star height of regular languages - uni-freiburg.de · 2015. 4. 1. · thomas lang star height of...
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Star height of regular languages
Thomas Lang
14 July 2014
![Page 2: Star height of regular languages - uni-freiburg.de · 2015. 4. 1. · Thomas Lang Star height of regular languages July 2014 4 / 29. Overview 1 Introduction 2 Star height 3 BMC algorithm](https://reader035.vdocuments.us/reader035/viewer/2022081407/604b0de879dbe2670e59f96f/html5/thumbnails/2.jpg)
Overview
1 Introduction
2 Star height
3 BMC algorithm
4 Loop complexity
5 Connection between star height and loop complexity
6 References
Thomas Lang Star height of regular languages July 2014 2 / 29
![Page 3: Star height of regular languages - uni-freiburg.de · 2015. 4. 1. · Thomas Lang Star height of regular languages July 2014 4 / 29. Overview 1 Introduction 2 Star height 3 BMC algorithm](https://reader035.vdocuments.us/reader035/viewer/2022081407/604b0de879dbe2670e59f96f/html5/thumbnails/3.jpg)
Overview
1 Introduction
2 Star height
3 BMC algorithm
4 Loop complexity
5 Connection between star height and loop complexity
6 References
Thomas Lang Star height of regular languages July 2014 3 / 29
![Page 4: Star height of regular languages - uni-freiburg.de · 2015. 4. 1. · Thomas Lang Star height of regular languages July 2014 4 / 29. Overview 1 Introduction 2 Star height 3 BMC algorithm](https://reader035.vdocuments.us/reader035/viewer/2022081407/604b0de879dbe2670e59f96f/html5/thumbnails/4.jpg)
Motivation
Let L1, L2 be regular languages.
Aim: Give meaning to the statement
L1 is more complicated than L2.
Attempt 1: L1 is more complicated than L2 if |L1| > |L2|I Problem: Infinite languages not comparable
Attempt 2: L1 is more complicated than L2 if the minimal automatonof L1 has more states than the minimal automaton of L2
I Problem: {a1000} more complicated than {an|n ∈ N0}
Thomas Lang Star height of regular languages July 2014 4 / 29
![Page 5: Star height of regular languages - uni-freiburg.de · 2015. 4. 1. · Thomas Lang Star height of regular languages July 2014 4 / 29. Overview 1 Introduction 2 Star height 3 BMC algorithm](https://reader035.vdocuments.us/reader035/viewer/2022081407/604b0de879dbe2670e59f96f/html5/thumbnails/5.jpg)
Motivation
Let L1, L2 be regular languages.
Aim: Give meaning to the statement
L1 is more complicated than L2.
Attempt 1: L1 is more complicated than L2 if |L1| > |L2|I Problem: Infinite languages not comparable
Attempt 2: L1 is more complicated than L2 if the minimal automatonof L1 has more states than the minimal automaton of L2
I Problem: {a1000} more complicated than {an|n ∈ N0}
Thomas Lang Star height of regular languages July 2014 4 / 29
![Page 6: Star height of regular languages - uni-freiburg.de · 2015. 4. 1. · Thomas Lang Star height of regular languages July 2014 4 / 29. Overview 1 Introduction 2 Star height 3 BMC algorithm](https://reader035.vdocuments.us/reader035/viewer/2022081407/604b0de879dbe2670e59f96f/html5/thumbnails/6.jpg)
Motivation
Let L1, L2 be regular languages.
Aim: Give meaning to the statement
L1 is more complicated than L2.
Attempt 1: L1 is more complicated than L2 if |L1| > |L2|I Problem: Infinite languages not comparable
Attempt 2: L1 is more complicated than L2 if the minimal automatonof L1 has more states than the minimal automaton of L2
I Problem: {a1000} more complicated than {an|n ∈ N0}
Thomas Lang Star height of regular languages July 2014 4 / 29
![Page 7: Star height of regular languages - uni-freiburg.de · 2015. 4. 1. · Thomas Lang Star height of regular languages July 2014 4 / 29. Overview 1 Introduction 2 Star height 3 BMC algorithm](https://reader035.vdocuments.us/reader035/viewer/2022081407/604b0de879dbe2670e59f96f/html5/thumbnails/7.jpg)
Motivation
Let L1, L2 be regular languages.
Aim: Give meaning to the statement
L1 is more complicated than L2.
Attempt 1: L1 is more complicated than L2 if |L1| > |L2|
I Problem: Infinite languages not comparable
Attempt 2: L1 is more complicated than L2 if the minimal automatonof L1 has more states than the minimal automaton of L2
I Problem: {a1000} more complicated than {an|n ∈ N0}
Thomas Lang Star height of regular languages July 2014 4 / 29
![Page 8: Star height of regular languages - uni-freiburg.de · 2015. 4. 1. · Thomas Lang Star height of regular languages July 2014 4 / 29. Overview 1 Introduction 2 Star height 3 BMC algorithm](https://reader035.vdocuments.us/reader035/viewer/2022081407/604b0de879dbe2670e59f96f/html5/thumbnails/8.jpg)
Motivation
Let L1, L2 be regular languages.
Aim: Give meaning to the statement
L1 is more complicated than L2.
Attempt 1: L1 is more complicated than L2 if |L1| > |L2|I Problem: Infinite languages not comparable
Attempt 2: L1 is more complicated than L2 if the minimal automatonof L1 has more states than the minimal automaton of L2
I Problem: {a1000} more complicated than {an|n ∈ N0}
Thomas Lang Star height of regular languages July 2014 4 / 29
![Page 9: Star height of regular languages - uni-freiburg.de · 2015. 4. 1. · Thomas Lang Star height of regular languages July 2014 4 / 29. Overview 1 Introduction 2 Star height 3 BMC algorithm](https://reader035.vdocuments.us/reader035/viewer/2022081407/604b0de879dbe2670e59f96f/html5/thumbnails/9.jpg)
Motivation
Let L1, L2 be regular languages.
Aim: Give meaning to the statement
L1 is more complicated than L2.
Attempt 1: L1 is more complicated than L2 if |L1| > |L2|I Problem: Infinite languages not comparable
Attempt 2: L1 is more complicated than L2 if the minimal automatonof L1 has more states than the minimal automaton of L2
I Problem: {a1000} more complicated than {an|n ∈ N0}
Thomas Lang Star height of regular languages July 2014 4 / 29
![Page 10: Star height of regular languages - uni-freiburg.de · 2015. 4. 1. · Thomas Lang Star height of regular languages July 2014 4 / 29. Overview 1 Introduction 2 Star height 3 BMC algorithm](https://reader035.vdocuments.us/reader035/viewer/2022081407/604b0de879dbe2670e59f96f/html5/thumbnails/10.jpg)
Motivation
Let L1, L2 be regular languages.
Aim: Give meaning to the statement
L1 is more complicated than L2.
Attempt 1: L1 is more complicated than L2 if |L1| > |L2|I Problem: Infinite languages not comparable
Attempt 2: L1 is more complicated than L2 if the minimal automatonof L1 has more states than the minimal automaton of L2
I Problem: {a1000} more complicated than {an|n ∈ N0}
Thomas Lang Star height of regular languages July 2014 4 / 29
![Page 11: Star height of regular languages - uni-freiburg.de · 2015. 4. 1. · Thomas Lang Star height of regular languages July 2014 4 / 29. Overview 1 Introduction 2 Star height 3 BMC algorithm](https://reader035.vdocuments.us/reader035/viewer/2022081407/604b0de879dbe2670e59f96f/html5/thumbnails/11.jpg)
Overview
1 Introduction
2 Star height
3 BMC algorithm
4 Loop complexity
5 Connection between star height and loop complexity
6 References
Thomas Lang Star height of regular languages July 2014 5 / 29
![Page 12: Star height of regular languages - uni-freiburg.de · 2015. 4. 1. · Thomas Lang Star height of regular languages July 2014 4 / 29. Overview 1 Introduction 2 Star height 3 BMC algorithm](https://reader035.vdocuments.us/reader035/viewer/2022081407/604b0de879dbe2670e59f96f/html5/thumbnails/12.jpg)
Regular expressions
Let A be an alphabet, then we have:
∅, ε and a ∈ A are regular expressions.
If e and e ′ are regular expressions, thenI (e + e′),
I (e · e′),
I 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
![Page 13: Star height of regular languages - uni-freiburg.de · 2015. 4. 1. · Thomas Lang Star height of regular languages July 2014 4 / 29. Overview 1 Introduction 2 Star height 3 BMC algorithm](https://reader035.vdocuments.us/reader035/viewer/2022081407/604b0de879dbe2670e59f96f/html5/thumbnails/13.jpg)
Regular expressions
Let A be an alphabet, then we have:
∅, ε and a ∈ A are regular expressions.
If e and e ′ are regular expressions, thenI (e + e′),
I (e · e′),
I 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
![Page 14: Star height of regular languages - uni-freiburg.de · 2015. 4. 1. · Thomas Lang Star height of regular languages July 2014 4 / 29. Overview 1 Introduction 2 Star height 3 BMC algorithm](https://reader035.vdocuments.us/reader035/viewer/2022081407/604b0de879dbe2670e59f96f/html5/thumbnails/14.jpg)
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
I (e + e′),
I (e · e′),
I 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
![Page 15: Star height of regular languages - uni-freiburg.de · 2015. 4. 1. · Thomas Lang Star height of regular languages July 2014 4 / 29. Overview 1 Introduction 2 Star height 3 BMC algorithm](https://reader035.vdocuments.us/reader035/viewer/2022081407/604b0de879dbe2670e59f96f/html5/thumbnails/15.jpg)
Regular expressions
Let A be an alphabet, then we have:
∅, ε and a ∈ A are regular expressions.
If e and e ′ are regular expressions, thenI (e + e′),
I (e · e′),
I 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
![Page 16: Star height of regular languages - uni-freiburg.de · 2015. 4. 1. · Thomas Lang Star height of regular languages July 2014 4 / 29. Overview 1 Introduction 2 Star height 3 BMC algorithm](https://reader035.vdocuments.us/reader035/viewer/2022081407/604b0de879dbe2670e59f96f/html5/thumbnails/16.jpg)
Regular expressions
Let A be an alphabet, then we have:
∅, ε and a ∈ A are regular expressions.
If e and e ′ are regular expressions, thenI (e + e′),
I (e · e′),
I 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
![Page 17: Star height of regular languages - uni-freiburg.de · 2015. 4. 1. · Thomas Lang Star height of regular languages July 2014 4 / 29. Overview 1 Introduction 2 Star height 3 BMC algorithm](https://reader035.vdocuments.us/reader035/viewer/2022081407/604b0de879dbe2670e59f96f/html5/thumbnails/17.jpg)
Regular expressions
Let A be an alphabet, then we have:
∅, ε and a ∈ A are regular expressions.
If e and e ′ are regular expressions, thenI (e + e′),
I (e · e′),
I 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
![Page 18: Star height of regular languages - uni-freiburg.de · 2015. 4. 1. · Thomas Lang Star height of regular languages July 2014 4 / 29. Overview 1 Introduction 2 Star height 3 BMC algorithm](https://reader035.vdocuments.us/reader035/viewer/2022081407/604b0de879dbe2670e59f96f/html5/thumbnails/18.jpg)
Regular expressions
Let A be an alphabet, then we have:
∅, ε and a ∈ A are regular expressions.
If e and e ′ are regular expressions, thenI (e + e′),
I (e · e′),
I 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
![Page 19: Star height of regular languages - uni-freiburg.de · 2015. 4. 1. · Thomas Lang Star height of regular languages July 2014 4 / 29. Overview 1 Introduction 2 Star height 3 BMC algorithm](https://reader035.vdocuments.us/reader035/viewer/2022081407/604b0de879dbe2670e59f96f/html5/thumbnails/19.jpg)
Star height of regular expressions
Let e be a regular expression over an alphabet A, then its star height isdefined 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 ′∗, thenh(e) := 1 + h(e ′).
Thomas Lang Star height of regular languages July 2014 7 / 29
![Page 20: Star height of regular languages - uni-freiburg.de · 2015. 4. 1. · Thomas Lang Star height of regular languages July 2014 4 / 29. Overview 1 Introduction 2 Star height 3 BMC algorithm](https://reader035.vdocuments.us/reader035/viewer/2022081407/604b0de879dbe2670e59f96f/html5/thumbnails/20.jpg)
Star height of regular expressions
Let e be a regular expression over an alphabet A, then its star height isdefined 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 ′∗, thenh(e) := 1 + h(e ′).
Thomas Lang Star height of regular languages July 2014 7 / 29
![Page 21: Star height of regular languages - uni-freiburg.de · 2015. 4. 1. · Thomas Lang Star height of regular languages July 2014 4 / 29. Overview 1 Introduction 2 Star height 3 BMC algorithm](https://reader035.vdocuments.us/reader035/viewer/2022081407/604b0de879dbe2670e59f96f/html5/thumbnails/21.jpg)
Star height of regular expressions
Let e be a regular expression over an alphabet A, then its star height isdefined 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 ′∗, thenh(e) := 1 + h(e ′).
Thomas Lang Star height of regular languages July 2014 7 / 29
![Page 22: Star height of regular languages - uni-freiburg.de · 2015. 4. 1. · Thomas Lang Star height of regular languages July 2014 4 / 29. Overview 1 Introduction 2 Star height 3 BMC algorithm](https://reader035.vdocuments.us/reader035/viewer/2022081407/604b0de879dbe2670e59f96f/html5/thumbnails/22.jpg)
Star height of regular expressions
Let e be a regular expression over an alphabet A, then its star height isdefined 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 ′∗, thenh(e) := 1 + h(e ′).
Thomas Lang Star height of regular languages July 2014 7 / 29
![Page 23: Star height of regular languages - uni-freiburg.de · 2015. 4. 1. · Thomas Lang Star height of regular languages July 2014 4 / 29. Overview 1 Introduction 2 Star height 3 BMC algorithm](https://reader035.vdocuments.us/reader035/viewer/2022081407/604b0de879dbe2670e59f96f/html5/thumbnails/23.jpg)
Star height of regular expressions
Let e be a regular expression over an alphabet A, then its star height isdefined 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 ′∗, thenh(e) := 1 + h(e ′).
Thomas Lang Star height of regular languages July 2014 7 / 29
![Page 24: Star height of regular languages - uni-freiburg.de · 2015. 4. 1. · Thomas Lang Star height of regular languages July 2014 4 / 29. Overview 1 Introduction 2 Star height 3 BMC algorithm](https://reader035.vdocuments.us/reader035/viewer/2022081407/604b0de879dbe2670e59f96f/html5/thumbnails/24.jpg)
Star height of regular expressions
Let e be a regular expression over an alphabet A, then its star height isdefined 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 ′∗, thenh(e) := 1 + h(e ′).
Thomas Lang Star height of regular languages July 2014 7 / 29
![Page 25: Star height of regular languages - uni-freiburg.de · 2015. 4. 1. · Thomas Lang Star height of regular languages July 2014 4 / 29. Overview 1 Introduction 2 Star height 3 BMC algorithm](https://reader035.vdocuments.us/reader035/viewer/2022081407/604b0de879dbe2670e59f96f/html5/thumbnails/25.jpg)
Star height of regular expressions
Let e be a regular expression over an alphabet A, then its star height isdefined 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
![Page 26: Star height of regular languages - uni-freiburg.de · 2015. 4. 1. · Thomas Lang Star height of regular languages July 2014 4 / 29. Overview 1 Introduction 2 Star height 3 BMC algorithm](https://reader035.vdocuments.us/reader035/viewer/2022081407/604b0de879dbe2670e59f96f/html5/thumbnails/26.jpg)
Star height of regular expressions
Let e be a regular expression over an alphabet A, then its star height isdefined 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 ′∗, thenh(e) := 1 + h(e ′).
Thomas Lang Star height of regular languages July 2014 7 / 29
![Page 27: Star height of regular languages - uni-freiburg.de · 2015. 4. 1. · Thomas Lang Star height of regular languages July 2014 4 / 29. Overview 1 Introduction 2 Star height 3 BMC algorithm](https://reader035.vdocuments.us/reader035/viewer/2022081407/604b0de879dbe2670e59f96f/html5/thumbnails/27.jpg)
Examples
e1 := a∗(ba∗)∗ ⇒ h(e1) = 2
e2 := a∗ + ((b∗ab∗)∗a)∗ ⇒ h(e2) = 3
Caution: L(a∗) = L((a∗)∗), but h(a∗) = 1 6= 2 = h((a∗)∗)
Thomas Lang Star height of regular languages July 2014 8 / 29
![Page 28: Star height of regular languages - uni-freiburg.de · 2015. 4. 1. · Thomas Lang Star height of regular languages July 2014 4 / 29. Overview 1 Introduction 2 Star height 3 BMC algorithm](https://reader035.vdocuments.us/reader035/viewer/2022081407/604b0de879dbe2670e59f96f/html5/thumbnails/28.jpg)
Examples
e1 := a∗(ba∗)∗ ⇒ h(e1) =
2
e2 := a∗ + ((b∗ab∗)∗a)∗ ⇒ h(e2) = 3
Caution: L(a∗) = L((a∗)∗), but h(a∗) = 1 6= 2 = h((a∗)∗)
Thomas Lang Star height of regular languages July 2014 8 / 29
![Page 29: Star height of regular languages - uni-freiburg.de · 2015. 4. 1. · Thomas Lang Star height of regular languages July 2014 4 / 29. Overview 1 Introduction 2 Star height 3 BMC algorithm](https://reader035.vdocuments.us/reader035/viewer/2022081407/604b0de879dbe2670e59f96f/html5/thumbnails/29.jpg)
Examples
e1 := a∗(ba∗)∗ ⇒ h(e1) = 2
e2 := a∗ + ((b∗ab∗)∗a)∗ ⇒ h(e2) = 3
Caution: L(a∗) = L((a∗)∗), but h(a∗) = 1 6= 2 = h((a∗)∗)
Thomas Lang Star height of regular languages July 2014 8 / 29
![Page 30: Star height of regular languages - uni-freiburg.de · 2015. 4. 1. · Thomas Lang Star height of regular languages July 2014 4 / 29. Overview 1 Introduction 2 Star height 3 BMC algorithm](https://reader035.vdocuments.us/reader035/viewer/2022081407/604b0de879dbe2670e59f96f/html5/thumbnails/30.jpg)
Examples
e1 := a∗(ba∗)∗ ⇒ h(e1) = 2
e2 := a∗ + ((b∗ab∗)∗a)∗ ⇒ h(e2) =
3
Caution: L(a∗) = L((a∗)∗), but h(a∗) = 1 6= 2 = h((a∗)∗)
Thomas Lang Star height of regular languages July 2014 8 / 29
![Page 31: Star height of regular languages - uni-freiburg.de · 2015. 4. 1. · Thomas Lang Star height of regular languages July 2014 4 / 29. Overview 1 Introduction 2 Star height 3 BMC algorithm](https://reader035.vdocuments.us/reader035/viewer/2022081407/604b0de879dbe2670e59f96f/html5/thumbnails/31.jpg)
Examples
e1 := a∗(ba∗)∗ ⇒ h(e1) = 2
e2 := a∗ + ((b∗ab∗)∗a)∗ ⇒ h(e2) = 3
Caution: L(a∗) = L((a∗)∗), but h(a∗) = 1 6= 2 = h((a∗)∗)
Thomas Lang Star height of regular languages July 2014 8 / 29
![Page 32: Star height of regular languages - uni-freiburg.de · 2015. 4. 1. · Thomas Lang Star height of regular languages July 2014 4 / 29. Overview 1 Introduction 2 Star height 3 BMC algorithm](https://reader035.vdocuments.us/reader035/viewer/2022081407/604b0de879dbe2670e59f96f/html5/thumbnails/32.jpg)
Examples
e1 := a∗(ba∗)∗ ⇒ h(e1) = 2
e2 := a∗ + ((b∗ab∗)∗a)∗ ⇒ h(e2) = 3
Caution: L(a∗) = L((a∗)∗), but
h(a∗) = 1 6= 2 = h((a∗)∗)
Thomas Lang Star height of regular languages July 2014 8 / 29
![Page 33: Star height of regular languages - uni-freiburg.de · 2015. 4. 1. · Thomas Lang Star height of regular languages July 2014 4 / 29. Overview 1 Introduction 2 Star height 3 BMC algorithm](https://reader035.vdocuments.us/reader035/viewer/2022081407/604b0de879dbe2670e59f96f/html5/thumbnails/33.jpg)
Examples
e1 := a∗(ba∗)∗ ⇒ h(e1) = 2
e2 := a∗ + ((b∗ab∗)∗a)∗ ⇒ h(e2) = 3
Caution: L(a∗) = L((a∗)∗), but h(a∗) = 1 6= 2 = h((a∗)∗)
Thomas Lang Star height of regular languages July 2014 8 / 29
![Page 34: Star height of regular languages - uni-freiburg.de · 2015. 4. 1. · Thomas Lang Star height of regular languages July 2014 4 / 29. Overview 1 Introduction 2 Star height 3 BMC algorithm](https://reader035.vdocuments.us/reader035/viewer/2022081407/604b0de879dbe2670e59f96f/html5/thumbnails/34.jpg)
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
![Page 35: Star height of regular languages - uni-freiburg.de · 2015. 4. 1. · Thomas Lang Star height of regular languages July 2014 4 / 29. Overview 1 Introduction 2 Star height 3 BMC algorithm](https://reader035.vdocuments.us/reader035/viewer/2022081407/604b0de879dbe2670e59f96f/html5/thumbnails/35.jpg)
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
![Page 36: Star height of regular languages - uni-freiburg.de · 2015. 4. 1. · Thomas Lang Star height of regular languages July 2014 4 / 29. Overview 1 Introduction 2 Star height 3 BMC algorithm](https://reader035.vdocuments.us/reader035/viewer/2022081407/604b0de879dbe2670e59f96f/html5/thumbnails/36.jpg)
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
![Page 37: Star height of regular languages - uni-freiburg.de · 2015. 4. 1. · Thomas Lang Star height of regular languages July 2014 4 / 29. Overview 1 Introduction 2 Star height 3 BMC algorithm](https://reader035.vdocuments.us/reader035/viewer/2022081407/604b0de879dbe2670e59f96f/html5/thumbnails/37.jpg)
Overview
1 Introduction
2 Star height
3 BMC algorithm
4 Loop complexity
5 Connection between star height and loop complexity
6 References
Thomas Lang Star height of regular languages July 2014 10 / 29
![Page 38: Star height of regular languages - uni-freiburg.de · 2015. 4. 1. · Thomas Lang Star height of regular languages July 2014 4 / 29. Overview 1 Introduction 2 Star height 3 BMC algorithm](https://reader035.vdocuments.us/reader035/viewer/2022081407/604b0de879dbe2670e59f96f/html5/thumbnails/38.jpg)
Description
Brzozowski-McCluskey algorithm
Operates on generalized automaton A, i.e. an automaton whoseedges 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
![Page 39: Star height of regular languages - uni-freiburg.de · 2015. 4. 1. · Thomas Lang Star height of regular languages July 2014 4 / 29. Overview 1 Introduction 2 Star height 3 BMC algorithm](https://reader035.vdocuments.us/reader035/viewer/2022081407/604b0de879dbe2670e59f96f/html5/thumbnails/39.jpg)
Description
Brzozowski-McCluskey algorithm
Operates on generalized automaton A, i.e. an automaton whoseedges 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
![Page 40: Star height of regular languages - uni-freiburg.de · 2015. 4. 1. · Thomas Lang Star height of regular languages July 2014 4 / 29. Overview 1 Introduction 2 Star height 3 BMC algorithm](https://reader035.vdocuments.us/reader035/viewer/2022081407/604b0de879dbe2670e59f96f/html5/thumbnails/40.jpg)
Description
Brzozowski-McCluskey algorithm
Operates on generalized automaton A, i.e. an automaton whoseedges 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
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Description
Brzozowski-McCluskey algorithm
Operates on generalized automaton A, i.e. an automaton whoseedges are labeled with regular expressions
Computes regular expression e with L(e) = L(A)
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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
3 Successively remove the states of A, updating the edges in thefollowing manner:
A X Be1
e2
e3
A Be1e∗2e3
4 Join all expressions on edges from i to t using ’+’
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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
3 Successively remove the states of A, updating the edges in thefollowing manner:
A X Be1
e2
e3
A Be1e∗2e3
4 Join all expressions on edges from i to t using ’+’
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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
3 Successively remove the states of A, updating the edges in thefollowing manner:
A X Be1
e2
e3
A Be1e∗2e3
4 Join all expressions on edges from i to t using ’+’
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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
3 Successively remove the states of A, updating the edges in thefollowing manner:
A X Be1
e2
e3
A Be1e∗2e3
4 Join all expressions on edges from i to t using ’+’
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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
3 Successively remove the states of A, updating the edges in thefollowing manner:
A X Be1
e2
e3
A Be1e∗2e3
4 Join all expressions on edges from i to t using ’+’
Thomas Lang Star height of regular languages July 2014 12 / 29
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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
3 Successively remove the states of A, updating the edges in thefollowing manner:
A X Be1
e2
e3
A Be1e∗2e3
4 Join all expressions on edges from i to t using ’+’
Thomas Lang Star height of regular languages July 2014 12 / 29
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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
3 Successively remove the states of A, updating the edges in thefollowing manner:
A X Be1
e2
e3
A Be1e∗2e3
4 Join all expressions on edges from i to t using ’+’
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Example
1start 2 3
a
b
a
istart X 2 3 tε
a
b
a ε
istart 2 3 ta
ba
a ε
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Example
1start 2 3
a
b
a
istart X 2 3 tε
a
b
a ε
istart 2 3 ta
ba
a ε
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Example
1start 2 3
a
b
a
istart X 2 3 tε
a
b
a ε
istart 2 3 ta
ba
a ε
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Example
1start 2 3
a
b
a
istart X 2 3 tε
a
b
a ε
istart 2 3 ta
ba
a ε
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Example continued
istart X 3 ta
ba
a ε
istart 3 ta(ba)∗a ε
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Example continued
istart X 3 ta
ba
a ε
istart 3 ta(ba)∗a ε
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Example continued
istart X ta(ba)∗a ε
istart ta(ba)∗a
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Example continued
istart X ta(ba)∗a ε
istart ta(ba)∗a
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Overview
1 Introduction
2 Star height
3 BMC algorithm
4 Loop complexity
5 Connection between star height and loop complexity
6 References
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Graph-theoretic concepts
A set of vertices V of a graph is called
strongly connected, if every vertex in V is reachable by every othervertex in V .
a ball, if V is strongly connected and has at least one edge.
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Graph-theoretic concepts
A set of vertices V of a graph is called
strongly connected, if every vertex in V is reachable by every othervertex in V .
a ball, if V is strongly connected and has at least one edge.
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Graph-theoretic concepts
A set of vertices V of a graph is called
strongly connected, if every vertex in V is reachable by every othervertex in V .
a ball, if V is strongly connected and has at least one edge.
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Graph-theoretic concepts
A set of vertices V of a graph is called
strongly connected, if every vertex in V is reachable by every othervertex in V .
a ball, if V is strongly connected and has at least one edge.
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Examples
Figure : A graph
Figure : Its strongly connected components
Figure : Its ball
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Examples
Figure : A graph
Figure : Its strongly connected components
Figure : Its ball
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Examples
Figure : A graph
Figure : Its strongly connected components
Figure : Its ball
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Loop complexity
Let G be a graph, then its loop complexity is defined as follows:
If G contains no balllc(G ) := 0
If G is not a ball
lc(G ) := max({lc(B)|B is a ball of G})
If G is a ball
lc(G ) := 1 + min({lc(G \ {v})|v is a vertex of G})
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Loop complexity
Let G be a graph, then its loop complexity is defined as follows:
If G contains no balllc(G ) := 0
If G is not a ball
lc(G ) := max({lc(B)|B is a ball of G})
If G is a ball
lc(G ) := 1 + min({lc(G \ {v})|v is a vertex of G})
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Loop complexity
Let G be a graph, then its loop complexity is defined as follows:
If G contains no ball
lc(G ) := 0
If G is not a ball
lc(G ) := max({lc(B)|B is a ball of G})
If G is a ball
lc(G ) := 1 + min({lc(G \ {v})|v is a vertex of G})
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Loop complexity
Let G be a graph, then its loop complexity is defined as follows:
If G contains no balllc(G ) := 0
If G is not a ball
lc(G ) := max({lc(B)|B is a ball of G})
If G is a ball
lc(G ) := 1 + min({lc(G \ {v})|v is a vertex of G})
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Loop complexity
Let G be a graph, then its loop complexity is defined as follows:
If G contains no balllc(G ) := 0
If G is not a ball
lc(G ) := max({lc(B)|B is a ball of G})
If G is a ball
lc(G ) := 1 + min({lc(G \ {v})|v is a vertex of G})
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Loop complexity
Let G be a graph, then its loop complexity is defined as follows:
If G contains no balllc(G ) := 0
If G is not a ball
lc(G ) := max({lc(B)|B is a ball of G})
If G is a ball
lc(G ) := 1 + min({lc(G \ {v})|v is a vertex of G})
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Loop complexity
Let G be a graph, then its loop complexity is defined as follows:
If G contains no balllc(G ) := 0
If G is not a ball
lc(G ) := max({lc(B)|B is a ball of G})
If G is a ball
lc(G ) := 1 + min({lc(G \ {v})|v is a vertex of G})
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Loop complexity
Let G be a graph, then its loop complexity is defined as follows:
If G contains no balllc(G ) := 0
If G is not a ball
lc(G ) := max({lc(B)|B is a ball of G})
If G is a ball
lc(G ) := 1 + min({lc(G \ {v})|v is a vertex of G})
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Examples
Figure : A graph G
We have lc(G ) = 1.
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Examples
Figure : A graph G
We have lc(G ) = 1.
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Overview
1 Introduction
2 Star height
3 BMC algorithm
4 Loop complexity
5 Connection between star height and loop complexity
6 References
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Theorem
The loop complexity of a trim automaton A is equal to the minimum ofthe star heights of the expressions obtained by the different possible runsof the BMC algorithm on A.
In general, given an automaton A and two total orders on its states ω1
and ω2, we have
h(BMC(A, ω1)) 6= h(BMC(A, ω2))
as the following example shows:
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Theorem
The loop complexity of a trim automaton A is equal to the minimum ofthe star heights of the expressions obtained by the different possible runsof the BMC algorithm on A.
In general, given an automaton A and two total orders on its states ω1
and ω2, we have
h(BMC(A, ω1)) 6= h(BMC(A, ω2))
as the following example shows:
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Theorem
The loop complexity of a trim automaton A is equal to the minimum ofthe star heights of the expressions obtained by the different possible runsof the BMC algorithm on A.
In general, given an automaton A and two total orders on its states ω1
and ω2, we have
h(BMC(A, ω1)) 6= h(BMC(A, ω2))
as the following example shows:
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Example
1start 2
34
a
b a
b
a
ba
b
Star height of result of BMC algorithm: 3
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Example
1start 2
34
a
b a
b
a
ba
b
Star height of result of BMC algorithm: 3
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Example continued
1start 3
24
a
b a
b
a
ba
b
Star height of result of BMC algorithm: 2
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Example continued
1start 3
24
a
b a
b
a
ba
b
Star height of result of BMC algorithm: 2
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Minimal automata
Consider now a regular language L.
Does the loop complexity of its minimal automaton correspond toh(L)?
Not necessarily, as the following example shows.
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Minimal automata
Consider now a regular language L.
Does the loop complexity of its minimal automaton correspond toh(L)?
Not necessarily, as the following example shows.
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Minimal automata
Consider now a regular language L.
Does the loop complexity of its minimal automaton correspond toh(L)?
Not necessarily, as the following example shows.
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Minimal automata
Consider now a regular language L.
Does the loop complexity of its minimal automaton correspond toh(L)?
Not necessarily, as the following example shows.
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Example
Figure : Minimal automata A, B, C for their respective languages1
lc(A) = 3, lc(B) = 2, lc(C) = 1
But: L(A) = L(B) ∪ L(C)
Therefore: h(L(A)) ≤ 2 < 3 = lc(A)
1LoSa00, On the star height of regular languagesThomas Lang Star height of regular languages July 2014 26 / 29
![Page 88: Star height of regular languages - uni-freiburg.de · 2015. 4. 1. · Thomas Lang Star height of regular languages July 2014 4 / 29. Overview 1 Introduction 2 Star height 3 BMC algorithm](https://reader035.vdocuments.us/reader035/viewer/2022081407/604b0de879dbe2670e59f96f/html5/thumbnails/88.jpg)
Example
Figure : Minimal automata A, B, C for their respective languages1
lc(A) = 3,
lc(B) = 2, lc(C) = 1
But: L(A) = L(B) ∪ L(C)
Therefore: h(L(A)) ≤ 2 < 3 = lc(A)
1LoSa00, On the star height of regular languagesThomas Lang Star height of regular languages July 2014 26 / 29
![Page 89: Star height of regular languages - uni-freiburg.de · 2015. 4. 1. · Thomas Lang Star height of regular languages July 2014 4 / 29. Overview 1 Introduction 2 Star height 3 BMC algorithm](https://reader035.vdocuments.us/reader035/viewer/2022081407/604b0de879dbe2670e59f96f/html5/thumbnails/89.jpg)
Example
Figure : Minimal automata A, B, C for their respective languages1
lc(A) = 3, lc(B) = 2,
lc(C) = 1
But: L(A) = L(B) ∪ L(C)
Therefore: h(L(A)) ≤ 2 < 3 = lc(A)
1LoSa00, On the star height of regular languagesThomas Lang Star height of regular languages July 2014 26 / 29
![Page 90: Star height of regular languages - uni-freiburg.de · 2015. 4. 1. · Thomas Lang Star height of regular languages July 2014 4 / 29. Overview 1 Introduction 2 Star height 3 BMC algorithm](https://reader035.vdocuments.us/reader035/viewer/2022081407/604b0de879dbe2670e59f96f/html5/thumbnails/90.jpg)
Example
Figure : Minimal automata A, B, C for their respective languages1
lc(A) = 3, lc(B) = 2, lc(C) = 1
But: L(A) = L(B) ∪ L(C)
Therefore: h(L(A)) ≤ 2 < 3 = lc(A)
1LoSa00, On the star height of regular languagesThomas Lang Star height of regular languages July 2014 26 / 29
![Page 91: Star height of regular languages - uni-freiburg.de · 2015. 4. 1. · Thomas Lang Star height of regular languages July 2014 4 / 29. Overview 1 Introduction 2 Star height 3 BMC algorithm](https://reader035.vdocuments.us/reader035/viewer/2022081407/604b0de879dbe2670e59f96f/html5/thumbnails/91.jpg)
Example
Figure : Minimal automata A, B, C for their respective languages1
lc(A) = 3, lc(B) = 2, lc(C) = 1
But: L(A) = L(B) ∪ L(C)
Therefore: h(L(A)) ≤ 2 < 3 = lc(A)
1LoSa00, On the star height of regular languagesThomas Lang Star height of regular languages July 2014 26 / 29
![Page 92: Star height of regular languages - uni-freiburg.de · 2015. 4. 1. · Thomas Lang Star height of regular languages July 2014 4 / 29. Overview 1 Introduction 2 Star height 3 BMC algorithm](https://reader035.vdocuments.us/reader035/viewer/2022081407/604b0de879dbe2670e59f96f/html5/thumbnails/92.jpg)
Example
Figure : Minimal automata A, B, C for their respective languages1
lc(A) = 3, lc(B) = 2, lc(C) = 1
But: L(A) = L(B) ∪ L(C)
Therefore: h(L(A)) ≤ 2 < 3 = lc(A)
1LoSa00, On the star height of regular languagesThomas Lang Star height of regular languages July 2014 26 / 29
![Page 93: Star height of regular languages - uni-freiburg.de · 2015. 4. 1. · Thomas Lang Star height of regular languages July 2014 4 / 29. Overview 1 Introduction 2 Star height 3 BMC algorithm](https://reader035.vdocuments.us/reader035/viewer/2022081407/604b0de879dbe2670e59f96f/html5/thumbnails/93.jpg)
Computability
Is the star height of a regular language computable?
Not in general
But: For a subset of the regular languages (pure-group languages) itis computable
Thomas Lang Star height of regular languages July 2014 27 / 29
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Computability
Is the star height of a regular language computable?
Not in general
But: For a subset of the regular languages (pure-group languages) itis computable
Thomas Lang Star height of regular languages July 2014 27 / 29
![Page 95: Star height of regular languages - uni-freiburg.de · 2015. 4. 1. · Thomas Lang Star height of regular languages July 2014 4 / 29. Overview 1 Introduction 2 Star height 3 BMC algorithm](https://reader035.vdocuments.us/reader035/viewer/2022081407/604b0de879dbe2670e59f96f/html5/thumbnails/95.jpg)
Computability
Is the star height of a regular language computable?
Not in general
But: For a subset of the regular languages (pure-group languages) itis computable
Thomas Lang Star height of regular languages July 2014 27 / 29
![Page 96: Star height of regular languages - uni-freiburg.de · 2015. 4. 1. · Thomas Lang Star height of regular languages July 2014 4 / 29. Overview 1 Introduction 2 Star height 3 BMC algorithm](https://reader035.vdocuments.us/reader035/viewer/2022081407/604b0de879dbe2670e59f96f/html5/thumbnails/96.jpg)
Computability
Is the star height of a regular language computable?
Not in general
But: For a subset of the regular languages (pure-group languages) itis computable
Thomas Lang Star height of regular languages July 2014 27 / 29
![Page 97: Star height of regular languages - uni-freiburg.de · 2015. 4. 1. · Thomas Lang Star height of regular languages July 2014 4 / 29. Overview 1 Introduction 2 Star height 3 BMC algorithm](https://reader035.vdocuments.us/reader035/viewer/2022081407/604b0de879dbe2670e59f96f/html5/thumbnails/97.jpg)
Overview
1 Introduction
2 Star height
3 BMC algorithm
4 Loop complexity
5 Connection between star height and loop complexity
6 References
Thomas Lang Star height of regular languages July 2014 28 / 29
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References
[LoSa00] Lombardy, S.; Sakarovitch, J.: On the star height of ra-tional languages. In: Proceedings of the 3rd InternationalConference on Words, Languages and Combinatorics, Kyoto(Japan), March 2000
Thomas Lang Star height of regular languages July 2014 29 / 29