[PPT] - Syntactic Monoids in a Category CALCO 2015 Ji r Ad amek, Stefan PowerPoint Presentation

SLIDE 1

Syntactic Monoids in a Category

CALCO 2015 Jiˇ r´ ı Ad´ amek, Stefan Milius and Henning Urbat June 25, 2015

SLIDE 2

Overview

Category theory has all the tools for studying automata theory: e.g. extensive work on minimization

Goguen, Arbib and Manes 1970’s Bonchi, Bonsangue, Rutten, Silva 2012 Bezhanishvili, Kupke, Panangaden 2013 Ad´ amek, Milius, Myers, Urbat 2014 . . .

universal algebra: Lawvere theories, monads, monoidal categories, . . . What about algebraic automata theory?

Syntactic Monoids in a Category June 25, 2015 2 / 17

SLIDE 3

Overview

Category theory has all the tools for studying automata theory: e.g. extensive work on minimization

Goguen, Arbib and Manes 1970’s Bonchi, Bonsangue, Rutten, Silva 2012 Bezhanishvili, Kupke, Panangaden 2013 Ad´ amek, Milius, Myers, Urbat 2014 . . .

universal algebra: Lawvere theories, monads, monoidal categories, . . . What about algebraic automata theory?

Syntactic Monoids in a Category June 25, 2015 2 / 17

SLIDE 4

Overview

Category theory has all the tools for studying automata theory: e.g. extensive work on minimization

Goguen, Arbib and Manes 1970’s Bonchi, Bonsangue, Rutten, Silva 2012 Bezhanishvili, Kupke, Panangaden 2013 Ad´ amek, Milius, Myers, Urbat 2014 . . .

universal algebra: Lawvere theories, monads, monoidal categories, . . . What about algebraic automata theory?

Syntactic Monoids in a Category June 25, 2015 2 / 17

SLIDE 5

Overview

Algebraic automata theory

Automata and languages vs. associated algebraic structures Rabin/Scott 1959: syntactic monoids Pol´ ak 2001: syntactic idempotent semirings Reutenauer 1980: syntactic associative algebras (for weighted languages)

Similar but non-uniform concepts, constructions, theorems, proofs.

Syntactic Monoids in a Category June 25, 2015 3 / 17

SLIDE 6

Overview

Algebraic automata theory

Automata and languages vs. associated algebraic structures Rabin/Scott 1959: syntactic monoids Pol´ ak 2001: syntactic idempotent semirings Reutenauer 1980: syntactic associative algebras (for weighted languages)

Similar but non-uniform concepts, constructions, theorems, proofs.

Syntactic Monoids in a Category June 25, 2015 3 / 17

SLIDE 7

Overview

Algebraic automata theory

Automata and languages vs. associated algebraic structures Rabin/Scott 1959: syntactic monoids Pol´ ak 2001: syntactic idempotent semirings Reutenauer 1980: syntactic associative algebras (for weighted languages)

Similar but non-uniform concepts, constructions, theorems, proofs.

Syntactic Monoids in a Category June 25, 2015 3 / 17

SLIDE 8

Overview

Algebraic automata theory

Automata and languages vs. associated algebraic structures Rabin/Scott 1959: syntactic monoids Pol´ ak 2001: syntactic idempotent semirings Reutenauer 1980: syntactic associative algebras (for weighted languages)

Similar but non-uniform concepts, constructions, theorems, proofs.

Syntactic Monoids in a Category June 25, 2015 3 / 17

SLIDE 9

Overview

Algebraic automata theory

Automata and languages vs. associated algebraic structures Rabin/Scott 1959: syntactic monoids Pol´ ak 2001: syntactic idempotent semirings Reutenauer 1980: syntactic associative algebras (for weighted languages)

Similar but non-uniform concepts, constructions, theorems, proofs.

Syntactic Monoids in a Category June 25, 2015 3 / 17

SLIDE 10

Overview

(Syntactic) monoids, idempotent semirings, associative algebras, . . . What do they have in common? They are monoid objects in different algebraic categories D

namely D = sets, semilattices and semimodules.

And these base categories D are symmetric monoidal closed.

Our goal

A uniform theory of algebraic recognition in a symmetric monoidal

closed (algebraic) category (D, ⊗, I).

Syntactic Monoids in a Category June 25, 2015 4 / 17

SLIDE 11

Overview

(Syntactic) monoids, idempotent semirings, associative algebras, . . . What do they have in common? They are monoid objects in different algebraic categories D

namely D = sets, semilattices and semimodules.

And these base categories D are symmetric monoidal closed.

Our goal

A uniform theory of algebraic recognition in a symmetric monoidal

closed (algebraic) category (D, ⊗, I).

Syntactic Monoids in a Category June 25, 2015 4 / 17

SLIDE 12

Overview

(Syntactic) monoids, idempotent semirings, associative algebras, . . . What do they have in common? They are monoid objects in different algebraic categories D

namely D = sets, semilattices and semimodules.

And these base categories D are symmetric monoidal closed.

Our goal

A uniform theory of algebraic recognition in a symmetric monoidal

closed (algebraic) category (D, ⊗, I).

Syntactic Monoids in a Category June 25, 2015 4 / 17

SLIDE 13

Overview

(Syntactic) monoids, idempotent semirings, associative algebras, . . . What do they have in common? They are monoid objects in different algebraic categories D

namely D = sets, semilattices and semimodules.

And these base categories D are symmetric monoidal closed.

Our goal

A uniform theory of algebraic recognition in a symmetric monoidal

closed (algebraic) category (D, ⊗, I).

Syntactic Monoids in a Category June 25, 2015 4 / 17

SLIDE 14

The Categorical Setting

A symmetric monoidal closed category (D, ⊗, I) g : A ⊗ B → C λg : A → [B, C] . . . with limits, colimits and (strong epi, mono)-factorizations. Two fixed D-objects: an input object Σ, and an output object O.

Observation (Goguen 1971)

Minimization of automata works at this level of generality.

Syntactic Monoids in a Category June 25, 2015 5 / 17

SLIDE 15

The Categorical Setting

A symmetric monoidal closed category (D, ⊗, I) g : A ⊗ B → C λg : A → [B, C] . . . with limits, colimits and (strong epi, mono)-factorizations. Two fixed D-objects: an input object Σ, and an output object O.

Observation (Goguen 1971)

Minimization of automata works at this level of generality.

Syntactic Monoids in a Category June 25, 2015 5 / 17

SLIDE 16

Automata and Languages in (D, ⊗, I)

Language: morphism L : Σ⊛ → O with Σ⊛ the free D-monoid on Σ. Σ⊛ =

n<ω

Σ⊗n D-automaton: Σ ⊗ Q

δ

I

i Q f

λδ
O

[Σ, Q] Hence at the same time

an algebra for the functor FQ = I + Σ ⊗ Q; a coalgebra for the functor TQ = O × [Σ, Q].

Initial F-algebra: Σ⊛ with structure I + Σ ⊗ Σ⊛ → Σ⊛ given by “pick empty word” and “right multiplication”. Final T-coalgebra: [Σ⊛, O] with structure [Σ⊛, O] → O × [Σ, [Σ⊛, O]] given by “evaluate at empty word” and “left derivative”.

Syntactic Monoids in a Category June 25, 2015 6 / 17

SLIDE 17

Automata and Languages in (D, ⊗, I)

Language: morphism L : Σ⊛ → O with Σ⊛ the free D-monoid on Σ. Σ⊛ =

n<ω

Σ⊗n D-automaton: Σ ⊗ Q

δ

I

i Q f

λδ
O

[Σ, Q] Hence at the same time

an algebra for the functor FQ = I + Σ ⊗ Q; a coalgebra for the functor TQ = O × [Σ, Q].

Initial F-algebra: Σ⊛ with structure I + Σ ⊗ Σ⊛ → Σ⊛ given by “pick empty word” and “right multiplication”. Final T-coalgebra: [Σ⊛, O] with structure [Σ⊛, O] → O × [Σ, [Σ⊛, O]] given by “evaluate at empty word” and “left derivative”.

Syntactic Monoids in a Category June 25, 2015 6 / 17

SLIDE 18

Automata and Languages in (D, ⊗, I)

Language: morphism L : Σ⊛ → O with Σ⊛ the free D-monoid on Σ. Σ⊛ =

n<ω

Σ⊗n D-automaton: Σ ⊗ Q

δ

I

i Q f

λδ
O

[Σ, Q] Hence at the same time

an algebra for the functor FQ = I + Σ ⊗ Q; a coalgebra for the functor TQ = O × [Σ, Q].

Initial F-algebra: Σ⊛ with structure I + Σ ⊗ Σ⊛ → Σ⊛ given by “pick empty word” and “right multiplication”. Final T-coalgebra: [Σ⊛, O] with structure [Σ⊛, O] → O × [Σ, [Σ⊛, O]] given by “evaluate at empty word” and “left derivative”.

Syntactic Monoids in a Category June 25, 2015 6 / 17

SLIDE 19

Automata and Languages in (D, ⊗, I)

Language: morphism L : Σ⊛ → O with Σ⊛ the free D-monoid on Σ. Σ⊛ =

n<ω

Σ⊗n D-automaton: Σ ⊗ Q

δ

I

i Q f

λδ
O

[Σ, Q] Hence at the same time

an algebra for the functor FQ = I + Σ ⊗ Q; a coalgebra for the functor TQ = O × [Σ, Q].

Initial F-algebra: Σ⊛ with structure I + Σ ⊗ Σ⊛ → Σ⊛ given by “pick empty word” and “right multiplication”. Final T-coalgebra: [Σ⊛, O] with structure [Σ⊛, O] → O × [Σ, [Σ⊛, O]] given by “evaluate at empty word” and “left derivative”.

Syntactic Monoids in a Category June 25, 2015 6 / 17

SLIDE 20

Automata and Languages in (D, ⊗, I)

Language: morphism L : Σ⊛ → O with Σ⊛ the free D-monoid on Σ. Σ⊛ =

n<ω

Σ⊗n D-automaton: Σ ⊗ Q

δ

I

i Q f

λδ
O

[Σ, Q] Hence at the same time

an algebra for the functor FQ = I + Σ ⊗ Q; a coalgebra for the functor TQ = O × [Σ, Q].

Initial F-algebra: Σ⊛ with structure I + Σ ⊗ Σ⊛ → Σ⊛ given by “pick empty word” and “right multiplication”. Final T-coalgebra: [Σ⊛, O] with structure [Σ⊛, O] → O × [Σ, [Σ⊛, O]] given by “evaluate at empty word” and “left derivative”.

Syntactic Monoids in a Category June 25, 2015 6 / 17

SLIDE 21

Automata and Languages in (D, ⊗, I)

Σ ⊗ Q

δ

I

i Q f

λδ
O

[Σ, Q] FQ = I + Σ ⊗ Q; TQ = O × [Σ, Q]. Language accepted by Q: Σ⊛

initial F-morphism

Q

f

O

Minimal automaton: Σ⊛

initial F-morphism

Q

final T-morphism

[Σ⊛, O]

Theorem (Goguen 1971)

Every language L : Σ⊛ → O is accepted by unique minimal D-automaton.

Syntactic Monoids in a Category June 25, 2015 7 / 17

SLIDE 22

Automata and Languages in (D, ⊗, I)

Σ ⊗ Q

δ

I

i Q f

λδ
O

[Σ, Q] FQ = I + Σ ⊗ Q; TQ = O × [Σ, Q]. Language accepted by Q: Σ⊛

initial F-morphism

Q

f

O

Minimal automaton: Σ⊛

initial F-morphism

Q

final T-morphism

[Σ⊛, O]

Theorem (Goguen 1971)

Every language L : Σ⊛ → O is accepted by unique minimal D-automaton.

Syntactic Monoids in a Category June 25, 2015 7 / 17

SLIDE 23

Automata and Languages in (D, ⊗, I)

Σ ⊗ Q

δ

I

i Q f

λδ
O

[Σ, Q] FQ = I + Σ ⊗ Q; TQ = O × [Σ, Q]. Language accepted by Q: Σ⊛

initial F-morphism

Q

f

O

Minimal automaton: Σ⊛

initial F-morphism

Q

final T-morphism

[Σ⊛, O]

Theorem (Goguen 1971)

Every language L : Σ⊛ → O is accepted by unique minimal D-automaton.

Syntactic Monoids in a Category June 25, 2015 7 / 17

SLIDE 24

The Algebraic Setting

So far: symmetric monoidal closed category (D, ⊗, I) with fixed

bjects Σ (inputs) and O (outputs).

Now: specialize to algebraic setting (Kock ∼1970)

a commutative variety D of algebras [B, C] = algebra of homomorphisms with pointwise structure ⊗ = usual tensor product (universal for bimorphisms) I = free one-generated algebra D-monoid = D-object M together with monoid multiplication

: |M| × |M| → |M| a bimorphism.

e.g. D = sets, semilattices, semimodules: D-monoids = classical monoids, idempotent semirings, associative algebras

Syntactic Monoids in a Category June 25, 2015 8 / 17

SLIDE 25

The Algebraic Setting

So far: symmetric monoidal closed category (D, ⊗, I) with fixed

bjects Σ (inputs) and O (outputs).

Now: specialize to algebraic setting (Kock ∼1970)

a commutative variety D of algebras [B, C] = algebra of homomorphisms with pointwise structure ⊗ = usual tensor product (universal for bimorphisms) I = free one-generated algebra D-monoid = D-object M together with monoid multiplication

: |M| × |M| → |M| a bimorphism.

e.g. D = sets, semilattices, semimodules: D-monoids = classical monoids, idempotent semirings, associative algebras

Syntactic Monoids in a Category June 25, 2015 8 / 17

SLIDE 26

The Algebraic Setting

So far: symmetric monoidal closed category (D, ⊗, I) with fixed

bjects Σ (inputs) and O (outputs).

Now: specialize to algebraic setting (Kock ∼1970)

a commutative variety D of algebras [B, C] = algebra of homomorphisms with pointwise structure ⊗ = usual tensor product (universal for bimorphisms) I = free one-generated algebra D-monoid = D-object M together with monoid multiplication

: |M| × |M| → |M| a bimorphism.

e.g. D = sets, semilattices, semimodules: D-monoids = classical monoids, idempotent semirings, associative algebras

Syntactic Monoids in a Category June 25, 2015 8 / 17

SLIDE 27

The Algebraic Setting

So far: symmetric monoidal closed category (D, ⊗, I) with fixed

bjects Σ (inputs) and O (outputs).

Now: specialize to algebraic setting (Kock ∼1970)

a commutative variety D of algebras [B, C] = algebra of homomorphisms with pointwise structure ⊗ = usual tensor product (universal for bimorphisms) I = free one-generated algebra D-monoid = D-object M together with monoid multiplication

: |M| × |M| → |M| a bimorphism.

e.g. D = sets, semilattices, semimodules: D-monoids = classical monoids, idempotent semirings, associative algebras

Syntactic Monoids in a Category June 25, 2015 8 / 17

SLIDE 28

Algebraic Recognition in (D, ⊗, I)

Definition (Classical)

A language L ⊆ Σ∗ is recognized by a monoid morphism f : Σ∗ → M if L = f −1[S] for some subset S ⊆ M. Σ∗

L f

M

{0, 1}

Definition (Categorical)

A language L : Σ⊛ → O is recognized by a D-monoid morphism f : Σ⊛ → M if L factorizes through f . Σ⊛

L

f M
O

Syntactic Monoids in a Category June 25, 2015 9 / 17

SLIDE 29

Algebraic Recognition in (D, ⊗, I)

Definition (Classical)

A language L ⊆ Σ∗ is recognized by a monoid morphism f : Σ∗ → M if L = f −1[S] for some subset S ⊆ M. Σ∗

L f

M

{0, 1}

Definition (Categorical)

A language L : Σ⊛ → O is recognized by a D-monoid morphism f : Σ⊛ → M if L factorizes through f . Σ⊛

L

f M
O

Syntactic Monoids in a Category June 25, 2015 9 / 17

SLIDE 30

Syntactic Monoids in (D, ⊗, I)

Definition (Classical)

Syntactic monoid of L ⊆ Σ∗ = smallest Σ-generated monoid recognizing L Σ∗

eL f

M

(rec. L)

Syn(L)

Definition (Categorical)

Syntactic D-monoid of L : Σ⊛ → O = smallest Σ-generated D-monoid recognizing L Σ⊛

eL f

M

(rec. L)

Syn(L)

Syntactic Monoids in a Category June 25, 2015 10 / 17

SLIDE 31

Syntactic Monoids in (D, ⊗, I)

Definition (Classical)

Syntactic monoid of L ⊆ Σ∗ = smallest Σ-generated monoid recognizing L Σ∗

eL f

M

(rec. L)

Syn(L)

Definition (Categorical)

Syntactic D-monoid of L : Σ⊛ → O = smallest Σ-generated D-monoid recognizing L Σ⊛

eL f

M

(rec. L)

Syn(L)

Syntactic Monoids in a Category June 25, 2015 10 / 17

SLIDE 32

Construction of Syntactic Monoids

Classically there are two standard constructions of syntactic monoids: (1) via the syntactic congruence; (2) via transition monoids. We will generalize both constructions to (D, ⊗, I). give a third (duality-based) construction, new even in D = Set.

Syntactic Monoids in a Category June 25, 2015 11 / 17

SLIDE 33

Construction of Syntactic Monoids

Classically there are two standard constructions of syntactic monoids: (1) via the syntactic congruence; (2) via transition monoids. We will generalize both constructions to (D, ⊗, I). give a third (duality-based) construction, new even in D = Set.

Syntactic Monoids in a Category June 25, 2015 11 / 17

SLIDE 34

Syntactic Monoids via the Syntactic Congruence

Theorem (Classical)

Let L ⊆ Σ∗. Then Syn(L) = Σ∗/ ∼L with ∼L the syntactic congruence u ∼L v iff ∀x, y ∈ Σ∗ : xuy ∈ L ⇔ xvy ∈ L

Theorem (Categorical)

Let L : Σ⊛ → O. Then Syn(L) = Σ⊛/ ∼L with ∼L the syntactic congruence u ∼L v iff ∀x, y ∈ Σ⊛ : L(xuy) = L(xvy)

Syntactic Monoids in a Category June 25, 2015 12 / 17

SLIDE 35

Syntactic Monoids via the Syntactic Congruence

Theorem (Classical)

Let L ⊆ Σ∗. Then Syn(L) = Σ∗/ ∼L with ∼L the syntactic congruence u ∼L v iff ∀x, y ∈ Σ∗ : xuy ∈ L ⇔ xvy ∈ L

Theorem (Categorical)

Let L : Σ⊛ → O. Then Syn(L) = Σ⊛/ ∼L with ∼L the syntactic congruence u ∼L v iff ∀x, y ∈ Σ⊛ : L(xuy) = L(xvy)

Syntactic Monoids in a Category June 25, 2015 12 / 17

SLIDE 36

Syntactic Monoids: Examples

Σ⊛

eL f

M

(rec. L)

Syn(L) Our generalized Syn(L) and ∼L covers three known constructions: D = sets: the classical syntactic monoid (Rabin/Scott 1959) D = semilattices: the syntactic idempotent semiring (Pol´ ak 2001) D = S-semimodules: syntactic associative algebra of an S-weighted language (Reutenauer 1980) Plus new syntactic algebras for free, e.g. D = pointed sets: syntactic partial monoid

Syntactic Monoids in a Category June 25, 2015 13 / 17

SLIDE 37

Syntactic Monoids: Examples

Σ⊛

eL f

M

(rec. L)

Syn(L) Our generalized Syn(L) and ∼L covers three known constructions: D = sets: the classical syntactic monoid (Rabin/Scott 1959) D = semilattices: the syntactic idempotent semiring (Pol´ ak 2001) D = S-semimodules: syntactic associative algebra of an S-weighted language (Reutenauer 1980) Plus new syntactic algebras for free, e.g. D = pointed sets: syntactic partial monoid

Syntactic Monoids in a Category June 25, 2015 13 / 17

SLIDE 38

Syntactic Monoids: Examples

Σ⊛

eL f

M

(rec. L)

Syn(L) Our generalized Syn(L) and ∼L covers three known constructions: D = sets: the classical syntactic monoid (Rabin/Scott 1959) D = semilattices: the syntactic idempotent semiring (Pol´ ak 2001) D = S-semimodules: syntactic associative algebra of an S-weighted language (Reutenauer 1980) Plus new syntactic algebras for free, e.g. D = pointed sets: syntactic partial monoid

Syntactic Monoids in a Category June 25, 2015 13 / 17

SLIDE 39

Syntactic Monoids: Examples

Σ⊛

eL f

M

(rec. L)

Syn(L) Our generalized Syn(L) and ∼L covers three known constructions: D = sets: the classical syntactic monoid (Rabin/Scott 1959) D = semilattices: the syntactic idempotent semiring (Pol´ ak 2001) D = S-semimodules: syntactic associative algebra of an S-weighted language (Reutenauer 1980) Plus new syntactic algebras for free, e.g. D = pointed sets: syntactic partial monoid

Syntactic Monoids in a Category June 25, 2015 13 / 17

SLIDE 40

Syntactic Monoids: Examples

Σ⊛

eL f

M

(rec. L)

Syn(L) Our generalized Syn(L) and ∼L covers three known constructions: D = sets: the classical syntactic monoid (Rabin/Scott 1959) D = semilattices: the syntactic idempotent semiring (Pol´ ak 2001) D = S-semimodules: syntactic associative algebra of an S-weighted language (Reutenauer 1980) Plus new syntactic algebras for free, e.g. D = pointed sets: syntactic partial monoid

Syntactic Monoids in a Category June 25, 2015 13 / 17

SLIDE 41

Transition Monoids in (D, ⊗, I)

Given an automaton Σ ⊗ Q

δ

I

i Q f

O

consider λδ : Σ → [Q, Q] (in Set: a → (δa : Q → Q)) extend to a D-monoid morphism (λδ)+ : Σ⊛ → [Q, Q] (in Set: a1 . . . an → (δan ◦ . . . ◦ δa1 : Q → Q)) Its image Σ⊛ ։ TransMon(Q) ֌ [Q, Q] is the transition D-monoid of Q.

Theorem

For any language L : Σ⊛ → O the syntactic D-monoid of L is the transition D-monoid of the minimal D-automaton for L: Syn(L) ∼ = TransMon(Min(L))

Syntactic Monoids in a Category June 25, 2015 14 / 17

SLIDE 42

Transition Monoids in (D, ⊗, I)

Given an automaton Σ ⊗ Q

δ

I

i Q f

O

consider λδ : Σ → [Q, Q] (in Set: a → (δa : Q → Q)) extend to a D-monoid morphism (λδ)+ : Σ⊛ → [Q, Q] (in Set: a1 . . . an → (δan ◦ . . . ◦ δa1 : Q → Q)) Its image Σ⊛ ։ TransMon(Q) ֌ [Q, Q] is the transition D-monoid of Q.

Theorem

For any language L : Σ⊛ → O the syntactic D-monoid of L is the transition D-monoid of the minimal D-automaton for L: Syn(L) ∼ = TransMon(Min(L))

Syntactic Monoids in a Category June 25, 2015 14 / 17

SLIDE 43

Transition Monoids in (D, ⊗, I)

Given an automaton Σ ⊗ Q

δ

I

i Q f

O

consider λδ : Σ → [Q, Q] (in Set: a → (δa : Q → Q)) extend to a D-monoid morphism (λδ)+ : Σ⊛ → [Q, Q] (in Set: a1 . . . an → (δan ◦ . . . ◦ δa1 : Q → Q)) Its image Σ⊛ ։ TransMon(Q) ֌ [Q, Q] is the transition D-monoid of Q.

Theorem

For any language L : Σ⊛ → O the syntactic D-monoid of L is the transition D-monoid of the minimal D-automaton for L: Syn(L) ∼ = TransMon(Min(L))

Syntactic Monoids in a Category June 25, 2015 14 / 17

SLIDE 44

Transition Monoids in (D, ⊗, I)

Given an automaton Σ ⊗ Q

δ

I

i Q f

O

consider λδ : Σ → [Q, Q] (in Set: a → (δa : Q → Q)) extend to a D-monoid morphism (λδ)+ : Σ⊛ → [Q, Q] (in Set: a1 . . . an → (δan ◦ . . . ◦ δa1 : Q → Q)) Its image Σ⊛ ։ TransMon(Q) ֌ [Q, Q] is the transition D-monoid of Q.

Theorem

For any language L : Σ⊛ → O the syntactic D-monoid of L is the transition D-monoid of the minimal D-automaton for L: Syn(L) ∼ = TransMon(Min(L))

Syntactic Monoids in a Category June 25, 2015 14 / 17

SLIDE 45

Transition Monoids in (D, ⊗, I)

Given an automaton Σ ⊗ Q

δ

I

i Q f

O

consider λδ : Σ → [Q, Q] (in Set: a → (δa : Q → Q)) extend to a D-monoid morphism (λδ)+ : Σ⊛ → [Q, Q] (in Set: a1 . . . an → (δan ◦ . . . ◦ δa1 : Q → Q)) Its image Σ⊛ ։ TransMon(Q) ֌ [Q, Q] is the transition D-monoid of Q.

Theorem

For any language L : Σ⊛ → O the syntactic D-monoid of L is the transition D-monoid of the minimal D-automaton for L: Syn(L) ∼ = TransMon(Min(L))

Syntactic Monoids in a Category June 25, 2015 14 / 17

SLIDE 46

Regular Languages in (D, ⊗, I)

D-regular languages: languages accepted by D-automata with finitely presentable carrier Assume: finitely presentable D-objects are closed under subalgebras, quotients and finite products

D = (pointed) sets, semilattices, abelian groups, vector spaces, . . .

Theorem

For any language L : Σ⊛ → O the following statements are equivalent: L is D-regular. The minimal D-automaton Min(L) has finitely presentable carrier. L is recognized by some D-monoid with finitely presentable carrier. The syntactic D-monoid Syn(L) has finitely presentable carrier.

Syntactic Monoids in a Category June 25, 2015 15 / 17

SLIDE 47

Regular Languages in (D, ⊗, I)

D-regular languages: languages accepted by D-automata with finitely presentable carrier Assume: finitely presentable D-objects are closed under subalgebras, quotients and finite products

D = (pointed) sets, semilattices, abelian groups, vector spaces, . . .

Theorem

For any language L : Σ⊛ → O the following statements are equivalent: L is D-regular. The minimal D-automaton Min(L) has finitely presentable carrier. L is recognized by some D-monoid with finitely presentable carrier. The syntactic D-monoid Syn(L) has finitely presentable carrier.

Syntactic Monoids in a Category June 25, 2015 15 / 17

SLIDE 48

Regular Languages in (D, ⊗, I)

D-regular languages: languages accepted by D-automata with finitely presentable carrier Assume: finitely presentable D-objects are closed under subalgebras, quotients and finite products

D = (pointed) sets, semilattices, abelian groups, vector spaces, . . .

Theorem

For any language L : Σ⊛ → O the following statements are equivalent: L is D-regular. The minimal D-automaton Min(L) has finitely presentable carrier. L is recognized by some D-monoid with finitely presentable carrier. The syntactic D-monoid Syn(L) has finitely presentable carrier.

Syntactic Monoids in a Category June 25, 2015 15 / 17

SLIDE 49

Syntactic Monoids via Duality

For L ⊆ Σ∗ regular, Syn(L) lives in Setf . Setf ≃ Boolop

f

– so what is the dual boolean algebra Syn(L)?

Theorem

Syn(L) is isomorphic to the smallest boolean subalgebra of P(Σ∗) that contains the reversed language Lrev; is closed under (both-sided) derivatives, i.e. K ∈ Syn(L) implies x−1Ky−1 := {w ∈ Σ∗ : xwy ∈ K} ∈ Syn(L) for all x, y ∈ Σ∗. Generalizes from Boolf /Setf to pairs Cf /Df with Stone-type duality.

e.g. semilattices/semilattices, vector spaces/vector spaces, boolean rings/pointed sets

Related Work on Duality: FOSSACS 2014, LICS 2015

Syntactic Monoids in a Category June 25, 2015 16 / 17

SLIDE 50

Syntactic Monoids via Duality

For L ⊆ Σ∗ regular, Syn(L) lives in Setf . Setf ≃ Boolop

f

– so what is the dual boolean algebra Syn(L)?

Theorem

Syn(L) is isomorphic to the smallest boolean subalgebra of P(Σ∗) that contains the reversed language Lrev; is closed under (both-sided) derivatives, i.e. K ∈ Syn(L) implies x−1Ky−1 := {w ∈ Σ∗ : xwy ∈ K} ∈ Syn(L) for all x, y ∈ Σ∗. Generalizes from Boolf /Setf to pairs Cf /Df with Stone-type duality.

e.g. semilattices/semilattices, vector spaces/vector spaces, boolean rings/pointed sets

Related Work on Duality: FOSSACS 2014, LICS 2015

Syntactic Monoids in a Category June 25, 2015 16 / 17

SLIDE 51

Syntactic Monoids via Duality

For L ⊆ Σ∗ regular, Syn(L) lives in Setf . Setf ≃ Boolop

f

– so what is the dual boolean algebra Syn(L)?

Theorem

Syn(L) is isomorphic to the smallest boolean subalgebra of P(Σ∗) that contains the reversed language Lrev; is closed under (both-sided) derivatives, i.e. K ∈ Syn(L) implies x−1Ky−1 := {w ∈ Σ∗ : xwy ∈ K} ∈ Syn(L) for all x, y ∈ Σ∗. Generalizes from Boolf /Setf to pairs Cf /Df with Stone-type duality.

e.g. semilattices/semilattices, vector spaces/vector spaces, boolean rings/pointed sets

Related Work on Duality: FOSSACS 2014, LICS 2015

Syntactic Monoids in a Category June 25, 2015 16 / 17

SLIDE 52

Conclusion

Symmetric monoidal closed categories form a suitable setting for algebraic automata theory. Uniform perspective on scattered concepts and results. Ongoing work:

Purely categorical setting? Syntactic ordered monoids (Pin 1995): enriched setting? Connection with Boj´ anczyk’s work (2015, unpublished): syntactic T-algebras, T a monad on sorted sets. Duality for classifying properties of languages, e.g. generalized Sch¨ utzenberger theorem?