Theory of Computer Science C4. Regular Languages: Minimal Automata, Closure Properties and Decidability Gabriele R¨oger

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Theory of Computer Science

C4. Regular Languages: Minimal Automata, Closure Properties and Decidability

Gabriele R¨oger

University of Basel

March 27, 2019

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Minimal Automata Closure Properties Decidability Summary

Overview

Automata &

Formal Languages

Languages

& Grammars

Regular Languages

Regular Grammars

DFAs

NFAs Regular Expressions

Pumping Lemma Minimal Automata Properties Context-free

Languages

Context-sensitive &

Type-0 Languages

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Minimal Automata

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Example

The following DFAs accept the same language:

q0 q1

q2

q3 0

1

0 1 0

1

0,1

q0 q1

q2

q3

q4 0

1

0

1

0 1

0,1

Question: What is the smallest DFA that accepts this language?

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Minimal Automaton: Definition

Definition

Aminimal automaton for a regular languageL is a DFAM =hQ,Σ, δ,q0,Ei with L(M) =L and aminimal number of states.

This means there is no DFAM⁰=hQ⁰,Σ, δ⁰,q⁰₀,E⁰i withL(M) =L(M⁰) and|Q⁰|<|Q|.

How to find a minimal automaton? Idea:

Start with any DFA that accepts the language. Merge states from which exactly the same words lead to an end state.

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Minimal Automaton: Definition

Definition

Aminimal automaton for a regular languageL is a DFAM =hQ,Σ, δ,q0,Ei with L(M) =L and aminimal number of states.

This means there is no DFAM⁰=hQ⁰,Σ, δ⁰,q⁰₀,E⁰i withL(M) =L(M⁰) and|Q⁰|<|Q|.

How to find a minimal automaton?

Idea:

Start with any DFA that accepts the language.

Merge states from which exactly the same words lead to an end state.

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Minimal Automaton: Algorithm

Input: DFAM

Input:

(without states that are unreachable from the start state) Output: list of states that have to be merged

Output:

to obtain an equivalent minimal automaton

1 Create table of all pairs of states {q,q⁰}with q 6=q⁰.

2 Mark all pairs {q,q⁰}with q ∈E and q⁰ ∈/ E.

3 If there is an unmarked pair {q,q⁰} where{δ(q,a), δ(q⁰,a)}

for somea∈Σ is already marked, then also mark {q,q⁰}.

4 Repeat the last step until there are no more changes.

5 All states in pairs that are still unmarked can be merged into one state.

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Minimal Automaton: Example

q0 q1

q2

q3

q4 0

1

0

1

0 1

0,1

q0 q1 q2 q3

q₄ q3

q₂ q1

States q0,q2 andq1,q3 can be merged into one state each.

Result: q0q2 q1q3 q4 0

1 0

1

0,1

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Minimal Automaton: Example

q0 q1

q2

q3

q4 0

1

0

1

0 1

0,1

q0 q1 q2 q3

q₄ q3

q₂ q1

1 0

1

0,1

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Minimal Automaton: Example

q0 q1

q2

q3

q4 0

1

0

1

0 1

0,1

q0 q1 q2 q3

q₄ q3

q₂ q1

× × × ×

1 0

1

0,1

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Minimal Automaton: Example

q0 q1

q2

q3

q4 0

1

0

1

0 1

0,1

q0 q1 q2 q3

q₄ q3

q₂ q1

× × × ×

1 0

1

0,1

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Minimal Automaton: Example

q0 q1

q2

q3

q4 0

1

0

1

0 1

0,1

q0 q1 q2 q3

q₄ q3

q₂ q1

× × × ×

×

1 0

1

0,1

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Minimal Automaton: Example

q0 q1

q2

q3

q4 0

1

0

1

0 1

0,1

q0 q1 q2 q3

q₄ q3

q₂ q1

× × × ×

×

1 0

1

0,1

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Minimal Automaton: Example

q0 q1

q2

q3

q4 0

1

0

1

0 1

0,1

q0 q1 q2 q3

q₄ q3

q₂ q1

× × × ×

×

1 0

1

0,1

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Minimal Automaton: Example

q0 q1

q2

q3

q4 0

1

0

1

0 1

0,1

q0 q1 q2 q3

q₄ q3

q₂ q1

× × × ×

×

1 0

1

0,1

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Computation and Uniqueness of Minimal Automata

Theorem

The algorithm described on the previous slides produces a minimal automaton for the language accepted by the given input DFA.

Theorem

All minimal automata for a language L are unique up to isomorphism (i.e., renaming of states).

Without proof.

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Questions

Questions?

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Overview

Automata &

Formal Languages

Languages

& Grammars

Regular Languages

Regular Grammars

DFAs

NFAs Regular Expressions

Pumping Lemma Minimal Automata Properties Context-free

Languages

Context-sensitive &

Type-0 Languages

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Closure Properties

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Closure Properties

How can you combine regular languages in a way to get

another regular language as a result?

Picture courtesy of stockimages / FreeDigitalPhotos.net

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Closure Properties: Operations

LetLand L⁰ be regular languages over Σ and Σ⁰, respectively.

We consider the following operations:

union L∪L⁰ ={w |w ∈Lor w ∈L⁰} over Σ∪Σ⁰

intersection L∩L⁰ ={w |w ∈Landw ∈L⁰} over Σ∩Σ⁰ complementL¯={w ∈Σ^∗ |w ∈/ L}over Σ

product LL⁰={uv |u ∈L andv ∈L⁰} over Σ∪Σ⁰ special case: Lⁿ=Lⁿ⁻¹L, whereL⁰={ε}

star L^∗=S

k≥0L^k over Σ

German: Abschlusseigenschaften, Vereinigung, Schnitt, Komplement, German: Produkt, Stern

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Closure Properties

Definition (Closure)

LetK be a class of languages.

ThenK is closed. . .

. . . under union if L,L⁰ ∈ Kimplies L∪L⁰ ∈ K . . . under intersection if L,L⁰ ∈ Kimplies L∩L⁰ ∈ K . . . under complement ifL∈ Kimplies ¯L∈ K . . . under product if L,L⁰ ∈ KimpliesLL⁰ ∈ K . . . under star if L∈ K impliesL^∗∈ K

German: Abgeschlossenheit,K ist abgeschlossen unter Vereinigung German: (Schnitt, Komplement, Produkt, Stern)

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Clourse Properties of Regular Languages

Theorem

The regular languages are closed under:

union intersection complement product star

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Closure Properties

Proof.

Closure underunion, product, and starfollows because for regular expressionsα andβ, the expressions

(α|β), (αβ) and (α^∗) describe the corresponding languages.

Complement: LetM =hQ,Σ, δ,q₀,Ei be a DFA withL(M) =L.

ThenM⁰ =hQ,Σ, δ,q₀,Q\Eiis a DFA withL(M⁰) = ¯L.

Intersection: LetM1 =hQ₁,Σ1, δ1,q01,E1i and

M₂=hQ₂,Σ₂, δ₂,q₀₂,E₂i be DFAs. The product automaton

M =hQ₁×Q2,Σ1∩Σ2, δ,hq₀₁,q02i,E1×E2i

with δ(hq₁,q2i,a) =hδ₁(q1,a), δ2(q2,a)i acceptsL(M) =L(M₁)∩ L(M₂).

German: Kreuzproduktautomat

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Closure Properties

Proof.

Complement: Let M =hQ,Σ, δ,q₀,Ei be a DFA withL(M) =L.

M₂=hQ₂,Σ₂, δ₂,q₀₂,E₂i be DFAs. The product automaton

M =hQ₁×Q2,Σ1∩Σ2, δ,hq₀₁,q02i,E1×E2i

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Closure Properties

Proof.

Complement: Let M =hQ,Σ, δ,q₀,Ei be a DFA withL(M) =L.

M₂=hQ₂,Σ₂, δ₂,q₀₂,E₂i be DFAs. The product automaton M =hQ₁×Q2,Σ1∩Σ2, δ,hq₀₁,q02i,E1×E2i

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Questions

Questions?

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Decidability

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Decision Problems and Decidability (1)

“Intuitive Definition:” Decision Problem, Decidability Adecision problemis an algorithmic problem where

for a given input

an algorithmdetermines if the input has a givenproperty and then produces the output“yes” or “no” accordingly.

A decision problem isdecidableif an algorithm for it (that always gives the correct answer) exists.

German: Entscheidungsproblem, Eingabe, Eigenschaft, Ausgabe, German: entscheidbar

Note: “exists”6= “is known”

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Decision Problems and Decidability (2)

Notes:

not a formal definition: we did not formally define

“algorithm”, “input”, “output” etc. (which is not trivial) lack of a formal definition makes it difficult to prove that something is notdecidable

studied thoroughly in the next part of the course

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Decision Problems: Example

For now we describe decision problems in a semi-formal

“given”/“question”way:

Example (Emptiness Problem for Regular Languages) Theemptiness problemP_∅ for regular languages is the following problem:

Given: regular grammarG Question: Is L(G) =∅?

German: Leerheitsproblem

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Word Problem

Definition (Word Problem for Regular Languages) Theword problem P∈ for regular languages is:

Given: regular grammarG with alphabet Σ and word w ∈Σ^∗

Question: Is w ∈ L(G)?

German: Wortproblem (f¨ur regul¨are Sprachen)

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Decidability: Word Problem

Theorem

The word problem for regular languages isdecidable.

Proof.

Construct a DFAM with L(M) =L(G).

(The proofs in Chapter C2 describe a possible method.) SimulateM on input w. The simulation ends after |w|steps.

The DFAM is an end state after this iff w ∈ L(G).

Print “yes” or “no” accordingly.

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Emptiness Problem

Definition (Emptiness Problem for Regular Languages) Theemptiness problemP∅ for regular languages is:

Given: regular grammarG Question: Is L(G) =∅?

German: Leerheitsproblem

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Decidability: Emptiness Problem

Theorem

The emptiness problem for regular languages isdecidable.

Proof.

We haveL(G) =∅iff in the transition diagram of M there is no path from the start state to any end state.

This can be checked with standard graph algorithms (e.g., breadth-first search).

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Finiteness Problem

Definition (Finiteness Problem for Regular Languages) Thefiniteness problem P∞ for regular languagesis:

Given: regular grammarG Question: Is |L(G)|<∞?

German: Endlichkeitsproblem

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Decidability: Finiteness Problem

Theorem

The finiteness problem for regular languages isdecidable.

Proof.

We have|L(G)|=∞iff in the transition diagram of M there is a cycle that is reachable from the start state and from which an end state can be reached.

This can be checked with standard graph algorithms.

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Intersection Problem

Definition (Intersection Problem for Regular Languages) Theintersection problem P∩ for regular languages is:

Given: regular grammars G andG⁰ Question: Is L(G)∩ L(G⁰) =∅?

German: Schnittproblem

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Decidability: Intersection Problem

Theorem

The intersection problem for regular languages isdecidable.

Proof.

Using the closure of regular languages under intersection, we can construct (e.g., by converting to DFAs, constructing the product automaton, then converting back to a grammar) a grammarG⁰⁰ with L(G⁰⁰) =L(G)∩ L(G⁰)

and use the algorithm for the emptiness problemP_∅.

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Equivalence Problem

Definition (Equivalence Problem for Regular Languages) Theequivalence problemP₌ for regular languages is:

Given: regular grammars G andG⁰ Question: Is L(G) =L(G⁰)?

German: Aquivalenzproblem¨

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Decidability: Equivalence Problem

Theorem

The equivalence problem for regular languages isdecidable.

Proof.

In general for languagesLandL⁰, we have

L=L⁰ iff (L∩L¯⁰)∪(¯L∩L⁰) =∅.

The regular languages are closed under intersection, union and complement, and we know algorithms for these operations.

We can therefore construct a grammar for (L∩L¯⁰)∪(¯L∩L⁰) and use the algorithm for the emptiness problemP_∅.

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Questions

Questions?

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Summary

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Summary

Minimal automata are the smallest possible DFAs for a given language and are unique for each language.

The regular languages are closed under all usual operations (union, intersection, complement, product, star).

All usual decision problems (word problem, emptiness, finiteness, intersection, equivalence) aredecidable for regular languages.