Modal Semirings with Operators for Knowledge Representation

Kim Solin

Dept. of Philosophy, Uppsala University, Uppsala, Sweden

Keywords:

Modal Semirings, Modal Algebra, Boolean Algebra with Operators, Modal Semirings with Operators,

Knowledge Representation, Dynamic Epistemic Logic.

Abstract:

Modal semirings are combined with modal algebra (Boolean algebra with operators) to form modal semirings

with operators. In turn, these are extended with a revision operator and used for knowledge representation.

1 INTRODUCTION

There are several ways of representing and reasoning

about changing knowledge. Two of these stand out.

First, one has the semiformal approach of Alchourron,

ardenfors and Makinson (Alchourr

on et al., 1985),

which became a landmark in this ﬁeld, and which

has spurred a colossal amount of research. Second,

one has a way of reasoning about changing knowl-

edge that is fully formal: dynamic epistemic logic,

see (van Ditmarsch et al., 2007) with references. But

this paper will be about a third style, namely, the alge-

braic. If one takes this approach, one strives to reason

about knowledge with the aid of algebra, see (Baltag

et al., 2007; Baltag et al., 2005; Baltag and Sadrzadeh,

2006; M

oller, 2008; Solin, 2010; Panangaden and

Sadrzadeh, 2010). Why would one take the algebraic

approach? The answer is that there are many advan-

tages to algebra, in particular clear-cut reasoning, and

efﬁcient automation and mechanisation (Aboul-Hosn

and Kozen, 2006; H

ofner and Struth, 2007; Foster and

Struth, 2012).

The algebraic approach can be further subdi-

vided into a concrete and an abstract approach. The

concrete-algebraic approach means that one reasons

in a concrete algebra of, for example, relations. The

abstract-algebraic approach means that one via ax-

ioms deﬁnes an abstract algebra suited for one’s pur-

poses; the approach could also be called axiomatic-

algebraic. I shall focus on the abstract-algebraic de-

velopment that has its roots in Dexter Kozen’s work

on Kleene algebra (Kozen, 1994).

1.1 Background

Dexter Kozen conceived of Kleene algebra with tests

in the late 1990s (Kozen, 1997). This structure, which

is an idempotent semiring extended with Kleene star,

and embedded with a Boolean subalgebra of tests,

was later developed into Kleene algebra with domain

by Desharnais, M

oller and Struth (Desharnais et al.,

2006). The last-mentioned structure allows the intro-

duction of modal operators in the sense of dynamic

logic (Harel et al., 2000), so it is also called modal

Kleene algebra.

Modal Kleene algebra was further augmented

with a revision operator (Solin, 2010) that allows

one to reason abstract-algebraically about chang-

ing knowledge in the style of dynamic epistemic

logic (van Ditmarsch et al., 2007). The tests were then

conceived of as actions that check whether the agent

believes some proposition or not. But since the tests

form a Boolean algebra, this had the unfortunate con-

sequence that ¬B¬p by double negation reduces to

p. In addition, one could not distinguish between the

agent’s beliefs and truth, since they were both mod-

elled by the same structure. Although this might be

all right in some circumstances, in others it is cer-

tainly not. The aim of this paper is to remedy that

situation by introducing the belief operator as an al-

gebraic operator in its own right, and to incorporate

that operator into modal Kleene algebra. I will focus

on the semiring reduct, that is, on modal semirings.

Since one has a Boolean subalgebra at hand in

any modal semiring, it is suitable to model the be-

lief operator with the aid of a Boolelan algebra with

operators. Boolean algebras with operators were

introduced by (J

onsson and Tarski, 1951; J

onsson

and Tarski, 1952) and – independently, it seems –

by (Lemmon, 1966a; Lemmon, 1966b). Lemmon

called them modal algebras. The work in this paper

will build mainly on Lemmon’s results. Indeed, I will

197

Solin K..

Modal Semirings with Operators for Knowledge Representation.

DOI: 10.5220/0004181001970202

In Proceedings of the 5th International Conference on Agents and Artiﬁcial Intelligence (ICAART-2013), pages 197-202

ISBN: 978-989-8565-39-6

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

propose that for modelling the belief operator one can

use what Lemmon calls a normal epistemic algebra.

The focus of this paper is the abstract algebra, and

not some speciﬁc set-theoretic model – such as rela-

tions or hypertheories (Segerberg, 1999) – that could

satisfy the axioms of the algebra; for appreciating the

paper, it is pivotal that the reader constantly bears this

in mind. The axioms that will be presented are mo-

tivated only by common sense, and there will not be

any model-theoretic justiﬁcations for them. The ab-

stract algebra is here considered as a ﬁrst-class con-

ceputal tool, and not merely as a calucaltional aid for

a model.

Nevertheless, we shall throughout this pa-

per consider a simple relational model that will be

given below. It is a model with which many are fa-

miliar and that is useful for getting a more tangible

technical-mathematical understanding of the axioms.

But I want to emphasise, that this model is only a

heuristic aid and a technical tool. I shall only use the

model to prove technical results, such as the fact that

the tentative axioms do not contradict each other.

1.2 Structure of the Paper

The paper has the following structure. First, the the-

ory leading up to modal semirings is reviewed. Then

the parts of Lemmon’s work on modal algebra that

are needed for this paper are presented, upon which –

the main contribution of this paper – modal semirings

and modal algebra are combined; this structure we’ll

call a modal semiring with operators. After combin-

ing the two algebraic structures, a revision operator is

also proposed, yielding dynamic epistemic modal al-

gebra. Then some basic properties in the algebra are

considered, and the paper concludes with a number

of interesting themes for further investigation, theo-

retical as well as practical.

2 MODAL SEMIRINGS

An idempotent semiring is a structure over the sig-

nature (+, ;, 0, 1) that satisﬁes the following axioms

(a, b and c in the carrier set, and ; left implicit):

In contrast, (Solin, 2010) viewed the abstract algebra

as something of a calculational tool intended for existing

models. The models were the driving force behind the ax-

ioms of the algebra. The perspective in this paper is the

opposite: the abstract algebra here speaks for itself, the ax-

ioms are motivated only by common sense, and the various

models are considered secondary. – There will of course be

relations to set-theoretic models, but these will not be in-

vestigated in this paper. And, needless to say, the algebraic

formulation can itself be seen as a sort of axiomatically de-

ﬁned set-theoretic model.

a + (b + c) = (a + b) + c, (1)

a + b = b + a, (2)

a + 0 = a, (3)

a + a = a, (4)

a(bc) = (ab)c, (5)

1a = a = a1, (6)

0a = 0 = a0, (7)

a(b + c) = ab + ac and (8)

(a + b)c = ac + bc. (9)

This means that (+, 0) is a commutative and idem-

potent monoid, and that (;, 1) is a monoid such that

; distributes over + , and that 0 annihilates to left

and right. The operator ; will often be left implicit.

The canonical order ≤ on a semiring is deﬁned by

a ≤ b ⇔

def

a + b = b for all a and b in the carrier set.

By this, 0 is the least element, + is join, and ; and +

are both monotone (isotone).

A test semiring (Desharnais et al., 2006; Kozen,

1997) is a two-sorted algebra

(S, test(S), +, ; , ¬, 0, 1)

such that

(i) (S, +, ;, 0, 1) is an idempotent semiring,

(ii) (test(S), +, ;, ¬, 0, 1) is a Boolean algebra (BA)

and

(iii) test(S) ⊆ S.

So join and meet in test(S) are + and ;, respectively,

and the complement is given by ¬. Naturally, 0 is

the least and 1 is the greatest element. For general

semiring elements it is the custom to use the notation

a, b, . . ., and for test elements p, q, . . . is the conven-

tion. On a test semiring one can axiomatise a domain

operator p : S → test(S) by

a ≤pa ; a, (10)

p(pa) ≤ p and (11)

p(apb) ≤ p(ab), (12)

for all a, b ∈ S and p ∈ test(S) (Desharnais et al.,

2006). It has been proved that (10) and (12) can be

strengthened to equalities.

A relational interpretation of domain and the other

operators is given below, and an epistemic interpreta-

tion will be given later (Section 5.1).

As mentioned in the introduction, one can deﬁne

modal operators with the aid of the domain operator

(Desharnais et al., 2006):

haip =

def

p(ap) and [a]p =

def

¬hai¬p . (13)

This is why test semirings with a domain operator are

also called modal semirings. The diamond should be

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read “it is possible to successfully perform a so that p

will hold” and the box as “after every successful way

of performing a it will be the case that p holds.” In

the terminology of Dijkstra (Dijkstra, 1976), the box

is the weakest liberal precondition of a with respect

to p: if a terminates, p will hold. What is here called

‘successful’ corresponds to what Dijkstra calls ‘ter-

minating’.

Let us list some basic properties of this structure

(Desharnais et al., 2006). One has stability of tests,

pp = p , (14)

and additivity,

p(a + b) = pa + pb . (15)

Of course, isotony of domain is immediate from addi-

tivity. If one lets the operator p → q be the syntactic

equivalent of ¬p + q, then

(p → q = 1) ⇔ p ≤ q

obtains.

2.1 A Relational Model

Below follows a concrete algebra that forms a rela-

tional model for the above axioms.

Let S, T ⊆ Σ × Σ, and deﬁne the binary operators ;

and +, and the unary operator p as follows:

(x, y) ∈ (S;T ) ⇔

def

∃z ∈ Σ

•

(x, z) ∈ S

and (z, y) ∈ T,

(x, y) ∈ (S + T ) ⇔

def

(x, y) ∈ S or (x, y) ∈ T,

(x, x) ∈ pS ⇔

def

(x, y) ∈ S for some y ∈ Σ.

Now let the identity relation be denoted by ∆ and the

empty relation by Ø, and let ¬ be the complement

relative to ∆. Then the structure

(℘(Σ × Σ),℘(∆), +, ;, ¬, p , Ø, ∆)

is a modal semiring (Desharnais et al., 2006).

The following example from (Solin, 2010) is in-

formative when ﬁguring out how the modal operators

work in this heuristic model. But keep in mind that

the model is only a technical tool in this paper, since

the axioms are motivated primarily by the common-

sense interpretation found in Section 5.1.

Example 2.1. To the letter from (Solin, 2010). Let

Σ = {1, 2, 3, 4, 5}, let

S = {(1, 2), (2, 2), (2, 5), (4, 3)}

and let

P = {(2, 2), (3, 3)}.

Think of Σ as a set of states, S as a state-changing

action and P as a predicate determining the states 2

and 3. According to the deﬁnitions above we then

have

hSiP = {(1, 1), (2, 2), (4, 4)}

and

[S]P = {(1, 1), (3, 3), (4, 4), (5, 5)}.

Note that both hSiP and [S]P are predicates. It is thus

possible to use the action S to reach one of the states

prescribed by P if the action is performed from one

of the states 1, 2 or 4; but it is not guaranteed since

from state 2 we could also end up in 5. From the

states 1, 3, 4 and 5 we are guaranteed to end up in

a state prescribed by P as long as the action is per-

formed successfully – but from states 3 and 5 there is

no succesful way of performing the action. 

3 MODAL ALGEBRA

A modal algebra is deﬁned by (Lemmon, 1966a) in

the following way.

Deﬁnition 3.1. A structure (M, ∪, ∩, ¬, 0, 1, P) is a

modal algebra iff M is a set of elements closed under

operations ∪, ∩, ¬, and P such that:

(i) M is a Boolean algebra with respect to

∪, ∩, ¬, 0, 1;

(ii) for x, y ∈ M, P(x ∪ y) = Px ∪ Py.

This is a Boolean algebra with operators in

onsson and Tarski’s sense, when the Boolean alge-

bra is endowed with only one function (J

onsson and

Tarski, 1951). The operator P may be read as ”pos-

sible”, and with it one can deﬁne the operator N for

”necessary” by Nx =

def

¬P¬x.

Lemmon calls a modal algebra that satisﬁes the

postulate

(iii) P1 = 1

a deontic algebra, and a modal algebra that satisﬁes

the postulate

(iv) x ≤ Px

an epistemic algebra. A modal algebra is normal if it

satisﬁes the postulate

(v) P0 = 0.

As Lemmon shows, it is easy to establish the follow-

ing proposition.

Proposition 1. All epistemic algebras are deontic,

and in a normal epistemic algebra it is the case that

(Nx → Px) = 1 (16)

ModalSemiringswithOperatorsforKnowledgeRepresentation

199

and

(Nx → x) = 1 (17)

hold. 

It is a normal epistemic algebra that will be used

later.

3.1 The Model

In the relational model, one can simply take the

Boolean algebra of

(℘(∆), ∪, ∩, ¬, Ø, ∆)

and let P be the identity function. This does not do

anything for intuition, but that is also not the inten-

tion, as explained above. Letting P be the identity

function is useful for proving the soundness of the

proposed axiomatisation, which will be made clear

below.

4 THE TWO COMBINED:

MODAL SEMIRINGS WITH

OPERATORS

The structure

(M, test(M), +, ; , ¬, p , P, 0, 1),

where

(M, test(M), +, ; , ¬, p 0, 1)

is a modal semiring, and

(test(M), +, ; , ¬, P, 0, 1)

is a modal algebra, we shall call a modal semiring

with one operator.

This structure can be generalised to form modal

semirings with multiple operators, which in turn can

be extended to modal Kleene algebra with multiple

operators. If a modal operator is normal, epistemic

or deontic, respectively, we shall use the terminology

modal semiring with a normal, epistemic, or deontic

operator, respectively.

The mathematical properties of these structures are, to

the best of my knowledge, completely uninvestigated, and

deserve proper attention along the lines of (J

onsson and

Tarski, 1951; J

onsson and Tarski, 1952; Lemmon, 1966a;

Lemmon, 1966b). I hope to return to this in future work.

5 ADDING A REVISION

OPERATOR

In this section we add a belief-revision operator to a

modal semiring with a normal epistemic operator. As

will be shown, this makes reasoning about dynamic

epistemic properties possible.

5.1 Intended Interpretation

This is how the constituents of the below algebra are

intended to be understood. The semiring axioms in

Section 2 are all to be interpreted against this back-

ground; the reader might want to have the axioms at

hand. The elements of the carrier set should be seen

as actions upon which the operators (a sort of meta-

actions) work. The operator + has the agent choose

between performing either the left or the right action,

and the operator ; has the agent perform the left action

ﬁrst, and then the right action. The constant 0 always

fails (creates a disaster) and the idle action 1 leaves

everything as it is. Tests are seen as actions checking

if some proposition is true. So 0 checks if the con-

tradiction holds, which it never does, so it fails, and

1 checks if the tautology holds, which it always does,

so why bother. The domain operator applied to an ac-

tion gives a test that is true whenever it is possible to

successfully perform the action. The modal operators

are interpreted as above (page 2).

A belief operator B will be given as the modal op-

erator of a normal epistemic algebra. For saying that

some proposition holds, one writes p = 1. For saying

that the agent believes that a proposition holds, one

writes B p = 1. In (Solin, 2010) it was not possible

to make this distinction, which amongst other things

meant that ¬B¬p couldn’t be expressed properly. Fi-

nally, ~p is an action that revises the agent’s beliefs

by p.

5.2 Dynamic Epistemic Modal Algebra

We are now ready to state the formal deﬁnition of a

dynamic epistemic modal algebra. It should be noted

that the axioms for the revision operator are only ten-

tative and that, depending on one’s goals, they can be

modiﬁed at will. The main point of this deﬁnition is to

show what something like a dynamic epistemic modal

algebra should look like. The axioms are inspired by

those of (Solin, 2010).

Deﬁnition 5.1. A dynamic epistemic modal algebra

(DEMA) is a two-sorted algebra

(D, test(D), +, ; , ¬, ~, p , B, 0, 1)

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such that

(D, test(D), +, ; , ¬, p , B, 0, 1)

is a modal semiring with a normal epistemic operator,

and ~ : test(D) → D satisﬁes

Bp ≤ ~p, (18)

~p;~p ≤ ~p, (19)

~p;Bq ≤ ~(p;q), (20)

~(p + q);Bp = ~p, (21)

~p ≤ ~q ⇒ p ≤ q, (22)

p~p = 1, when p 6= 0. (23)

This concludes the deﬁnition. C

The axioms for the revision operator can all be jus-

tiﬁed according to the interpretation given in Sec-

tion 5.1.

5.3 The Model and Soundness

We can now use our relational model to show that the

above axioms will not contradict each other, and the

that the axiomatisation is sound in that sense. In the

relational model, let ~P =

def

∇;P, for any relation

P ∈ ℘(∆), where ∇ = Σ × Σ. Now, if one lets all

the operators of the modal semiring be interpreted as

suggested above, and lets the modal operator B be in-

terpreted as the identity, then the resulting relational

structure is exactly that of (Solin, 2010, Section 4).

From this, it follows that the above axiomatisation is

sound, in the sense that the axioms will not lead to

contradictions (granted that the relational structure is

noncontradictory, which shouldn’t be controversial).

A completeness theorem for this model would not be

very interesting in our context, since the model is not

what justiﬁes the axioms.

6 A FEW CLASSICAL

PROPERTIES

In the algebra, it is easy to prove some classical prop-

erties that stem from formal reasoning about belief

change (Alchourr

on et al., 1985; Segerberg, 1999;

Cantwell, 2000; van Ditmarsch et al., 2007). The

proofs closely follow those for dynamic epistemic

semirings (Solin, 2010), but the modal operator B

must be taken into account.

The next proposition is one of Segerberg’s success

conditions (Segerberg, 1999). After successfully re-

vising by p, the agent believes p.

Cf. also (Solin, 2010, p. 598).

Proposition 2. In any DEMA the property

[~p]Bp = 1 (24)

holds.

The following property is an encoding of one of

the AGM axioms (Alchourr

on et al., 1985; Cantwell,

2000). If the agent believes q after successfully revis-

ing by p, then the agent believes that q follows from

Proposition 3. In any DEMA the property

[~p]Bq ≤ B(p → q) (25)

holds.

The proposition below, which also relates to an

AGM axiom, says that a successful revision will never

have the agent believe a contradiction.

Proposition 4. In any DEMA the property

[~p]¬B0 = 1 (26)

holds.

From the proof of the last proposition, it is evident

that a successful revision (or any successul action)

will never stop the agent from believing the tautology.

7 CONCLUSIONS

This paper opens up a wealth of interesting themes to

investigate.

7.1 Theoretical Themes

On the purely mathematical side, the general modal

semirings with operators deserve proper attention

along the lines of (J

onsson and Tarski, 1951; J

onsson

and Tarski, 1952; Lemmon, 1966a; Lemmon, 1966b).

Another interesting theme is the investigation of dif-

ferent forms of dynamic epistemic modal algebras, in

which, for example, a different axiomatisation of re-

vision is given, or in which only a deontic operator is

embedded. As in (Solin, 2010), iteration could be in-

troduced as Kleene star. And it would be interesting

to sort out the connections between modal semirings,

modal algebras and the systems employed by (Baltag

et al., 2005; Baltag et al., 2007; Baltag and Sadrzadeh,

2006; Panangaden and Sadrzadeh, 2010).

How these algebras relate to various set-theoretic

models for belief revision and the alike could also

be investigated, following Lemmon’s investigation of

correspondences between his algebras and the stan-

dard models for modal logic – but as noted in earlier,

such an investigation is in no way a prerequisite for

using the abstract algebra as a conceptual tool.

ModalSemiringswithOperatorsforKnowledgeRepresentation

201

7.2 Practical Themes

As shown by (H

ofner and Struth, 2007; Foster and

Struth, 2012), the level of abstraction that semirings

and Kleene algebra provide is very well-suited for au-

tomation. It would therefore be very interesting to

mechanise and automate the algebraic theory of this

paper. With the aid of such a mechanisation, more

elaborate applications of the algebra would become

feasible.

I have emphasised earlier, that the abstract algebra

is the pivot of the framework presented in this paper –

not a model (a conrete algebra). The axioms were jus-

tiﬁed by common sense, and the relational model was

only used for proving technical results. Reasoning in

this framework is therefore genuinely about pushing

symbols, especially since the operators in Kleene al-

gebra and semirings all have a low, ﬁxed arity.

This

means that a robot, say, that would be constructed on

the basis of the theory in this paper, would not have a

sole, involved model as its theoretical underpinning,

but just the axioms of a very simple abstract algebra,

and with them the whole associated class of models

(such as sets with only a few members, the relational

model, etc.). In combination with the tools avail-

able for automated proof and counterexample genera-

tion, this could make for very efﬁcient and pragmatic

knowledge engineering.

ACKNOWLEDGEMENTS

Most of this work was done at the Department of Phi-

losophy, Uppsala University, but some was done at the

Computing Department, Macquarie University, Syd-

ney, and a signiﬁcant part at the University of Queens-

land. A grant from Sven och Dagmar Sal

ens Stif-

telse made the stay in Sydney possible. The work

in Queensland was supported by the Australian Re-

search Council (ARC) Discovery Grant DP0987452.

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