On Kleene Algebras and Closed Semirings

Home , Kleene algebra, Semiring

Dexter Kozen

Department of Computer Science

Cornell University

Ithaca New York USA

May

Abstract

Kleene algebras are an imp ortant class of algebraic structures that arise in diverse

areas of computer science program logic and semantics relational algebra automata

theory and the design and analysis of algorithms The literature contains several

inequivalent denitions of Kleene algebras and related algebraic structures

In this pap er we establish some new relationships among these structures Our

main results are

There is a Kleene algebra in the sense of that is not continuous

The categories of continuous Kleene algebras closed semirings and

Salgebras are strongly related by adjunctions

The axioms of Kleene algebra in the sense of are not complete for the universal

Horn theory of the regular events This refutes a conjecture of Conway p

Righthanded Kleene algebras are not necessarily lefthanded Kleene algebras

This veries a weaker version of a conjecture of Pratt

In Rovan ed Proc Math Found Comput Scivolume of Lect Notes in Comput Sci pages

Springer

Supp orted by NSF grant CCR

Intro duction

Kleene algebras are algebraic structures with op erators and satisfying certain

prop erties They have b een used to mo del programs in Dynamic Logic to prove

the equivalence of regular expressions and nite automata to give fast algorithms for

transitive closure and shortest paths in directed graphs and to axiomatize relational

algebra

There has b een some disagreement regarding the prop er denition of Kleene algebras

At least ten inequivalent denitions of Kleene algebras or related structures have app eared

in the literature all serving roughly the same purp ose It is imp ortantto

understand the relationships b etween these classes in order to extract the axiomatic essence

of Kleene algebra Perhaps it is not to o unreasonable to hop e that wemight one day converge

on a standard denition

In this pap er we discuss some of the relationships among these classes For the purp oses of

this pap er we adopt the denition of which denes a Kleene algebra to b e an idemp otent

semiring with a op erator satisfying

aa a

a a a

ax x a x x

xa x xa x

where refers to the natural order in the idemp otent semiring It is shown in that these

axioms completely characterize the algebra of regular events giving a nitary axiomatization

of that equational theory and thus improving Salomaas completeness theorem A similar

and in fact stronger theorem is stated by Conway without pro of p but to his

knowledge and the authors no pro of has ever app eared in the literature

A Kleene algebra is called continuous if it satises the axiom

ab c ab c

where refers to the supremum in the natural order This innitary prop ertyof implies

the previous four These algebras have b een used to mo del programs in Dynamic Logic

In the design and analysis of algorithms a related family of structures called closed

semirings form an imp ortant algebraic abstraction They give a unied framework for de

riving ecient algorithms for transitive closure and allpairs shortest paths in graphs and

constructing regular expressions from nite automata Very fast algorithms for all

these problems can b e derived as sp ecial cases of a single general algorithm over an arbitrary

closed semiring Closed semirings are dened in terms of a countable summation op erator

P P

as well as and the op erator is dened in terms of Under the op erations of

nite andany closed semiring is a continuous Kleene algebra In fact in

the treatment of the sole purp ose of seems to b e to dene A more descriptive

name for closed semirings mightbe complete idempotent semirings

These algebras are strongly related to several classes of algebras dened byConwayin

his monograph Conways Salgebras are similar to closed semirings except that

arbitrary sums not just countable ones are p ermitted A b etter name for Salgebras might

be complete idempotent semirings The op eration is dened as in closed semirings in terms

P P

of and again this seems to b e the sole purp ose of The Nalgebras are algebras of

signature that are subsets of Salgebras containing and and closed under

nite and Combining results of and Theorem p it can b e shown

that the classes of Nalgebras and continuous Kleene algebras coincide An Ralgebra is

any algebra of signature satisfying the equational theory of the Nalgebras

In Pratt gives two denitions of Kleene algebras in the context of dynamic

algebra In Kleene algebras are dened to b e the Kleenean comp onent of the separable

dynamic algebras in this class is enlarged to contain all subalgebras of such algebras

Generalizations of Kleenes and Parikhs Theorems havebeengiven by Kuich in

complete semirings which are similar to Salgebras in all resp ects except that idemp otence

of is replaced bya weaker condition called completeness

In this pap er we establish some new relationships among some of these structures Our

main results are as follows

It is known that the Ralgebras Kleene algebras continuous Kleene algebras aka

Nalgebras closed semirings and Salgebras eachcontain the next in the list All

inclusions were known to b e strict except Kleene algebras and continuous Kleene

algebras We complete the picture by constructing a Kleene algebra that is not

continuous Such an algebra must necessarily b e innite since all nite Kleene algebras

are continuous

Conway gives a construction that embeds every continuous Kleene algebra in an S

algebra Theorem p This construction can b e describ ed as a completion

by ideals We show that this construction extends to a functor from the category

KA of continuous Kleene algebras and Kleene algebra morphisms to the category

SA of Salgebras and continuous semiring morphisms and that this functor is a left

adjoint for the forgetful functor that assigns to each Salgebra its Kleene algebra struc

ture Moreover this adjunction factors into two adjunctions one relating KA and

the category CS of closed semirings and continuous semiring morphisms and one

relating CS and SA The functor from KA to CS can b e describ ed as a completion

by countably generated ideals

We show that the axioms for Kleene algebra are not complete for the universal Horn

theory of the regular events This refutes a conjecture of Conway p

We showthatrighthanded Kleene algebras do not have to b e lefthanded This veries

aweaker version of a conjecture of Pratt Pratts conjecture referred to Boolean

semirings yet another related structure in which the natural order comes from a

Bo olean algebra

Denitions

Kleene Algebras

For the purp oses of this pap er we adopt the denition of There a Kleene algebra is

dened to b e a structure K with binary op erations and unary op eration and constants

and such that

is an idemp otent semiring ie

is asso ciative commutative and idemp otenta a awithidentity

is asso ciativewithtwosided identity

distributes over on b oth sides and

isatwosided annihilator for

and the op eration satises the following prop erties

aa a

a a a

ax x a x x

x xa x xa

where refers to the natural partial order on K

a b a b b

We often abbreviate a b to ab and avoid parentheses by assigning the priority to

the op erators

A somewhat stronger axiom for is the socalled continuity condition

ab c ab c

n n

where b b bb and refers to the supremum with resp ect to the natural order

A Kleene algebra satisfying is said to b e continuousInany idemp otent semiring

implies

An idemp otent semiring satisfying is called a righthanded Kleene algebra An

idemp otent semiring satisfying and is called a lefthandedKleene algebraThus an

idemp otent semiring is a Kleene algebra i it is b oth left and righthanded

The category of Kleene algebras and Kleene algebra homomorphisms will b e denoted

KA The full sub category of continuous Kleene algebras will b e denoted KA

Examples of Kleene Algebras

An imp ortant example of a Kleene algebra is Reg the family of regular sets over a nite

alphab et The equational theory of this structure is called the algebraofregular events

This theory was axiomatized by Salomaa but his axiomatization involved rules that

were not sound in general when interpreted over other Kleene algebras It was shown in

that the Kleene algebra axioms completely axiomatize the algebra of regular events In

other words Reg is the free Kleene algebra on free generators Conway Theorem

p claimed without pro of that the axioms for righthanded Kleene algebra completely

axiomatize the algebra of regular events but to his knowledge and the authors no pro of

has ever app eared in the literature A weaker completeness pro of involving instead of

was previously given in

Another imp ortant class of Kleene algebras is the class of algebras of binary relations A

family of binary relations with op erations of for relational comp osition

R S fx z jy x y R and y z S g

for the empty relation for the identity relation for and reexive transitive closure for

is a Kleene algebra Kleene algebras of binary relations are used to mo del programs in

Dynamic Logic and other logics of programs

All these examples are continuous In xwe will construct a Kleene algebra that is not

continuous

The n n matrices over a Kleene algebra again form a Kleene algebra

Elementary Prop erties of Kleene Algebras

We list here some elementary consequences of the Kleene algebra axioms See for

pro ofs

In any Kleene algebra K the relation is a partial order and K is an upp er semilattice

with join and minimum element Also

a b ac bc

a b ca cb

a b a c b c

a b a b

a a a a

a a

aa a

a a a

b ax x a b x

b xa x ba x

ax xb a x xb

cd c cdc

a b a ba

pp p p p a p p ap

Closed Semirings

According to a closed semiring is an idemp otent semiring equipp ed with an innitary

summation op erator that applies to countable sequences and satises innitary asso cia

tivity commutativity idemp otence and distributivitylaws In b oth the nitary

and innitary summation op erators and are taken as primitive and no explicit axioms

connecting them are given However it is clear from subsequent arguments what was meant

and how to complete the axiomatization

Since is asso ciative commutative and idemp otent its value on a given sequence is

indep endent of the order and multiplicity of elements o ccurring in the sequence Thus we

P P

mightaswell dene on nite or countable subsets instead of sequences In this view

gives the supremum with resp ect to the natural order To see this let A b e a nonempty

nite or countable subset of a closed semiring C Ifx Athen

X X

x A A fxg

thus x A and if x y for all x Athen x y y for all x Athus

X X X

x y Ay

xA xA

x y

therefore

A y

Thus wemay regard as a supremum op erator for countable sets The innite distributivity

laws then say exactly that is bicontinuous with resp ect to countable suprema Note also

that is the supremum of the empty set

In a closed semiring C one can dene by

b b

n n

where b and b bb Thenby innite distributivity

ab c ab c

therefore C is also a continuous Kleene algebra KC Moreover if C and C are two closed

semirings and h C C is an contin uous semiring morphism ie a semiring morphism

that preserves suprema of countable sets then h must preserve therefore constitutes a

Kleene algebra morphism Kh KC KC Wethus have a forgetful functor

K CS KA

from the category CS of closed semirings and continuous semiring morphisms to the

category KA of continuous Kleene algebras and Kleene algebra morphisms We will

showinx that this functor has a left adjoint

Examples of Closed Semirings

The regular sets Reg do not form a closed semiring if A is nonregular the countable set

ffxgjx Ag has no supremum However the p ower set of do es form a closed semiring

Similarly the family of all binary relations on a set forms a closed semiring under the

relational op erations describ ed in x and innite union for

A rather bizarre example of a closed semiring is the socalled min algebra used to

derive an ecient algorithm for nding the shortest paths b etween all pairs of vertices in a

directed graph This algebra consists of the nonnegative reals with an innite element

adjoined The Kleene algebra op erations and are given bymin and

resp ectively The op eration is the constant function

As with Kleene algebras the n n matrices over a closed semiring again form a closed

semiring

Conways Algebras

According to the denition in an Salgebra

is similar to a closed semiring except that is dened not on sequences but on multisets

of elements of S moreover there is no cardinality restriction on the multiset thus is to o

big to b e represented in ZF set theory However as with closed semirings the value that

P P

takes on a given multiset is indep endentofthemultiplicity of the elements so might

as well b e dened on subsets of S instead of multisets So dened A gives the supremum

of A with resp ect to the order We assume the axiom fag a which is omitted in

Thus the only essential dierence b etween Salgebras and closed semirings is that closed

semirings are only required to contain suprema of countable sets whereas Salgebras must

contain suprema of all sets Thus every Salgebra is automatically a closed semiring and every

continuous semiring morphism semiring morphism preserving all suprema is automatically

con tinuous and these notions coincide on countable algebras This gives a forgetful functor

G SA CS from the category SA of Salgebras and continuous semiring morphisms to

A Kleene algebra that is not continuous

Recall the denitions of the categories

category objects morphisms

SA Salgebras continuous semiring morphisms

CS closed semirings continuous semiring morphisms

KA continuous Kleene algebras Kleene algebra morphisms

KA Kleene algebras Kleene algebra morphisms

RA Ralgebras maps preserving

From results of the equational theories of all these algebras coincide The

following inclusions are also known

SA CS KA KA RA

All inclusions were known to b e strict except KA KA Conway p gives a

fourelement Ralgebra R that is not a continuous Kleene algebra It is easily shown

that all nite Kleene algebras are continuous thus R is not a Kleene algebra either The

family Reg of regular events over an alphab et gives an example of a continuous Kleene

algebra that is not a closed semiring as shown in x It is also easy to construct closed

semirings that are not Salgebras for example the countable and cocountable subsets of

the rst uncoutable ordinal with op erations of set union for setintersection for for

for and A

To complete the picture we need to construct a Kleene algebra that is not continuous

Let denote the set of ordered pairs of natural numb ers and let and b e new elements

Order these elements such that is the minimum element is the maximum element and

is ordered lexicographically in b etween Dene to give the supremum in this order

Dene according to the following table

x x

x x x

a b c d a c b d

Then is the additiveidentityand is the multiplicativeidentity Finally dene

if a or a

otherwise

It is easily checked that this is a Kleene algebra Weverify the axiom

ax x a x x

explicitly Assuming ax xwewishtoshow a x x If a or a then

a and we are done since is the multiplicativeidentityIfx or x

u v in whichcase ax xcontradicting the we are done Otherwise a and x

assumption

This Kleene algebra is not continuous since but

X X

n n

Some categorical connections

In this section we establish strong relationships among the categories KA CS and SA

Recall from x that KA is the category of Kleene algebras and Kleene algebra morphisms

KA is the full sub category of continuous Kleene algebras and Kleene algebra morphisms

CS is the category of closed semirings and continuous semiring morphisms and SA is the

category of Salgebras and continuous semiring morphisms

In Theorem p Conway gives a construction that shows that every continuous

Kleene algebra is emb edded in an Salgebra This construction can b e describ ed as a com

pletion by ideals Conwaygives the construction but omits the pro of claiming that the

details are rather subtleT o his knowledge and the authors no pro of has app eared in

the literature We will show in this section that Conways construction constitutes a functor

KA SA that is a left adjoint for the forgetful functor SA KA Moreover this

adjunction factors into a comp osition of adjunctions

C S

KA CS SA

K G

where C KA CS is a left adjoint for the forgetful functor K CS KA and

S CS SA is a left adjoint for the forgetful functor G SA CSConways construction

is then the comp osition S C

The construction C whichshows that every continuous Kleene algebra is embedded

in a closed semiring can b e describ ed as a completion by countably generated ideals

Kleene algebras and closed semirings

Denition Conway Let K be a continuous Kleene algebra A ideal is a subset

A of K such that

A is nonempty

A is closed under

A is closed downward under

if ab c A for all n then ab c A

Wesay that a nonempty set A generates a ideal I if I is the smallest ideal containing

AWe write hAi to denote the ideal generated by AA ideal is countably generated if

it has a countable generating set If A is a singleton fxgthenwe abbreviate hfxgi by hxi

Such an ideal is called principal with generator x

Let K be a continuous Kleene algebra For subsets A B K dene

A B fa b j a A b B g

A fy jx Ay xg

A fab c j ab c A for all n g

Note that for principal ideals

hxi fxg

Note also that

A B C A B A C

A B C A C B C

A B A B

Dene the set op erator on nonempty subsets A Kas follows

A A A A A

Then A is a ideal i it is nonempty and closed under Note that A A and that

is monotone in the sense that if A B then A B Using dene for any A K

the transnite sequence

A A

A A a limit ordinal

A A

Note that A A where is the least ordinal such that A A Sucha

exists since is monotone In fact it can b e shown that the rst uncountable

ordinal By the KnasterTarski Theorem

A hAi

Lemma

A B A B

Proof Weprove the rst statement the second is symmetric

A B A B A A A

A A B A B A B

A B A B A B A B

A B

Lemma

hA B i hhAihB ii

Proof The inclusion follows from monotonicityFor the reverse inclusion we prove

by transnite induction that for all ordinals

A B hA B i

Certainly

A B A B hA B i

Also

A B A B

A B by Lemma

hA B i by induction hyp othesis and monotonicityof

hA B i

and similarly

A B hA B i

For limit ordinals

A B A B

A B

hA B i

and similarly

A B hA B i

Thus

hhAihB ii h A B i

hA B i

Nowwe dene a closed semiring CK as follows The elements of CK will b e the countably

generated ideals of K For any countable set of countably generated ideals I dene

I h I i

n n

This ideal is countably generated since if A is countable and generates I for n then

n n

S P P

A is countable and generates I The op erator is asso ciative commutative and

n n

idemp otent since is

For any pair of elements I J dene

I J hI J i

This ideal is countably generated if I and J are since

hAihB i hhAihB ii

hA B i

by Lemma and A B is countable if A and B are

The ideal hi fg is included in every ideal and is thus an additive identity It is also

amultiplicative annihilator

hiI hhiI i

hfgI i

The ideal hi isamultiplicative identity

hiI hhiI i

hfgI i by Lemma

hI i

Finally the distributivelaws hold

X X

J I J i hI

n n

hI h J ii

hI J i by Lemma

h I J i

h hI J ii

h I J i

I J

and symmetrically

Using the fact hxi fxg it is straightforwardtoverify that the map

x hxi

is onetoone and preserves and thus constitutes an isomorphic emb edding of

K into the continuous Kleene algebra KCK This map will b e the unit of our adjunction

If all references to countability are removed from the foregoing construction we get

an emb edding of K into an Salgebra consisting of not necessarily countably generated

ideals This gives a pro of of Theorem p

Wenow wish to establish an adjunction b etween KA and CS The rst matter at hand

is to establish a bijection b etween the homsets KKC and C KC for a continuous

Kleene algebra K and closed semiring C

Let g KKC a Kleene algebra morphism For a subset A of K dene

g A fg x j x Ag

Lemma For any A Kand ordinal the fol lowing hold

g A g A

g hAi hg Ai

Ai hg hAii hg

Proof Toprove the rst statement note that

g A A fg x g y j x y Ag

g A g A

g A fg y jx Ay xg

fg y jx Agy g xg

g A

g A fg xy z j xy z A n g

fg xg y g z j xy z A n g

fg xg y g z j g xg y g z g An g

g A

Then

A A A A g A g

g A A g A g A

g A g A g A g A

g A

The second statement follows from the rst by transnite induction

The third follows from the second by taking suciently large

The inclusion of the last statement follows from the third statement and the inclusion

follows from the monotonicityof g and hi

P P

Lemma If I CK then g I exists and is equal to g A for any generating set A of

Proof By Lemma

hg I i hg hAii hg Ai

for any generating set A of I In particular since I CK I has a countable generating set

B and since CK is closed under countable suprema g B exists Then

X X

hg Ai hg B i h g B i g B

thus g B is an upp er b ound for g A F or any other upp er b ound y

g B hg Ai hy i

thus g B y

There is a subtlety here it is tempting to conclude that hg I i h g I i in Lemma

This app ears not to b e true in general

Now dene the map g CKC by

g I g I

This is well dened by Lemma

Wehavenotyet sp ecied the action of the functor C on morphisms of KA Thiscan

b e b e derived from the adjunction For anypairK K of continuous Kleene algebras and

Kleene algebra morphism h KK dene

hih Ch

Theorem The map g is an continuous semiring morphism The construction g g

gives a natural bijection between the homsets KKC and CKC

g is an continuous semiring morphism wemust show that Proof To showthat

and are preserved

X X X

g I g I

n n

g h I i

g I by Lemma

g I

X X

g I

g I J g I J

g hI J i

g I J by Lemma

g I g J

X X

g I g J by innite distributivity

g I g J

g preserves and is straightforward to verify That

To showthatg g is a bijection dene the map h h for h CKC by

h x hhxi

Straightforward arrowchasing shows that and are inverses

Strictly sp eaking in order to establish an adjunction wemust also check a naturality

and interacts nicely with morphisms condition that ensures that the bijection given by

in KA and CS see Sp ecicallywemust show that for all morphisms h K K

g KKC and f CC

Kf g h f g Ch

Again this can b e veried by straightforward arrowchasing whichwe leave to the reader

Closed semirings and Salgebras

The other half of the factorization of Conways construction emb eds an arbitrary closed

semiring into an Salgebra In comparison to the construction of previous section this

construction is muchlessinteresting Wegive the main construction and omit formal details

Recall that closed semirings and Salgebras are b oth idemp otent semirings with an innite

summation op erator satisfying innitary asso ciativitycommutativity idemp otence and

distributivitylaws The only dierence is that closed semirings allow only countable sums

whereas Salgebras allow arbitrary sums Morphisms of closed semirings are the continuous

semiring morphisms and those of Salgebras are the continuous semiring morphisms

To construct an emb edding of a given closed semiring C in an Salgebra SC we complete

C by ideals An ideal is a subset A Csuchthat

A is nonempty

A is closed under countable sum

A is closed downward under

Take SC to b e the set of ideals of C with the following op erations

I i I h

I J hI J i

where is dened as in x

The arguments from here on are quite analogous to those of x

The universal Horn theory of regular events

The universal Horn theory of a class of structures is the set of universally quantied equa

tional implications of the form

n n

true in all structures in the class In this section wegive a brief argument showing that

the axioms for Kleene algebra given in x whichcharacterize the universal Horn theory of

Kleene algebras and the equational theory of Reg donotcharacterize the universal Horn

theory of Reg This refutes a conjecture of Conway p

Consider the equational implication

a a

This formula is true in Reg for all values of a in other words for any regular event Aif

A fg then A fgHowever this formula is not a theorem of Kleene algebra since

there exists a Kleene algebra in which it fails interpret a as the matrix

in the Kleene algebra of matrices over Reg This algebra was shown to b e a Kleene

algebra in

It is an op en problem to give a complete axiomatization of the universal Horn theory of

the regular events

Righthanded Kleene algebras are not necessarily

lefthanded

Recall the right and lefthanded Kleene algebra rules

ax x a x x

xa x xa x

In Pratt asks whether these two rules are equivalen t and conjectures that they are

not His conjecture actually refers to the more sp ecic case of Bo olean monoids in which

comes from a Bo olean algebra However the question makes sense in the more general

context of Kleene algebras In this section we construct a righthanded Kleene algebra that

is not left handed this answers Pratts question in the more general case of Kleene algebras

and provides evidence in favor of his conjecture The conjecture as stated in is still op en

Consider the set F of strict nitely additive set functions f P X PX for any

innite set X that is functions f for which

f A B f A f B

Dene

f g A f A g A

f g A f g A

f A A

f A f f A

f A f A a limit ordinal

f A f A

A A

Note that nite additivity implies monotonicity

Prop osition The structure

is a righthandedKleene algebra but not a lefthandedKleene algebra

Proof To show that F is a righthanded Kleene algebra we need to verify all the Kleene

algebra axioms except The additive asso ciativity commutativity idemp otence and

identitylaws hold since they hold for set union Function comp osition is surely asso ciative

with twosided identity The function is a left annihilator by denition and a right

annihilator b ecause of strictness of functions in F The left distributivitylaw holds by

denition of and and the right distributivitylaw holds b ecause of nite additivityof

functions in F Thus F is an idemp otent semiring

The axioms for can b e veried by transnite induction Let us verify the right

handed rule Supp ose f g g Then for any A X f g A g A Wehave

f g A g A

f g A f f g A

f g A by induction hyp othesis and monotonicityof f

g A

f g A f g A

g A

g A a limit ordinal

f g A f g A

g A

All we need to show that is not satised is the existence of a strict nitely additive

function g that is not continuous ie suchthat

g g

Then taking f g wehave

f g g g g f

but

f g f g g g

g g

contradicting There are many g that satisfy for example pick out a countable

prop er subset Y of X and call them the natural numb ers then dene

A fx j x Ag if A is a nite subset of Y

g A

X otherwise

Then g fgY but g fgg Y X

Acknowledgments

I am grateful to John Horton ConwayVaughan Pratt and Dana Scott for their valuable

comments

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