1. Chapter 5 Context-Free Grammars

2. Grammar (Review) A grammar is a 4-tuple G = (NT, T, S, P), where
NT: a finite set of non-terminal symbols
T: a finite set of terminal (alphabet) symbols
S: is the starting symbol S  NT
P: a finite set of production rules: x → y
x  NT y  (NT  T)*
Derivation of w  T*:
S  w1 ···  wn  w (written as S * w)
The language generated by the grammar is
L(G) = {w  T* : S * w}

3. 5.1: Context-Free Grammars (1)
Context-Free Grammars (CFG) are language-description mechanisms used to generate the strings of a language
A grammar is said to be context-free if every rule has a single non-terminal on the left-hand side
A  
A  NT: single non-terminal on the left hand side
  (NT  T)*: string of terminals and non-terminals
A language L is called context-free language (CFL) if and only if there is a context-free grammar G (CFG) that generates it (i.e., L = L(G))

4. 5.1: Context-Free Grammars (3)
Example 5.1
G = ({S}, {a, b}, S, {S → aSa | bSb | })
For aabbaa, typical derivation might be:
S  aSa  aaSaa  aabSbaa  aabbaa
Grammar generates language
L(G)={wwR : w  {a, b}*}

5. 5.1: Context-Free Grammars (5)
What is the language of this grammar?
G = ({S}, {a, b}, S, {S → aSb, S → ab})
L(G) = {anbn | n  1}
G = ({S}, {a, b}, S, {S → aSb, S → e})
L(G) = {anbn | n  0}

6. 5.1: Context-Free Grammars (6)
What is the language of this grammar?
G = ({S, A}, {a, b}, S, {S → aSa | aAa
A → bA | b }
L(G) = {anbman | n  1, m  1}
G1 = ({S, A, B}, {a, b}, S, {S → AB
A → aA | a
B → bB | e }
G2 = ({S, B}, {a, b}, S, {S → aS | aB
L(G) = {a+b*} = {anbm | n  1, m  0}

7. 5.1: Context-Free Grammars (7)
What is the language of this grammar?
G = ({S, B}, {a, b, c}, S, {S → abScB | e
B → bB | b }
L = {(ab)n(cbm)n | n  0, m  1}
What is the grammar of this language?
L = {anbmcmdn | n  1, m  1}
G = ({S, A}, {a, b, c, d}, S, { S → aSd | aAd
A → bAc | bc }
L = {anbmcmd2n | n  0, m  1}
G = ({S, A}, {a, b, c, d}, S, { S → aSdd | A

8. 5.1: Context-Free Grammars (8)
What is the grammar of this language?
L = {a*ba*ba*}
G1 = ({S, A}, {a, b}, S, {S → AbAbA
A → aA | e }
G2 = ({S, A, C}, {a, b}, S, {S → aS | bA
A → aA | bC }
C → aC | e

9. 5.1: Context-Free Grammars (9)
What is the grammar of this language?
A language over {a, b} with at least 2 b’s
G1 = ({S, A}, {a, b}, S, {S → AbAbA
A → aA | bA | e }
G2 = ({S, A, C}, {a, b}, S, {S → aS | bA
A → aA | bC
C → aC | bC | e }

10. 5.1: Context-Free Grammars (10)
What is the grammar of this language?
A language over {a, b} of even-length strings
G = ({S, O}, {a, b}, S, {S → aO | bO | e
O → aS | bS }
A language over {a, b} of an even no. of b’s
G = ({S, B, C}, {a, b}, S, { S → aS | bA | e
A → aA | bS }

11. 5.1: Context-Free Grammars (11)
What is the grammar of this language?
A language over {a, b, c} that do not contain the substring abc
G = ({S, B, C}, {a, b, c}, S, {S → bS | cS | aA | e
A → aA | bB | cS | e
B → aA | bS | e}

12. 5.1: Context-Free Grammars (12)
Write a CFG for the following Languages
L1 = {wcwR | w  {a, b}*}
S  aSa | bSb | c
L2 = {anbncmdm | n  1, m  1}
S  XY
X  aXb | ab
Y  cYd | cd

13. 5.1: Context-Free Grammars (13)
The following languages are NOT CFLs
L1 = {wcw | w  {a, b}*}
L2 = {anbmcndm | n  1, m  1}
L3 = {anbncn | n  0}
G = ({S}, {a, b}, S, {S → aSb | bSa | SS | })
is this grammar context-free?
yes, there is a single variable on the left hand side

14. 5.1: Context-Free Grammars (14)
Are regular languages context-free? Why?
yes, SEE NEXT SLIDE
But, not all context-free grammars are regular.
Regular languages are a proper subset of the class of context-free languages.
Regular grammars are a proper subset of context-free grammars.
Non-regular languages can be generated by context-free grammars,
so the term context-free generally includes non-regular languages and regular languages.

15. 5.1: Context-Free Grammars (14)
Constructing (right linear) CFG from DFA
Each state of the DFA will be represented by a nonterminal
The initial state will correspond to the start nonterminal
For each transition d(qi, a) = qj , add a rule
qi → aqj
For each accepting state qf , add a rule
qf → e

16. 5.1: Context-Free Grammars (15) Leftmost & Rightmost Derivations
Given the grammar
S → aAB, A → bBb, B → A | 
String abbbb can be derived in different ways:
Left-most derivation:
always replace the leftmost NT in each sentential form
S  aAB  abBbB  abAbB  abbBbbB  abbbbB  abbbb
Right-most derivation:
always replace the rightmost NT in each sentential form
S  aAB  aA  abBb  abAb  abbBbb  abbbb

17. 5.1: Context-Free Grammars (16)
DEFINITION
 derives in one step
 derives in  one step
 indicates that the derivation utilizes the rules of grammar G + * G

18. 5.1: Context-Free Grammars (17)
Derivation Order
Leftmost: Replace the leftmost non-terminal symbol
Rightmost: Replace the rightmost non-terminal symbol lm lm
Important Notes: A  y
If xAz xyz , what’s true about x ? terminals
If xAz xyz , what’s true about z ? terminals  lm  rm

19. 5.1: Context-Free Grammars (18)
Example
S  S + E | E
E  number | ( S )
Left-most derivation
S  S+E  E+E  (S)+E  (S+E)+E (S+E+E)+E (E+E+E)+E (1+E+E)+E (1+2+E)+E (1+2+(S))+E (1+2+(S+E))+E (1+2+(E+E))+E
(1+2+(3+E))+E (1+2+(3+4))+E (1+2+(3+4))+5
Right-most derivation
S  S+E  E+5  (S)+5  (S+E)+5 (S+(S))+5 (S+(S+E))+5 (S+(S+4))+5 (S+(E+4))+5 (S+(3+4))+5 (S+E+(3+4))+5 (S+2+(3+4))+5 (E+2+(3+4))+5 (1+2+(3+4))+5

20. 5.1: Context-Free Grammars (19) Derivation (Parsing) Trees
Given the grammar
S → aaSB | , B → bB | b
A leftmost derivation
S  aaSB  aaB  aab
A rightmost derivation
S  aaSB  aaSb  aab
Both derivations correspond to the parse (or derivation) tree above.

21. 5.1: Context-Free Grammars (20)
In a parse tree, the nodes labeled with NTs correspond to the left side of a production rule and the children of that node correspond to the right side of the rule, e.g., S → aaSB.
The tree structure shows the rule that is applied to each NT (for a specific derivation), without showing the order of rule application.
Each internal node of the tree corresponds to a NT, and the leaves of the derivation tree represent the string of terminals.
Note the tree applies to a specific derivation and may not include all rules, e.g., B → bB is not shown above.

22. 5.1: Context-Free Grammars (21)
Definition 5.3: Let G = (NT, T, S, P) be a context-free grammar. An ordered tree is a derivation tree for G if and only if it has the following properties:
1. the root is labeled S
2. every leaf has a label from T  {}
3. every non-leaf (interior) vertex has a label from NT.
4. if a vertex has label A  NT, and its children are labeled (left to right) a1, a2, ..., an, then P contain a production of the form A → a1a2…an
5. a leaf labeled  has no siblings; that is, a vertex with a child labeled  can have no other children
Partial derivation tree: a tree having properties 3–5, but for which property 1 may not hold, and for which property 2 is replaced with:
2a. every leaf has a label from NT  T  {}

23. 5.1: Context-Free Grammars (22)
Derivation  Parse Tree S
Parse
Tree
S  S + E | E
E  number | ( S )
S + E E 5
( S )
S + E
S + E
( S ) E
S + E 2 4 1 E 3
Derivation
S  S +E  E+E  (S)+E  (S+E)+E (S+E+E)+E (E+E+E)+E (1+E+E)+E (1+2+E)+E … (1+2+(3+4))+E (1+2+(3+4))+5

24. 5.1: Context-Free Grammars (23)op
E  E op E E op
E  E op E | ( E ) | - E | id op  + | - | * | / | 
 id op E id E op id *
 id * E E op id *
 id * id

25. 5.1: Context-Free Grammars (24)
A derivation tree for grammar G yields a sentence of the language L(G)
A partial derivation tree yields a sentential form
The yield of a parse tree is the string of leaf symbols obtained by reading the tree left-to-right
the order they are encountered when the tree is traversed in a depth-first manner, always taking the leftmost unexplored branch

26. 5.1: Context-Free Grammars (25)
Given the grammar
S → aAB A → bBb B → A | ,
the yield of the partial derivation tree is the sentential form abBbB

27. 5.1: Context-Free Grammars (26)
Given the grammar
S → aAB A → bBb B → A | ,
the yield of the derivation tree is the sentence abbbb  L(G).

28. 5.1: Context-Free Grammars (27)
Theorem 5.1 (Relationship between Sentential Forms & Derivation Trees)
Let G = (NT, T, S, P) be a CFG. Then
for every w  L(G), there exists a derivation tree of G whose yield is w.
the yield, w, of any derivation tree of G is such that w  L(G).
if tG is any partial derivation tree for G whose root is labeled S, then the yield of tG is a sentential form of G.
As a side note, any w  L(G) has a leftmost and a rightmost derivation.

29. 5.2: Parsing and Ambiguity (1)
In practical applications, it is usually not enough to decide whether a string belongs to a language.
It is also important to know how to derive the string from the language.
Parsing uncovers the syntactical structure of a string, which is represented by a parse tree.
The syntactical structure is important for assigning semantics to the string -- for example, if it is a computer program.

30. 5.2: Parsing and Ambiguity (2)
Application of parsing (Compilers)
Let G be a context-free grammar for the C language. Let the string w be a C program.
One thing a compiler does - in particular, the part of the compiler called the “parser” - is determine whether w is a syntactically correct C program.
It also constructs a parse tree for the program that is used in code generation.
There are many sophisticated and efficient algorithms for parsing. You may study them in more advanced classes (for example, on compilers).

31. 5.2: Parsing and Ambiguity (3)
S-grammars
Definition 5.4: A context-free grammar G = (NT, T, S, P) is said to be a simple grammar or s-grammar if all of its productions are of the form
A → ax where A  NT, a  T, x  NT*, and any pair (A, a) occurs at most once in P.
Examples:
S → aS | bSS | c is an s-grammar.
S → aS | bSS | aSB | c is not an s-grammar.
because pair (S, a) occurs in two productions:
S → aS, and S → aSB

32. 5.2: Parsing and Ambiguity (4)
Example:
given s-grammar G with production
S → aS | bSS | c
show the derivation of the string w = abcc
Since G is an s-grammar,
the only way to get the a up front is via rule
S → aS
the only way to get the b is via rule
S → bSS
the only way to get each c is via rule
S → c
Thus, we have parsed the string in 4 steps as
S  aS  abSS  abcS  abcc

33. 5.2: Parsing and Ambiguity (5)
Find an s-grammar for
aaa*b | b
We need to start with an a or have only the b.
S → aA
S → b
A → aB
B → aB
B → b

34. 5.2: Parsing and Ambiguity (6)
A terminal string may be generated by a number of different derivations
Let G be a CFG. A string w is in L(G), iff there is
a leftmost derivation of w from S. (S  w)
Question
Is there a unique leftmost derivation of every sentence (string) in the language of a grammar? NO * lm

35. 5.2: Parsing and Ambiguity (7)
Consider the expression grammar:
E  E+E | E*E | (E) | -E | id
Two different leftmost derivations of id + id * id
E  E + E
 id + E
 id + E * E
 id + id * E
 id + id * id E * + id

36. 5.2: Parsing and Ambiguity (8)
Consider the expression grammar:
E  E+E | E*E | (E) | -E | id
Two different leftmost derivations of id + id * id
E  E * E
 E + E * E
 id + E * E
 id + id * E
 id + id * id E + * id

37. 5.2: Parsing and Ambiguity (9) Ambiguity
Definition 5.5:
a grammar G is ambiguous if there is a string with at least two parse trees,
two or more leftmost or rightmost derivations.
Example: CFG G with productions S → aSb | SS |  is ambiguous because there are two parse tree for w = aabb as shown below.

38. 5.2: Parsing and Ambiguity (10)
A CFG is ambiguous if there is a string w  L(G) that can be derived by two distinct leftmost derivations
A grammar G is ambiguous if there exists a sentence in G with more than one derivation (parsing) tree
A grammar that is not ambiguous is called unambiguous
If G is ambiguous then L(G) is not necessarily ambiguous
A language L is inherently ambiguous if there is no unambiguous grammar that generates it

39. 5.2: Parsing and Ambiguity (11)
Let G be S → aS | Sa | a
G is ambiguous since the string aa has 2 distinct leftmost derivation
S  aS  aa
S  Sa  aa
L(G) = a+
This language is also generated by the unambiguous grammar S → aS | a
L(G) is not ambiguous. Why?

40. 5.2: Parsing and Ambiguity (12)
Let G be S → bS | Sb | a L(G) = b*ab*
G is ambiguous since the string bab has 2 distinct leftmost derivation
S  bS  bSb  bab
S  Sb  bSb  bab
The ability to generate the b’s in either order must be eliminated to obtain an unambiguous grammar
This language is also generated by the unambiguous grammars
G2: S → bS | A
A → Ab | a
G1: S → bS | aA
A → bA | e

41. 5.2: Parsing and Ambiguity (13)
Eliminating Ambiguity
Consider the following grammar
E  E + E | E * E | num
Consider the sentence * 3
Leftmost Derivation 1
E  E + E
 1 + E
 1 + E * E
 * E
 * 3
Leftmost Derivation 2
E  E * E
 E + E * E E * + 1 3 2 E + * 3 1 2

42. 5.2: Parsing and Ambiguity (14)
Different parse trees correspond to different evaluations (meaning)
LMD-1
LMD-2 * + + * 1 2 3 1 2 3
= 9
= 7

43. 5.2: Parsing and Ambiguity (15)
Ambiguous: E → E + E | E * E | number
Both + and * have the same precedence
To remove ambiguity, you have to give + and * different precedence
Let us say * has higher precedence than +
E  E + T | T
T  T * num | num

44. 5.2: Parsing and Ambiguity (16)
Eliminating Ambiguity
E  E + T | T
T  T * num | num
Consider the sentence * 3
Leftmost Derivation (only one LMD)
E  E + T
 T + T
 num + T
 1 + T
 1 + T * num
 1 + num * num
 * num
 * 3
num T * E + 1 3 2
= 7

45. 5.2: Parsing and Ambiguity (17)
Let us say + has higher precedence than *
E  E * T | T
T  T + num | num
Consider the sentence * 3
Leftmost Derivation (only one LMD)
E  E * T
 T * T
 T + num * T
 num + num * T
 1 + num * T
 * T
 * num
 * 3 E T * T
num 3
num T + 2
num
= 9 1

46. 5.2: Parsing and Ambiguity (18)
A derivation tree…
Corresponds to exactly one leftmost derivation.
Corresponds to exactly one rightmost derivation.
CFG ambiguous  any of following equivalent statements:
 string w with multiple derivation trees.
 string w with multiple leftmost derivations.
 string w with multiple rightmost derivations.
Note: Defining grammar, not language, ambiguity.

47. 5.3: CFGs & Programming Languages (1)
Programming languages are context-free, but not regular
Programming languages have the following features that require infinite “stack memory”
matching parentheses in algebraic expressions
nested if .. then .. else statements
nested loops
block structure

48. 5.3: CFGs & Programming Languages (2)
<unsigned constant>  <unsigned number>
<constant>  <unsigned number> | <sign> <unsigned number>
<unsigned number>  <unsigned integer> | <unsigned real>
<unsigned integer>  <digit> <unsigned integer> | <digit>
<unsigned real>  <unsigned integer> . <unsigned integer> |
<unsigned integer> . <unsigned integer> E <exp> |
<unsigned integer> E <exp>
<exp>  <unsigned integer> | <sign> <unsigned integer>
<digit>  0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
<letter>  a | b | c | … | y | z | A | B | C | … | Y | Z
<sign>  + | -
<identifier>  <letter> <identifier tail>
<identifier tail>  <letter> <identifier tail> | <digit> <identifier tail>

49. 5.3: CFGs & Programming Languages (3)
<expression>  <simple expression>
<simple expression>  <term> | <sign term> |
<simple expression> <adding operator> <term>
<term>  <factor> |
<term> <multiplying operator> <factor>
<factor>  <variable> | <unsigned constant> | (<expression>)
<adding operator>  + | -
<multiplying operator>  * | / | div | mod