Regular Languages Sequential Machine Theory Prof. K. J. Hintz Department of Electrical and Computer Engineering Lecture 3 Comments, additions and modifications by Marek Perkowski

Languages Informal Languages –English –Body –Bureaucratize Formal Languages –Rule-based –Elements are decidable –No deeper understanding required

Formal Language All the Rules of the Language Are Explicitly Stated in Terms of the Allowed Strings of Symbols, e.g., –Programming languages, e.g., C, Lisp, Ada –Military communications (formal “informal” L ) –Digital network protocols

A to... Alphabet: a finite set of symbols, aka I,  –Roman: { a, b, c,..., z } –Binary:{ 0, 1 } –Greek:{  } –Cyrillic:{ Ж, Й, Њ, С, Р,... }

String String, word: a finite ordered sequence of symbols from the alphabet, usually written with no intervening punctuation –x 1 = “ t h e “ –x 2 = “ “ –x 3 = “ “ –x 4 = “ Ж Й С Р “

String Reverse of String –The sequence of symbols written backwards Reverse of Concatenation –Strings themselves must be reversed

String Length or Size of String –The number of symbols

Strings Null String, Empty String, e,  –A string of length or size zero –The symbols e or , meant to denote the null string, are not allowed to be part of the language

Substring A String, v, Is a Substring of a String, w, iff There Are Strings x and y Such That –w = x v y –x is called the prefix –y is called the suffix –x and/or y could be 

Kleene Closure Set of All Strings,  *, I* –Order IS important –Not the same as, the powerset of the alphabet, since order is NOT important in the powerset

Concatenation Operator If x, y  I*, then the concatenation of x and y is written as –z = x  y –e.g., if x = “Red”| x | = 3 y = “skins”| y | = 5 z = x  y = “Redskins“| z | = 8

Concatenation Operator Concatenation of Any String With the Null String Results in the Original String –x  e = e  x = x –x   =   x = x Concatenation is Associative –x = abcy=defz= ghi –( x  y )  z = x  ( y  z ) –abcdefghi = abcdefghi

Language Language, L : Any Subset of the Set of All Strings of an Alphabet I*I* L1L1 L2L2

Classes of Languages Enumerated Languages –Defined by a List of All Words in the Language L e = { “quidditch”, “nimbus 2000”, } not very interesting Rule-based –Defined by Properties or a Set of Rules

Rule Based Languages A Test to Determine Whether a String Is a Member of a Language A Means of Constructing Strings That Are in the Language –Must be able to construct ALL strings in the language –Must be able to construct ONLY strings in the language

Rule-Based Language Example Let I = { a, b } A Language That Consists of All Two Letter Strings – L = { aa, ab, ba, bb } –  is not an element of the language

Empty Language Null Language, Empty Language,  : The Language With No Words in It –Not the same as  –  can be made into a language with words –A language consisting only of  is still a language

Kleene Star If is a language, then L * Is the Set of All Strings Obtained by Concatenating Zero or More Strings of L. Concatenation of Zero Strings Is  Concatenation of One String Is the String Itself L + = L * - {  }

Kleene Closure Example L = { 0, 1} L * = { , 0, 00, 000,..., 0*, 1, 11, 111,..., 1*, 01, 001, 0001,..., 0*1,... }

Kleene Closure Examples L = { ab, f } L * = { , ab, abf, fab, ffab, ffabf,... }  * = {  } if L = {  } then L * = {  }

Kleene Closure Examples Let I = { a, b } L = Language ( ( ab )* ) { , ab, abab, ababab,... } which is not the same as L = Language ( a* b* ) { , a, b, ab, aab, abb,... } The language of all strings of a’s and b’s in which the a’s, if any, come before the b’s

Recursive Language Definition Variation of Rule-Based Three-step Process 1. Specify some basic elements of the set 2. Specify the rules for forming new elements from old elements of the set 3. Specify that elements not in 1 or 2 above are NOT elements of the set

Recursive Example Two Equivalent Recursive Definitions of Rational Numbers –Rational #1 1. Rat_1 = { - ,... -3, -2, -1, 1, 2, 3,...,  } 2. if p, q  Rat_1, then p/q  Rat_1 3. the only rational numbers are those generated by 1 and 2 above.

Recursive Example –Rational #2 1. Rat_2 = { -1, +1 } 2. if p, q  Rat_2, p,q != 0, then (p+q)/p  Rat_2 3. the only rational numbers are those generated by 1 and 2 above. e.g., generates all integers

Interest in Recursive Definitions Allows Us to Prove Some Statements About What Is Computable. Leads to Proof by Induction

Principle of Mathematical Induction* Let A Be a Subset of the Natural Numbers 0  A, and for each natural number, n, –if { 0, 1,..., n }  A, –implies(n + 1)  A –then A = N * Lewis & Papadimitriou, pg. 24

Mathematical Induction In practice, mathematical induction is used to prove assertions of the form For all natural numbers, n, property P is true

Mathematical Induction Practice To prove statements of the form A = { n : P is true of n }, three steps 1. Basis Step: show that 0  A, i.e., P is true of n = 0 2. Induction Hypothesis: assume that for some arbitrary, but fixed n > 0, P holds for each natural number 0, 1,..., n

Mathematical Induction Practice 3. Induction Step: use the induction hypothesis (that P is true of n) to show that P is true of (n + 1) By the Induction Principle, Then A=N and Hence, P Is True of Every Natural Number.

Induction Example* * Lewis & Papadimitriou, pg Basis Step

Induction Example 2. Induction Hypothesis

Induction Example 3. Induction Step

Induction Example

Another Induction Example Define EVEN as 1. 0 is in EVEN 2. if x  EVEN then so is x The only elements of EVEN are those produced by 1 & 2 above. Prove by induction that all of elements of EVEN end in either 0, 2, 4, 6, or 8.

Induction Example (cont) Proof 1. Basis Step 0  EVEN by definition, therefore the property is true of the zero’th step since 0  { 0, 2, 4, 6, 8 } 2. Induction Hypothesis Assume that the last digit of (m+2)  { 0, 2, 4, 6, 8 } for 0 < m < n

Induction Example 3. Induction Step n  EVEN n2n n+1(2n+2)+2 38n+12(n+1) ends in {0,2,4,6,8} by step 2 0+2=2, 2+2=4,4+2=6 6+2=8, 8+2=0  {0,2,4,6,8}

Regular Expressions Shorthand Notation for Concisely Expressing Languages Defined Recursively Lead to a Definition of Regular Languages Provide Finite Representation of Possibly Infinite Languages Lead to Lexical Analyzers

Regular Expressions Notation

Regular Expressions Over I  and  are regular expressions a is a regular expression for each a  I If r and s are regular expressions, then so are r  s, r  s, and r* No other sequences of symbols are regular expressions

Regular Expressions Alternative 1. L (  ) = {  } L ( a ) = { a } If p and q are regular expressions, then 2. L ( pq ) = L ( p ) L ( q ) 3. L ( p  q ) = L ( p )  L ( q ) 4. L ( p* ) = L ( p )*

Regular Expressions Example What is L 3 ( ( a  b )* a ) ?

Regular Expressions Boolean OR Distributes over Concatenation –which is the language of all strings beginning with a, ending with b, and having none or more c’s in the middle, and, –all strings beginning and ending with b and having at least one c in the middle

Regular Expressions The Boolean OR Operator Can Distribute When It Is Inside a Kleene Starred Expression, but Only in Certain Ways

Regular Expressions Useful String –( a + b )* = the set of all strings of a and b of any length – L = Language ( ( a + b )* ) –{ , ab, abab, abaab, abbaab, babba, bbb,... }

Regular Languages If L  I* is finite, then L is regular. If L 1 and L 2 are regular, so are – L 3 = L 1  L 2 – L 4 = L 1  L 2 = {x 1  x 2 | x 1  L 1, x 2  L 2 } If L is regular, then so is L *, where * is the Kleene Star

Regular Languages If L Is a Finite Language, Then L Can Be Defined by a Regular Expression. The Converse Is Not True. That Is, Not All Regular Expressions Represent Finite Languages. L = Language( ( a + b )* ) Is Infinite Yet Regular