From NFA to minimal DFA

Bas Ploeger1 Rob van Glabbeek2 Jan Friso Groote1

1 Department of Mathematics and Computer ScienceTechnische Universiteit Eindhoven

The Netherlands

2 National ICT AustraliaSydney, 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 forevery a ∈ Σ

Language semantics

I Language of a state s: L(s) = {σ ∈ Σ∗ | ∃f ∈ F . sσ−→ f }

I Language preorder and equivalence on states s, s ′:

s vL s ′ ⇔ L(s) ⊆ L(s ′)s ≡L s ′ ⇔ L(s) = L(s ′)

Preliminaries

Finite automata

I NFA is a tuple (S ,Σ,→, i ,F )

I DFA: every state has at most one outgoing a-transition forevery a ∈ Σ

Language semantics

I Language of a state s: L(s) = {σ ∈ Σ∗ | ∃f ∈ F . sσ−→ f }

I Language preorder and equivalence on states s, s ′:

s vL s ′ ⇔ L(s) ⊆ L(s ′)s ≡L s ′ ⇔ L(s) = L(s ′)

Canonization

ProblemGiven an NFA, find the smallest, language equivalent DFA.

Solution

1. Determinize NFA (subset construction)

2. Minimize DFA (Hopcroft)

Minimal DFA can be exponentially larger than NFA

Canonization

ProblemGiven an NFA, find the smallest, language equivalent DFA.

Solution

1. Determinize NFA (subset construction)

2. Minimize DFA (Hopcroft)

Minimal DFA can be exponentially larger than NFA

Canonization

ProblemGiven 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

Canonization

ProblemGiven 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

ExampleSubset construction

NFA

0

1 2

3

a a,b

a

b

a

a

b

DFA

{0}

{1, 2}

a

{2}{2}

b

a

{3}

b

a

b

ExampleSubset construction

NFA

0

1 2

3

a a,b

a

b

a

a

b

DFA

{0}

{1, 2}

a

{2}{2}

b

a

{3}

b

a

b

ExampleSubset construction

NFA

0

1 2

3

a a,b

a

b

a

a

b

DFA

{0}

{1, 2}

a

{2}

{2}

b

a

{3}

b

a

b

ExampleSubset construction

NFA

0

1 2

3

a a,b

a

b

a

a

b

DFA

{0}

{1, 2}

a

{2}

{2}

b

a

{3}

b

a

b

ExampleSubset construction

NFA

0

1 2

3

a a,b

a

b

a

a

b

DFA

{0}

{1, 2}

a

{2}

{2}

b

a

{3}

b

a

b

ExampleSubset construction

NFA

0

1 2

3

a a,b

a

b

a

a

b

DFA

{0}

{1, 2}

a

{2}

{2}

b

a

{3}

b

a

b

ExampleSubset construction

NFA

0

1 2

3

a a,b

a

b

a

a

b

DFA

{0}

{1, 2}

a

{2}

{2}

b

a

{3}

b

a

b

ExampleMinimization

NFA

0

1 2

3

a a,b

a

b

a

a

b

DFA

{0}

{1, 2}

a

{2}

{2}

b

a

{3}

b

a

b

ExampleMinimization

NFA

0

1 2

3

a a,b

a

b

a

a

b

DFA

{0}

{1, 2}

a,b

a

{3}

b

Relevance to Process Theory (1)

Labelled Transition Systems

I Process modelled by LTS (S ,Σ,→, i)

I No final states (computation “never” stops)

Trace semanticsI Traces of a state s: Tr(s) = {σ ∈ Σ∗ | ∃f ∈ S . s

σ−→ f }I Trace equivalence: s ≡T s ′ ⇔ Tr(s) = Tr(s ′)

Bisimulation semanticsI Bisimulation is a relation R on states satisfying, for a ∈ Σ:

I if s R t and sa−→ s ′, then ∃t ′ . t

a−→ t ′ and s ′ R t ′;I if s R t and t

a−→ t ′, then ∃s ′ . sa−→ s ′ and s ′ R t ′;

I Bisimulation equivalence: s↔ s ′ if there exists a bisimulationR with s R s ′

Relevance to Process Theory (1)

Labelled Transition Systems

I Process modelled by LTS (S ,Σ,→, i)

I No final states (computation “never” stops)

Trace semanticsI Traces of a state s: Tr(s) = {σ ∈ Σ∗ | ∃f ∈ S . s

σ−→ f }I Trace equivalence: s ≡T s ′ ⇔ Tr(s) = Tr(s ′)

Bisimulation semanticsI Bisimulation is a relation R on states satisfying, for a ∈ Σ:

I if s R t and sa−→ s ′, then ∃t ′ . t

a−→ t ′ and s ′ R t ′;I if s R t and t

a−→ t ′, then ∃s ′ . sa−→ s ′ and s ′ R t ′;

I Bisimulation equivalence: s↔ s ′ if there exists a bisimulationR with s R s ′

Relevance to Process Theory (1)

Labelled Transition Systems

I Process modelled by LTS (S ,Σ,→, i)

I No final states (computation “never” stops)

Trace semanticsI Traces of a state s: Tr(s) = {σ ∈ Σ∗ | ∃f ∈ S . s

σ−→ f }I Trace equivalence: s ≡T s ′ ⇔ Tr(s) = Tr(s ′)

Bisimulation semanticsI Bisimulation is a relation R on states satisfying, for a ∈ Σ:

I if s R t and sa−→ s ′, then ∃t ′ . t

a−→ t ′ and s ′ R t ′;I if s R t and t

a−→ t ′, then ∃s ′ . sa−→ s ′ and s ′ R t ′;

I Bisimulation equivalence: s↔ s ′ if there exists a bisimulationR with s R s ′

Relevance to Process Theory (2)

ProblemGiven an LTS, minimize it under trace semantics

Facts

I Deciding trace equivalence is PSPACE-complete

I Deciding bisimulation equivalence is in PTIME

I If LTS is deterministic: ≡T equals ↔

Solution

1. Determinize LTS (subset construction)

2. Minimize LTS under bisimulation semantics (Paige-Tarjan)

Relevance to Process Theory (2)

ProblemGiven an LTS, minimize it under trace semantics

Facts

I Deciding trace equivalence is PSPACE-complete

I Deciding bisimulation equivalence is in PTIME

I If LTS is deterministic: ≡T equals ↔

Solution

1. Determinize LTS (subset construction)

2. Minimize LTS under bisimulation semantics (Paige-Tarjan)

Relevance to Process Theory (2)

ProblemGiven an LTS, minimize it under trace semantics

Facts

I Deciding trace equivalence is PSPACE-complete

I Deciding bisimulation equivalence is in PTIME

I If LTS is deterministic: ≡T equals ↔

Solution

1. Determinize LTS (subset construction)

2. Minimize LTS under bisimulation semantics (Paige-Tarjan)

Relevance to Process Theory (2)

ProblemGiven an LTS, minimize it under trace semantics

Facts

I Deciding trace equivalence is PSPACE-complete

I Deciding bisimulation equivalence is in PTIME

I If LTS is deterministic: ≡T equals ↔

Solution

1. Determinize LTS (subset construction)

2. Minimize LTS under bisimulation semantics (Paige-Tarjan)

Relevance to Process Theory (2)

ProblemGiven an LTS, minimize it under trace semantics

Facts

I Deciding trace equivalence is PSPACE-complete

I Deciding bisimulation equivalence is in PTIME

I If LTS is deterministic: ≡T equals ↔

Solution

1. Determinize LTS (subset construction)

2. Minimize LTS under bisimulation semantics (Paige-Tarjan)

Problem

Overview

NFA mDFADFAsubset construction minimize

Questions

I What if DFA is much larger than mDFA?

I Can we avoid the generation of redundant states?

I Space efficiency (average case)

Problem

Overview

NFA mDFADFAsubset construction minimize

Questions

I What if DFA is much larger than mDFA?

I Can we avoid the generation of redundant states?

I Space efficiency (average case)

Problem

Overview

NFA mDFADFAsubset construction minimize

Questions

I What if DFA is much larger than mDFA?

I Can we avoid the generation of redundant states?

I Space efficiency (average case)

Problem

Overview

NFA mDFA?

Questions

I What if DFA is much larger than mDFA?

I Can we avoid the generation of redundant states?

I Space efficiency (average case)

Problem

Overview

NFA mDFA?

Questions

I What if DFA is much larger than mDFA?

I Can we avoid the generation of redundant states?

I Space efficiency (average case)

Solution 1

Subset construction

Input: NFA N = (SN ,ΣN ,−→N , iN ,FN )Output: DFA D = (SD,ΣD,−→D, iD,FD)

Every DFA state P ∈ SD is a set of NFA states

Basic ideaI 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) =⋃

p∈P LN (p)

I Language in DFA: LD(P), defined as usual

I Lemma: LN (P) = LD(P) for every P ⊆ SN

Solution 1

Subset construction

Input: NFA N = (SN ,ΣN ,−→N , iN ,FN )Output: DFA D = (SD,ΣD,−→D, iD,FD)

Every DFA state P ∈ SD is a set of NFA states

Basic ideaI 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) =⋃

p∈P LN (p)

I Language in DFA: LD(P), defined as usual

I Lemma: LN (P) = LD(P) for every P ⊆ SN

Solution 1

Subset construction

Input: NFA N = (SN ,ΣN ,−→N , iN ,FN )Output: DFA D = (SD,ΣD,−→D, iD,FD)

Every DFA state P ∈ SD is a set of NFA states

Basic ideaI 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) =⋃

p∈P LN (p)

I Language in DFA: LD(P), defined as usual

I Lemma: LN (P) = LD(P) for every P ⊆ SN

Solution 1

Subset construction

Input: NFA N = (SN ,ΣN ,−→N , iN ,FN )Output: DFA D = (SD,ΣD,−→D, iD,FD)

Every DFA state P ∈ SD is a set of NFA states

Basic ideaI 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) =⋃

p∈P LN (p)

I Language in DFA: LD(P), defined as usual

I Lemma: LN (P) = LD(P) for every P ⊆ SN

Solution 1

Subset construction

Input: NFA N = (SN ,ΣN ,−→N , iN ,FN )Output: DFA D = (SD,ΣD,−→D, iD,FD)

Every DFA state P ∈ SD is a set of NFA states

Basic ideaI 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) =⋃

p∈P LN (p)

I Language in DFA: LD(P), defined as usual

I Lemma: LN (P) = LD(P) for every P ⊆ SN

Solution 1

Subset construction

Input: NFA N = (SN ,ΣN ,−→N , iN ,FN )Output: DFA D = (SD,ΣD,−→D, iD,FD)

Every DFA state P ∈ SD is a set of NFA states

Basic ideaI 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) =⋃

p∈P LN (p)

I Language in DFA: LD(P), defined as usual

I Lemma: LN (P) = LD(P) for every P ⊆ SN

Solution 1

Closure

I For any set P ⊆ SN define: P = {p ∈ SN | LN (p) ⊆ LN (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 TheoremGiven an NFA, the algorithm constructs the minimal, languageequivalent DFA

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 TheoremGiven an NFA, the algorithm constructs the minimal, languageequivalent DFA

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 TheoremGiven an NFA, the algorithm constructs the minimal, languageequivalent DFA

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 TheoremGiven an NFA, the algorithm constructs the minimal, languageequivalent DFA

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 TheoremGiven an NFA, the algorithm constructs the minimal, languageequivalent DFA

Example

NFA

0 (a|b)a∗b

1a∗b

2a∗b

3 {λ}

a a,b

a

b

a

a

b

DFA

{0}

{1, 2} {2}{1, 2}

{3}

ab

a

b

Example

NFA

0 (a|b)a∗b

1a∗b

2a∗b

3 {λ}

a a,b

a

b

a

a

b

DFA

{0}

{1, 2}

{2}{1, 2}

{3}

a

b

a

b

Example

NFA

0 (a|b)a∗b

1a∗b

2a∗b

3 {λ}

a a,b

a

b

a

a

b

DFA

{0}

{1, 2} {2}

{1, 2}

{3}

ab

a

b

Example

NFA

0 (a|b)a∗b

1a∗b

2a∗b

3 {λ}

a a,b

a

b

a

a

b

DFA

{0}

{1, 2}

{2}

{1, 2}

{3}

ab

a

b

Example

NFA

0 (a|b)a∗b

1a∗b

2a∗b

3 {λ}

a a,b

a

b

a

a

b

DFA

{0}

{1, 2}

{2}{1, 2}

{3}

a,b

b

a

b

Example

NFA

0 (a|b)a∗b

1a∗b

2a∗b

3 {λ}

a a,b

a

b

a

a

b

DFA

{0}

{1, 2}

{2}{1, 2}

{3}

a,b

b

a

b

Example

NFA

0 (a|b)a∗b

1a∗b

2a∗b

3 {λ}

a a,b

a

b

a

a

b

DFA

{0}

{1, 2}

{2}{1, 2}

{3}

a,b

b

a

b

Complexity issues

Closure

I Language inclusion is PSPACE-complete

Simulation semantics

I Simulation is a relation R on states satisfying, for a ∈ ΣN :I if s R t and s

a−→ s ′, then ∃t ′ . ta−→ t ′ and s ′ R t ′

I if s R t then s ∈ F ⇒ t ∈ F

I Simulation preorder: s ⊂→ s ′ if there exists a simulation R withs R s ′.

I Simulation is in PTIME

Complexity issues

Closure

I Language inclusion is PSPACE-complete

Simulation semantics

I Simulation is a relation R on states satisfying, for a ∈ ΣN :I if s R t and s

a−→ s ′, then ∃t ′ . ta−→ t ′ and s ′ R t ′

I if s R t then s ∈ F ⇒ t ∈ F

I Simulation preorder: s ⊂→ s ′ if there exists a simulation R withs R s ′.

I Simulation is in PTIME

Complexity issues

Closure

I Language inclusion is PSPACE-complete

Simulation semantics

I Simulation is a relation R on states satisfying, for a ∈ ΣN :I if s R t and s

a−→ s ′, then ∃t ′ . ta−→ t ′ and s ′ R t ′

I if s R t then s ∈ F ⇒ t ∈ F

I Simulation preorder: s ⊂→ s ′ if there exists a simulation R withs R s ′.

I Simulation is in PTIME

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 subsetconstruction

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 subsetconstruction

Preliminary results

NFA min. DFAstates transitions states transitions

1 236 456 1.367 2.690

2 1.438 2.821 18.925 37.615

DFAstates transitions memory time

1 - normal 42.665 83.416 5,90 MB 0,200 sec1 - solution 1 4.712 9.252 4,39 MB 0,844 sec

2 - normal 7.403.224 14.616.424 468,05 MB 72,468 sec2 - solution 1 176.105 348.005 63,57 MB 561,022 sec

Preliminary results

NFA min. DFAstates transitions states transitions

1 236 456 1.367 2.690

2 1.438 2.821 18.925 37.615

DFAstates transitions memory time

1 - normal 42.665 83.416 5,90 MB 0,200 sec1 - solution 1 4.712 9.252 4,39 MB 0,844 sec

2 - normal 7.403.224 14.616.424 468,05 MB 72,468 sec2 - solution 1 176.105 348.005 63,57 MB 561,022 sec

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:

T = {(a, q) ∈ Σ× S | ∃p ∈ P . pa−→ q}

I Remove irrelevant transitions from T

I A t ∈ T is irrelevant if: ∃u ∈ T . t 6= u ∧ t vL u

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:

T = {(a, q) ∈ Σ× S | ∃p ∈ P . pa−→ q}

I Remove irrelevant transitions from T

I A t ∈ T is irrelevant if: ∃u ∈ T . t 6= u ∧ t vL u

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:

T = {(a, q) ∈ Σ× S | ∃p ∈ P . pa−→ q}

I Remove irrelevant transitions from T

I A t ∈ T is irrelevant if: ∃u ∈ T . t 6= u ∧ t vL u

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:

T = {(a, q) ∈ Σ× S | ∃p ∈ P . pa−→ q}

I Remove irrelevant transitions from T

I A t ∈ T is irrelevant if: ∃u ∈ T . t 6= u ∧ t vL u

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:

T = {(a, q) ∈ Σ× S | ∃p ∈ P . pa−→ 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)}

I t vL u ⇔ a = b ∧ q vL r

Compression

For any set of transitions T define:

↓T =

T if ¬∃t, u ∈ T . t 6= u ∧ t vL u↓(T − {t}) where t ∈ T and ∃u ∈ T . t 6= u ∧ t vL u,

otherwise

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:

↓T =

T if ¬∃t, u ∈ T . t 6= u ∧ t vL u↓(T − {t}) where t ∈ T and ∃u ∈ T . t 6= u ∧ t vL u,

otherwise

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:

↓T =

T if ¬∃t, u ∈ T . t 6= u ∧ t vL u↓(T − {t}) where t ∈ T and ∃u ∈ T . t 6= u ∧ t vL u,

otherwise

Compression

Language equivalence

For a set of transitions T :

I Language of T : L(T ) =⋃

t∈T L(t)

I Proposition: 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 languageequivalent

Compression

Language equivalence

For a set of transitions T :

I Language of T : L(T ) =⋃

t∈T L(t)

I Proposition: 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 languageequivalent

Compression

Language equivalence

For a set of transitions T :

I Language of T : L(T ) =⋃

t∈T L(t)

I Proposition: 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 languageequivalent

Compression

Language equivalence

For a set of transitions T :

I Language of T : L(T ) =⋃

t∈T L(t)

I Proposition: 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 languageequivalent

Sets of transitions

Language equivalence

I For any state p, Tp is the set of outgoing transitions of p

I L(p) = L(Tp)

∪{{λ} if p ∈ F∅ if p 6∈ F

I For any set of states P: L(P) =⋃

p∈P L(p)

DFA state

I A set of transitions T

I T does not determine whether the state is final

I Add a boolean

Sets of transitions

Language equivalence

I For any state p, Tp is the set of outgoing transitions of p

I L(p) = L(Tp)

∪{{λ} if p ∈ F∅ if p 6∈ F

I For any set of states P: L(P) =⋃

p∈P L(p)

DFA state

I A set of transitions T

I T does not determine whether the state is final

I Add a boolean

Sets of transitions

Language equivalence

I For any state p, Tp is the set of outgoing transitions of p

I L(p) = L(Tp) ∪{{λ} if p ∈ F∅ if p 6∈ F

I For any set of states P: L(P) =⋃

p∈P L(p)

DFA state

I A set of transitions T

I T does not determine whether the state is final

I Add a boolean

Sets of transitions

Language equivalence

I For any state p, Tp is the set of outgoing transitions of p

I L(p) = L(Tp) ∪{{λ} if p ∈ F∅ if p 6∈ F

I For any set of states P: L(P) =⋃

p∈P L(p)

DFA state

I A set of transitions T

I T does not determine whether the state is final

I Add a boolean

Sets of transitions

Language equivalence

I For any state p, Tp is the set of outgoing transitions of p

I L(p) = L(Tp) ∪{{λ} if p ∈ F∅ if p 6∈ F

II For any set of states P:

L(P) =⋃

p∈P L(Tp) ∪{{λ} if p ∈ F∅ 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

Sets of transitions

Language equivalence

I For any state p, Tp is the set of outgoing transitions of p

I L(p) = L(Tp) ∪{{λ} if p ∈ F∅ if p 6∈ F

II For any set of states P:

L(P) =⋃

p∈P L(Tp) ∪{{λ} if p ∈ F∅ 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)

Implementation

I Minimize NFA under simulation semantics

I Use ⊂→ for compression

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 Minimize NFA under simulation semantics

I Use ⊂→ for compression

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

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

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

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

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