1. Formal Languages, Automata and Models of Computation
CDT314 FABER
Formal Languages, Automata and Models of Computation
Lecture 8
Mälardalen University
2012

2. Content Context-Free Languages Push-Down Automata, PDA NPDA: Non-Deterministic PDA Formal Definitions for NPDAs NPDAs Accept Context-Free Languages Converting NPDA to Context-Free Grammar

3. Non-regular languages
Context-Free Languages
Regular Languages

4. Context-Free Languages
Based on C Busch, RPI, Models of Computation

5. Context-Free Languages
Grammars
Pushdown
Automata
stack
automaton
(CF grammars are
defined as generalized Regular Grammars)

6. Definition: Context-Free Grammars
Variables
Terminal
symbols
Start
variables
Productions of the form:
is string of variables and terminals

7. Pushdown Automata PDAs

8. Pushdown Automaton - PDA
Input String
Stack
States

9. The Stack A PDA can write symbols on stack and read them later on.
POP reading symbol PUSH writing symbol
All access to the stack is only on the top!
(Stack top is written leftmost in the string, e.g. yxz)
A stack is valuable as it can hold an unlimited amount of information (but it is not random access!).
The stack allows pushdown automata to recognize some non-regular languages.

10. The States Pop old - reading Push new Input symbol stack symbol
- writing
stack symbol
Input
symbol

11. input stack top Replace
(An alternative is to either start and finish with empty stack or with a stack bottom symbol such as $)

12. input
Push
top
stack

13. input
Pop
top
stack

14. input
No Change
top
stack

15. NPDAs Non-deterministic Push-Down Automata

16. Non-Determinism

17. A string is accepted if:
All the input is consumed
The last state is a final state
Stack is in the initial condition
(either: empty (when we started with empty stack),
or: bottom symbol reached, depending on convention)

18. Example NPDA
is the language accepted by the NPDA:

19. Example NPDA NPDA M (Even-length palindromes)
Example : aabaaabbblbbbaaabaa

20. Pushing Strings
Pop
symbol
Input
symbol
Push
string

21. Example
input
pushed
string
stack
top
Push

22. Another NPDA example
NPDA M

23. Execution Example
Time 0
Input
Stack
Current state

24. Time 1
Input
Stack

25. Time 2
Input
Stack

26. Time 3
Input
Stack

27. Time 4
Input
Stack

28. Time 5
Input
Stack

29. Time 6
Input
Stack

30. Time 7
Input
Stack
accept

31. Formal Definitions for NPDAs

32. Transition function

33. Transition function new state current state current stack top
new stack top
current input symbol
An unspecified transition function is to the null set and represents a dead configuration for the NPDA.

34. Formal Definition Non-Deterministic Pushdown Automaton NPDA Final
States
Input
alphabet
Stack
Transition
function
Final
states
start
symbol

35. Instantaneous Description
Current
stack
contents
Current
state
Remaining
input

36. Instantaneous Description
Example
Instantaneous Description
Input
Time 4:
Stack

37. Instantaneous Description
Example
Instantaneous Description
Input
Time 5:
Stack

38. We write
Time 4
Time 5

39. A computation example

40. A computation example

41. A computation example

42. A computation example

43. A computation example

44. A computation example

45. A computation example

46. A computation example

47. For convenience we write
A computation example
For convenience we write

48. Formal Definition
Language of NPDA M
Initial state
Final state

49. Example
NPDA M

50. NPDA M

51. Therefore:
NPDA M

52. NPDAs Accept Context-Free Languages

53. Theorem
Context-Free
Languages
(Grammars)
Accepted by
NPDAs

54. Proof - Step 1: Context-Free Languages Languages Accepted by
(Grammars)
Languages
Accepted by
NPDAs
Convert any context-free grammar G to a NPDA M with L(G) = L(M)

55. Proof - Step 2: Context-Free Languages Languages Accepted by
(Grammars)
Languages
Accepted by
NPDAs
Convert any NPDA M to a context-free grammar G with L(M) = L(G)

56. Converting Context-Free Grammars to NPDAs

57. An example grammar:
What is the equivalent NPDA?

58. Grammar NPDA For each production add transition: For each terminal

59. L(Grammar) = L(NPDA) The NPDA simulates
the leftmost derivations of the grammar
L(Grammar) = L(NPDA)

60. Grammar:
A leftmost derivation:

61. NPDA execution:
Time 0
Input
Stack
Start

62. Time 1
Input
Stack

63. Time 2
Input
Stack

64. Time 3
Input
Stack

65. Time 4
Input
Stack

66. Time 5
Input
Stack

67. Time 6
Input
Stack

68. Time 7
Input
Stack

69. Time 8
Input
Stack

70. Time 9
Input
Stack

71. Time 10
Input
Stack
accept

72. In general
Given any grammar G
we can construct a NPDA M with

73. Constructing NPDA M from grammar G
Top-down parser
For any production
For any terminal

74. Grammar G generates string w
if and only if
NPDA M accepts w

75. For any context-free language
there is an NPDA
that accepts the same language

76. Context-Free Languages (Grammars)
Which means
Languages
Accepted by
NPDAs
Context-Free Languages (Grammars)

77. Converting NPDAs to Context-Free Grammars

78. For any NPDA M
we will construct
a context-free grammar G with

79. The grammar simulates the machine
A derivation in Grammar
terminals
variables
Input processed
Stack contents
in NPDA M

80. Some Simplifications First we modify the NPDA so that
It has a single final state qf and
It empties the stack when it accepts the input.
Original NPDA
Empty Stack

81. Second we modify the NPDA transitions. All transitions will have form:
which means that each move
increases/decreases stack by a single symbol.

82. Those simplifications do not affect generality of our argument.
It can be shown that for any NPDA there exists an equivalent one having the above two properties
i.e.
the equivalent NPDA with a single final state which empties its stack when it accepts the input, and which for each move increases/decreases stack by a single symbol.

83. The Grammar Construction
In grammar G
Terminals:
Input symbols of NPDA
states
Stack symbol
Variables:

84. For each transition:
we add production:

85. For each transition:
we add production:
for all states qk , ql

86. Stack bottom symbol
Start Variable
Start state
(Single) Final state

87. From NPDA to CFG, in short:
When we write a grammar, we can use any variable names we choose. As in programming languages, we like to use "meaningful" variable names.
Translating an NPDA into a CFG, we will use variable names that encode information about both the state of the NPDA and the stack contents. Variable names will have the form [qiAqj], where qi and qj are states and A is a variable.
The "meaning" of the variable [qiAqj] is that the NPDA can go from state qi with Ax on the stack to state qj with x on the stack. Each transition of the form (qi, a, A) = (qj,l ) results in a single grammar rule.

88. From NPDA to CFG
Each transition of the form (qi, a, A) = (qj, BC) results in a multitude of grammar rules, one for each pair of states qx and qy in the NPDA.
This algorithm results in a lot of useless (unreachable) productions, but the useful productions define the context-free grammar recognized by the NPDA.
using JFLAP

89. For any NPDA
there is an context-free grammar
that generates the same language.

90. We have the procedure to convert
any NPDA M to a context-free
grammar G with L(M) = L(G)
which means:
Context-Free
Languages
(Grammars)
Accepted by
NPDAs

91. Context-Free Languages (Grammars)
We have already shown that for any context-free language
there is an NPDA
that accepts the same language. That is:
Languages
Accepted by
NPDAs
Context-Free Languages (Grammars)

92. Therefore:
Context-Free
Languages
(Grammars)
Accepted by
NPDAs
END OF PROOF

93. An example of a NPDA in an appropriate form

94. Example
Grammar production:

95. Grammar productions:

96. Grammar production:

97. Resulting Grammar

98. Resulting Grammar, cont.

99. Resulting Grammar, cont.

100. Derivation of string

101. In general, in grammar:
if and only if
is accepted by the NPDA

102. Explanation By construction of Grammar: if and only if
in the NPDA going from qi to qj
the stack doesn’t change below
and A is removed from stack

103. Example (Sudkamp 8.1.2)
Language consisting solely of a’s or an equal number of a´s and b´s.

104. Concerning examination in the course:
Exercises  are voluntary
Labs are voluntary
Midterms are voluntary
Lectures are voluntary…
All of them are recommended!
JFLAP demo