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The Theory of Languages Highlights in London September 12-15, 2017 Paul Brunet University College London Introduction Outline I. Introduction II. Free Representation III. Main results IV. Outlook Paul Brunet 2/18 Language Algebra

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The Theory of Languages

Highlights in London September 12-15, 2017 Paul Brunet

University College London

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SLIDE 2

Introduction

Outline

  • I. Introduction
  • II. Free Representation
  • III. Main results
  • IV. Outlook

Paul Brunet Language Algebra 2/18

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SLIDE 3

Introduction

Universal laws

a ∪ b = b ∪ a (commutativity of union) a · (b · c) = (a · b) · c (associativity of concatenation)

Paul Brunet Language Algebra 3/18

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SLIDE 4

Introduction

Universal laws

∀Σ, ∀a, b, c ⊆ Σ⋆ a ∪ b = b ∪ a (commutativity of union) a · (b · c) = (a · b) · c (associativity of concatenation)

Paul Brunet Language Algebra 3/18

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SLIDE 5

Introduction

Universal laws

∀Σ, ∀a, b, c ⊆ Σ⋆ a ∪ b = b ∪ a (commutativity of union) a · (b · c) = (a · b) · c (associativity of concatenation) e, f ∈ EX ::= 0 | 1 | a | e ∪ f | e ∩ f | e · f | e | e⋆.

Paul Brunet Language Algebra 3/18

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SLIDE 6

Introduction

Universal laws

∀Σ, ∀a, b, c ⊆ Σ⋆ a ∪ b = b ∪ a (commutativity of union) a · (b · c) = (a · b) · c (associativity of concatenation) e, f ∈ EX ::= 0 | 1 | a | e ∪ f | e ∩ f | e · f | e | e⋆.

Language equivalence

Lang | = e ≃ f iff ∀Σ, ∀σ : X → P (Σ⋆) , σ (e) = σ (f ).

Paul Brunet Language Algebra 3/18

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SLIDE 7

Introduction

representation

: EX → r R

Paul Brunet Language Algebra 4/18

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SLIDE 8

Introduction

Effective representation

: EX → r R Computable Decidable

Paul Brunet Language Algebra 4/18

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SLIDE 9

Introduction

Effective free representation

: EX → r R Computable Decidable r(e) = r(f ) Lang | = e ≃ f

Paul Brunet Language Algebra 4/18

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SLIDE 10

Introduction

Effective free representation

: EX → r R Computable Decidable r(e) = r(f ) Lang | = e ≃ f Ax ⊢ e = f

Paul Brunet Language Algebra 4/18

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SLIDE 11

Introduction

Effective free representation

: EX → r R Computable Decidable r(e) = r(f ) Lang | = e ≃ f Ax ⊢ e = f

Kleene Algebra, KA with Tests, Kleene lattices, Allegories, Monoids,...

Paul Brunet Language Algebra 4/18

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SLIDE 12

Introduction

Outline

  • I. Introduction
  • II. Free Representation
  • III. Main results
  • IV. Outlook

Paul Brunet Language Algebra 5/18

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SLIDE 13

Free Representation

Outline

  • I. Introduction
  • II. Free Representation
  • III. Main results
  • IV. Outlook

Paul Brunet Language Algebra 6/18

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SLIDE 14

Free Representation

Example

Lang | = (1 ∩ a) · b ≃ b · (1 ∩ a)

Paul Brunet Language Algebra 7/18

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SLIDE 15

Free Representation

Example

Lang | = (1 ∩ a) · b ≃ b · (1 ∩ a)

  • Proof. Let σ : {a, b} → P (Σ⋆).

Paul Brunet Language Algebra 7/18

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SLIDE 16

Free Representation

Example

Lang | = (1 ∩ a) · b ≃ b · (1 ∩ a)

  • Proof. Let σ : {a, b} → P (Σ⋆).

Paul Brunet Language Algebra 7/18

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SLIDE 17

Free Representation

Example

Lang | = (1 ∩ a) · b ≃ b · (1 ∩ a)

  • Proof. Let σ : {a, b} → P (Σ⋆).

◮ If ε ∈ σ (a):

  • σ ((1 ∩ a) · b) =
  • σ (b · (1 ∩ a)) =

◮ If ε /

∈ σ (a):

  • σ ((1 ∩ a) · b) =
  • σ (b · (1 ∩ a)) =

Paul Brunet Language Algebra 7/18

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SLIDE 18

Free Representation

Example

Lang | = (1 ∩ a) · b ≃ b · (1 ∩ a)

  • Proof. Let σ : {a, b} → P (Σ⋆).

◮ If ε ∈ σ (a): then

σ (1 ∩ a) = {ε}, thus:

  • σ ((1 ∩ a) · b) =
  • σ (b · (1 ∩ a)) =

◮ If ε /

∈ σ (a):

  • σ ((1 ∩ a) · b) =
  • σ (b · (1 ∩ a)) =

Paul Brunet Language Algebra 7/18

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SLIDE 19

Free Representation

Example

Lang | = (1 ∩ a) · b ≃ b · (1 ∩ a)

  • Proof. Let σ : {a, b} → P (Σ⋆).

◮ If ε ∈ σ (a): then

σ (1 ∩ a) = {ε}, thus:

  • σ ((1 ∩ a) · b) = {ε} · σ (b) = σ (b) .
  • σ (b · (1 ∩ a)) = σ (b) · {ε} = σ (b) .

◮ If ε /

∈ σ (a):

  • σ ((1 ∩ a) · b) =
  • σ (b · (1 ∩ a)) =

Paul Brunet Language Algebra 7/18

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SLIDE 20

Free Representation

Example

Lang | = (1 ∩ a) · b ≃ b · (1 ∩ a)

  • Proof. Let σ : {a, b} → P (Σ⋆).

◮ If ε ∈ σ (a): then

σ (1 ∩ a) = {ε}, thus:

  • σ ((1 ∩ a) · b) = {ε} · σ (b) = σ (b) .
  • σ (b · (1 ∩ a)) = σ (b) · {ε} = σ (b) .

◮ If ε /

∈ σ (a): then σ (1 ∩ a) = ∅, thus:

  • σ ((1 ∩ a) · b) =
  • σ (b · (1 ∩ a)) =

Paul Brunet Language Algebra 7/18

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SLIDE 21

Free Representation

Example

Lang | = (1 ∩ a) · b ≃ b · (1 ∩ a)

  • Proof. Let σ : {a, b} → P (Σ⋆).

◮ If ε ∈ σ (a): then

σ (1 ∩ a) = {ε}, thus:

  • σ ((1 ∩ a) · b) = {ε} · σ (b) = σ (b) .
  • σ (b · (1 ∩ a)) = σ (b) · {ε} = σ (b) .

◮ If ε /

∈ σ (a): then σ (1 ∩ a) = ∅, thus:

  • σ ((1 ∩ a) · b) = ∅ · σ (b) = ∅.
  • σ (b · (1 ∩ a)) = σ (b) · ∅ = ∅.

Paul Brunet Language Algebra 7/18

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SLIDE 22

Free Representation

Example

Lang | = (1 ∩ a) · b ≃ b · (1 ∩ a)

  • Proof. Let σ : {a, b} → P (Σ⋆).

◮ If ε ∈ σ (a): then

σ (1 ∩ a) = {ε}, thus:

  • σ ((1 ∩ a) · b) = {ε} · σ (b) = σ (b) .
  • σ (b · (1 ∩ a)) = σ (b) · {ε} = σ (b) .

◮ If ε /

∈ σ (a): then σ (1 ∩ a) = ∅, thus:

  • σ ((1 ∩ a) · b) = ∅ · σ (b) = ∅.
  • σ (b · (1 ∩ a)) = σ (b) · ∅ = ∅.
  • Paul Brunet

Language Algebra 7/18

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SLIDE 23

Free Representation

Example

Lang | = (1 ∩ a) · b ≃ b · (1 ∩ a)

  • Proof. Let σ : {a, b} → P (Σ⋆).

◮ If ε ∈ σ (a): then

σ (1 ∩ a) = {ε}, thus:

  • σ ((1 ∩ a) · b) = {ε} · σ (b) = σ (b) .
  • σ (b · (1 ∩ a)) = σ (b) · {ε} = σ (b) .

◮ If ε /

∈ σ (a): then σ (1 ∩ a) = ∅, thus:

  • σ ((1 ∩ a) · b) = ∅ · σ (b) = ∅.
  • σ (b · (1 ∩ a)) = σ (b) · ∅ = ∅.
  • Idea

Compare 1-free terms under the assumption that certain variables contain ε.

Paul Brunet Language Algebra 7/18

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SLIDE 24

Free Representation

Weak graphs

Definition

A weak graph is a pair of a graph and a set of tests.

Paul Brunet Language Algebra 8/18

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Free Representation

Weak graphs

Definition

A weak graph is a pair of a graph and a set of tests.

Weak graph preorder

G, A ◭ H, B if B ⊆ A and there is an A-weak morphism from H to G.

Paul Brunet Language Algebra 8/18

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SLIDE 26

Free Representation

Weak graphs

Definition

A weak graph is a pair of a graph and a set of tests.

Weak graph preorder

G, A ◭ H, B if B ⊆ A and there is an A-weak morphism from H to G. A = {a}

a b a c a c b H : G :

Paul Brunet Language Algebra 8/18

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SLIDE 27

Free Representation

Characterisation Theorem

u, v ∈ TX ::= 1 | a | u · v | u ∩ v

Paul Brunet Language Algebra 9/18

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Free Representation

Characterisation Theorem

u, v ∈ TX ::= 1 | a | u · v | u ∩ v For every term u ∈ TX we can build a weak graph G (u).

Paul Brunet Language Algebra 9/18

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Free Representation

Characterisation Theorem

u, v ∈ TX ::= 1 | a | u · v | u ∩ v For every term u ∈ TX we can build a weak graph G (u).

Corollary

Lang | = u ⊆ v ⇔ G (u) ◭ G (v) .

Paul Brunet Language Algebra 9/18

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Free Representation

Free representation of expressions

e, f ∈ EX ::= 0 | 1 | a | e ∪ f | e ∩ f | e · f | e | e⋆.

Paul Brunet Language Algebra 10/18

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SLIDE 31

Free Representation

Free representation of expressions

e, f ∈ EX ::= 0 | 1 | a | e ∪ f | e ∩ f | e · f | e | e⋆.

EX

Paul Brunet Language Algebra 10/18

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SLIDE 32

Free Representation

Free representation of expressions

e, f ∈ EX ::= 0 | 1 | a | e ∪ f | e ∩ f | e · f | e | e⋆.

EX P (TX∪X ′) T

Paul Brunet Language Algebra 10/18

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SLIDE 33

Free Representation

Free representation of expressions

e, f ∈ EX ::= 0 | 1 | a | e ∪ f | e ∩ f | e · f | e | e⋆.

EX P (TX∪X ′) T P (WeakGraphX∪X ′) P (G)

Paul Brunet Language Algebra 10/18

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SLIDE 34

Free Representation

Free representation of expressions

e, f ∈ EX ::= 0 | 1 | a | e ∪ f | e ∩ f | e · f | e | e⋆.

EX P (TX∪X ′) T P (WeakGraphX∪X ′) P (G) P (WeakGraphX∪X ′)

◭_

Paul Brunet Language Algebra 10/18

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SLIDE 35

Free Representation

Free representation of expressions

e, f ∈ EX ::= 0 | 1 | a | e ∪ f | e ∩ f | e · f | e | e⋆.

EX P (TX∪X ′) T P (WeakGraphX∪X ′) P (G) P (WeakGraphX∪X ′)

◭_

R

Paul Brunet Language Algebra 10/18

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SLIDE 36

Free Representation

Free representation of expressions

e, f ∈ EX ::= 0 | 1 | a | e ∪ f | e ∩ f | e · f | e | e⋆.

EX P (TX∪X ′) T P (WeakGraphX∪X ′) P (G) P (WeakGraphX∪X ′)

◭_

R

Theorem

Lang | = e ≃ f ⇔ R (e) = R (f )

Paul Brunet Language Algebra 10/18

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SLIDE 37

Free Representation

Free representation of expressions

e, f ∈ EX ::= 0 | 1 | a | e ∪ f | e ∩ f | e · f | e | e⋆.

EX P (TX∪X ′) T P (WeakGraphX∪X ′) P (G) P (WeakGraphX∪X ′)

◭_

R

Lemma

If e doesn’t use the Kleene star, then T (e) is finite.

Theorem

Lang | = e ≃ f ⇔ R (e) = R (f )

Paul Brunet Language Algebra 10/18

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SLIDE 38

Main results

Outline

  • I. Introduction
  • II. Free Representation
  • III. Main results
  • IV. Outlook

Paul Brunet Language Algebra 11/18

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SLIDE 39

Main results

Decidability and Complexity

Representation Theorem

Lang | = e ≃ f ⇔ R (e) = R (f ).

Paul Brunet Language Algebra 12/18

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SLIDE 40

Main results

Decidability and Complexity

Representation Theorem

Lang | = e ≃ f ⇔ R (e) = R (f ).

Half-Kleene Theorem

L (A (e)) = R (e) and |A (e)| = |e| × 2|X|.

Paul Brunet Language Algebra 12/18

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SLIDE 41

Main results

Decidability and Complexity

Representation Theorem

Lang | = e ≃ f ⇔ R (e) = R (f ).

Half-Kleene Theorem

L (A (e)) = R (e) and |A (e)| = |e| × 2|X|.

Simulation algorithm

Comparing Weighted Petri automata is ExpSpace.

Paul Brunet Language Algebra 12/18

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SLIDE 42

Main results

Decidability and Complexity

Representation Theorem

Lang | = e ≃ f ⇔ R (e) = R (f ).

Half-Kleene Theorem

L (A (e)) = R (e) and |A (e)| = |e| × 2|X|.

Simulation algorithm

Comparing Weighted Petri automata is ExpSpace.

Main Theorem

The equational theory of languages over the signature EX is ExpSpace-complete.

Paul Brunet Language Algebra 12/18

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SLIDE 43

Main results

Axiomatization for series parallel terms

u, v, w ∈ ❙PX ::= a | u v | u · v, A ⊆ X. A ⊢sp u ≤ u A ⊢sp u ≤ v A ⊢sp v ≤ w A ⊢sp u ≤ w A ⊢sp u ≤ v A ⊢sp u′ ≤ v ′ A ⊢sp u · u′ ≤ v · v ′ A ⊢sp u ≤ v A ⊢sp u′ ≤ v ′ A ⊢sp u ∩ u′ ≤ v ∩ v ′ A ⊢sp u · (v · w) = (u · v) · w A ⊢sp u ≤ u ∩ u A ⊢sp u ∩ v ≤ v ∩ u A ⊢sp u ∩ v ≤ u var (u) ⊆ A A ⊢sp v ≤ u · v var (u) ⊆ A A ⊢sp v ≤ v · u

Paul Brunet Language Algebra 13/18

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SLIDE 44

Main results

Axiomatization for series parallel terms

u, v, w ∈ ❙PX ::= a | u v | u · v, A ⊆ X. Preorder A ⊢sp u ≤ u A ⊢sp u ≤ v A ⊢sp v ≤ w A ⊢sp u ≤ w A ⊢sp u ≤ v A ⊢sp u′ ≤ v ′ A ⊢sp u · u′ ≤ v · v ′ A ⊢sp u ≤ v A ⊢sp u′ ≤ v ′ A ⊢sp u ∩ u′ ≤ v ∩ v ′ A ⊢sp u · (v · w) = (u · v) · w A ⊢sp u ≤ u ∩ u A ⊢sp u ∩ v ≤ v ∩ u A ⊢sp u ∩ v ≤ u var (u) ⊆ A A ⊢sp v ≤ u · v var (u) ⊆ A A ⊢sp v ≤ v · u

Paul Brunet Language Algebra 13/18

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SLIDE 45

Main results

Axiomatization for series parallel terms

u, v, w ∈ ❙PX ::= a | u v | u · v, A ⊆ X. Monotonicity A ⊢sp u ≤ u A ⊢sp u ≤ v A ⊢sp v ≤ w A ⊢sp u ≤ w A ⊢sp u ≤ v A ⊢sp u′ ≤ v ′ A ⊢sp u · u′ ≤ v · v ′ A ⊢sp u ≤ v A ⊢sp u′ ≤ v ′ A ⊢sp u ∩ u′ ≤ v ∩ v ′ A ⊢sp u · (v · w) = (u · v) · w A ⊢sp u ≤ u ∩ u A ⊢sp u ∩ v ≤ v ∩ u A ⊢sp u ∩ v ≤ u var (u) ⊆ A A ⊢sp v ≤ u · v var (u) ⊆ A A ⊢sp v ≤ v · u

Paul Brunet Language Algebra 13/18

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SLIDE 46

Main results

Axiomatization for series parallel terms

u, v, w ∈ ❙PX ::= a | u v | u · v, A ⊆ X. Associativity A ⊢sp u ≤ u A ⊢sp u ≤ v A ⊢sp v ≤ w A ⊢sp u ≤ w A ⊢sp u ≤ v A ⊢sp u′ ≤ v ′ A ⊢sp u · u′ ≤ v · v ′ A ⊢sp u ≤ v A ⊢sp u′ ≤ v ′ A ⊢sp u ∩ u′ ≤ v ∩ v ′ A ⊢sp u · (v · w) = (u · v) · w A ⊢sp u ≤ u ∩ u A ⊢sp u ∩ v ≤ v ∩ u A ⊢sp u ∩ v ≤ u var (u) ⊆ A A ⊢sp v ≤ u · v var (u) ⊆ A A ⊢sp v ≤ v · u

Paul Brunet Language Algebra 13/18

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SLIDE 47

Main results

Axiomatization for series parallel terms

u, v, w ∈ ❙PX ::= a | u v | u · v, A ⊆ X. Meet-semilattice A ⊢sp u ≤ u A ⊢sp u ≤ v A ⊢sp v ≤ w A ⊢sp u ≤ w A ⊢sp u ≤ v A ⊢sp u′ ≤ v ′ A ⊢sp u · u′ ≤ v · v ′ A ⊢sp u ≤ v A ⊢sp u′ ≤ v ′ A ⊢sp u ∩ u′ ≤ v ∩ v ′ A ⊢sp u · (v · w) = (u · v) · w A ⊢sp u ≤ u ∩ u A ⊢sp u ∩ v ≤ v ∩ u A ⊢sp u ∩ v ≤ u var (u) ⊆ A A ⊢sp v ≤ u · v var (u) ⊆ A A ⊢sp v ≤ v · u

Paul Brunet Language Algebra 13/18

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SLIDE 48

Main results

Axiomatization for series parallel terms

u, v, w ∈ ❙PX ::= a | u v | u · v, A ⊆ X. Super-units A ⊢sp u ≤ u A ⊢sp u ≤ v A ⊢sp v ≤ w A ⊢sp u ≤ w A ⊢sp u ≤ v A ⊢sp u′ ≤ v ′ A ⊢sp u · u′ ≤ v · v ′ A ⊢sp u ≤ v A ⊢sp u′ ≤ v ′ A ⊢sp u ∩ u′ ≤ v ∩ v ′ A ⊢sp u · (v · w) = (u · v) · w A ⊢sp u ≤ u ∩ u A ⊢sp u ∩ v ≤ v ∩ u A ⊢sp u ∩ v ≤ u var (u) ⊆ A A ⊢sp v ≤ u · v var (u) ⊆ A A ⊢sp v ≤ v · u

Paul Brunet Language Algebra 13/18

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SLIDE 49

Main results

Axiomatization for ⋆-free terms

e, f ∈ EX ::= 0 | 1 | a | e ∪ f | e ∩ f | e · f | e.

Paul Brunet Language Algebra 14/18

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SLIDE 50

Main results

Axiomatization for ⋆-free terms

e, f ∈ EX ::= 0 | 1 | a | e ∪ f | e ∩ f | e · f | e.

◮ 0, 1, ·, ∪ is an idempotent semi-ring;

Paul Brunet Language Algebra 14/18

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SLIDE 51

Main results

Axiomatization for ⋆-free terms

e, f ∈ EX ::= 0 | 1 | a | e ∪ f | e ∩ f | e · f | e.

◮ 0, 1, ·, ∪ is an idempotent semi-ring; ◮ ∪, ∩ is a distributive lattice;

Paul Brunet Language Algebra 14/18

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SLIDE 52

Main results

Axiomatization for ⋆-free terms

e, f ∈ EX ::= 0 | 1 | a | e ∪ f | e ∩ f | e · f | e.

◮ 0, 1, ·, ∪ is an idempotent semi-ring; ◮ ∪, ∩ is a distributive lattice; ◮ mirror image laws:

0 = 0 1 = 1 e = e e · f = f · e e ∩ f = e ∩ f e ∪ f = e ∪ f

Paul Brunet Language Algebra 14/18

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SLIDE 53

Main results

Axiomatization for ⋆-free terms

e, f ∈ EX ::= 0 | 1 | a | e ∪ f | e ∩ f | e · f | e.

◮ 0, 1, ·, ∪ is an idempotent semi-ring; ◮ ∪, ∩ is a distributive lattice; ◮ mirror image laws:

0 = 0 1 = 1 e = e e · f = f · e e ∩ f = e ∩ f e ∪ f = e ∪ f

◮ sub-unit laws:

Paul Brunet Language Algebra 14/18

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SLIDE 54

Main results

Axiomatization for ⋆-free terms

e, f ∈ EX ::= 0 | 1 | a | e ∪ f | e ∩ f | e · f | e.

◮ 0, 1, ·, ∪ is an idempotent semi-ring; ◮ ∪, ∩ is a distributive lattice; ◮ mirror image laws:

0 = 0 1 = 1 e = e e · f = f · e e ∩ f = e ∩ f e ∪ f = e ∪ f

◮ sub-unit laws:

1 ∩ (e · f ) = 1 ∩ (e ∩ f )

Paul Brunet Language Algebra 14/18

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SLIDE 55

Main results

Axiomatization for ⋆-free terms

e, f ∈ EX ::= 0 | 1 | a | e ∪ f | e ∩ f | e · f | e.

◮ 0, 1, ·, ∪ is an idempotent semi-ring; ◮ ∪, ∩ is a distributive lattice; ◮ mirror image laws:

0 = 0 1 = 1 e = e e · f = f · e e ∩ f = e ∩ f e ∪ f = e ∪ f

◮ sub-unit laws:

1 ∩ (e · f ) = 1 ∩ (e ∩ f ) 1 ∩

  • e

= 1 ∩ e

Paul Brunet Language Algebra 14/18

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SLIDE 56

Main results

Axiomatization for ⋆-free terms

e, f ∈ EX ::= 0 | 1 | a | e ∪ f | e ∩ f | e · f | e.

◮ 0, 1, ·, ∪ is an idempotent semi-ring; ◮ ∪, ∩ is a distributive lattice; ◮ mirror image laws:

0 = 0 1 = 1 e = e e · f = f · e e ∩ f = e ∩ f e ∪ f = e ∪ f

◮ sub-unit laws:

1 ∩ (e · f ) = 1 ∩ (e ∩ f ) 1 ∩

  • e

= 1 ∩ e (1 ∩ e) · f = f · (1 ∩ e)

Paul Brunet Language Algebra 14/18

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SLIDE 57

Main results

Axiomatization for ⋆-free terms

e, f ∈ EX ::= 0 | 1 | a | e ∪ f | e ∩ f | e · f | e.

◮ 0, 1, ·, ∪ is an idempotent semi-ring; ◮ ∪, ∩ is a distributive lattice; ◮ mirror image laws:

0 = 0 1 = 1 e = e e · f = f · e e ∩ f = e ∩ f e ∪ f = e ∪ f

◮ sub-unit laws:

1 ∩ (e · f ) = 1 ∩ (e ∩ f ) 1 ∩

  • e

= 1 ∩ e (1 ∩ e) · f = f · (1 ∩ e) ((1 ∩ e) · f ) ∩ g = (1 ∩ e) · (f ∩ g)

Paul Brunet Language Algebra 14/18

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SLIDE 58

Outlook

Outline

  • I. Introduction
  • II. Free Representation
  • III. Main results
  • IV. Outlook

Paul Brunet Language Algebra 15/18

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SLIDE 59

Outlook

Open problems

  • I. Can we axiomatize with e⋆?

Paul Brunet Language Algebra 16/18

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SLIDE 60

Outlook

Open problems

  • I. Can we axiomatize with e⋆?
  • II. Is there a free representation with residuals?

b ≤ a \ c ⇔ a · b ≤ c ⇔ a ≤ c / b

Paul Brunet Language Algebra 16/18

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SLIDE 61

Outlook

Open problems

  • I. Can we axiomatize with e⋆?
  • II. Is there a free representation with residuals?

b ≤ a \ c ⇔ a · b ≤ c ⇔ a ≤ c / b

  • III. Can we have tests? or names?

Paul Brunet Language Algebra 16/18

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SLIDE 62

Outlook

Open problems

  • I. Can we axiomatize with e⋆?
  • II. Is there a free representation with residuals?

b ≤ a \ c ⇔ a · b ≤ c ⇔ a ≤ c / b

  • III. Can we have tests? or names?
  • IV. What about ⊤?

Paul Brunet Language Algebra 16/18

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SLIDE 63

Outlook

That’s all folks!

Thank you!

Freyd & Scedrov, Categories, Allegories, 1990 Andréka & Bredikhin, The equational theory of union-free algebras of relations, 1995 Andréka, Mikulás & Németi, The equational theory of Kleene lattices, 2011 Bloom, Ésik & Stefanescu, Notes on equational theories of relations, 1995

  • B. & Pous, Petri Automata for Kleene Allegories, 2015

B., Reversible Kleene lattices, 2017

See more at: http://paul.brunet-zamansky.fr

Paul Brunet Language Algebra 17/18

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SLIDE 64

Outlook

Outline

  • I. Introduction
  • II. Free Representation
  • III. Main results
  • IV. Outlook

Paul Brunet Language Algebra 18/18