1. Classification of Languages

2. Strings and Regular Expression
Alphabet: The set A is not a set of numbers, but some set of symbols.
Strings (words) : The set A* consisting of all finite sequences of letters or other symbols from the set A, written without the commas, are referred as strings.
Empty String: The set A* contains empty string or empty sequence, containing no symbols, and we denote this string by Λ.

3. Catenation: If w1 = s1s2…sn and w2 = t1t2…tk are elements of A
Catenation: If w1 = s1s2…sn and w2 = t1t2…tk are elements of A* for some set A, then the catenation of w1 and w2 as the sequence s1s2…snt1t2…tk.
i.e. w1 * w2 = w1w2 = s1s2…snt1t2…tk.

4. Regular expression
A regular expression over A is a string constructed from the elements of A and the symbols (, ), *, Λ.
The symbol Λ is a regular expression.
If x Є A, the symbol x is a regular expression.
If α and β are regular expressions, then the expression αβ is regular.
If α and β are regular expressions, then the expression αvβ is regular.
If α is a regular expression, then (α)* is regular.

5. Example Following expressions are all regular expressions over A.
(i) 0*(0v1)* (ii) 00*(0v1)*1

6. Eg 1. Let A = {a, b, c}. Then the regular expression a
Eg 1. Let A = {a, b, c}. Then the regular expression a* corresponds to the set of all finite sequence of a’s, such as aaaa, aa, aaaaaaaa so on. The regular expression a(bvc) corresponds to the set {ab, ac} ⊆ A*. The regular expression ab(bc)* corresponds to abbc, abbcbcbc, and so on.

7. Regular Set
Associated with each regular expression over A, there is a corresponding subset A*. Such sets are called regular subsets of A*or just regular sets if no response of A is needed.

8. Languages S: Set of words
S*: The collection of all possible sentences.
Syntax: The specification of the proper construction of sentences is called the syntax of a language.
Phase structure grammar: The syntax of a class of languages

9. Grammar
A Phase structure grammar G is defined to be a 4 –tuple (V,S, v0 and ↦), where V is a finite set, S is a subset of V, v0 Є V-S and ↦ is a finite relation on V*.
V consists of S together with some other symbols.
v0: The element v0 of V is a starting point of substitution.
The relation ↦ on V* specifies allowable replacements, in the sense that, if w ↦ w’, we may replace w by w’ whenever the string w occurs.

10. Production of G: The statement w ↦ w’, is called a production of G
Production of G: The statement w ↦ w’, is called a production of G. Then w and w’ are called left and right sides of the production, respectively.
Terminal and Non terminal symbols: If G = (V, S, v0, ↦) is a phase structure grammar, then set S is the set of terminal symbols and N = V-S is the set of non terminal symbols.
Note: V = S U N.
Language: The set of all properly constructed sentences that can be produced using a grammar G is called the language of G and is denoted by L(G).

11. Eg1: Let S = { Ramesh, Seema, drives, jogs, carelessly, rapidly, frequently} & N = { sentence, noun, verbphase, verb, adverb} and let V = S U N. Let v0 = sentence, and suppose that the relation ↦ on V* is described by
sentence ↦ noun verbphase
noun ↦ Ramesh
noun ↦ Seema
verbphase ↦ verb adverb
verb ↦ drives
verb ↦ jogs
adverb ↦ carelessly
adverb ↦ rapidly
adverb ↦ frequently
The set S contains all the allowed words in the language; N consists of words that describe parts of sentences but that are not actually contained in the language. Write the derivation of the sentence “ Seema drives rapidly”. Also draw the derivation tree.

12. Eg2: Let V = { v0, w, a, b, c}, S = { a, b, c} and let ↦ be the relation on V* given by
1. v0 ↦ aw 2. w ↦ bbw 3. w ↦ c.
Consider the phase structure grammar G = (V, S, v0, ↦).
Derive the sentence ab6c. Also draw the derivation tree.
Derive the sentence ab4c. Also draw the derivation tree.

13. Eg3: Let V = { v0, w, a, b, c}, S = { a, b, c} and let ↦ be the relation on V* given by
1. v0 ↦ aw 2. w ↦ bbw 3. w ↦ c.
Consider the phase structure grammar G = (V, S, v0, ↦). Determine the form of allowable sentences in L(G).
Eg: In examples 4 and 5, a grammar G is specified. In each case describe precisely the language, L(G), produced by this grammar; that is, describe all syntactically correct “sentences”.

14. 4. G = (v, S, v0, ↦ )
V = {v0, v1, x, y, z}, S = {x, y, z}
↦: vo ↦ xvo, vo ↦ yv1, v1 ↦yv1, v1 ↦ z.
5. G = (v, S, v0, ↦ )
V = {v0,a, b}, S = {a, b}
↦: vo ↦ aavo, vo ↦ a, v0 ↦b
Eg 6. Construct a phrase structure grammar G such that the language, L(G), of G is equal to the language L.
(i) L = {anbn/ n≥ 3} (ii) L = { strings of 0’s and 1’s with an equal number n≥0}.

15. Classification of Phase structure Grammar
Let G = (V, S, v0, ↦) be a phase structure grammar. Then we say that G is
(i) Type 0: If no restrictions are placed on the production of G.
Type 1. If for any production w1 ↦ w2, the length of w1 is less than or equal to the length of w2.
Type 2. If the left hand side of each production is a single, non-terminal symbol and the right hand side consists of one or more symbols.
Type 3.: If the left hand side of each production is a single, non-terminal symbol and the right hand side consists of one or more symbols, including at most one non-terminal symbol.

16. Type-2 or Type-3 Language: A language will be called typa-2 or type-3 if there is a grammar of type-2 or type-3 that produces it.
BNF Notation: For Type-2 grammars, there are some useful alternative methods of displaying the productions.(Backus-Naur form)
Step1: The symbol w, remains on the left and all RHS associated with w are listed together, separated by the symbol (∣).
Step2.The relational symbol ↦ is replaced by the symbol (::=) .
Step3. The non-terminal symbols, where ever they occur, are enclosed in pointed brackets < > .

17. Eg: The following grammar describes the syntax of decimal numbers and can be viewed as a mini-grammar whose corresponding language consists precisely of all properly formed decimal numbers.
Let S = { 0,1,2,3,4,5,6,7,8,9,.}
Let V be the union of S with the set
N= {decimal-number, decimal fraction, unsigned-integer, digit}
Let G be a grammar with symbol sets V and S, with starting symbol “decimal-number” and with productions given in BNF forms as follows.

18. 1. <decimal-number> ::= <unsigned integer>∣
<decimal-fraction>∣ <unsigned integer>
<decimal-fraction>
2. <decimal-fraction> ::= . <unsigned integer>
3. <unsigned integer> ::= <digit>∣<digit> < unsigned integer>
4. <digit> ::=0 ∣ 1 ∣ 2 ∣ 3 ∣4 ∣ 5 ∣ 6 ∣ 7 ∣ 8 ∣ 9
Example: By using above grammar draw derivation tree for 23.14

19. Syntax diagram: A second alternative method for displaying the production in some type-2 grammar is the syntax diagram

20. Finite State Machines
Suppose that we have a finite set S = {s0, s1, …, sn}, a finite set I, and for each x є I, a function fx: S→S. Let F = {fx/ xє I}. The triplet (S, I, F) is called a finite state machine, S is called the state set of the machine and the elements of S are called states. The set I is called the input set of the machine. Thus ,if the machine is in the state si and input x occurs, the next state of the machine will be fx(si).
Since the next fx(si) is uniquely determined by the pair (si, x), there is a function F: S x I → S given by F(si, x) = fx(si).

21. Moore Machine (Recognition Machine)
It is defined as a sequence (S, I, F, s0, T), where (S, I, F) constitute a finite-state machine, s0 Є S and T⊆ S. The state s0 is called the starting state of M, and it will be used to represent the condition of the machine before it receives any input. The set T is called the set of acceptance state of M. These states will be used in language recognition.
Note: In the digraph of Moore Machine, the acceptance states are indicated with two concentric circles.