St Andrews, September 59, 2006 Interpreting graphs in 0-simple - - PowerPoint PPT Presentation

St Andrews, September 5–9, 2006 Interpreting graphs in 0-simple semigroups with involution with applications to computational complexity and the finite basis problem

Mikhail Volkov (joint work with Marcel Jackson)

Ural State University, Ekaterinburg, Russia

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Overview

We are interested in:

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Overview

We are interested in:

• Computational complexity of the (Pseudo)variety

Membership Problem (PMP)

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Overview

We are interested in:

Computational complexity of the (Pseudo)variety Membership Problem (PMP) and

the Finite Basis Problem (FBP)

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Overview

We are interested in:

Computational complexity of the (Pseudo)variety Membership Problem (PMP) and

the Finite Basis Problem (FBP) In Lecture 1 I’ll explain our motivation while in Lectures 2 and 3 I’ll present some new techniques that has proved to be very efficient for semigroups equipped with an additional unary operation

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Motivation

For motivation, we look in some detail at a classic application of semigroup theory to computer science:

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Motivation

For motivation, we look in some detail at a classic application of semigroup theory to computer science: Imre Simon’s characterization of piecewise testable languages

St Andrews 2006 – p.3/32

Motivation

For motivation, we look in some detail at a classic application of semigroup theory to computer science: Imre Simon’s characterization of piecewise testable languages

is easy to explain;

St Andrews 2006 – p.3/32

Motivation

For motivation, we look in some detail at a classic application of semigroup theory to computer science: Imre Simon’s characterization of piecewise testable languages

is easy to explain;

is hard to prove;

St Andrews 2006 – p.3/32

Motivation

For motivation, we look in some detail at a classic application of semigroup theory to computer science: Imre Simon’s characterization of piecewise testable languages

is easy to explain;

is hard to prove;

has many surprising connections,

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Motivation

For motivation, we look in some detail at a classic application of semigroup theory to computer science: Imre Simon’s characterization of piecewise testable languages

is easy to explain;

is hard to prove;

has many surprising connections,

is related to my recent research.

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Hydra automata

An

• head hydra automaton

is a very simple device consisting of:

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Hydra automata

An

• head hydra automaton

is a very simple device consisting of: a tape divided into cells filled with letters of a finite input alphabet

(the numbers of cells is not bounded);

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Hydra automata

An

• head hydra automaton

is a very simple device consisting of: a tape divided into cells filled with letters of a finite input alphabet

(the numbers of cells is not bounded);

reading heads that can move along the tape independently of each other (but preserving the relative order of the heads: the first head always remains on the left of the second etc) and read symbols in the cells that they observe;

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Hydra automata

An

• head hydra automaton

is a very simple device consisting of: a tape divided into cells filled with letters of a finite input alphabet

(the numbers of cells is not bounded);

reading heads that can move along the tape independently of each other (but preserving the relative order of the heads: the first head always remains on the left of the second etc) and read symbols in the cells that they observe; finite read-only memory that contains two lists of words of length

• ver

:

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Hydra automata

An

• head hydra automaton

is a very simple device consisting of: a tape divided into cells filled with letters of a finite input alphabet

(the numbers of cells is not bounded);

reading heads that can move along the tape independently of each other (but preserving the relative order of the heads: the first head always remains on the left of the second etc) and read symbols in the cells that they observe; finite read-only memory that contains two lists of words of length

• ver

: passwords

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Hydra automata

An

• head hydra automaton

is a very simple device consisting of: a tape divided into cells filled with letters of a finite input alphabet

(the numbers of cells is not bounded);

reading heads that can move along the tape independently of each other (but preserving the relative order of the heads: the first head always remains on the left of the second etc) and read symbols in the cells that they observe; finite read-only memory that contains two lists of words of length

• ver

: passwords and taboos.

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Hydra automata

Figure 1: A 9-head hydra

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Hydra automata

Figure 3: A 7-head hydra automaton

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Hydra automata

Tape Figure 2: A 7-head hydra automaton

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Hydra automata

Word a l g e b r a m p e u l y l k y u m b Figure 2: A 7-head hydra automaton

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Hydra automata

m p e u l y l k y u m b Heads a l g e b r a

Figure 2: A 7-head hydra automaton

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Hydra automata

m p e u l y l k y u m b

Memory a l g e b r a a l g e b r a

Figure 2: A 7-head hydra automaton

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Hydra automata

m p e u l y l k y u m b

a l g e b r a a l g e b r a algebra

Figure 2: A 7-head hydra automaton

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Hydra automata

m p e u l y l k y u m b

a l g e b r a a l g e b r a algebra viagra

Figure 2: A 7-head hydra automaton

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Hydra automata

A hydra automaton accepts a word

if it finds in

• ne of the passwords but none of the
• taboos. Otherwise it rejects

.

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Hydra automata

A hydra automaton accepts a word

if it finds in

• ne of the passwords but none of the
• taboos. Otherwise it rejects

. For instance the automaton on Fig. 2 accepts the word written on the tape (AmpleUglyElkByRumba) as it finds in it the password algebra but not the tabooed word viagra.

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Hydra automata

A hydra automaton accepts a word

if it finds in

• ne of the passwords but none of the
• taboos. Otherwise it rejects

. For instance the automaton on Fig. 2 accepts the word written on the tape (AmpleUglyElkByRumba) as it finds in it the password algebra but not the tabooed word viagra. A language

is said to be recognized by a hydra automaton if accepts exactly words that are members of

. Such languages are called piecewise testable.

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Piecewise testable languages

More precisely, a language

• ❍

is called piecewise testable of height

if

can be recognized by a hydra automaton with

heads.

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Piecewise testable languages

More precisely, a language

is called piecewise testable of height

if

can be recognized by a hydra automaton with

heads. Let

[resp.

] denote the family of all piecewise testable languages [of height

] over a fixed alphabet

.

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Piecewise testable languages

More precisely, a language

is called piecewise testable of height

if

can be recognized by a hydra automaton with

heads. Let

[resp.

] denote the family of all piecewise testable languages [of height

] over a fixed alphabet

. Simon’s hierarchy of piecewise testable languages:

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Piecewise testable languages

Question 1. Given a language

, how to decide whether or not

is piecewise testable?

St Andrews 2006 – p.9/32

Piecewise testable languages

Question 1. Given a language

, how to decide whether or not

is piecewise testable? Question 2. Given a piecewise testable language

, how to determine its height (the least

such that

belongs to

but not to

)?

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Piecewise testable languages

Question 1. Given a language

, how to decide whether or not

is piecewise testable? Question 2. Given a piecewise testable language

, how to determine its height (the least

such that

belongs to

but not to

)?

• Exercise. Is the language

(

) piecewise testable?

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Piecewise testable languages

Question 1. Given a language

, how to decide whether or not

is piecewise testable? Question 2. Given a piecewise testable language

, how to determine its height (the least

such that

belongs to

but not to

)?

• Exercise. Is the language

(

) piecewise testable?

Yes

No

Don’t know

It depends

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Syntactic monoids

For a language

its syntactic congruence

is defined by

if, for any

Thus,

and

• ccur in

in the same contexts.

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Syntactic monoids

For a language

its syntactic congruence

is defined by

if, for any

Thus,

and

• ccur in

in the same contexts. One can check that

is the largest congruence on

for which

is a union of classes. The quotient monoid

is called the syntactic monoid of the language

.

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Syntactic monoids

For a language

its syntactic congruence

is defined by

if, for any

Thus,

and

• ccur in

in the same contexts. One can check that

is the largest congruence on

for which

is a union of classes. The quotient monoid

is called the syntactic monoid of the language

. For a regular language

, the syntactic monoid

can be also defined as the transition monoid of the minimal automaton of

.

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Syntactic monoids

Rather than formal definitions from the previous slide, the following crucial ideas are to be understood:

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Syntactic monoids

Rather than formal definitions from the previous slide, the following crucial ideas are to be understood: For a regular language

, its syntactic monoid

is always finite (and vice versa) — this is Myhill’s form of Kleene’s theorem.

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Syntactic monoids

Rather than formal definitions from the previous slide, the following crucial ideas are to be understood: For a regular language

, its syntactic monoid

is always finite (and vice versa) — this is Myhill’s form of Kleene’s theorem. The syntactic monoid

can be efficiently calculated whenever

is efficiently presented — say, by a regular expression or by a finite automaton.

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Syntactic monoids

Rather than formal definitions from the previous slide, the following crucial ideas are to be understood: For a regular language

, its syntactic monoid

is always finite (and vice versa) — this is Myhill’s form of Kleene’s theorem. The syntactic monoid

can be efficiently calculated whenever

is efficiently presented — say, by a regular expression or by a finite automaton. Thus, whenever

is “given”, so is

.

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Simon’s theorem

A monoid is said to be

• trivial if every principal

ideal of has a unique generator:

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Simon’s theorem

A monoid is said to be

• trivial if every principal

ideal of has a unique generator:

In different terms, being

• trivial amounts to saying

that the (bilateral) divisibility relation

is an order relation on .

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Simon’s theorem

A monoid is said to be

• trivial if every principal

ideal of has a unique generator:

In different terms, being

• trivial amounts to saying

that the (bilateral) divisibility relation

is an order relation on .

• Theorem. (Imre Simon, 1972) A language

is piecewise testable if and only if its syntactic monoid

is

• trivial.

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Simon’s theorem

1

1 1

The monoid

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Simon’s theorem

1

• 1

1

• ý

• þ

• ÿ

• þ

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Simon’s theorem

1

1 1

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Simon’s theorem

1

1 1

Similarly one can verify that

whence the monoid

is not

• trivial.

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Simon’s theorem

Example: solution to the above Exercise.

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Simon’s theorem

Example: solution to the above Exercise.

• Exercise. Is the language

(

) piecewise testable?

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Simon’s theorem

Example: solution to the above Exercise.

• Exercise. Is the language

(

) piecewise testable?

Yes

No

Don’t know

It depends

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Simon’s theorem

Example: solution to the above Exercise.

• Exercise. Is the language

(

) piecewise testable?

Yes

No

Don’t know

It depends

Yes

No

Don’t know

!

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Simon’s theorem

Example: solution to the above Exercise.

• Exercise. Is the language

(

) piecewise testable?

Yes

No

Don’t know

It depends

Yes

No

Don’t know

! If

, the language

is piecewise testable.

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Simon’s theorem

Example: solution to the above Exercise.

• Exercise. Is the language

• ❉

(

• ■

) piecewise testable?

Yes

No

Don’t know

It depends

Yes

No

Don’t know

! If

• ◆

, the language

• ❉

is piecewise testable. If

• ◆

, the language

• ❉

is not piecewise testable.

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Simon’s theorem

If

, the minimal automaton of

looks as follows:

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Simon’s theorem

If

, the minimal automaton of

looks as follows:

Using this, one readily calculates that

subject to the relations

and that

is

• trivial. In fact,

.

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Simon’s theorem

If

, the minimal automaton of

• nly slightly changes:

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Simon’s theorem

If

, the minimal automaton of

• nly slightly changes:

One gets

and we already know that

is not

• trivial. Thus, the

language

is not piecewise testable.

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Simon’s theorem

Nice: relates a very natural combinatorial property to a very natural semigroup-theoretic property.

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Simon’s theorem

Nice: relates a very natural combinatorial property to a very natural semigroup-theoretic property. Efficient: given a monoid (by its Cayley table, say), one can easily (in

time ) verify whether or not is

• trivial.

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Simon’s theorem

Nice: relates a very natural combinatorial property to a very natural semigroup-theoretic property. Efficient: given a monoid (by its Cayley table, say), one can easily (in

time ) verify whether or not is

• trivial.

Very efficient: There are polynomial time algorithms to verify if the syntactic monoid

is

• trivial

when presented the minimal automaton of

.

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Simon’s theorem

Nice: relates a very natural combinatorial property to a very natural semigroup-theoretic property. Efficient: given a monoid (by its Cayley table, say), one can easily (in

time ) verify whether or not is

• trivial.

Very efficient: There are polynomial time algorithms to verify if the syntactic monoid

is

• trivial

when presented the minimal automaton of

. Such a description of

is much more compact than the Cayley table — recall that the transition monoid of an automaton with

states may consist of as many as

elements!

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Simon’s theorem

Deep: a crossing where many ideas meet.

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Simon’s theorem

Deep: a crossing where many ideas meet. Proofs come from:

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Simon’s theorem

Deep: a crossing where many ideas meet. Proofs come from: Combinatorics on words — Simon’s original proof, 1972, 1975;

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Simon’s theorem

Deep: a crossing where many ideas meet. Proofs come from: Combinatorics on words — Simon’s original proof, 1972, 1975; Model theory — Stern, 1985;

St Andrews 2006 – p.20/32

Simon’s theorem

Deep: a crossing where many ideas meet. Proofs come from: Combinatorics on words — Simon’s original proof, 1972, 1975; Model theory — Stern, 1985; Ordered monoids — Straubing and Thérien, 1988;

St Andrews 2006 – p.20/32

Simon’s theorem

Deep: a crossing where many ideas meet. Proofs come from: Combinatorics on words — Simon’s original proof, 1972, 1975; Model theory — Stern, 1985; Ordered monoids — Straubing and Thérien, 1988; Profinite topology — Almeida, 1990;

St Andrews 2006 – p.20/32

Simon’s theorem

Deep: a crossing where many ideas meet. Proofs come from: Combinatorics on words — Simon’s original proof, 1972, 1975; Model theory — Stern, 1985; Ordered monoids — Straubing and Thérien, 1988; Profinite topology — Almeida, 1990; Transformation semigroups — Higgins, 1997.

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Recognizing height

A pseudovariety of finite monoids is a class of finite monoids closed under taking submonoids, morphic images and finite direct products.

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Recognizing height

A pseudovariety of finite monoids is a class of finite monoids closed under taking submonoids, morphic images and finite direct products. Fact (Eilenberg). Each pseudovariety is generated by syntactic monoids it contains.

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Recognizing height

A pseudovariety of finite monoids is a class of finite monoids closed under taking submonoids, morphic images and finite direct products. Fact (Eilenberg). Each pseudovariety is generated by syntactic monoids it contains. Simon’s theorem means that the pseudovariety

• f all

finite

• trivial monoids is generated by syntactic

monoids of piecewise testable languages.

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Recognizing height

Let

denote the pseudovariety of finite monoids generated by the syntactic monoids of piecewise testable languages of height

. We have

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Recognizing height

Let

denote the pseudovariety of finite monoids generated by the syntactic monoids of piecewise testable languages of height

. We have

Thus, the algebraic counterpart of Question 2 is: Question 3. Given a finite monoid and a number

, how to determine whether or not belongs to

?

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Recognizing height

Let

denote the pseudovariety of finite monoids generated by the syntactic monoids of piecewise testable languages of height

. We have

Thus, the algebraic counterpart of Question 2 is: Question 3. Given a finite monoid and a number

, how to determine whether or not belongs to

? This is a typical instance of the PMP (Pseudovariety Membership Problem). The PMP has proved to systematically arise whenever one translates a “real world” (computer science) question into algebra.

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Straubing’s theorem

— the monoid of all reflexive binary relations on a set with

• elements. It can be thought of as the

monoid of all

matrices whose diagonal entries are 1 over the boolean semiring

.

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Straubing’s theorem

— the monoid of all reflexive binary relations on a set with

• elements. It can be thought of as the

monoid of all

matrices whose diagonal entries are 1 over the boolean semiring

.

— the submonoid of

consisting of upper triangular matrices.

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Straubing’s theorem

— the monoid of all reflexive binary relations on a set with

• elements. It can be thought of as the

monoid of all

matrices whose diagonal entries are 1 over the boolean semiring

.

— the submonoid of

consisting of upper triangular matrices.

— the monoid of all order preserving and extensive transformations of a chain with

elements.

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Straubing’s theorem

— the monoid of all reflexive binary relations on a set with

• elements. It can be thought of as the

monoid of all

• ✁
• matrices whose diagonal entries

are 1 over the boolean semiring

.

— the submonoid of

consisting of upper triangular matrices.

— the monoid of all order preserving and extensive transformations of a chain with

• elements.

A transformation

• f a chain

is order preserving if

implies

for all

and extensive if

for every

.

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Straubing’s theorem

• Theorem. (Howard Straubing, 1980) For a finite

monoid the following are equivalent:

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Straubing’s theorem

• Theorem. (Howard Straubing, 1980) For a finite

monoid the following are equivalent: (i) is

• trivial;

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Straubing’s theorem

• Theorem. (Howard Straubing, 1980) For a finite

monoid the following are equivalent: (i) is

• trivial;

(ii) divides (is a morphic image of a submonoid of)

for some

;

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Straubing’s theorem

• Theorem. (Howard Straubing, 1980) For a finite

monoid the following are equivalent: (i) is

• trivial;

(ii) divides (is a morphic image of a submonoid of)

for some

; (iii) divides

for some

;

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Straubing’s theorem

• Theorem. (Howard Straubing, 1980) For a finite

monoid the following are equivalent: (i) is

• trivial;

(ii) divides (is a morphic image of a submonoid of)

for some

; (iii) divides

for some

; (iv) divides

for some

.

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Straubing’s theorem

• Theorem. (Howard Straubing, 1980) For a finite

monoid the following are equivalent: (i) is

• trivial;

(ii) divides (is a morphic image of a submonoid of)

for some

; (iii) divides

for some

; (iv) divides

for some

. This looks as a quite innocent Cayley-type theorem but in fact the proof heavily depends on Simon’s theorem, and moreover, it can be shown relatively easily that the two theorems are equivalent.

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Straubing’s theorem

• Corollary. Each of the three sequences

,

and

(

) generates the pseudovariety

• f all finite
• trivial monoids.

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Straubing’s theorem

• Corollary. Each of the three sequences

,

and

(

) generates the pseudovariety

• f all finite
• trivial monoids.

We thus have four stratifications for

:

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Straubing’s theorem

• Corollary. Each of the three sequences

,

and

• ❊

(

) generates the pseudovariety

• f all finite
• trivial monoids.

We thus have four stratifications for

:

• ◗

• ❘

• ❊

St Andrews 2006 – p.25/32

Straubing’s theorem: a refinement

Surprisingly enough, the four stratifications coincide:

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Straubing’s theorem: a refinement

Surprisingly enough, the four stratifications coincide:

• Theorem. (

, 2004) For every

, each of the monoids

,

,

generates the pseudovariety

.

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Straubing’s theorem: a refinement

Surprisingly enough, the four stratifications coincide:

• Theorem. (

, 2004) For every

, each of the monoids

,

,

generates the pseudovariety

. Thus, for each

the pseudovariety

is generated by a single finite monoid. It easily follows from some basic universal algebra that the PMP for a (pseudo)variety generated by a single finite algebra is always decidable.

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Straubing’s theorem: a refinement

Surprisingly enough, the four stratifications coincide:

• Theorem. (

, 2004) For every

, each of the monoids

,

,

generates the pseudovariety

. Thus, for each

the pseudovariety

is generated by a single finite monoid. It easily follows from some basic universal algebra that the PMP for a (pseudo)variety generated by a single finite algebra is always decidable.

• Corollary. (Jean-Eric Pin, 1984) For each

, the membership problem for the pseudovariety

is decidable, and hence, given a piecewise testable language, its height can be algorithmically determined.

St Andrews 2006 – p.26/32

Recognizing height

Is this an efficient solution?

St Andrews 2006 – p.27/32

Recognizing height

Is this an efficient solution? It doesn’t seem so — even if we allow ourselves the luxury of the language being presented by its syntactic monoid rather than by its minimal automaton.

St Andrews 2006 – p.27/32

Recognizing height

Is this an efficient solution? It doesn’t seem so — even if we allow ourselves the luxury of the language being presented by its syntactic monoid rather than by its minimal automaton. Indeed, if

and

, then the only known time bound for the algorithm that recognizes whether or not

belongs to

is

— so requires doubly exponential time (as a function of

).

St Andrews 2006 – p.27/32

Recognizing height

Is this an efficient solution? It doesn’t seem so — even if we allow ourselves the luxury of the language being presented by its syntactic monoid rather than by its minimal automaton. Indeed, if

and

, then the only known time bound for the algorithm that recognizes whether or not

belongs to

is

— so requires doubly exponential time (as a function of

). Can we do better?

St Andrews 2006 – p.27/32

Recognizing height via identities

• Lemma. (Eilenberg-Sch¨

utzenberger, 1976) Every pseudovariety generated by a single finite monoid is equational, that is, it consists precisely of finite monoids satisfying a certain system of usual monoid identities.

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Recognizing height via identities

• Lemma. (Eilenberg-Sch¨

utzenberger, 1976) Every pseudovariety generated by a single finite monoid is equational, that is, it consists precisely of finite monoids satisfying a certain system of usual monoid identities. A monoid is said to be finitely based if all identities holding in follow from a finite set of such identities (an identity basis of ).

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Recognizing height via identities

• Lemma. (Eilenberg-Sch¨

utzenberger, 1976) Every pseudovariety generated by a single finite monoid is equational, that is, it consists precisely of finite monoids satisfying a certain system of usual monoid identities. A monoid is said to be finitely based if all identities holding in follow from a finite set of such identities (an identity basis of ). If we know a finite identity basis

• f a monoid

then we can use it to efficiently decide the membership in

.

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Recognizing height via identities

• Lemma. (Eilenberg-Sch¨

utzenberger, 1976) Every pseudovariety generated by a single finite monoid is equational, that is, it consists precisely of finite monoids satisfying a certain system of usual monoid identities. A monoid is said to be finitely based if all identities holding in follow from a finite set of such identities (an identity basis of ). If we know a finite identity basis

• f a monoid

then we can use it to efficiently decide the membership in

. Indeed, given a finite monoid

, we can simply check if it satisfies each identity in

, and this requires polynomial time (as a function of

).

St Andrews 2006 – p.28/32

Recognizing height via identities

Example: The pseudovariety

• f all
• trivial

monoids of height 1 is generated by the monoid

• f

extensive endomorphisms of the chain

.

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Recognizing height via identities

Example: The pseudovariety

• f all
• trivial

monoids of height 1 is generated by the monoid

• f

extensive endomorphisms of the chain

. Therefore

is nothing but the 2-element semilattice.

St Andrews 2006 – p.29/32

Recognizing height via identities

Example: The pseudovariety

• f all
• trivial

monoids of height 1 is generated by the monoid

• f

extensive endomorphisms of the chain

. Therefore

is nothing but the 2-element semilattice. It is obvious that its identity basis consists of the two identities: the commutative law

and the idempotency law

.

St Andrews 2006 – p.29/32

Recognizing height via identities

Example: The pseudovariety

• f all
• trivial

monoids of height 1 is generated by the monoid

• f

extensive endomorphisms of the chain

. Therefore

is nothing but the 2-element semilattice. It is obvious that its identity basis consists of the two identities: the commutative law

and the idempotency law

. Thus, in order to check whether or not a given language

is piecewise testable of height 1, it suffices to verify if its syntactic monoid

is commutative and idempotent.

St Andrews 2006 – p.29/32

Recognizing height via identities

Does this approach apply to heights

?

St Andrews 2006 – p.30/32

Recognizing height via identities

Does this approach apply to heights

?

• Theorem. (

, 2004) a) The identities

,

form an identity basis of the monoid

.

St Andrews 2006 – p.30/32

Recognizing height via identities

Does this approach apply to heights

?

• Theorem. (

, 2004) a) The identities

,

form an identity basis of the monoid

. b) The identities

form an identity basis of the monoid

.

St Andrews 2006 – p.30/32

Recognizing height via identities

Does this approach apply to heights

?

• Theorem. (

, 2004) a) The identities

,

form an identity basis of the monoid

. b) The identities

form an identity basis of the monoid

. c) The identities

form an identity basis of the monoid

.

St Andrews 2006 – p.30/32

Recognizing height via identities

Does this approach apply to heights

?

• Theorem. (

, 2004) a) The identities

,

form an identity basis of the monoid

. b) The identities

form an identity basis of the monoid

. c) The identities

form an identity basis of the monoid

. d) The monoids

with

are nonfinitely based.

St Andrews 2006 – p.30/32

Recognizing height via identities

Thus, there is an efficient algorithm to check if a given piecewise testable language can be recognized by a hydra automaton with 1, 2 or 3 heads, but this approach fails for larger numbers of heads.

St Andrews 2006 – p.31/32

Recognizing height via identities

Thus, there is an efficient algorithm to check if a given piecewise testable language can be recognized by a hydra automaton with 1, 2 or 3 heads, but this approach fails for larger numbers of heads. One may conclude that the optimal number of heads is equal to 3!

St Andrews 2006 – p.31/32

Conclusions and directions

Computational complexity of (some instances of) PMP is of interest and “practical” importance

St Andrews 2006 – p.32/32

Conclusions and directions

Computational complexity of (some instances of) PMP is of interest and “practical” importance

try to find upper and lower bounds for it!

St Andrews 2006 – p.32/32

Conclusions and directions

Computational complexity of (some instances of) PMP is of interest and “practical” importance

try to find upper and lower bounds for it!

A positive answer to some instance of FBP gives an efficient algorithm for the corresponding instance of PMP

St Andrews 2006 – p.32/32

Conclusions and directions

Computational complexity of (some instances of) PMP is of interest and “practical” importance

try to find upper and lower bounds for it!

A positive answer to some instance of FBP gives an efficient algorithm for the corresponding instance of PMP

systematically investigate FBP!

St Andrews 2006 – p.32/32

Conclusions and directions

Computational complexity of (some instances of) PMP is of interest and “practical” importance

try to find upper and lower bounds for it!

A positive answer to some instance of FBP gives an efficient algorithm for the corresponding instance of PMP

systematically investigate FBP!

For many important instances, the answer to FBP is negative

St Andrews 2006 – p.32/32

Conclusions and directions

Computational complexity of (some instances of) PMP is of interest and “practical” importance

try to find upper and lower bounds for it!

A positive answer to some instance of FBP gives an efficient algorithm for the corresponding instance of PMP

systematically investigate FBP!

For many important instances, the answer to FBP is negative

develop some “equational” approach to PMP for nonfinitely based pseudovarieties!

St Andrews 2006 – p.32/32