Regular Expressions CIS 361

Need finite descriptions of infinite sets of strings. Discover and specify “regularity”. The set of languages over a finite alphabet is uncountable, while the set of descriptions is countable Fundamental Problems

Regular Expressions Language L is regular if there exists a finite acceptor for it Any language that is described by a regular expression can be accepted by some finite automaton

Regular Expressions Regular expressions Combination of strings of symbols from some alphabet, parentheses and operators U,., * U is union (some literature uses +). (or nothing) is concatenation * is star closure or Kleene star superscripted repetition, 0 or more times + is closure superscripted repetition, 1 or more times

Specifying Lexical Structure Using Regular Expressions Have some alphabet  = set of symbols Regular expressions are built from:  - empty string Any letter from  r 1 r 2 – String r 1 followed by r 2 (concatenation) r 1 U r 2 (r 1 + r 2 ) – either regular expression r 1 or r 2 (union) r* - iterated sequence and choice  | r | r r | … Parentheses to indicate grouping/precedence

Regular Expressions Operations Union Complement Intersection Difference Concatenation Repetition Kleene star Plus operator

Regular Expressions Union L  M The union of two regular expressions Q and R is Q U R In terms of automata A and B, respectively create a new initial state q connect it to the initial states of A and B by  transitions

Regular Expressions Complement  * - L To construct the complement of a regular expression L, inspect the automaton that accepts its strings convert the automaton for L to a deterministic automaton flips favorable and nonfavorable states construct a regular expression for strings accepted by the updated automaton

Regular Expressions Complement of bit strings with at least one “1” = bit strings containing no “1”s = 0* Complement of bit strings with exactly one “1” = bit strings containing no “1”s U bit strings with at least two “1”s = 0* U (0* 1 0* 1 0*)(0 U 1)*

Regular Expressions Intersection L  M Apply DeMorgan’s law Union of the complements of L and M

Regular Expressions Difference L – M Can be expressed as the intersection of languages L and  * - M

Regular Expressions Concatenation Strings u and v over alphabet  is string uv Languages L 1 and L 2 concatenated L 1 L 2 ={uv|u  L 1, v  L 2 } Can be extended to any finite number of languages

Regular Expressions Concatenation LM Algorithm connects every favorable state of L to the initial state of M by an arrow labeled  Favorable states of L become non-favorable Favorable states of M become favorable states of the new automaton

Regular Expressions Kleene star L * In terms of automaton connect every favorable state of L to the initial state of L by a transition labeled  create a new initial state s, make it the only favorable state and connect it to the old initial state by  transition

Regular Expressions Plus (+) L + In terms of automaton connect every favorable state of L to the initial state of L by a transition labeled  That’s it. This gets one or more times to a favorable state

Naming Languages Regular sets can be named using the derivation in terms of the seed elements and the closure operations. Regular expressions formalize this approach. Regular sets  Regular Expressions Numbers  Numerals Semantics  Syntax

Regular expressions for strings over {a,b} containing at least one “a”. Focus on the one “a” (a u b)*a(a u b)* Focus on the leftmost “a” b*a(a u b)* Focus on the “a”s b*ab*(ab*)* Further optimization b*(ab*) + Example

Two regular expressions are equivalent if they represent the same regular set. Equivalence of regular expressions

Concept of Language Generated by Regular Expressions Set of all strings generated by a regular expression is the language of the regular expression In general, a language may be (countably) infinite A string in a language is often called a token

Examples of Languages and Regular Expressions  = { 0, 1,. } (0 U 1)*.(0 U 1)* - Binary floating point numbers (00)* - even-length all-zero strings 1*(01*01*)* - strings with even number of zeros  = {A,…,Z, a,…,z, 0,…,9,_ } (A U … U z)(A U … U z U 0 U … U 9 U _) * identifiers (1 U … U 9)(0 U … U 9) * natural numbers (no negatives) (0|1|2)* - trinary (base 3) numbers

Finite-State Automata Alphabet  Set of states with initial and accepting states Transitions between states, labeled with symbol(s) (0 | 1)*.(0|1)*