CPSC 121: Models of Computation Instructor: Bob Woodham woodham@cs.ubc.ca Department of Computer Science University of British Columbia Lecture Notes 2009/2010, Section 203 CPSC 121: Models of Computation
Menu March 15, 2010 Topics: DFAs and Regular Expressions Pre-class reading: Today: Epp 12.1 Lab 7 (Pre-Lab) Reminders: Lab 8 this week (pre-lab worth marks) 2nd Midterm Wednesday, March 24, 7:00–8:00pm Final exam Thursday, Apr 29, noon, OSBO A Bob’s course office hours, Friday, 10:30–noon, ICCS 119 Bring iClickers to class READ the WebCT Vista course announcements board www: http://www.ugrad.cs.ubc.ca/~cs121/ WebCT Vista: http://www.vista.ubc.ca CPSC 121: Models of Computation
Formal Languages: Terminology An alphabet, Σ , is a finite set of characters A string over Σ is an ordered n-tuple of elements of Σ , written without parentheses or commas, or the null (i.e., empty) string ǫ A formal language over Σ is a set of strings over the alphabet, Σ CPSC 121: Models of Computation
Formal Languages: Notation Let Σ be a given alphabet. Then Σ n is the set of all strings over Σ with length n 1 Σ ∗ is the set of all strings over Σ . 2 Σ ∗ is called the Kleene closure of Σ Σ + is the set of all strings over Σ with length at least 1. 3 Σ + is called the positive closure of Σ CPSC 121: Models of Computation
Operations on Formal Languages Let L and L ′ be languages over Σ . Three new languages are defined as follows: The concatentation of L and L ′ , denoted LL ′ , is the set 1 LL ′ = { xy | x ∈ L and y ∈ L ′ } The union of L and L ′ , denoted L ∪ L ′ , is the set 2 L ∪ L ′ = { x | x ∈ L or x ∈ L ′ } The Kleene closure of L , denoted L ∗ , is the set 3 L ∗ = { x | x is the concatentation of any number of strings in L } NOTE: ǫ ∈ L ∗ CPSC 121: Models of Computation
Definition: Regular Expression Let Σ be a given (finite) alphabet. The following defines the regular expressions over Σ : Base cases: ∅ , ǫ and each individual character in Σ are 1 regular expressions over Σ Recursive cases: If r and s are regular expressions over Σ , 2 then so are the following: ( rs ) , the concatenation of r and s ( r | s ) , read as r or s ( r ∗ ) , the Kleene closure of R Exclusion: Nothing is a regular expression over Σ except 3 for the objects defined in the 1. Base cases and the 2. Recursive cases given above CPSC 121: Models of Computation
Language Defined by a Regular Expression Let Σ be a given (finite) alphabet. The function L that associates a language to each regular expression, r , over Σ is defined below. L ( r ) is called the language defined by r Base cases: L ( ∅ ) = ∅ , L ( ǫ ) = { ǫ } , ∀ a ∈ Σ , L ( a ) = { a } 1 Recursive cases: If L ( r ) and L ( s ) are languages defined by 2 regular expressions r and s over Σ , then L ( rs ) = L ( r ) L ( s ) L ( r | s ) = L ( r ) ∪ L ( s ) L ( r ∗ ) = ( L ( r )) ∗ CPSC 121: Models of Computation
Result: Regular Languages and DFAs By definition, regular languages over a given alphabet, Σ , are (formal) languages that can be defined by a regular expression, r over Σ Kleene’s theorem: The set of languages defined by regular expressions is identical to the set of languages accepted by a deterministic finite state automaton (DFA). That is, Given any language that is accepted by a DFA, there is a regular expression that defines the same language Given any language defined by a regular expression, there is a DFA that accepts the same language CPSC 121: Models of Computation
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