From NFA to minimal DFA
Bas Ploeger1
1
Rob van Glabbeek2
Jan Friso Groote1
Department of Mathematics and Computer Science
Technische Universiteit Eindhoven
The Netherlands
2
National ICT Australia
Sydney, Australia
ProSe, 19 October 2006
Overview
Introduction
Problem
Solution 1
Solution 2
Conclusions
Preliminaries
Finite automata
I
NFA is a tuple (S, Σ, →, i, F )
I
DFA: every state has at most one outgoing a-transition for
every a ∈ Σ
Preliminaries
Finite automata
I
NFA is a tuple (S, Σ, →, i, F )
I
DFA: every state has at most one outgoing a-transition for
every a ∈ Σ
Language semantics
I
I
σ
Language of a state s: L(s) = {σ ∈ Σ∗ | ∃f ∈ F . s −
→f}
0
Language preorder and equivalence on states s, s :
s vL s 0
s ≡L s 0
⇔
⇔
L(s) ⊆ L(s 0 )
L(s) = L(s 0 )
Canonization
Problem
Given an NFA, find the smallest, language equivalent DFA.
Canonization
Problem
Given an NFA, find the smallest, language equivalent DFA.
Solution
1. Determinize NFA (subset construction)
2. Minimize DFA (Hopcroft)
Canonization
Problem
Given an NFA, find the smallest, language equivalent DFA.
Solution
1. Determinize NFA (subset construction) EXPTIME
2. Minimize DFA (Hopcroft) PTIME
Canonization
Problem
Given an NFA, find the smallest, language equivalent DFA.
Solution
1. Determinize NFA (subset construction) EXPTIME
2. Minimize DFA (Hopcroft) PTIME
Minimal DFA can be exponentially larger than NFA
Example
Subset construction
DFA
NFA
{0}
0
a,b
a
a
1
2
a
b
b
3
a
Example
Subset construction
DFA
NFA
{0}
0
a
a,b
a
a
1
2
a
b
b
3
a
{1, 2}
Example
Subset construction
DFA
NFA
{0}
0
a
a,b
a
b
a
1
2
a
b
b
3
a
{1, 2}
{2}
Example
Subset construction
DFA
NFA
{0}
0
a
a,b
a
b
a
1
2
a
b
b
3
a
a
{1, 2}
{2}
Example
Subset construction
DFA
NFA
{0}
0
a
a,b
a
b
a
1
2
a
a
{1, 2}
{2}
a
b
b
3
b
{3}
Example
Subset construction
DFA
NFA
{0}
0
a
a,b
a
a
1
2
a
a
{1, 2}
b
a
a
b
b
3
b
{3}
{2}
Example
Subset construction
DFA
NFA
{0}
0
a
a,b
a
a
1
2
a
a
{1, 2}
b
a
{2}
a
b
b
3
b
b
{3}
Example
Minimization
DFA
NFA
{0}
0
a
a,b
a
a
1
2
a
a
{1, 2}
b
a
{2}
a
b
b
3
b
b
{3}
Example
Minimization
DFA
NFA
{0}
0
a,b
a
a,b
a
1
2
a
a
{1, 2}
a
b
b
3
b
{3}
Relevance to Process Theory (1)
Labelled Transition Systems
I
Process modelled by LTS (S, Σ, →, i)
I
No final states (computation “never” stops)
Relevance to Process Theory (1)
Labelled Transition Systems
I
Process modelled by LTS (S, Σ, →, i)
I
No final states (computation “never” stops)
Trace semantics
σ
I
Traces of a state s: Tr(s) = {σ ∈ Σ∗ | ∃f ∈ S . s −
→f}
I
Trace equivalence:
s ≡T s 0
⇔
Tr(s) = Tr(s 0 )
Relevance to Process Theory (1)
Labelled Transition Systems
I
Process modelled by LTS (S, Σ, →, i)
I
No final states (computation “never” stops)
Trace semantics
σ
I
Traces of a state s: Tr(s) = {σ ∈ Σ∗ | ∃f ∈ S . s −
→f}
I
Trace equivalence:
s ≡T s 0
⇔
Tr(s) = Tr(s 0 )
Bisimulation semantics
I
Bisimulation is a relation R on states satisfying, for a ∈ Σ:
I
I
I
a
a
if s R t and s −
→ s 0 , then ∃t 0 . t −
→ t 0 and s 0 R t 0 ;
a 0
a
0
if s R t and t −
→ t , then ∃s . s −
→ s 0 and s 0 R t 0 ;
Bisimulation equivalence: s ↔ s 0 if there exists a bisimulation
R with s R s 0
Relevance to Process Theory (2)
Problem
Given an LTS, minimize it under trace semantics
Relevance to Process Theory (2)
Problem
Given an LTS, minimize it under trace semantics
Facts
I
Deciding trace equivalence is PSPACE-complete
Relevance to Process Theory (2)
Problem
Given an LTS, minimize it under trace semantics
Facts
I
Deciding trace equivalence is PSPACE-complete
I
Deciding bisimulation equivalence is in PTIME
Relevance to Process Theory (2)
Problem
Given an LTS, minimize it under trace semantics
Facts
I
Deciding trace equivalence is PSPACE-complete
I
Deciding bisimulation equivalence is in PTIME
If LTS is deterministic: ≡T equals ↔
I
Relevance to Process Theory (2)
Problem
Given an LTS, minimize it under trace semantics
Facts
I
Deciding trace equivalence is PSPACE-complete
I
Deciding bisimulation equivalence is in PTIME
If LTS is deterministic: ≡T equals ↔
I
Solution
1. Determinize LTS (subset construction)
2. Minimize LTS under bisimulation semantics (Paige-Tarjan)
Problem
Overview
NFA
subset construction
DFA
minimize
mDFA
Problem
Overview
NFA
subset construction
DFA
minimize
Questions
I
What if DFA is much larger than mDFA?
mDFA
Problem
Overview
NFA
subset construction
DFA
minimize
Questions
I
What if DFA is much larger than mDFA?
I
Can we avoid the generation of redundant states?
mDFA
Problem
Overview
NFA
?
Questions
I
What if DFA is much larger than mDFA?
I
Can we avoid the generation of redundant states?
mDFA
Problem
Overview
NFA
?
Questions
I
What if DFA is much larger than mDFA?
I
Can we avoid the generation of redundant states?
I
Space efficiency (average case)
mDFA
Solution 1
Subset construction
Input:
Output:
NFA N = (SN , ΣN , −
→N , iN , FN )
DFA D = (SD , ΣD , −
→D , iD , FD )
Every DFA state P ∈ SD is a set of NFA states
Solution 1
Subset construction
Input:
Output:
NFA N = (SN , ΣN , −
→N , iN , FN )
DFA D = (SD , ΣD , −
→D , iD , FD )
Every DFA state P ∈ SD is a set of NFA states
Basic idea
I
Add “irrelevant” NFA states to P
I
State is irrelevant if it does not alter L(P)
Solution 1
Subset construction
Input:
Output:
NFA N = (SN , ΣN , −
→N , iN , FN )
DFA D = (SD , ΣD , −
→D , iD , FD )
Every DFA state P ∈ SD is a set of NFA states
Basic idea
I
Add “irrelevant” NFA states to P
I
State is irrelevant if it does not alter L(P)
L(P)?
Solution 1
Subset construction
Input:
Output:
NFA N = (SN , ΣN , −
→N , iN , FN )
DFA D = (SD , ΣD , −
→D , iD , FD )
Every DFA state P ∈ SD is a set of NFA states
Basic idea
I
Add “irrelevant” NFA states to P
I
State is irrelevant if it does not alter L(P)
L(P)?
I
Language in NFA: LN (P) =
S
p∈P
LN (p)
Solution 1
Subset construction
Input:
Output:
NFA N = (SN , ΣN , −
→N , iN , FN )
DFA D = (SD , ΣD , −
→D , iD , FD )
Every DFA state P ∈ SD is a set of NFA states
Basic idea
I
Add “irrelevant” NFA states to P
I
State is irrelevant if it does not alter L(P)
L(P)?
S
I
Language in NFA: LN (P) =
I
Language in DFA: LD (P), defined as usual
p∈P
LN (p)
Solution 1
Subset construction
Input:
Output:
NFA N = (SN , ΣN , −
→N , iN , FN )
DFA D = (SD , ΣD , −
→D , iD , FD )
Every DFA state P ∈ SD is a set of NFA states
Basic idea
I
Add “irrelevant” NFA states to P
I
State is irrelevant if it does not alter L(P)
L(P)?
S
I
Language in NFA: LN (P) =
I
Language in DFA: LD (P), defined as usual
I
Lemma: LN (P) = LD (P) for every P ⊆ SN
p∈P
LN (p)
Solution 1
Closure
I
For any set P ⊆ SN define: P = {p ∈ SN | LN (p) ⊆ LN (P)}
Solution 1
Closure
I
For any set P ⊆ SN define: P = {p ∈ SN | p vL P}
Solution 1
Closure
I
For any set P ⊆ SN define: P = {p ∈ SN | p vL P}
I
Proposition: P ≡L P for any P ⊆ SN
Solution 1
Closure
I
For any set P ⊆ SN define: P = {p ∈ SN | p vL P}
I
Proposition: P ≡L P for any P ⊆ SN
Algorithm
I
Normal subset construction, but . . .
I
Replace every generated set P by P
Solution 1
Closure
I
For any set P ⊆ SN define: P = {p ∈ SN | p vL P}
I
Proposition: P ≡L P for any P ⊆ SN
Algorithm
I
Normal subset construction, but . . .
I
Replace every generated set P by P
Main Theorem
Given an NFA, the algorithm constructs the minimal, language
equivalent DFA
Example
DFA
NFA
0
{0}
(a|b)a∗ b
a,b
a
a
2
1
a
a∗ b
b
a∗ b
b
3
{λ}
a
Example
DFA
NFA
0
{0}
(a|b)a∗ b
a,b
a
a
a
2
1
a
a∗ b
b
a∗ b
b
3
{λ}
a
{1, 2}
Example
DFA
NFA
0
{0}
(a|b)a∗ b
a,b
a
a
b
a
2
1
a
a∗ b
b
a∗ b
b
3
{λ}
a
{1, 2}
{2}
Example
DFA
NFA
0
{0}
(a|b)a∗ b
a,b
a
a
b
a
2
1
a
a∗ b
b
a∗ b
b
3
{λ}
a
{1, 2}
{1, 2}
Example
DFA
NFA
0
{0}
(a|b)a∗ b
a,b
a
a,b
a
2
1
a
a∗ b
b
a∗ b
b
3
{λ}
a
{1, 2}
Example
DFA
NFA
0
{0}
(a|b)a∗ b
a,b
a
a,b
a
2
1
a
a∗ b
b
a∗ b
b
3
{λ}
a
a
{1, 2}
Example
DFA
NFA
0
{0}
(a|b)a∗ b
a,b
a
a,b
a
2
1
a
a∗ b
b
b
3
a
a
{1, 2}
a∗ b
{λ}
b
{3}
Complexity issues
Closure
I
Language inclusion is PSPACE-complete
Complexity issues
Closure
I
Language inclusion is PSPACE-complete
Simulation semantics
I
Simulation is a relation R on states satisfying, for a ∈ ΣN :
I
I
I
a
a
if s R t and s −
→ s 0 , then ∃t 0 . t −
→ t 0 and s 0 R t 0
if s R t then s ∈ F ⇒ t ∈ F
0
Simulation preorder: s ⊂
→ s if there exists a simulation R with
s R s 0.
Complexity issues
Closure
I
Language inclusion is PSPACE-complete
Simulation semantics
I
Simulation is a relation R on states satisfying, for a ∈ ΣN :
I
I
a
a
if s R t and s −
→ s 0 , then ∃t 0 . t −
→ t 0 and s 0 R t 0
if s R t then s ∈ F ⇒ t ∈ F
I
0
Simulation preorder: s ⊂
→ s if there exists a simulation R with
s R s 0.
I
Simulation is in PTIME
Implementation
Trade-off
I
Use ⊂
→ instead of vL in closure
Implementation
Trade-off
I
Use ⊂
→ instead of vL in closure
I
Resulting DFA is not minimal
I
But at most as large as the DFA produced by subset
construction
Preliminary results
1
2
NFA
states transitions
236
456
1.438
2.821
min. DFA
states transitions
1.367
2.690
18.925
37.615
Preliminary results
NFA
states transitions
236
456
1.438
2.821
1
2
1
1
2
2
-
normal
solution 1
normal
solution 1
min. DFA
states transitions
1.367
2.690
18.925
37.615
DFA
states
transitions
42.665
83.416
4.712
9.252
7.403.224 14.616.424
176.105
348.005
memory
5,90 MB
4,39 MB
468,05 MB
63,57 MB
time
0,200 sec
0,844 sec
72,468 sec
561,022 sec
Solution 2
Idea
I
Why not remove irrelevant states from a set P?
Solution 2
Idea
I
Why not remove irrelevant states from a set P?
I
A p ∈ P is irrelevant if:
∃q ∈ P . p 6= q ∧ p vL q
Solution 2
Idea
I
Why not remove irrelevant states from a set P?
I
A p ∈ P is irrelevant if:
∃Q ⊆ P . p 6∈ Q ∧ p vL Q
Solution 2
Idea
I
Why not remove irrelevant states from a set P?
I
A p ∈ P is irrelevant if:
∃Q ⊆ P . p 6∈ Q ∧ p vL Q
Better idea
I
Replace P by the set of transitions of the states in P:
a
T = {(a, q) ∈ Σ × S | ∃p ∈ P . p −
→ q}
Solution 2
Idea
I
Why not remove irrelevant states from a set P?
I
A p ∈ P is irrelevant if:
∃Q ⊆ P . p 6∈ Q ∧ p vL Q
Better idea
I
Replace P by the set of transitions of the states in P:
a
T = {(a, q) ∈ Σ × S | ∃p ∈ P . p −
→ q}
I
Remove irrelevant transitions from T
I
A t ∈ T is irrelevant if:
∃u ∈ T . t 6= u ∧ t vL u
Definitions
Language semantics
For transitions t = (a, q) and u = (b, r ):
I
Language of t: L(t) = {aσ ∈ Σ+ | σ ∈ L(q)}
Definitions
Language semantics
For transitions t = (a, q) and u = (b, r ):
I
Language of t: L(t) = {aσ ∈ Σ+ | σ ∈ L(q)}
I
t vL u
⇔
a = b ∧ q vL r
Definitions
Language semantics
For transitions t = (a, q) and u = (b, r ):
I
Language of t: L(t) = {aσ ∈ Σ+ | σ ∈ L(q)}
I
t vL u
⇔
a = b ∧ q vL r
Compression
For any set of transitions T define:

if ¬∃t, u ∈ T . t 6= u ∧ t vL u
 T
↓(T − {t}) where t ∈ T and ∃u ∈ T . t 6= u ∧ t vL u,
↓T =

otherwise
Compression
Language equivalence
For a set of transitions T :
I
Language of T : L(T ) =
S
t∈T
L(t)
Compression
Language equivalence
For a set of transitions T :
I
Language of T : L(T ) =
I
Proposition: T ≡L ↓T
S
t∈T
L(t)
Compression
Language equivalence
For a set of transitions T :
I
Language of T : L(T ) =
I
Proposition: T ≡L ↓T
S
t∈T
Uniqueness?
I
↓T is not unique for a given T
L(t)
Compression
Language equivalence
For a set of transitions T :
I
Language of T : L(T ) =
I
Proposition: T ≡L ↓T
S
t∈T
L(t)
Uniqueness?
I
↓T is not unique for a given T
I
↓T is unique if no two states in the NFA are language
equivalent
Sets of transitions
Language equivalence
I
For any state p, Tp is the set of outgoing transitions of p
Sets of transitions
Language equivalence
I
For any state p, Tp is the set of outgoing transitions of p
I
L(p) = L(Tp )
Sets of transitions
Language equivalence
I
I
For any state p, Tp is the set of outgoing transitions of p
{λ} if p ∈ F
L(p) = L(Tp ) ∪
∅
if p 6∈ F
Sets of transitions
Language equivalence
I
I
I
For any state p, Tp is the set of outgoing transitions of p
{λ} if p ∈ F
L(p) = L(Tp ) ∪
∅
if p 6∈ F
S
For any set of states P: L(P) = p∈P L(p)
Sets of transitions
Language equivalence
I
I
I
For any state p, Tp is the set of outgoing transitions of p
{λ} if p ∈ F
L(p) = L(Tp ) ∪
∅
if p 6∈ F
For any set of states P:
S
{λ} if p ∈ F
L(P) = p∈P L(Tp ) ∪
∅
if p 6∈ F
Sets of transitions
Language equivalence
I
I
I
For any state p, Tp is the set of outgoing transitions of p
{λ} if p ∈ F
L(p) = L(Tp ) ∪
∅
if p 6∈ F
For any set of states P:
S
{λ} if p ∈ F
L(P) = p∈P L(Tp ) ∪
∅
if p 6∈ F
DFA state
I
A set of transitions T
I
T does not determine whether the state is final
I
Add a boolean
Solution 2
Algorithm
I
Minimize NFA under language semantics
I
Subset construction; DFA states are tuples (T , b)
I
Replace every (T , b) by (↓T , b)
Solution 2
Algorithm
I
Minimize NFA under language semantics
I
Subset construction; DFA states are tuples (T , b)
I
Replace every (T , b) by (↓T , b)
Implementation
I
I
Minimize NFA under simulation semantics
Use ⊂
→ for compression
Conclusions
Done:
I
Two algorithms for NFA −
→ minimal DFA
I
Aim: improve space efficiency (average case)
Conclusions
Done:
I
Two algorithms for NFA −
→ minimal DFA
I
Aim: improve space efficiency (average case)
I
Implementation trade-off: simulation semantics
I
Suboptimal results
Conclusions
Done:
I
Two algorithms for NFA −
→ minimal DFA
I
Aim: improve space efficiency (average case)
I
Implementation trade-off: simulation semantics
I
Suboptimal results
To do:
I
Implement Solution 2
Conclusions
Done:
I
Two algorithms for NFA −
→ minimal DFA
I
Aim: improve space efficiency (average case)
I
Implementation trade-off: simulation semantics
I
Suboptimal results
To do:
I
Implement Solution 2
I
Investigate performance gain in practice (benchmarking!)
I
Compare to other tools
Conclusions
Done:
I
Two algorithms for NFA −
→ minimal DFA
I
Aim: improve space efficiency (average case)
I
Implementation trade-off: simulation semantics
I
Suboptimal results
To do:
I
Implement Solution 2
I
Investigate performance gain in practice (benchmarking!)
I
Compare to other tools
I
Finish paper