Strings and Languages

CS 130: Theory of Computation HMU textbook, Chapter 1 (Sec 1.5) Strings and Languages

Alphabet An alphabet  is a finite nonempty set of symbols. Each symbol is a non-divisible or atomic object used in the construction of strings. Examples 1= { a, b, c, d, e, … , z } 2 = { 0, 1 }

Strings • A string w of length k on an alphabet  is a sequence of k symbolsw = a1a2a3 … akwhere each symbol aj comes from . • Examples • aabakkejaa is a string of length 10 from 1 • 00011001 is a string of length 8 from 2

Length of a string The string w = a1a2a3 … ak has length kand we write: |w| = k The unique string containing no symbols is called the empty string, denoted by . Thus || = 0. Examples:|bababaako| = 9|01101001| = 8

Set of all strings on  If  is an alphabet, we denote by k the set of all strings of length k on :k = {all w such that |w| = k }. We denote by 0 the set containing just the empty string 0 = {  }. The set of all strings on  is *, and is defined by * =  k where k = 0 to . Thus* = 0123 …Also, + = 123 …

Concatenation on strings Concatenation is a binary operation on * which takes two strings w1 and w2 and produces a third string w1w2 by juxtaposing w2 after w1. Example: If  = {0, 1} and ifw1 = 00011, w2 = 100000, thenw1w2 = 00011100000.

Prefixes and suffixes If  and  are two strings from *, and if  = , then we say that  is a prefix of , and  is a suffix of . If  is not empty, then  is a proper prefix of . Similarly, if  is not empty, then  is a proper suffix of .

Properties of concatenation * is closed under concatenation.If *, and *, then *. Concatenation is associative. If , , *, then () = (). The empty string  is the identity element. For each string  in *,  =   = .

More properties Concatenation is not commutative.If , *, then , in general. For all , *, the lengths are related as follows: || = || + ||.

Other notation wk is the concatenation of k copies of w, that is wk = www … w, k times. If w = a1a2a3 … ak, thenwR = akak-1 … a3a2a1.wR is called the reverse of w.

Languages A language L over an alphabet  is any set L of strings on . That is L Í*. The set of all languages over  is the set of all subsets of *, that is P (*) the powerset of *. Note: * is countably infinite, butP (*) is uncountably infinite.

Examples of languages  = { 0, 1 }L = { 00, 01, 10, 11 }  = { a, b }L = {  , a , a2, a3, a4, ... } = { w | w=ak, k ³ 0 }  = { 0, 1 }L = { 1, 01, 001, 0001, ... } = { w | w=0k1, k ³ 0 }

Describing a language using “set-formers” • L = { w | say something about w } • Examples •  = { 0, 1 }; L = { 000, 001, 010, 011 }L = { w | w starts with a 0 and |w|=3 } •  = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 }L = { w | w is a prime number } •  = set of ASCII charactersL = { w | w is a valid Java program }

Operations on languages Set operations on languages A and B union A È B intersection A Ç B complement Ac = W – A Concatenation of languages A and BAB = { wj | w Î A, j Î B } Kleene closure A*

Concatenation of languages Example:A = { a, ab }B = { c, bc }AB = { ac, abc, abbc }BA = { ca, cab, bca, bcab }Thus AB ¹ BA

Kleene closure A* If  is an alphabet, * the set of all strings on , A Í*, A a language on V, then the Kleene closure of A, denoted by A*, is defined as follows:A0 = {} = , A1 = A, A2 = AAA3 = A2A, ... , Ak+1 = AkA, ...A* = A0È A1È A2È A3È ...

Languages and problems • Consider the following language over = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 }:L = { w | w is a number divisible by 3 } • Consider the following problem: • Given a number, determine if the number is divisible by 3 • Problems correspond to languages • The above problem can be restated as“determine if a given number/string is in L”

Strings and Languages