01; S -> 0S1.

• S => 0S1 => 00S11 => 000111.

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Iterated Derivation

• =>* means “zero or more derivation steps.”

• Basis: =>* for any string .

• Induction: if =>* and => , then =>* .

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Example: Iterated Derivation

• S -> 01; S -> 0S1.

• S => 0S1 => 00S11 => 000111.

• Thus S =>* S; S =>* 0S1;

S =>* 00S11; S =>* 000111.

Sample CFG

1. EI // Expression is an identifier2. EE+E // Add two expressions3. EE*E // Multiply two expressions4. E(E) // Add parenthesis5. I L // Identifier is a Letter6. I ID // Identifier + Digit7. I IL // Identifier + Letter8. D 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 // Digits9. L a | b | c | … A | B | … Z // Letters

Note Identifiers are regular; could describe as (letter)(letter + digit)* 19

Derivation

• Expand start symbol first and work way down in such a way that it matches the input string

• For example, given a*(a+b1) we can derive this by:– E E*E I*E L*E a*E a*(E) a*(E+E) a*(I+E)

a*(L+E) a*(a+E) a*(a+I) a*(a+ID) a*(a+LD) a*(a+bD) a*(a+b1)

• Note that at each step of the productions we could have chosen any one of the variables to replace with a more specific rule.

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Multiple Derivation

• We saw an example of in deriving a*(a+b1)

• We could have used * to condense the derivation. – E.g. we could just go straight to E * E*(E+E) or

even straight to the final step• E * a*(a+b1)• Going straight to the end is not recommended on a

homework or exam problem if you are supposed to show the derivation

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Leftmost and Rightmost Derivations

• Derivations allow us to replace any of the variables in a string.– Leads to many different derivations of the same

string.

• By forcing the leftmost variable (or alternatively, the rightmost variable) to be replaced, we avoid these “distinctions without a difference.”

Leftmost Derivation

• In the previous example we used a derivation called a leftmost derivation. We can specifically denote a leftmost derivation using the subscript “lm”, as in:

lm or *lm

• A leftmost derivation is simply one in which we replace the leftmost variable in a production body by one of its production bodies first, and then work our way from left to right.

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Example: Leftmost Derivations

• Balanced-parentheses grammar:

S -> SS | (S) | ()

• S =>lm SS =>lm (S)S =>lm (())S =>lm (())()

• Thus, S =>*lm (())()

• S => SS => S() => (S)() => (())() is a derivation, but not a leftmost derivation.

Rightmost Derivation

• Not surprisingly, we also have a rightmost derivation which we can specifically denote via:

• rm or *rm

• A rightmost derivation is one in which we replace the rightmost variable by one of its production bodies first, and then work our way from right to left.

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Rightmost Derivation Example

• a*(a+b1) was already shown previously using a leftmost derivation.

• We can also come up with a rightmost derivation, but we must make replacements in different order:– E rm E*E rm E * (E) rm E*(E+E) rm E*(E+I) rm

E*(E+ID) rm E*(E+I1) rm E*(E+L1) rm E*(E+b1) rm E*(I+b1) rm E*(L+b1) rm E*(a+b1) rm I*(a+b1) rm L*(a+b1) rm a*(a+b1)

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Example: Rightmost Derivations

• Balanced-parentheses grammar:

S -> SS | (S) | ()

• S =>rm SS =>rm S() =>rm (S)() =>rm (())()

• Thus, S =>*rm (())()

• S => SS => SSS => S()S => ()()S => ()()() is neither a rightmost nor a leftmost derivation.

Left or Right?

• Does it matter which method you use?

• Answer: No

• Any derivation has an equivalent leftmost and rightmost derivation. That is, A * . iff A *lm and A *rm .

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Sentential Forms

• Any string of variables and/or terminals derived from the start symbol is called a sentential form.

• Formally, is a sentential form iff

S =>* .

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Language of a Grammar

• If G is a CFG, then L(G), the language of G, is {w | S =>* w}.

• Example: G has productions

S -> ε and S -> 0S1.

• L(G) = {0n1n | n > 0}.

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Context-Free Languages

• A language that is defined by some CFG is called a context-free language.

• There are CFL’s that are not regular languages, such as the example just given.

• But not all languages are CFL’s.

• Intuitively: CFL’s can count two things, not three.

CFG Exercise

• {aibjck | i ≠ j or j ≠ k}

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