cas 738 relation algebra and kleene algebra and their ... · cas 738 relation algebra and kleene...

CAS 738 RelationAlgebra

and Kleene Algebraand their

Applications

Dr. R. Khedri

Outline

Introduction

Binary relations

Relation algebras

Atom structures ofRAs

(Slide 1 of 38)

CAS 738 Relation Algebraand Kleene Algebra

and their Applications

Dr. Ridha Khedri

Department of Computing and Software, McMaster UniversityCanada L8S 4L7, Hamilton, Ontario



Applications

Dr. R. Khedri

Outline

Introduction

Binary relations

Relation algebras


(Slide 2 of 38)

1 Introduction2 Binary relations

Proper relation algebrasSquare proper relation algebras

3 Relation algebras

Definition of relation algebras

Peircean law4

Atom structures of relation algebras

Boolean algebras with operator

Atom structures of RAs

Consistent and forbidden triples of atoms

Representations of relation algebras

Examples of relation algebra



Applications

Dr. R. Khedri

Outline

Introduction

Binary relations

Relation algebras


(Slide 3 of 38)Relation Algebras Introduction

We start with a very basic mathematical object, arelation of a certain rank (arity)

We actually look at a field of relations:

a set of relations, over some fixed domainclosed under certain concrete operations: union,intersection, complement of a relation (relative to somebiggest relation over the domain)

So we have a field of sets

We need other operations pertaining to the relationalproperties

identity relation, the converse of a relation, and thecomposition of two binary relationsa field of binary relations should then include theidentity and be closed under conversion andcomposition



Applications

Dr. R. Khedri

Outline

Introduction

Binary relations

Relation algebras



Then, we write out some axioms for these operations

The set of axioms ⊇ the set of axioms for booleanalgebra ∪ a set of other axioms involving the relationaloperators

The axioms will almost invariably be valid over fields ofrelations of the appropriate type

However, the question of completeness is more tricky

Let us call an abstract structure that obeys theseaxioms an algebraThe question of completeness can now be thought of inanother way: is an arbitrary algebra isomorphic to afield of relations of the appropriate type?An isomorphism from an algebra to a field of relationsis called a representation



Applications

Dr. R. Khedri

Outline

Introduction

Binary relations

Relation algebras



What are the advantages of working with the algebrasrather than the concrete structures?

The technique of focusing our whole attention on asmall number of properties to the exclusion of allothers is quite general in scientific analysis

Because we are ignoring many of the concrete featuresof our structures, we may find that our approach ismore general

Algebra is a very thoroughly studied discipline andthere is a range of methods and results that we canapply to the study of relations

Universal algebra and model theory have techniquesand theorems that can be exploited fruitfully here



Applications

Dr. R. Khedri

Outline

Introduction

Binary relations

Proper relation algebras

Square proper R.A.

Relation algebras


(Slide 6 of 38)Relation Algebras Binary relations

We considered fields of unary relations (or fields ofsets), and saw that the corresponding algebra wasBoolean algebra

We start our investigation of binary relations bydefining a field of binary relations

A binary relation on a set B is a subset of B × BB is said to be the base set

on B we can define a (proper) relation algebra



Applications

Dr. R. Khedri

Outline

Introduction

Binary relations


Square proper R.A.

Relation algebras


(Slide 7 of 38)Relation Algebras Binary relations Properrelation algebras

Definition

Let B be a set. A proper relation algebra (PRA) with baseset B is an algebra of the form S = (S , ∅, U,∪, \, IB ,` , ;),where S is a non-empty set of binary relations over B, andthe following hold:

1 (S , ∅, U,∪, \) forms a field of sets

2 IBdef= (b, b) | b ∈ B ∈ S (the identity over B)

3 S is closed under taking converses

s ∈ S =⇒ s` ∈ S , where s` = (c , b) | (b, c) ∈ s.

4 S is closed under composition of binary relations:r , s ∈ S =⇒ r ; s ∈ S , where

r ; s = (c , b) | ∃(d |: (b, d) ∈ r ∧ (d , c) ∈ s ).



Applications

Dr. R. Khedri

Outline

Introduction

Binary relations


Square proper R.A.

Relation algebras


(Slide 8 of 38)Relation Algebras Binary relations Properrelation algebras

The unit U does not need to be B × B

Example

B = 0, 1, 2S contains only the following relations:

∅ (0, 0), (1, 1) (2, 2)(0, 1), (1, 0) (0, 0), (1, 1), (2, 2) = IB (0, 0), (1, 1), (0, 1), (1, 0)(2, 2), (0, 1), (1, 0) (0, 0), (1, 1), (2, 2), (0, 1), (1, 0) = U



Applications

Dr. R. Khedri

Outline

Introduction

Binary relations


Square proper R.A.

Relation algebras


(Slide 9 of 38)Relation Algebras Binary relations Squareproper relation algebras

To remember

the complement \ in a proper relation algebra is alwaystaken relative to the unit UU of a proper relation algebra with base B need not beB × B

In terms of the previous definition, (S , ∅, U,∪, \) isNOT necessarily a subalgebra of the field of sets(P(B × B), ∅,B × B,∪, \)

It is a subalgebra of the field of sets (P(U), ∅, U,∪, \)



Applications

Dr. R. Khedri

Outline

Introduction

Binary relations


Square proper R.A.

Relation algebras



Several reasons for not insisting that U = B × B

relativising the operations to the unit, leads to otherclasses of algebras which are in many waysbetter-behaved

to ensure that the class of algebras isomorphic toproper relation algebras is a variety (has nice properties)

Nonetheless, those proper relation algebras whereU = B × B are very important



Applications

Dr. R. Khedri

Outline

Introduction

Binary relations


Square proper R.A.

Relation algebras



Definition (Proper relation algebra)

A proper relation algebra S with base B is said to be squareif its unit U is equal to B × B.

An arbitrary proper relation algebra S need not besquare

But its unit will always be an equivalence relation

Lemma

Let S = (S , ∅, U,∪, \, IB ,` , ;) be a proper relation algebrawith base B. Then the unit U is an equivalence relationover B.

Proof: . . .



Applications

Dr. R. Khedri

Outline

Introduction

Binary relations


Square proper R.A.

Relation algebras



New construction: Restricted proper relation algebra (RPA)

Definition (restricted PRA)

Let S = (S , ∅, U,∪, \, IB ,` , ;) be a proper relation algebrawith base B. Let E ⊆ B be an equivalence class of U.Define the restricted proper relation algebra

S↓E

def= (S↓

E, ∅,E × E ,∪, \

E, IE ,` , ;), where

S↓E

def= s ∩ E × E | s ∈ S

\E(s) = (E × E ) \ s, for s ∈ S .

As example (S↓E= s | s ⊂ IB, for a given B)



Applications

Dr. R. Khedri

Outline

Introduction

Binary relations


Square proper R.A.

Relation algebras



Another new construction: Proper relation algebra withbase B = ∪(i | i ∈ I : Bi )

Definition

Let Si = (Si , ∅, Ui ,∪, \i , IBi,` , ;) be a proper relation

algebra with base Bi for each i ∈ I (some index set I ), andsuppose that if i 6= j in I then Bi ∩ Bj = ∅. We define

⊗i∈ISi = (S , ∅, U,∪, \, IB ,` , ;)

to be the proper relation algebra with baseB = ∪(i | i ∈ I : Bi ), where:

S = ∪(i | i ∈ I : si ) | si ∈ Si for each i ∈ I(we are including arbitrary unions of the relations)

U = ∪(i | i ∈ I : Ui )



Applications

Dr. R. Khedri

Outline

Introduction

Binary relations


Square proper R.A.

Relation algebras



Lemma

Let S = (S , ∅, U,∪, \, IB ,` , ;) be a proper relation algebrawith base B, and we write E for the set B/U of equivalenceclasses of the unit U. Then

S is a subalgebra of ⊗E∈E(S↓E), and

for each E ∈ E , there is a homomorphism mapping Sonto S↓

E.

Proof: . . .



Applications

Dr. R. Khedri

Outline

Introduction

Binary relations

Relation algebras

Definition of RAs

Peircean law


(Slide 15 of 38)Relation Algebras Relation algebras

Having defined a proper relation algebra, we now seekan algebraic counterpart

We present certain axioms defining abstract relationalgebras

We try to show, among other things, the relationshipof relation algebras to Boolean algebras



Applications

Dr. R. Khedri

Outline

Introduction

Binary relations

Relation algebras

Definition of RAs

Peircean law


(Slide 16 of 38)Relation Algebras Relation algebrasDefinition of relation algebras

Definition

A relation algebra is an algebra A = (A, 0, 1,+,−, 1′,` , ;)satisfying the following axioms, for all a, b, c ∈ A:

R1 The equations defining a Boolean algebra

R2 a ; (b ; c) = (a ; b) ; c

R3 (a + b) ; c = a ; c + b ; c

R4 a ; 1′ = a

R5 a`` = a

R6 (a + b)` = a` + b`

R7 (a ; b)` = b` ; a`

R8 a` ; (−(a ; b)) ≤ −b

We use RA to denote the class of all relation algebras



Applications

Dr. R. Khedri

Outline

Introduction

Binary relations

Relation algebras

Definition of RAs

Peircean law



1′ is the identity

` is the conversion

; is the composition

Binding conventions:` is tighter than −all unary operations are tighter than binary ones; tighter · tighter +

The operations in descending order of priority are` ,−, ;, ·,+



Applications

Dr. R. Khedri

Outline

Introduction

Binary relations

Relation algebras

Definition of RAs

Peircean law



What is the language that we use to describe relationalgebras?

Definition

LRA is the functional signature consisting of constants0, 1, 1′, unary function symbols −,` , and the binaryfunction symbols +, ;.

LRA-Structures are often called relation-type algebras

The interpretations of the symbols 0, 1, 1′ in anLRA-structure A are denoted by 0A, 1A, 1′A

Recall LRA-term, LRA-formulas . . .

The axioms for relation algebras are LRA-equations



Applications

Dr. R. Khedri

Outline

Introduction

Binary relations

Relation algebras

Definition of RAs

Peircean law



` is bijective` is order preserving

1` = 1 and 0` = 0

(−a)` = −a`

Lemma

In any relation algebra A, ` is an automorphism of theBoolean reduct bool(A).

Proof: We must show that ` is bijective and that 1` = 1,0` = 0, and∀(a, b | a, b ∈ A : (a+B)` = a` +b` ∧ (−a)` = −a` ).



Applications

Dr. R. Khedri

Outline

Introduction

Binary relations

Relation algebras

Definition of RAs

Peircean law


(Slide 20 of 38)Relation Algebras Relation algebrasPeircean law

How to understand[a` ; (−(a ; b)) ≤ −b

](i.e., R8) in a

more intuitive and useful form?

R8 is known as the Peircean law or De Morgan’sTheorem K

Definition

The Peircean law is the LRA-sentence

∀(x , y , z |: (x ; y) · z` = 0 ⇐⇒ (y ; z) · x` = 0 ) (PL)

We can continue and add a third equivalence,(z ; x) · y` = 0

Lemma

Axioms R1, R3, R5, R6, R7 |= (axiom R8 ⇐⇒ PL),



Applications

Dr. R. Khedri

Outline

Introduction

Binary relations

Relation algebras

Definition of RAs

Peircean law


(Slide 21 of 38)Relation Algebras Relation algebrasPeircean law



Applications

Dr. R. Khedri

Outline

Introduction

Binary relations

Relation algebras


BAO


Consistent andforbidden triples ofatoms

Representations of RAs

Examples of relationalgebra

(Slide 22 of 38)

Relation AlgebrasAtom structures of relation algebrasBoolean algebras with operator

Boolean algebras with operators

We now generalise the notion of Boolean algebras byadding extra functions (or operators)

What results is called a Boolean algebra withoperators, or BAO

This more general setting was proposed by Jonsson andTarski in 1951,

BAOs are helpful when we come to consider algebras ofbinary and higher-order relations



Applications

Dr. R. Khedri

Outline

Introduction

Binary relations

Relation algebras


BAO





(Slide 23 of 38)

Relation AlgebrasAtom structures of relation algebrasBoolean algebras with operatorBoolean algebras with operators

Definition (Operator)

Let B = 〈B, 0, 1,+,−〉 be a boolean algebra. An operatorΩ with arity or rank rk(Ω) = n < ω onB is a functionΩ : Bn −→ B such that:

1 Ω is normal:

∀(b0, · · · , bn−1 | b0, · · · , bn−1 ∈ B ∧ i < n :

bi = 0 =⇒ Ω(b0, · · · , bn−1) = 0 )

2 Ω is additive:

∀(b0, · · · , bn−1, b, b′ | b0, · · · , bn−1, b, b′ ∈ B ∧ i < n :

Ω (b0, · · · , bi−1, (b + b′), · · · , bn−1)

= Ω (b0, · · · , bi−1, b, · · · , bn−1)

+Ω (b0, · · · , bi−1, b′, · · · , bn−1) )

We may call Ω an n-ary operator.



Applications

Dr. R. Khedri

Outline

Introduction

Binary relations

Relation algebras


BAO





(Slide 24 of 38)


Definition (BAO)

A Boolean algebra with operators (BAO) is an algebra〈B, 0, 1,+,−,Ωλ : λ ∈ Λ〉, where B = 〈B, 0, 1,+,−〉 is aboolean algebra, Λ is a set (perhaps uncountable), andΩλ(λ ∈ Λ) are operators on B.

Definition

Let B = 〈B, 0, 1,+,−〉 be a boolean algebra, and f an n-aryfunction on B. f is said to be completely additive if itsatisfies,

∀(b0, · · · , bn−1,X | b0, · · · , bn−1 ∈ B∧ i < n ∧ X ⊆ B ∧ ΣX exists in B :

f (b0, · · · ,ΣX , · · · bn−1) = Σx∈X f (b0, · · · , x , · · · bn−1) )

for all a, b ∈ B. So f is normal and additive, and hence anoperator on B.



Applications

Dr. R. Khedri

Outline

Introduction

Binary relations

Relation algebras


BAO





(Slide 25 of 38)


A BAO is said to be completely additive if each of itsoperators is completely additive

A class K of BAOs is completely additive if each BAOin K is completely additive

Lemma

Any normal completely additive n-ary function f on aBoolean algebra B is an operator on B.

Proof: . . .



Applications

Dr. R. Khedri

Outline

Introduction

Binary relations

Relation algebras


BAO





(Slide 26 of 38)


Definition

Let L be a functional signature containing the signatureLBA of boolean algebras. We regard constants of L asnullary function symbols.

1 For each n-ary operator symbol Ω ∈ L \ LBA, introducean (n + 1)-ary relation symbol RΩ. Let La be therelational signature RΩ : Ω ∈ L \ LBA.

2 Let B be an atomic BAO of signature L. The atomstructure of B is defined to be the La-structure withdomain the set At(B) of atoms of B and with relationsdefined by:At(B) |= RΩ(a0, ..., an−1, b)

⇐⇒ B |= (b ≤ Ω(a0, ..., an−1))for each n-ary Ω ∈ L \ LBA and atoms a0, ..., an−1, b ofB.



Applications

Dr. R. Khedri

Outline

Introduction

Binary relations

Relation algebras


BAO





(Slide 27 of 38)


For a completely additive operator Ω over an atomicBAO, we can calculate Ω if we know how it is definedon the atoms

The behaviour of the operators on the atoms can bedefined by specifying the atom structure

This is one reason why atoms are so important

The following definitions and propositions are just a(long-winded) way of saying that you can work equallywell with an atomic, completely additive BAO

It’s easier to work with the atom structure (working atthe frame level)

They also allow us to construct various new BAOs fromold ones, by building atom structures



Applications

Dr. R. Khedri

Outline

Introduction

Binary relations

Relation algebras


BAO





(Slide 28 of 38)

Relation AlgebrasAtom structures of relation algebrasAtom structures of RAs

Relation algebras are (completely additive) BAOs

We can apply our definitions and results about atomstructures

The most convenient way to specify an atomic relationalgebra is by giving the atom structure

This means listingthe atoms underneath the identity,the pairs of atoms (a, b) such that b ≤ a`

the triples of atoms (a, b, c) such that c ≤ a ; b

We end with a structure 〈X ,R1′ ,R` ,R;〉



Applications

Dr. R. Khedri

Outline

Introduction

Binary relations

Relation algebras


BAO





(Slide 29 of 38)


Because we are dealing with relation algebras, we cansimplify the format of atom structures slightly

We will replace the binary relation symbol R` by theunary function symbol ` , and read R` (x , y) as x` = y

Definition

The atom structure, written At(A), of an atomic relationalgebra A is the structure

〈S , a ∈ S | a ≤ 1′, (a 7→ a`)a∈S , (a, b, c) ∈ S | c ≤ a ; b〉,

where S is the set of atoms ofA. The signature of thisstructure consists of a unary relation symbol R1′ , a unaryfunction symbol ` , and a ternary relation symbol R;.



Applications

Dr. R. Khedri

Outline

Introduction

Binary relations

Relation algebras


BAO





(Slide 30 of 38)


Working in the other direction:

Given any structure A in the signature R1′ ,` ,R;, wecan define a conventional La

RA-strueture A ′ by letting

(R` )A ′ = (a, a`) | a ∈ A

From A ′, we can define an LRA-BAO

Problem: it may fail to be a relation algebra (e.g.,a`` 6= a)

So we want to find axioms for those structures in thesignature R1′ ,` ,R; that are the atom structures ofrelation algebras



Applications

Dr. R. Khedri

Outline

Introduction

Binary relations

Relation algebras


BAO





(Slide 31 of 38)


Definition

A structure S for the signature R1′ ,` ,R; is called arelation algebra atom structure if it satisfies:

1 identity: ∀(x , y | x , y ∈ S : ∃(e | e ∈ S :R1′(e) ∧ R;(x , e, y) ) )

2 ∀(x , y , z | x , y , z ∈ S : R;(x , y , z)=⇒ R;(x`, z , y) ∧ R;(y`, x`, z`) )

3 Associativity:

∀(x , y , z , t | x , y , z , t ∈ S :

∃(u | u ∈ S : R;(x , y , u) ∧ R;(u, z , t) )

⇐⇒ ∃(v | v ∈ S : R;(y , z , v) ∧ R;(x , v , t) ) ).



Applications

Dr. R. Khedri

Outline

Introduction

Binary relations

Relation algebras


BAO





(Slide 32 of 38)

Relation AlgebrasAtom structures of relation algebrasConsistent and forbidden triples of atoms

In practice, we will often specify the compositionrelation R; of a relation algebra atom structure bylisting its consistent triples

Aternatively, we can list the inconsistent or forbiddentriples instead



Applications

Dr. R. Khedri

Outline

Introduction

Binary relations

Relation algebras


BAO





(Slide 33 of 38)

Relation AlgebrasAtom structures of relation algebrasConsistent and forbidden triples of atoms

Definition

Let A be an atomic relation algebra, and S a relationalgebra atom structure.

A triple (a, b, c) of atoms of A is said to be consistentif c` ≤ a ; b.

A triple (a, b, c) of atoms of S is said to be consistentif R;(a, b, c`)

A triple (a, b, c) of atoms of A or elements of S is saidto be inconsistent or forbidden, if it is not consistent(i.e., a ; b · c` = 0 or ¬R;(a, b, c`), resp.)

The six Peircean transforms of a triple (a, b, c) ofatoms are

(a, b, c), (b, c , a), (c , a, b), (a`, c`, b`), (b`, a`, c`), (c`, b`, a`)



Applications

Dr. R. Khedri

Outline

Introduction

Binary relations

Relation algebras


BAO





(Slide 34 of 38)

Relation AlgebrasAtom structures of relation algebrasRepresentations of relation algebras

Definition

Let A be a relation algebra.

A representation of A is an isomorphism from A to aproper relation algebra

A is said to be representable if it has a representation

A representation h : A −→ P is said to be square if Pis a square proper relation algebra



Applications

Dr. R. Khedri

Outline

Introduction

Binary relations

Relation algebras


BAO





(Slide 35 of 38)

Relation AlgebrasAtom structures of relation algebrasExamples of relation algebra

The smallest relation algebra

This is of course the one-element (degenerate) relationalgebra

It is unique up to isomorphism

A representation with base 0 is obtained by mapping itsunique element to 0

The algebra is isomorphic to F(∅)



Applications

Dr. R. Khedri

Outline

Introduction

Binary relations

Relation algebras


BAO





(Slide 36 of 38)


The smallest non-degenerate relation algebra

It has one atom (i.e., 1) and two elements, 0, 1

It is based on the two element boolean algebra with thefollowing operators:

1′ = 1both elements are self-converseand composition is defined by 1 ; 1 = 1, 0 ; a = a ; 0 = 0for all a ∈ 0, 1

For a representation, take a set X with exactly onepoint x

Let h(0) = ∅ and h(1) = (x , x)The algebra is isomorphic to F(x)



Applications

Dr. R. Khedri

Outline

Introduction

Binary relations

Relation algebras


BAO





(Slide 37 of 38)


Two-atom algebra

This relation algebra has two atoms 1′ and # (therefore has four elements)

Both atoms are self-converse

Composition is defined by; 1′ #

1′ 1′ #

# # 1 (or 1′)

Alternatively, the forbidden triples of atoms are

(1′, 1′,#), (1′,#, 1′), (#, 1′, 1′), (#,#,#)



Applications

Dr. R. Khedri

Outline

Introduction

Binary relations

Relation algebras


BAO





(Slide 38 of 38)

cas 738 relation algebra and kleene algebra and their ... · cas 738 relation algebra and kleene...

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