229: CST Introduction to Grammars Instructor: Sherry Yang
Regular Expressions
Language Concepts Review We use language definitions as: 1) Testers: given a word, w, is w in the language? 2) Generators: generate all the words in the langauge. Language definition formats: 1) Set notation: a) List of elements. Example: L1 = {aaa, bbb, ccc, ....} b) Math formulas. Example: L2 = {anbn| n>= 0} 2) Recursive definition. 3) Regular expressions Regular Expressions Using only the letters of the alphabet and these operators (kleene closure *, positive closure +, OR + and ( ) for grouping), we can form expressions to describe words of a class of language called Regular languages. Examples: Given = {a, b, c} Regular Expressions a ab a+b a* a*b* (ab)* ((a+b) c)* Words in the Langauge as described by the regular expressions {a} {ab} <- concatenation {a, b} {, a, aa, aaa, aaaa, .....} {, a, b, ab, aab, aabb, .....} {, ab, abab, ababab.....} {, ac, bc, acac, acbc, bcbc....}
Definition of Regular Expressions: Given : 1) , , and all elements of are Regular Expressions (primitive expressions) if 2) r1 and r2 are regular expressions, so are r1+r2, r1r2, r1* and (r1) Example problems Two types of problems: 1) Given a regular expression, describe the language defined by the regular expression. 2) Given a regular language, use the regular expression to describe the language.
Every regular language MUST have a regular expression which describes it. A language is NOT regular if it can't be described using a regular expression. Examples of non-regular languages: a) Lp - palindrome b) Lbalance = {anbn| n>= 0} Problem sets: L(r) --> r (given a language L(r), find regular expression r) L(r) = {w {0, 1}*, w has at least 1 pair of consecutiv...