Classification of Languages

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.

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.

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.

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

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.

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.

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

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.

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).

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.

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.

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”.

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}.

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.

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 < > .

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.

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

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

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).

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.