Corresponding author: mkucherov@sfu-kras.ru

1 Introduction

By access control we understand methods or

mechanisms that decide whether requests to access some

resource should be granted or denied. For example,

operating systems need to control which users and

applications can read, write, or delete which files;

networks need to govern which packets can pass through

a physical or logical perimeter; and managers need to

control which employees can perform which workflows

within an organization.

We take responses of an access control system to be

the values ’T, F, Both, None’, according to Belnap logic.

The meaning of these values can be described as follows:

• An atomic sentence is stated to be true only (T),

• An atomic sentence is stated to be false only (F),

• An atomic sentence is stated to be both true and false,

for instance, by different sources, or in different points

of time (Both), and

• An atomic sentence status is unknown. That is, neither

true, nor false (None).

These values and their two orderings form a

distributive, interlaced bilattice [2, 3]. Thus, bilattice

FOUR2 has many convenient properties, for example: ∧

and ∨ distribute as in propositional logic, as do ⊗ and

⊕, and all these operators are also monotone to both the

information and logic orderings.

In this paper we extend use of multivalued logics for

dealing with incomplete and inconsistent information to

SIXTEEN3 [5, 10]. The five information levels, five

logical levels and five levels of falsity in trilattice of

sixteen truth values are arranged in three meet-

distributive lattices. One of the interesting observations

involves computation of ’an access is granted ∧ an

access is not rejectable’. The conclusion is based on the

rule that a conjunction is true if and only if both

conjuncts are true. Since there is no truth value such that

both an access is granted, and an access is not rejectable

has it, the result is the empty set.

Motivation. Imagine two access control policies, each

of which reports a Boolean representing whether an

access should be permitted or not. We can form four

possible sets by collecting the statements: {grant},

conflict − {grant, deny}, {deny}, and undefined {}. For

example, the set {grant, deny} arises when one policy

permits the access and the other denies it. The empty set

Trilattice-Based Access Control Models: How to Secure Current

Computer Network Mikhail

M. KUCHEROV*, NINA A. BOGULSKAYA

Siberian Federal University, School of Space and Information Technologies, 660074 26B

Kirenskogo Krasnoyarsk, RUSSIA

Abstract: Designing security, from the hardware level, is essential to ensure the integrity of the intelligent cyber-

physical infrastructure that is the Industrial Internet of Things (IIoT). If intelligent cyber-physical infrastructure

fails to do the right things because it is insecure and vulnerable, then there will be negative social consequences

[1]. Security is, in a sense, the access control to IIoT systems, which increasingly relies on the ability to compose

different policies. Therefore, the advantage in any framework for compiling policies is that it is intuitive, formal,

expressive, application-independent, as well as expandable to create domain-specific instances. Recently, such

a scheme was proposed based on Belnap logic FOUR2 [2]. Four values of the Belnap bilattice have been

interpreted as grant, deny, conflict, or unspecified with respect to access-control policy. Belnap's four-valued

logic has found a variety of applications in various fields, such as deductive database theory, distributed logic

programming, and other areas. However, it turns out that the truth order in FOUR2 is a truth-and-falsity order

at the same time [3]. The smallest lattice, where the orders of truth and falsity are independent of each other,

which is especially important for security policy, is that of Shramko-Wansing’s SIXTEEN3. This generalization

is well-motivated and leads from the bilattice FOUR2 with an information and a truth-and-falsity ordering to

another algebraic structure, namely the trilattice SIXTEEN3 with an information ordering together with a truth

ordering and a (distinct) falsity ordering. Based on SIXTEEN3 and new Boolean predicates to control access

[4], we define an expressive access-control policy language, having composition statements based on the

statements of Schramko-Wansing’s logic. Natural orderings on politics are obtained by independent lifting the

orders of truth and falsity of trilattice, which results in a query language in which conflict freedom analysis can

be developed. The reduction of formal verification of queries to that on predicates over access requests enables

to carry out policy analysis. We evaluate our approach through examples of control access model policy.

Keywords: access control, multivalued logic, computer network

Received: April 29, 2022. Revised: May 26, 2023. Accepted: June 25, 2023. Published: August 3, 2023.

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{} arises when the access is outside the domain of both

policies. We can order these sets in two different ways.

If we order by the degree to which the access is

permitted, we have that {grant} is greatest and {deny} is

least. We can also order by the amount of information

we have obtained. In this ordering conflict is greatest and

undefined is least. We write ≤t for the truth ordering and

≤k for the knowledge (information) ordering.

But if we give formal definitions for the partial order

relations (≤k and ≤t), we get following [3]: The definition

of ≤k is very simple: for any x, y ∈ FOUR2, x ≤k y iff x ⊆

y. With the ≤t ratio, the situation is not so trivial. If we

define for each element of FOUR2 its "truth-containing

part" and "lie-containing part": xt := {z ∈ x | z = T}; xf

:= {z ∈ x | z = F}, then we can make sure that: x ≤t y iff xt

⊆ yt and yf ⊆ xf. This suggests that truth and lies in

Belnap's logic are not completely independent concepts:

Belnap's logic suggests that the very meaning of "lie" is

not only less true than the meaning of "truth," but also

the meaning of "neither truth, not a lie ".

Next consider the issue of access control policies

containing both “permit rules” and “deny rules”. A

permit rule returns true(+), if the access is granted, and

false(+), if the access is not granted (but not obligatorily

rejectable). Deny rules work symmetrically, i.e. return

true(−), if the access is denied, and false(−), if there is no

explicit denial (but is not granted). Much recent work on

access control has considered “permit rules”. However,

the logic of generalized truth values that comes with

trilattice SIXTEEN3 and the propositional language Ltf

allows to solve using “deny rules” many problems in the

most natural way. Moreover, if the three partial

orderings ≤t, ≤f, and ≤i are used to define orderings on

valuations, different kinds of non-monotonic inference

relations can be defined in terms of minimizing or

maximizing truth, falsity, or information.

Contributions. We consider equivalence between

different presentations of multivalued data, and obtain a

parametric expression for an access policy as a

combination of “permit rules” and “deny rules”. Since

the set SIXTEEN3 is the smallest multivalued lattice

having two independent orderings: in true and in falsity,

it is necessary to consider the presentation of access

models based on the SIXTEEN3. The role of falsity order

in consequence relations important for the control of

communication was determined distinctly, which

allowed to overcome shortcomings of the use of bilattice

FOUR2.

The rest of the paper is organized as follows. Related

work and statements from many-valued logic are

discussed in Section 2 and 3, respectively. The main

results of the paper are placed in Section 3– 8. The paper

ends with discussions in Section 9.

2 Related works

Already Aristotle was aware of the problem of future

contingencies. Łukasiewicz devised a three- valued

calculus whose third value,

, can be ’intermediate’ or

’neutral’ or ’indeterminate’ [6, 7].

Another three-valued logic, known as Kleene’s logic

is developed in [8] and has three truth values, truth,

unknown and false, where unknown indicates a state of

partial vagueness. These truth values represent the states

of a world that does not change.

The semantic characterization of a four-valued logic

for expressing practical deductive processes is presented

in [9]. In most information systems the management of

databases is not considered to include neither explicit nor

hidden inconsistencies. In real life situation information

often come from different contradicting sources. Thus,

different sources can provide inconsistent data while

deductive reasoning may result in hidden inconsistencies.

The idea in Belnap’s approach is to develop a logic that is

not dependable on inconsistencies.

Sixteen generalized truth values obtained as a power

set of the initial truth values of Belnap’s logic are

arranged in a trilattice [10].

Fuzzy logic is a form of multi-valued logic originally

studied by Łukasiewicz and Tarski [12] and later

popularized as logic of vagueness by Zadeh [13]. It is

based on replacing the standard set of Boolean truth

values with a different lattice, most often, like in the

present paper, the unit interval. Saying that a formula ϕ

has truth value r ∈ [0,1] then means that ϕ holds with

degree r, which would apply to typical vague

qualiﬁcations such as a given person being tall (in

contrast to assigning a probability p ∈ [0,1] to ϕ, which

would be read as saying that ϕ is either completely true

with probability p or completely false with probability 1

− p, as in ‘the die under the cup shows a 3 with

probability p’).

3 Meet-distributive lattices

Let P be a non-empty ordered set. If sup {x, y} and inf

{x, y} exist for all x, y ∈ P, then P is called a lattice [11].

A complete lattice is a partially ordered set in which all

subsets have both a supremum (join) and an infimum

(meet). A lattice L is meet-distributive if every coatomic

interval is Boolean. A billatice is a set equipped with two

partial orderings ≤t and ≤k.

The four truth values are arranged in a logical lattice

[2] on Figure 1. A logical conjunction and logical

disjunction are related to the meet operation and to the

join operation respectively.

Fig. 1. The Belnap bilattice: with synonyms for access

control decisions in parentheses.

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Let us, for example, consider the following bounded

lattices: fuzzy data [14, 15, 16]: then C [0, 1] is the

infinite set of real numbers from 0 to 1. For any ground

atom r (d) ∈ H the p = I(r (d)) represents its plausibility.

The quantity p is determined as a closed subinterval

[17, 18, 19]: W = C [0, 1] is the set of all closed

subintervals. For any ground atom r (d) ∈ H the (L, U) =

Imv(r (d)) represents the lower and upper bounds for an

expert’s evaluation of r (d). The generalized truth-value

space has as a base a set I = (T, F, t, f) containing the

initial truth values: T − an access is granted,

   

( )

1,1 , 0,1

;

F − an access is denied,

   

( )

0,0 , 0,1

; t − an access is not

rejectable,

   

( )

0,1 , 0,0

, and f − an access is rejectable,

   

( )

0,1 , 1,1

. The power set P(I) gives sixteen values with

the presentation as closed subintervals in Table 1:

Table 1. Levels of trilattice SIXTEEN3.

Information

   

( )

0,1 , 0,1

   

( )

1,1 , 0,1

   

( )

0,0 , 0,1

   

( )

0,1 , 0,0

   

( )

0,1 , 1,1

   

( )

1,0 , 0,1

   

( )

1,1 , 0,0

   

( )

1,1 , 1,1

   

( )

0,0 , 0,0

   

( )

0,0 , 1,1

   

( )

0,1 , 1,0

   

( )

1,0 , 0,0

   

( )

1,0 , 1,1

   

( )

1,1 , 1,0

   

( )

0,0 , 1,0

   

( )

1,0 , 1,0

For example, {T, t} =

   

( )

1,1 , 0,1

i

   

( )

0,1 , 0,0

   

( )

1,1 , 0,0

The three partial orderings ≤i, ≤t, ≤f arrange elements

according to the possessed degree of information, truth

and falsity respectively. The ≤f is the falsity order:

‘falsity’ decreases if the access is granted. The sixteen

truth values are arranged in the falsity meet-distributive

lattice on Figure 2. The details relative to the three

partial orderings are shown in Appendix (Table 4).

Table 2. Policy language: the rp are access predicates,

and b {F, T}.

p::=([x,y],[z,v])

Policy

b if rp

Basic policy [2]

([1-y,1-x], [1-v,1-z])

Logical negation

([min {x, x1}, min {y, y1}],

[max {z, z1}, max {v, v1}])

Logical meet

([1-x,1-y], [1-z,1-v])

Falsity negation

([max {x, x1}, max {y, y1}],

[max {z, z1}, max {v, v1}])

Disjunction; p  q

Fig. 2. Falsity meet-distributive lattice.

4 Policies

We take responses of an access control system to be the

values ([x,y],[z,v]) of the sixteen generalized truth values

space, which we write as SIXTEEN3. These values and

their three orderings form a distributive, interlaced

bilattice.

Consequence relations |=t and |=f are extensions of

implication to SIXTEEN3. For each logical order a

separate entailment relation between any sentences A, B

 Ltf was defined in [20]:

A|=t B iff

( ) ( )

( )

16 16 16

v v A v B

, and

A|=f B iff

( ) ( )

( )

16 16 16

v v B v A

There it was proved that both of these restrictions

coincide in fact with Belnap’s four-valued logic.

Expression a  b with a, b ∈ SIXTEEN3 yields b if a

is less or equal to T in the falsity ordering and yields T

otherwise. Table 2 gives the syntax of access control

policy language. Informally, an expression for every

access a:: = {(r(d), I(r(d))) | r(d)  A, where A  H} is

interpreted as a mapping from accesses to elements of

Ltf. If policy p produces result



on access a and policy q

produces result  on a, then p ∧ q produces result  ∧ 

on a. The other Belnap operators are similarly

interpreted on policies. The intuition behind these policy

operators is that:

•  p denies an access iff p grants it (and vice versa)

• p ∧ q grants an access iff both p and q grant it, and

denies an access iff at least one of p and q denies it

• p ⊃ q grants an access iff p does not grant it or q does

grant it, and p ⊃ q denies an access if p grants it and q

denies it

• p ⊕ q grants (resp. denies) an access iff p or q grants

(resp. denies) it.

The only operator in the policy language not derived

from Belnap logic is the basic expression b if rp, where b

ranges over {F, T}, and rp is an access predicate.

Informally, the b if rp “rule” gives result b for an access

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satisfying rp and gives result ⊥ otherwise. The idea is

that rp deﬁnes the domain of accesses covered by the

rule, so that ⊥ is produced on accesses outside of the

rule’s domain.

5 Queries

Properties about policies and their relationships can be

expressed as queries, propositional formulas in which

the atomic propositions concern the truth or falsity

ordering among policies. Table 3 gives the syntax for

queries.

Table 3. Query language: p and q range over policy

expressions.





::=

Query

p t q

Truth ordering

p f q

Falsity ordering

rpc (a,b) = ( a  b)

Confidentiality

request

rpi (a,b) = (a   b)

Integrity request

In [4], we defined expressions for basic policies in a

natural way, based on material implication, as a rigorous

expression of the information flow operator [21], as

logical formulas for some predicate query symbols:

− confidentiality request (Bell–LaPadula),

− integrity request (Biba),

where a − the class of confidentiality /integrity of

subject, and b – of object (a, b {0,1}).

6 Example for analysis

Access control refers to mechanism by which the access

of principals to resources is regulated, as can be seen in

the following example of the access referring to the

transport resources.

Generally, access control is implemented by an

authorization service, which includes an authorization

decision function (ADF) for deciding whether a user

request to access a resource should be permitted or not.

The output of ADF is usually determined by evaluating

the request with respect to authorization state (AS). The

authorization policy (AP), which is implemented by the

ADF, is to authorize a request if it is listed in the

authorization state [22].

A group of students of Institute of Space and

Information Technology in Krasnoyarsk is planning a

trip to see the Pillars Nature Sanctuary (Stolby).

Normally, it takes about one hour to get to the Right

Bank across the Yenisei River by car, but the students

know that no-one of locals needs to be told that

Kopylovski Bridge Expressway (KBE) is “the longest

parking lot”.

The students consult a traffic service, which

integrates information from several independent

information sources to provide traffic advisory along

various travel routes. They have a combination of

several sources. Let us assume that these sources are:

• weather forecast (rain, snow, fog);

• social activity;

• police activity;

• road reparation.

The service uses the following rules to generate

advisories:

If the weather is bad, and there is road reparation

along the route, the probability of a delay is 0.9.

If there is road reparation, and social activities along

the route, the likelihood of a delay is 0.8.

If there is road reparation, and police activities along

the route, the likelihood of a delay is 0.99.

These rules are expressed as levels of evaluation, as

[0.9, 1], [0.8, 1] and [0.99, 1]. The service generates

advisories expressed as the likelihood of delays along the

routes of interest (AS). Students do not want to miss the

excursion due to traffic, but they also have conference

deadlines and so do not want to leave too early.

They decide that if the advisory says that the

likelihood of delays is between 0.2 and 0.4, then they

add one extra hour to the trip time. If the likelihood is

between 0.4 and 0.6, then they add two hours, and if the

likelihood is over 0.6 then they take a river-train (AP). In

Section 8, we will return to this example and show that

our approach improves the quality of the advisory and

could have helped the students avoid unnecessary

trouble.

7 Combining answers

We can learn levels for compound events, which include

basic objects, having the values for basic objects [5].

7.1 Case 1: Independence

Since in the model an event can be true, false, or

unknown, (i.e., we are modeling knowledge and

competency independently) we have:

Cl1 (A t B) = ([xx1, yy1], [1− (1− z)(1− z1), 1−

(1−v)(1−v1)]).

Cl1 (A t B) = ([1−(1− x)(1− x1), 1− (1−y)(1−y1)], [zz1,

vv1]). (3)

7.2 Case 2: Positive correlation

Two events A and B are positively correlated if they

overlap as much as possible. One can obtain that the

solution can be shown to be

Cl2 (AtB)=([min{x,x1}, min{y, y1}], [max{z,z1},

max{v,v1}]).

Cl2 (AtB) =([max{ x,x1}, max{ y, y1}], [min{ z,z1},

min{ v,v1}]) (4)

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8 Example for analysis (continued)

Returning to the example in Section 7, suppose that our

information sources predict 50% chance of bad weather,

parades with 50% certainty, roadwork along the

Kopylovski Bridge with certainty 80%, and police

activity with the probability of 40%. This information is

expressed in this way:

− roadwork (RW): ([0.8, 0.8], [0.2, 0.2]);

− social_act (SA): ([0.5, 0.5], [0.5, 0.5]);

− bad_weather (BW): ([0.5, 0.5], [0.5, 0.5]);

− police_act (PA): ([0.4, 0.4], [0.6, 0.6]).

The traffic service fetches the above information

from four different information sources and integrates

them independently using these rules as in Eq. (3):

− delay, if there are roadworks and bad weather, for

example:

Cl1 (RW t BW) =([0.90.80.5,0.80.5], [0.6,0.64]).

Eventually, one can obtain

Cl1 (RW t BW) = ([0.36, 0.4], [0.6,0.64]) (5)

− delay, if there are roadworks and the marathon:

Cl1 (RW t SA) = ([0.32, 0.4], [0.6,0.68]) (6)

If there are road works and traffic accident:

Cl1 (RW t PA) = ([0.32,0.32], [0.68,0.68]) (7)

Using the second rule of Eq. (3) one can obtain from

Eqs. (5) and (6):

Cltot [(RW t BW) t (RW t SA)] = ([0.56,0.64],

[0.36, 0.44]). (8)

Adding Eq. (7) one has finally:

([0.7, 0.76], [0.24, 0.3]) (9)

When correlation is not considered, the evaluation of

delay (KBE) is [0.36, 1], which means that the available

information predicts traffic delay with certainty 0.36 and

smooth traffic with certainty [0, 0.64]. Considering the

possibilities of parades and accidents, it is reasonable to

up the expectation of delays.

In contrast, our method computes the confidence

level for traffic delays to be [0.7,0.76], which is

narrower then the certainty factor according Belief Logic

Programming theory [0.63,1] [23]. Thus, we see that the

theory can predict better expert’s evaluation of the

combined information.

9 Discussions

In fact, Belnap’s interpretation suits perfectly well when

applied to a single computer. In addition, it presupposes

that this computer receives information from classical

sources, i.e., from sources which can operate exclusively

with classical truth values (T, F). And it appears that

Belnap’s interpretation cannot adequately be applied

when we deal not just with one computer but with

several interconnected computers, i.e., a computer

network. Indeed, consider four Belnap computers (C1,

C2, C3, C4) connected to some central computer (С’1), a

server, to which they are supposed to supply information

(Figure 3). Incidentally, it is interesting to observe that if

we wish to extend our network and connect our server to

some “higher” computer (С’’1), then generalized truth

values of the third order (the set P (SIXTEEN3) comes

into question (and so on).

Fig. 3. Computer network.

Apparently, we should stop at the 16-digit logic, for

example, such as SIXTEEN3. If we look at the set

SIXTEEN3 and consider its algebraic structure, it turns

out that within the framework of this set it is possible to

make an effective distinction between the increase in the

truth of the facts and the decrease of their falsity, and

therefore one can determine the order in truth and the

falsity order as two different and mutually independent

orders [3]. Since the set SIXTEEN3 is the smallest

multivalued lattice having this property, it is necessary to

refer to the presentation of access models based on the

SIXTEEN3.

In conclusion, we extend a formal framework for

authorization policies based on trilattice SIXTEEN3; all

possible complex patterns for base policies are supported

in the presence of conflict or lack of information; and

authorization state and policy specifications are cleanly

separated. Policy composition is achieved through

combination modes (independence, positive correlation,

etc.), that is familiar in the theory of deductive databases.

They are formally analyzable and functionally complete

for policy coordination. The parameterized method not

only helps on analysis of policies, but these types and

their analysis can also certify in the future investigations

important run-time behavior of policy evaluation.

Appendix

A.1 The algebra of confidence levels

We shall establish the structure and properties of

trilattices here, which is used in previous sections.

Definition 1. Denote by C [0,1] the set of all closed

subintervals over [0,1]. Consider the set Cc =def C [0,1] 

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C [0,1]. We denote the elements of Cc as ([x, y][z, v]).

Define the following orders on this set. Let ([x, y][z, v]),

([x1, y1][z1, v1]) be any two elements of Cc.

([x, y][z, v]) t ([x1, y1][z1, v1]) iff x  x1, y  y1 and z1  z, v1  v

([x, y][z, v]) f ([x1, y1][z1, v1]) iff x  x1, y  y1 and z  z1, v  v1

([x, y][z, v]) i ([x1, y1][z1, v1]) iff x  x1, y1  y and z  z1, v1  v

Definition 2. Let Cc, t, f, i be as defined in Definition

1. Then the meet and join corresponding to the truth,

falsity and knowledge (information) orders are defined

as follows. The symbols  and  denote meet and join,

and the subscripts t, f, and i represent truth, falsity, and

information (knowledge), respectively.

1. ([x, y][z, v]) t ([x1, y1][z1, v1]) =

([min{x, x1}, min{y, y1}], [max{z, z1}, max{v, v1}])

2. ([x, y][z, v]) t ([x1, y1][z1, v1]) =

([max{x, x1}, max{y, y1}], [min{z, z1}, min{v, v1}])

3. ([x, y][z, v]) f ([x1, y1][z1, v1]) =

([min{x, x1}, min{y, y1}], [min{z, z1}, min{v, v1}])

4. ([x, y][z, v]) f ([x1, y1][z1, v1]) =

([max{x, x1}, max{y, y1}], [max{z, z1}, max{v, v1}])

5. ([x, y][z, v]) i ([x1, y1][z1, v1]) =

([min{x, x1}, max{y, y1}], [min{z, z1}, max{v, v1}])

6. ([x, y][z, v]) i ([x1, y1][z1, v1]) =

([max{x, x1}, min{y, y1}], [max{z, z1}, min{v, v1}])

The top and bottom elements with respect to the various orders

are as follows.

Table 4. Bounds relative to the three partial orderings.

Top value

Bottom value

Тt =

   

( )

1,1 , 0,0

⊥t =

   

( )

0,0 , 1,1

Тf =

   

( )

0,0 , 0,0

⊥f =

   

( )

1,1 , 1,1

Тi =

   

( )

1,0 , 1,0

⊥i =

   

( )

0,1 , 0,1

Тt corresponds to {T, t}, ⊥t corresponds to {F, f}.

Тf corresponds to {F, t}, ⊥f corresponds to {T, f}.

Тi corresponds to {TF, tf}, ⊥i corresponds to { }.

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WSEAS TRANSACTIONS on COMPUTERS

DOI: 10.37394/23205.2023.22.12

M. Kucherov, Nina A. Bogulskaya

E-ISSN: 2224-2872

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