1. Three Basic Concepts
Languages
Grammars
Automata

2. Languages Alphabet: a finite and nonempty set of symbols  = {a, b}
String: finite sequence of symbols from 
w = abaaa
: empty string
*: the set of all strings on  (+ = *  {})

3. Languages
Language: a subset L of *
Sentence: a string in L

4. Languages Example 1:  = {a, b} * = {, a, b, aa, ab, ba, aaa, ...}
L1 = {a, aa, aab} (finite language)
L2 = {anbn | n  0} = {, ab, aabb, ...}

5. Languages Language concatenation: L1L2 = {xy | xL1, yL2}
Ln = L L ... L (n times)
L0 = {}

6. Languages
Example 2:
L = {anbn | n  0}

7. Languages
Example 2:
L = {anbn | n  0}
L2 = {anbnambm | n  0, m  0}

8. Languages Star-closure: L* = L0  L1  L2 ... Positive closure:

9. Grammars
A grammar for a natural language tells us whether a particular sentence is well-formed or not.
<sentence>  <noun-phrase><predicate>
<noun-phrase>  <article><noun>
<predicate>  <verb>
<article>  a | the
<noun>  boy | dog
<verb>  runs | walks

10. Grammars Formal grammar: G = (V, T, S, P) V: finite set of variables
T: finite set of terminal symbols
SV: start variable
P: finite set of productions

11. Grammars Productions: x  y x(VT)+ y(VT)* w = uxv derives z = uyv
w  z
w1 * wn (w1  w2  ...  wn | w1 = wn)
w1 + wn

12. Grammars Generated language: G = (V, T, S, P) L(G) = {wT* | S * w}
Derivation:
S  w1  w2  ...  wn  wL(G)
Sentential forms: S, w1, w2, ..., wn (containing variables)

13. Grammars Example 3: G = ({S}, {a, b}, S, P) P: S  aSb S  
S  aSb  aaSbb  aabb
aabb: sentence aaSbb: sentential form

14. Grammars
Example 3:
G = ({S}, {a, b}, S, P)
P: S  aSb
S  

15. Grammars Example 3: G = ({S}, {a, b}, S, P) P: S  aSb S  
L(G) = {anbn | n  0}

16. Grammars Example 4: G1 = ({A, S}, {a, b}, S, P1) P1: S  aAb | 
A  aAb | 

17. Grammars Example 4: G1 = ({A, S}, {a, b}, S, P1) P1: S  aAb | 
A  aAb | 
L(G1) = {anbn | n  0}
G and G1 are equivalent

18. Grammars Example 5: G2 = ({S}, {a, b}, S, P2) P2: S  SS S   S  bSa
S  aSb
S  bSa

19. Grammars Example 5: G2 = ({S}, {a, b}, S, P2) P2: S  SS S   S  bSa
S  aSb
S  bSa
L(G2) = {w | na(w) = nb(w)}

20. Automata An abstract model of digital computer: Input file
Control unit
Storage
Output

21. Automata Input file: is divided into squares.
Input is a string over a given alphabet.
Each input square holds a symbol.
The symbols are read from left to right, one at a time.
The end of the input string can be detected.

22. Automata Storage: consists of an unlimited number of cells.
Each cell can hold a symbol from an alphabet (which can be different from the input alphabet).
The contents of the storage cells can be read and changed.

23. Automata Control unit: has a finite number of internal states.
Can be in any one of the internal states.
Can change state in some defined manner.

24. current state  input symbol  storage info  next state
Automata
Transition function:
current state  input symbol  storage info  next state
Output may be produced
Info in the storage may be changed
Configuration: current state  input symbol  storage info
Move: current configuration  next configuration

25. Automata General types of automata: Accepter: yes/no output
Transducer: string of symbols as output
Deterministic: single move
Non-deterministic: multiple moves

26. Homework
Exercises: 4, 5, 6, 8, 9, 12, 15, 17 of Section Linz’s book.
Reading: Section Linz’s book.
Presentation: Section Linz’s book (procedures mark and reduce).