Star height of regular languages Thomas Lang 14 July 2014 Overview - PowerPoint PPT Presentation

Star height of regular languages Thomas Lang 14 July 2014

Overview Introduction 1 Star height 2 BMC algorithm 3 Loop complexity 4 Connection between star height and loop complexity 5 References 6 Thomas Lang Star height of regular languages July 2014 2 / 29

Overview Introduction 1 Star height 2 BMC algorithm 3 Loop complexity 4 Connection between star height and loop complexity 5 References 6 Thomas Lang Star height of regular languages July 2014 3 / 29

Motivation Thomas Lang Star height of regular languages July 2014 4 / 29

Motivation Let L 1 , L 2 be regular languages. Thomas Lang Star height of regular languages July 2014 4 / 29

Motivation Let L 1 , L 2 be regular languages. Aim: Give meaning to the statement L 1 is more complicated than L 2 . Thomas Lang Star height of regular languages July 2014 4 / 29

Motivation Let L 1 , L 2 be regular languages. Aim: Give meaning to the statement L 1 is more complicated than L 2 . Attempt 1: L 1 is more complicated than L 2 if | L 1 | > | L 2 | Thomas Lang Star height of regular languages July 2014 4 / 29

Motivation Let L 1 , L 2 be regular languages. Aim: Give meaning to the statement L 1 is more complicated than L 2 . Attempt 1: L 1 is more complicated than L 2 if | L 1 | > | L 2 | ◮ Problem: Infinite languages not comparable Thomas Lang Star height of regular languages July 2014 4 / 29

Motivation Let L 1 , L 2 be regular languages. Aim: Give meaning to the statement L 1 is more complicated than L 2 . Attempt 1: L 1 is more complicated than L 2 if | L 1 | > | L 2 | ◮ Problem: Infinite languages not comparable Attempt 2: L 1 is more complicated than L 2 if the minimal automaton of L 1 has more states than the minimal automaton of L 2 Thomas Lang Star height of regular languages July 2014 4 / 29

Motivation Let L 1 , L 2 be regular languages. Aim: Give meaning to the statement L 1 is more complicated than L 2 . Attempt 1: L 1 is more complicated than L 2 if | L 1 | > | L 2 | ◮ Problem: Infinite languages not comparable Attempt 2: L 1 is more complicated than L 2 if the minimal automaton of L 1 has more states than the minimal automaton of L 2 ◮ Problem: { a 1000 } more complicated than { a n | n ∈ N 0 } Thomas Lang Star height of regular languages July 2014 4 / 29

Overview Introduction 1 Star height 2 BMC algorithm 3 Loop complexity 4 Connection between star height and loop complexity 5 References 6 Thomas Lang Star height of regular languages July 2014 5 / 29

Regular expressions Let A be an alphabet, then we have: Thomas Lang Star height of regular languages July 2014 6 / 29

Regular expressions Let A be an alphabet, then we have: ∅ , ε and a ∈ A are regular expressions. Thomas Lang Star height of regular languages July 2014 6 / 29

Regular expressions Let A be an alphabet, then we have: ∅ , ε and a ∈ A are regular expressions. If e and e ′ are regular expressions, then Thomas Lang Star height of regular languages July 2014 6 / 29

Regular expressions Let A be an alphabet, then we have: ∅ , ε and a ∈ A are regular expressions. If e and e ′ are regular expressions, then ◮ ( e + e ′ ), Thomas Lang Star height of regular languages July 2014 6 / 29

Regular expressions Let A be an alphabet, then we have: ∅ , ε and a ∈ A are regular expressions. If e and e ′ are regular expressions, then ◮ ( e + e ′ ), ◮ ( e · e ′ ), Thomas Lang Star height of regular languages July 2014 6 / 29

Regular expressions Let A be an alphabet, then we have: ∅ , ε and a ∈ A are regular expressions. If e and e ′ are regular expressions, then ◮ ( e + e ′ ), ◮ ( e · e ′ ), ◮ e ∗ are regular expressions. Thomas Lang Star height of regular languages July 2014 6 / 29

Regular expressions Let A be an alphabet, then we have: ∅ , ε and a ∈ A are regular expressions. If e and e ′ are regular expressions, then ◮ ( e + e ′ ), ◮ ( e · e ′ ), ◮ e ∗ are regular expressions. The language described by a regular expression e is denoted by L ( e ). Thomas Lang Star height of regular languages July 2014 6 / 29

Star height of regular expressions Thomas Lang Star height of regular languages July 2014 7 / 29

Star height of regular expressions Let e be a regular expression over an alphabet A , then its star height is defined as Thomas Lang Star height of regular languages July 2014 7 / 29

Star height of regular expressions Let e be a regular expression over an alphabet A , then its star height is defined as If e = ∅ , e = ε or e = a ∈ A , then Thomas Lang Star height of regular languages July 2014 7 / 29

Star height of regular expressions Let e be a regular expression over an alphabet A , then its star height is defined as If e = ∅ , e = ε or e = a ∈ A , then h( e ) := 0 . Thomas Lang Star height of regular languages July 2014 7 / 29

Star height of regular expressions Let e be a regular expression over an alphabet A , then its star height is defined as If e = ∅ , e = ε or e = a ∈ A , then h( e ) := 0 . If e = e ′ + e ′′ or e = e ′ · e ′′ , then Thomas Lang Star height of regular languages July 2014 7 / 29

Star height of regular expressions Let e be a regular expression over an alphabet A , then its star height is defined as If e = ∅ , e = ε or e = a ∈ A , then h( e ) := 0 . If e = e ′ + e ′′ or e = e ′ · e ′′ , then h( e ) := max( h ( e ′ ) , h ( e ′′ )) . Thomas Lang Star height of regular languages July 2014 7 / 29

Star height of regular expressions Let e be a regular expression over an alphabet A , then its star height is defined as If e = ∅ , e = ε or e = a ∈ A , then h( e ) := 0 . If e = e ′ + e ′′ or e = e ′ · e ′′ , then h( e ) := max( h ( e ′ ) , h ( e ′′ )) . If e = e ′∗ , then Thomas Lang Star height of regular languages July 2014 7 / 29

Star height of regular expressions Let e be a regular expression over an alphabet A , then its star height is defined as If e = ∅ , e = ε or e = a ∈ A , then h( e ) := 0 . If e = e ′ + e ′′ or e = e ′ · e ′′ , then h( e ) := max( h ( e ′ ) , h ( e ′′ )) . If e = e ′∗ , then h( e ) := 1 + h ( e ′ ) . Thomas Lang Star height of regular languages July 2014 7 / 29

Examples Thomas Lang Star height of regular languages July 2014 8 / 29

Examples e 1 := a ∗ ( ba ∗ ) ∗ ⇒ h( e 1 ) = Thomas Lang Star height of regular languages July 2014 8 / 29

Examples e 1 := a ∗ ( ba ∗ ) ∗ ⇒ h( e 1 ) = 2 Thomas Lang Star height of regular languages July 2014 8 / 29

Examples e 1 := a ∗ ( ba ∗ ) ∗ ⇒ h( e 1 ) = 2 e 2 := a ∗ + (( b ∗ ab ∗ ) ∗ a ) ∗ ⇒ h( e 2 ) = Thomas Lang Star height of regular languages July 2014 8 / 29

Examples e 1 := a ∗ ( ba ∗ ) ∗ ⇒ h( e 1 ) = 2 e 2 := a ∗ + (( b ∗ ab ∗ ) ∗ a ) ∗ ⇒ h( e 2 ) = 3 Thomas Lang Star height of regular languages July 2014 8 / 29

Examples e 1 := a ∗ ( ba ∗ ) ∗ ⇒ h( e 1 ) = 2 e 2 := a ∗ + (( b ∗ ab ∗ ) ∗ a ) ∗ ⇒ h( e 2 ) = 3 Caution: L ( a ∗ ) = L (( a ∗ ) ∗ ), but Thomas Lang Star height of regular languages July 2014 8 / 29

Examples e 1 := a ∗ ( ba ∗ ) ∗ ⇒ h( e 1 ) = 2 e 2 := a ∗ + (( b ∗ ab ∗ ) ∗ a ) ∗ ⇒ h( e 2 ) = 3 Caution: L ( a ∗ ) = L (( a ∗ ) ∗ ), but h( a ∗ ) = 1 � = 2 = h(( a ∗ ) ∗ ) Thomas Lang Star height of regular languages July 2014 8 / 29

Star height of regular languages Thomas Lang Star height of regular languages July 2014 9 / 29

Star height of regular languages Let L be a regular language, then its star height is defined as Thomas Lang Star height of regular languages July 2014 9 / 29

Star height of regular languages Let L be a regular language, then its star height is defined as h( L ) := min( { h( e ) | L ( e ) = L } ) . Thomas Lang Star height of regular languages July 2014 9 / 29

Overview Introduction 1 Star height 2 BMC algorithm 3 Loop complexity 4 Connection between star height and loop complexity 5 References 6 Thomas Lang Star height of regular languages July 2014 10 / 29

Description Thomas Lang Star height of regular languages July 2014 11 / 29

Description Brzozowski-McCluskey algorithm Thomas Lang Star height of regular languages July 2014 11 / 29

Description Brzozowski-McCluskey algorithm Operates on generalized automaton A , i.e. an automaton whose edges are labeled with regular expressions Thomas Lang Star height of regular languages July 2014 11 / 29

Description Brzozowski-McCluskey algorithm Operates on generalized automaton A , i.e. an automaton whose edges are labeled with regular expressions Computes regular expression e with L ( e ) = L ( A ) Thomas Lang Star height of regular languages July 2014 11 / 29

BMC algorithm Thomas Lang Star height of regular languages July 2014 12 / 29

BMC algorithm 1 Insert new state i and ε -transitions from i to all initial states Thomas Lang Star height of regular languages July 2014 12 / 29

BMC algorithm 1 Insert new state i and ε -transitions from i to all initial states 2 Insert new state t and ε -transitions from all final states to t Thomas Lang Star height of regular languages July 2014 12 / 29