Star height of regular languages
Thomas Lang
14 July 2014
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Example
1start 2 3
a
b
a
istart X 2 3 tε
a
b
a ε
istart 2 3 ta
ba
a ε
Thomas Lang Star height of regular languages July 2014 13 / 29
Example
1start 2 3
a
b
a
istart X 2 3 tε
a
b
a ε
istart 2 3 ta
ba
a ε
Thomas Lang Star height of regular languages July 2014 13 / 29
Example
1start 2 3
a
b
a
istart X 2 3 tε
a
b
a ε
istart 2 3 ta
ba
a ε
Thomas Lang Star height of regular languages July 2014 13 / 29
Example
1start 2 3
a
b
a
istart X 2 3 tε
a
b
a ε
istart 2 3 ta
ba
a ε
Thomas Lang Star height of regular languages July 2014 13 / 29
Example continued
istart X 3 ta
ba
a ε
istart 3 ta(ba)∗a ε
Thomas Lang Star height of regular languages July 2014 14 / 29
Example continued
istart X 3 ta
ba
a ε
istart 3 ta(ba)∗a ε
Thomas Lang Star height of regular languages July 2014 14 / 29
Example continued
istart X ta(ba)∗a ε
istart ta(ba)∗a
Thomas Lang Star height of regular languages July 2014 15 / 29
Example continued
istart X ta(ba)∗a ε
istart ta(ba)∗a
Thomas Lang Star height of regular languages July 2014 15 / 29
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 16 / 29
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.
Thomas Lang Star height of regular languages July 2014 17 / 29
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.
Thomas Lang Star height of regular languages July 2014 17 / 29
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.
Thomas Lang Star height of regular languages July 2014 17 / 29
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.
Thomas Lang Star height of regular languages July 2014 17 / 29
Examples
Figure : A graph
Figure : Its strongly connected components
Figure : Its ball
Thomas Lang Star height of regular languages July 2014 18 / 29
Examples
Figure : A graph
Figure : Its strongly connected components
Figure : Its ball
Thomas Lang Star height of regular languages July 2014 18 / 29
Examples
Figure : A graph
Figure : Its strongly connected components
Figure : Its ball
Thomas Lang Star height of regular languages July 2014 18 / 29
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})
Thomas Lang Star height of regular languages July 2014 19 / 29
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})
Thomas Lang Star height of regular languages July 2014 19 / 29
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})
Thomas Lang Star height of regular languages July 2014 19 / 29
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})
Thomas Lang Star height of regular languages July 2014 19 / 29
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})
Thomas Lang Star height of regular languages July 2014 19 / 29
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})
Thomas Lang Star height of regular languages July 2014 19 / 29
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})
Thomas Lang Star height of regular languages July 2014 19 / 29
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})
Thomas Lang Star height of regular languages July 2014 19 / 29
Examples
Figure : A graph G
We have lc(G ) = 1.
Thomas Lang Star height of regular languages July 2014 20 / 29
Examples
Figure : A graph G
We have lc(G ) = 1.
Thomas Lang Star height of regular languages July 2014 20 / 29
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 21 / 29
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:
Thomas Lang Star height of regular languages July 2014 22 / 29
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:
Thomas Lang Star height of regular languages July 2014 22 / 29
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:
Thomas Lang Star height of regular languages July 2014 22 / 29
Example
1start 2
34
a
b a
b
a
ba
b
Star height of result of BMC algorithm: 3
Thomas Lang Star height of regular languages July 2014 23 / 29
Example
1start 2
34
a
b a
b
a
ba
b
Star height of result of BMC algorithm: 3
Thomas Lang Star height of regular languages July 2014 23 / 29
Example continued
1start 3
24
a
b a
b
a
ba
b
Star height of result of BMC algorithm: 2
Thomas Lang Star height of regular languages July 2014 24 / 29
Example continued
1start 3
24
a
b a
b
a
ba
b
Star height of result of BMC algorithm: 2
Thomas Lang Star height of regular languages July 2014 24 / 29
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.
Thomas Lang Star height of regular languages July 2014 25 / 29
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.
Thomas Lang Star height of regular languages July 2014 25 / 29
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.
Thomas Lang Star height of regular languages July 2014 25 / 29
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.
Thomas Lang Star height of regular languages July 2014 25 / 29
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
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
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
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
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
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
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
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
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
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
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
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