Reference Data Abstraction and Causal Relation

based on Algebraic Expressions

Susumu Yamasaki and Mariko Sasakura

Department of Computer Science, Okayama University, Tsushima-Naka, Okayama, Japan

Keywords:

Reference Data, Algebraic Structure, 3-Valued Model Theory.

Abstract:

This paper is related to algebraic aspects of referential relations in distributed systems, where the sites as

states are assumed to contain pages, and each page as reference data involves links to others as well as its own

contents. The links among pages are abstracted into causal relations in terms of algebraic expressions. As an

algebra for the representation basis of causal relations, more abstract Heyting algebra (a bounded lattice with

Heyting implication) is taken rather than the Boolean algebra with classical implication, where the meanings

of negatives are different in the two algebras. A standard form may be obtained from any Heyting algebra

expression, which may denote causal relations with Heyting negatives. If the evaluation domain is taken from

the 3-valued, then the algebraic expressions are abstract enough to represent referential links of pages in a

distributed system, where the link may be interpreted as active, inactive and unknown. There is a critical

problem to be solved in such a framework as theoretical basis. The model theory is relevant to nonmonotonic

function or reasoning in AI, with respect to the mapping associated with the causal relations, such that ﬁxed

point theory cannot be always routines. This paper presents a method to inductively construct models of

algebraic expressions conditioned in accordance to reference data characters. Then we examine the traverse

of states with models of algebraic expressions clustering at states, for metatheory regarding searching the

reference data in a distributed system. With abstraction from state transitions, an algebraic structure is reﬁned

such that operational aspect of traversing may be well formulated.

1 INTRODUCTION

As in a distributed system, we virtually assume sites

as states where data and contents are involved in,

which are associated with abstract state machine. Re-

garding existing data and contents at a state, they

contain static informations, however, references may

have another aspect to be captured as dynamic in the

sense of being linked with others and having search

effects. It can be also regarded as relational, in the

sense of linkage. As traverses from a state to another,

the state transition behaviors (like actions) should be

formalized abstractly for a whole distributed system.

The traverses of states have been examined as state

transitions from algebraic views. As backgrounds, we

have seen actions in abstract state machine structures

and traverses.

(a) The action is formulated as a key role in strate-

gic reasoning of abstract state machine, as well as

concretized actions as programs in dynamic logic,

acting and sensing failures are discussed as ad-

vanced works (Spalazzi and Traverso, 2000).

(b) Actions are also captured in logical systems from

the viewpoints of sequential process, as in the pa-

pers (Giordano et al., 2000; Hanks and McDer-

mott, 1987).

(c) Procedural action is expressed by denotational ap-

proach in the book (Mosses, 1992). The procedu-

ral method is in accordance with operational im-

plementation for programs to be executed. The

actions may be abstracted, with functional pro-

grams (Bertolissi et al., 2006).

(d) As regards transitions, abstract state machine is

discussed, in the paper (Reps et al., 2005). Re-

garding structure of streams possibly caused by

abstract state transitions, there is the note (Rutten,

2001).

In reasoning with semantics for AI programs and

data representations, logical approaches are often

taken:

(1) Logics with knowledge (Reiter, 2001) are classi-

cal. Based on beliefs and intentions, modal operations

have been applied to mental states (Dragoni et al.,

Yamasaki, S. and Sasakura, M.

Reference Data Abstraction and Causal Relation based on Algebraic Expressions.

DOI: 10.5220/0009825602070214

In Proceedings of the 9th International Conference on Data Science, Technology and Applications (DATA 2020), pages 207-214

ISBN: 978-989-758-440-4

Copyright

c

2020 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

207

1985).

(2) From model theoretic views, the argumentation

may be expressed by means of 3-valued logic to

the semantics for defeasible reasonings to implement

(Governatori et al., 2004).

(3) Mobile ambients (Cardelli and Gordon, 2000;

Merro and Nardelli, 2005) have been formulated with

environments to make communication reasonable.

(4) The papers (Dam and Gurov, 2002; Kozen, 1983)

are classical enough to formulate the proof systems

with ﬁxed points and their approximations.

(5) Compared with default or defeasible logic in AI

programming, defeasibility is beforehand assumed

in the given rules, but the causal relation consisting

of rules by Heyting algebra expressions is simpler

without treating ambiguity of rules containing default

negation.

Following such backgrounds of algebraic and log-

ical perspectives, this paper is motivated to exam-

ine algebraic approach to the representations of ref-

erence data. Through simple structure of reference

data, algebraic expressions are studied and captured

as causal relations, whose model theories are required

in Heyting algebra. As regards traverses through

states, semiring is characterized for abstract state tran-

sition. The theoretical aspect of Heyting algebra ex-

pressions in this paper contains a speciﬁc property in

postﬁx modal operator of modal mu-calculus (S. Ya-

masaki, in C0MPLEXIS 2020):

(a) Reference data in distributed systems are ana-

lyzed. A simple inductive structure of reference

links is abstracted from the view of deepening al-

gebraic aspects.

(b) Heyting algebra expressions are adopted for ap-

plication to abstractly represent referential struc-

tures. The model theory of algebraic expressions

is hard, since the mapping associated with expres-

sions is nonmonotonic such that classical ﬁxed

point theory cannot be always adopted. Some

preﬁxpoint may be inductively constructed as a

model. This paper makes reﬁnements in the con-

structions of models for given algebraic expres-

sions. The model theory can be applied to clas-

sical logic, but it is discussed in non-classical

framework over a 3-valued domain. In addition to

non-classical discussions, we just mention a clas-

sical tool to get negation by failure: Qeuries for

algebraic expressions are approximately realized,

by negation as failure rule being sound with re-

spect to models.

(c) State transitions are abstracted into semiring

structure, caused by models of algebraic expres-

sions. In accordance with ﬁnite state automata,

star semiring may be given, on one hand (S. Ya-

masaki, in COMPLEXIS 2017). From the view

on nondeterministic alternation of traverses, more

complex semiring may be deﬁned, on the other

hand.

The paper is organized as follows. Section 2 is

concerned with simple observation of structures in

reference data of distributed systems. In Section 3,

Heyting algebra expressions, and the standardization

of expressions are summarized, such that model the-

ory problems may be mentioned and solved. Section

4 presents an algebraic structure in terms of semir-

ing, by constructing the models of algebraic expres-

sion for concatenation and alternation regarding state

transitions. Concluding remarks and related topics are

described in Section 5.

2 STRUCTURE FOR REFERENCE

DATA

2.1 Reference Data

A distributed system is assumed to consist of sites,

where:

(i) the site contains pages, and

(ii) the page denotes references which are linked to

others.

With abstraction from the system containing data

with references to others, assume an abstract and sim-

ple system, where

(a) the sites are denoted as states,

(b) the pages in a site are deﬁned in each state, and

(c) the reference data is organized in each page by a

recursive way, as illustrated below.

Reference name Contents

Reference name

1

···

···

Reference name

n

Note that the content and recursive references are

separated such that the page is viewed as following

Frege-ontology. It is formally in Backus-Naur Form

described:

Syst ::= null

Syst

| s : P;Syst

P ::= null

P

| p;P

p ::= r; Con; re f

re f ::= null

re f

| r; re f

where:

DATA 2020 - 9th International Conference on Data Science, Technology and Applications

208

(a) null

Syst

, null

P

and null

re f

are the empty strings on

the domain of systems, pages and references, re-

spectively.

(b) Syst is a system variable, and s is a state variable.

(c) Con is a variable denoting a content.

(d) the semicolon “;” denotes a concatenation opera-

tion.

(e) p and r are page and reference variables, respec-

tively, such that P and re f denote a sequence of

pages and a sequence of references, respectively.

Rather than the content of each page, the relation

among references may be compiled in a page, that is,

the page involves a reference followed by a sequence

of references.

We now pay attention to the case assumption of

the page to be active, inactive or unknown, in terms

of the page to be linked, not linked or unknown

in a distributed system of sites. To a reference

(variable) r, the 3-valued domain may be taken to

make assignments:

Reference activity Values

Active (linked) 1

Unknown 1/2

Inactive (not linked) 0

A reference followed by a sequence of references

(possibly the empty) may suggests a causal relation

between the sequence (as a cause) and the reference

(as an effect). The relation is modeled by some pro-

graming, as well.

Related Programming:

A “logic program” with respect to its Herbrand base

may be regarded as containing the predicate pr (with

or without “

∼

” as a procedure) preceded by the

conjunction of:

pr

1

,..., pr

m

(as a procedural body)

for pr

1

,..., pr

m

(predicates or their negations). The

program may be dealt with in 3-valued logic, where

the negation is interpreted as default negation.

2.2 Causal Relation in Terms of

Algebraic Expressions

Heyting algebra (HA) (A,

W

,

V

,⊥,>) equipped with

the partial order v and an implication ⇒ is assumed

as follows:

(i) ⊥ and > are the least and the greatest elements

of the algebra (set) A, respectively, with respect to

the partial order v.

(ii) the join

W

and meet

V

are deﬁned for any two

elements of A.

(iii) as regards the implication ⇒,

c v (a ⇒ b) iff a

V

c v b.

The element a ⇒ ⊥ is denoted as “not a” (a nega-

tive) for a ∈ A, where not ⊥ = > and not > = ⊥. As

is well known, we note some algebraic properties on

the HA with parentheses of operation-priority repre-

sentation:

((a ⇒ b)

V

(b ⇒ c)) v (a ⇒ c),

a v not (not a),

as well as

(a

V

(a ⇒ b)) ⇒ b

are always equivalent to the top element, such that

not (a

W

b) = not a

V

not b.

Therefore the expression a ⇒ b may be regarded as

representing a causal relation between the cause de-

noted a and the effect denoted b. The implication ⇒

is more abstract than the classical (e.g. propositional)

logic one.

The causal relation is now taken into considera-

tion, for the denotation of reference data, by more

general form of the HA expression. The expression

F (over the underlined set A of the algebra) of the fol-

lowing form is regarded as a causal relation to abstract

reference data:

V

j

(l

j

1

V

...

V

l

j

n

j

⇒ l

j

)

where l

j

i

denotes a or a ⇒ ⊥ (not a) for a ∈ A.

Assume in the following that (a) the implication

is based on Heyting algebra, and (b) the evaluation

of not a (with respect to the value of a for a ∈ A)

follows the rule:

a not a

1 0

1/2 0

0 1

where 0 v 1/2 v 1.

Transformation of Expressions:

In an Heyting algebra (A,

W

,

V

,⊥,>), any expression

Ex

1

derives some expression Ex

2

of the form:

V

j

(l

j

1

V

...

V

l

j

n

j

⇒ l

j

),

Reference Data Abstraction and Causal Relation based on Algebraic Expressions

209

where l

j

i

and l

j

are an expression a or not a (denoting

a v ⊥), for a ∈ A, such that

Ex

2

v Ex

1

.

By the method in a language system (S. Yamasaki,

in COMPLEXIS 2020), we may have got such a stan-

dardization to transform a given expression Ex

1

to

Ex

2

. If there is some model of Ex

2

in 3-valued do-

main, then it may be also the model of Ex

1

. In this

sense, the expression Ex

2

is worthwhile being ob-

tained, as a standard form.

3 MODELS OF ALGEBRAIC

EXPRESSIONS

With respect to a denotation of pages in a state (site)

containing refernce data, the expression F (over the

underlined set A of the algebra) of the form:

V

j

(l

j

1

V

...

V

l

j

n

j

⇒ l

j

)

is represented as a set of the rules

{l

j

1

V

...

V

l

j

n

j

⇒ l

j

| j = 1,2,. . .}.

It may be regarded as a set of causal relations of the

form l

j

1

V

...

V

l

j

n

j

⇒ l

j

, with the HA implication ⇒,

where the outer meet is assumed in the evaluation of

the set (the whole expression). The set of rules is also

referred to by the same name F in the following.

3.1 Conditioned Algebraic Expressions

in 3-Valued Domain

To have a theory of HA expressions applicable to de-

notations of reference data, we here have restrictions

on the set of rules:

(a) Given a set, the left hand of ⇒ for a or not a (with

its right hand) is unique, if the rule “... ⇒ a” or

“... ⇒ not a” exists.

(b) For each a ∈ A, there is no case that both the rules

“... ⇒ a” and “... ⇒ not a” are deﬁned.

(c) The model of a given expression (a set) is consid-

ered in 3-valued domain.

We present preﬁxed point as model of the expres-

sion F, over the 3-valued domain {0, 1/2, 1}. With

the set A for a conditioned expression F, a mapping

Ψ

F

: 2

A

× 2

A

→ 2

A

× 2

A

,

Ψ

F

(I

1

,J

1

) = (I

2

,J

2

),

can be deﬁned with order of componentwise subset

inclusion.

The Mapping Ψ

F

:

Note that the left hand part is unique for each right

hand of the implication ⇒. Because the conditioned

form is assumed, in order to denote reference data

structure.

Assume (I

1

,J

1

) for a given set of rules, where I

1

is

regarded as the set of elements assigned to 1, and J

1

is as the set of elements assigned to 0. For each a ∈ A,

within the rules of F:

(1) In case that there is a rule (in the set)

b

1

V

...

V

b

n

V

not c

1

V

...

V

not c

m

⇒ a:

if any b

i

is in I

1

(1 ≤ i ≤ n), and any c

j

is in J

1

(1 ≤ j ≤ m), then a ∈ I

2

.

(2)(a) In case that there is no rule, whose right hand

of the implication ⇒ is a (that is, a may be only

in the left hands of rules): a ∈ J

2

.

(b) In case that there is a rule

b

1

V

...

V

b

n

V

not c

1

V

...

V

not c

m

⇒ a:

if some b

i

is in J

1

(1 ≤ i ≤ n), or some c

j

is not

in J

1

(1 ≤ j ≤ m), then a ∈ J

2

.

(c) In case that there is a rule

d

1

V

...

V

d

l

V

not e

1

V

...

V

not e

k

⇒ not a:

if any d

i

is not in J

1

(1 ≤ i ≤ l), and any e

j

is in

J

1

(1 ≤ j ≤ k), then a ∈ J

2

.

If Ψ

F

(I,J) ⊆

c

(I,J) (with the componentwise sub-

set inclusion ⊆

c

) and I ∩ J =

/

0, then (I,J) can be a

model of F, that is, F is evaluated as 1.

Note: If I ∩ J =

/

0, then I

0

∩ J

0

=

/

0 for (I

0

,J

0

) =

Ψ

F

(I,J). It is because of the restriction of the expres-

sion F. That is, both a and not a are not deﬁnable.

Since the mapping Ψ

F

is not monotonic, the method

by (pre-)ﬁxpoint of Ψ

F

is not always available as a

modelling of the given expression F.

Proposition 1. Assume a pair (I,J) ∈ 2

A

× 2

A

for a

given expression (a set of rules) with the element set

A. If Ψ

F

(I,J) ⊆

c

(I,J), then the pair (I,J) is a model

of F.

Proof. Let Ψ

F

(I,J) = (I

0

,J

0

). Following the deﬁni-

tion of the mapping Ψ

F

, we make the exhaustive ex-

amination. For any a occurring in F, there are three

types of rules.

(i) In case of (1), if the left hand of the rule (where

the left hand may be the empty) is evaluated as 1 by

(I,J), then the right hand a ∈ I

0

is in I, evaluated as 1.

(ii) In case of (2): (a) if no a may occur in the right

hand of a rule, a ∈ J

0

is evaluated as 0 for the pair

(I,J) to consistently be a model.

(b) if the left hand of the rule is evaluated as 0 (with

case of (b)), then the right hand a ∈ J

0

⊆ J is in J,

DATA 2020 - 9th International Conference on Data Science, Technology and Applications

210

evaluated as 0. (c) if the left hand of the rule (where

the left hand may be the empty) is evaluated as not 0,

then the right hand not a (a ∈ J

0

) is evaluated as 1.

Because a ∈ J.

Thus all the rules are evaluated as 1, with respect

to the relations between left and right hands of the

implication. This concludes the proposition.

In what follows, we suppose the set A (for HA

expressions) and the expression F in a set of rules.

We have got a procedure with respect to construc-

tion of some model (I,J), if Ψ

F

(I,J) ⊆

c

(I,J).

Predicates of Success and Failure for Query:

With respect to query a to be an effect for the expres-

sion, the predicates of success Suc

F

(a) and failure

Fail

F

(a) may be inductively deﬁned for a ∈ A and

a given expression (a set of rules) F as follows.

(1) If there is a rule

b

1

V

...

V

b

n

V

not c

1

V

...

V

not c

m

⇒ a

such that Suc

F

(b

i

) for any 1 ≤ i ≤ n, and

Fail

F

(c

j

) for any 1 ≤ j ≤ m, then Suc

F

(a).

(2)(a) If there is no rule, whose right hand of the im-

plication ⇒ is a, then Fail

F

(a).

(b) If there is a rule

b

1

V

... ∧ b

n

V

not c

1

V

...

V

not c

m

⇒ a

such that Fail

F

(b

i

) for some 1 ≤ i ≤ n, or not

Fail

F

(c

j

) for some 1 ≤ j ≤ m, then Fail

F

(a).

(c) If there is a rule

d

1

V

...

V

d

l

V

not e

1

V

...

V

not e

k

⇒ not a

such that not Fail

F

(d

i

) for any 1 ≤ i ≤ l, and

Fail

F

(e

j

) for any 1 ≤ j ≤ k, then Fail

F

(a).

Proposition 2. . Assume a pair (I,J) ∈ 2

A

× 2

A

for

an expression F over the set A, such that

I = {a | Suc

F

(a)} and J = {b | Fail

F

(b)}.

Then Ψ

F

(I,J) ⊆

c

(I,J).

Proof. Let Ψ

F

(I,J) = (I

0

,J

0

). (1) Assume that a ∈ I

0

.

Then there is a rule

b

1

V

...

V

b

n

V

not c

1

V

...

V

not c

m

⇒ a

such that b

i

∈ I for any 1 ≤ i ≤ n, and c

j

∈ J for any

1 ≤ j ≤ m. By the assumed deﬁnitions of I and J,

Suc

F

(b

i

) for any 1 ≤ i ≤ n, and Fail

F

(c

j

) for any 1 ≤

j ≤ m. It follows that Suc

F

(a). That is, a ∈ I. Thus

I

0

⊆ I.

(2) When a ∈ J

0

, then there are cases as follows.

(a) In case that there is no rule, whose right hand of

the implication is a, a ∈ J.

(b) In case that there is a rule

b

1

V

...

V

b

n

V

not c

1

V

...

V

not c

m

⇒ a

such that some b

i

is in J (1 ≤ i ≤ n), or some c

j

is

not in J (1 ≤ j ≤ m): It follows that there is some

Fail

F

(b

i

) (1 ≤ i ≤ n), or not Fail

F

(c

j

) for some 1 ≤

j ≤ m. Then Fail

F

(a), and a ∈ J.

(c) In case that there is a rule

d

1

V

...

V

d

l

V

not e

1

V

...

V

not e

k

⇒ not a

such that any d

i

is not in J (1 ≤ i ≤ l), and any e

j

is in J (1 ≤ j ≤ k): By the deﬁnitions of I and J,

not Fail

F

(d

i

) for any 1 ≤ i ≤ l, and Fail

F

(e

j

) for any

1 ≤ j ≤ k. Therefore Fail

F

(a), and a ∈ J.

In the above cases, a ∈ J on the assumption that

a ∈ J

0

. Therefore J

0

⊆ J. This completes that

(I

0

,J

0

) ⊆ (I,J)

The signiﬁcance of the above proposition is just

soundness of the predicates of Suc

F

(a) and Fail

F

(b)

with respect to a model (I,J) of the given expression

F, where the pair (I, J) is really organized by the pred-

icates.

3.2 Procedural Query

The predicate “not Fail

F

(a)” is not so practical,

where it is of use in the inductive deﬁnition of the

predicate Fail

F

(a). It is primarily from nonmono-

tonicity of the mapping Ψ

F

associated with a given

expression F as causal relation. To make it more

practical, we have simple predicates for queries con-

cerning the expression F. The predicates suc

F

(a) and

f ail

F

(a) are deﬁnable, such that

(i) if suc

F

(a) then Suc

F

(a) and “not Fail

F

(a)”, and

(ii) if f ail

F

(a) then Fail

F

(a).

Formally, the predicates are deﬁned inductively in

a similar manner.

(1) If there is a rule

b

1

V

b

n

V

not c

1

V

...

V

not c

m

⇒ a

such that suc

F

(b

i

) for any 1 ≤ i ≤ n, and f ail

F

(c

j

)

for any 1 ≤ j ≤ m, then suc

F

(a).

(2)(a) If there is no rule, whose right hand of the im-

plication ⇒ is a, then f ail

F

(a).

(b) If there is a rule

b

1

V

...

V

b

n

V

not c

1

V

...

V

not c

m

⇒ a

such that f ail

F

(b

i

) for some 1 ≤ i ≤ m, or

suc

F

(c

j

) for some 1 ≤ j ≤ m, then f ail

F

(a).

(c) If there is a rule

d

1

V

...

V

d

l

V

not e

1

V

...

V

not e

k

⇒ not a

such that suc

F

(d

i

) for any 1 ≤ i ≤ l, and

f ail

F

(e

j

) for any 1 ≤ j ≤ k, then f ail

F

(a).

Reference Data Abstraction and Causal Relation based on Algebraic Expressions

211

The predicates are in accordance with reasoning

of “negation as failure”.

(a) If a query of a succeeds, then a query of “not a”

fails.

(b) If a query a fails, then a query “not a” succeeds.

These predicates are sound with respect to a

model (I,J) constructed by the predicates Suc

F

(a)

and Fail

F

(b), and related by the mapping Ψ

F

.

Proposition 3. Assume an expression F over the set

A of algebraic elements. Let

I = {a | Suc

F

(a)} and J = {b | Fail

F

(b)}.

We have that:

(i) if suc

F

(a) then a ∈ I.

(ii) if f ail

F

(a) then a ∈ J.

Proof. By the inductive deﬁnition and induction on

proof,

(a) if suc

F

(a) then Suc

F

(a),

(b) if f ail

F

(a) then Fail

F

(a), and

(c) Suc

F

(a) and Fail

F

(a) are exclusive.

Thus if suc

F

(a) then not Fail

F

(a). This may conclude

the proposition.

Adjusting (Procedure for Query):

A procedure may be constructed, in accordance

with the deﬁnitions of the predicates suc

F

(a) and

f ail

F

(b).

By induction on the deﬁnitions of predicates suc

F

(a)

and f ail

F

(b), and on the following derivations

ha? suci and hb? f aili, we can see that:

suc

F

(a) iff ha? suci, and

f ail

F

(b) iff hb? f aili.

(1) With a given expression F, query of the sequence

“X?” is assumed, where X = y

1

;...; y

n

(n ≥ 0) with y

i

being a or not a for a ∈ A and with the concatenation

operation “;” (which is treated as

V

), where in case

of “n = 0”, X is null (the empty query). A sequence

query may be denoted as y;X? (with y being b or not b

for b ∈ A, and with X a sequence query), or Y ;X? with

Y and X sequence queries.

(2) The notations hX? suci, and hX? f aili stand for

the cases of the query X? to be a success, and a fail-

ure, respectively.

There are 2 routines of succeeding, and failing deriva-

tions for queries to be analyzed.

(i) hnull? suci.

(ii) hx; X? suci, if x = a (for a ∈ A) and there is Y ⇒ x

in F such that hY ; X? suci.

(iii) (a) hx; X? suci, if ha? f aili for x = not a, and

hX? suci.

(b) hx; X ? suci, if ha? f aili for x = not a where

there is some Y ⇒ x in F with hY ? suci, and

hX? suci.

(iv) hx; X? f aili, if there is no part with x = a ∈ A be-

ing the right hand of “⇒” in F, or hX? f aili.

(v) hx; X? f aili, if x = a (a ∈ A) such that hY ? f aili

for some Y (where Y ⇒ x in F), or hX? f aili.

(vi) (a) hx; X? f aili, if x = not a such that ha? suci, or

hX? f aili.

(b) hx; X ? f aili, if x = a such that hY ? suci for

some Y (where Y ⇒ not a in F), or hX? f aili.

4 SEMIRING STRUCTURE

As in the setting of a language system (S. Yamasaki,

in COMPLEXIS 2020), when traversing the states

(sites), pages from a state are concatenated to other

pages of another state. This observation can be ab-

stracted to some algebraic structure from expressions

F

1

, F

2

, ... .

Given a logical or algebraic expression F over the

set A, we may have a pair

(I,J) ∈ 2

A

× 2

A

,

which is assumed as a 3-valued model of F, and can

be regarded as deﬁning state changes (transitions).

With the set A, we can have denumerable expres-

sions F

1

, F

2

, . .., causing state changes, which are in

accordance with causal relations in a distributed sys-

tem. (state transitions). Then the 3-valued models of

expressions F

1

, F

2

, ... may be assumed as the pairs

(I

1

,J

1

), (I

2

,J

2

), ....

By means of the set concatenation “·” (which gets

the set of sequences obtained from taken elements of

sets), we might have

(I

1

,J

1

) • ... • (I

n

,J

n

)

= (I

1

· ... · I

n

− {w | some element of w of I

1

· ... · I

n

is in J

1

∪ ... ∪ J

n

},

J

1

∪ ... ∪ J

n

),

with multiplication “•” to express the sequence

formation.

In this paper, we newly have a semiring with

respect to the view of human computer interaction. It

is different from the star semiring constructed in the

case (S. Yamasaki, in COMPLXIS 2017). Reﬂecting

state transitions with models, alternations are denoted

DATA 2020 - 9th International Conference on Data Science, Technology and Applications

212

in terms of algebraic aspects. With alternation

aspects, let R

A

(R, for short with the assumption of

the set A) be the set of “direct sums” of the form

Σ

l

(pSeq

l

,nSet

l

) with l ranging indexes, where each

pair (pSeq

k

,nSet

k

) is supposedly consistent, with

a model of an expression F

k

over A. From imple-

mentation views, the “sum” means nondeterministic

selections as alternation so that it contains more

complexity. Therefore human computer interaction

may be of use, to control determinations. It is a

compact representation that this section is to aim at,

with respect to nondeterministic complexity. Keeping

such complexity of what the sum contains, we newly

have a formality of semiring structure as follows.

The operations + (addition–alternation) and ◦

(multiplication–composition) on R are deﬁned:

(1)

Σ

i

(pSeq

i

,nSet

i

) + Σ

j

(pSeq

j

,nSet

j

)

= Σ

k=i, j

(pSeq

k

,nSet

k

).

(2)

Σ

i

(pSeq

i

,nSet

i

) ◦ Σ

j

(pSeq

j

,nSet

j

)

= Σ

i, j

(pSeq

i

· pSeq

j

− {uv| u ∈ pSeq

i

,v ∈ pSeq

j

,

u is not consistent to nSet

j

},

− {uv| u ∈ pSeq

i

,v ∈ pSeq

j

,

v is not consistent to nSet

i

},

nSet

i

∪ nSet

j

),

where

(i) the sequence uv is constructed by concatenation

of sequences u and v,

(ii) by saying that u and v are not consistent to

(the sets) nSet

j

and nSet

i

, respectively, it means

that u and v contain some element in nSet

j

and

nSet

i

, respectively, and

(iii) the operation · is the set concatenation consist-

ing of concatenated sequences.

Note: The operation ◦ preserves “consistency” of

each pair in a resultant direct sum.

Identities with respect to + and ◦:

We can have identities with respect to the addition

and multiplication in terms of alternation and compo-

sition, respectively, if we care the direct sum of the

“form” Σ

i

(pSeq

i

,nSet

i

).

(i) Σ

i

(pSeq

i

,nSet

i

) is denoted

/

0

Σ

, if the direct sum is

the empty. It is the identity with respect to +.

(ii) The empty sequence in A

∗

is represented by ε.

({ε},

/

0) is the identity with respect to ◦.

We ﬁnally have a semiring R regarding consistent

sequences caused by models of expressions, in terms

of the following propositions:

Proposition 4. The structure hR

A

,+,◦,

/

0

Σ

,({ε},

/

0)i

is a semiring.

Proof. We can see the conditions of a semiring as fol-

lows.

(i) The operation + is deﬁned so that commutative

and associative laws may obviously hold. With

the identity

/

0

Σ

, hR,+,

/

0

Σ

i is a commutative

monoid (a commutative semigroup with the

identity).

(ii) The operation ◦ is associative, so that

hR,◦,({ε},

/

0)i is a semigroup with the iden-

tity ({ε},

/

0), that is, a monoid.

(iii) Left and right multiplications over addition are

both distributive:

Σ

i

(pSeq

i

,nSet

i

)

◦(Σ

j

(pSeq

j

,nSet

j

) + Σ

k

(pSeq

k

,nSet

k

))

= (Σ

i

(pSeq

i

,nSet

i

) ◦ Σ

j

(pSeq

j

,nSet

j

))

+(Σ

i

(pSeq

i

,nSet

i

) ◦ Σ

k

(pSeq

k

,nSet

k

)).

(Σ

j

(pSeq

j

,nSet

j

) + Σ

k

(pSeq

k

,nSet

k

))

◦Σ

i

(pSeq

i

,nSet

i

)

= (Σ

j

(pSeq

j

,nSet

j

) ◦ Σ

i

(pSeq

i

,nSet

i

))

+(Σ

k

(pSeq

k

,nSet

k

) ◦ Σ

i

(pSeq

i

,nSet

i

)).

(iv)

/

0

Σ

◦ Σ

i

(pSeq

i

,nSet

i

) = Σ

i

(pSeq

i

,nSet

i

) ◦

/

0

Σ

=

/

0

Σ

.

That is, annihilation holds for ◦, with the identity

/

0

Σ

regarding +.

5 CONCLUSION

The primary contribution of this paper is to take Heyt-

ing algebra expressions as representations of causal

relations with model theories to be newly established.

(a) The form of the expressions is restricted to a rep-

resentation of reference data as static link, which is

speciﬁc in the class of expressions as terms possi-

bly occurring in postﬁx modal operator of modal mu-

calculus, or as programs in a language system (S. Ya-

masaki, in COMPLEXIS 2020). In this sense, this

paper is viewed as an application of the algebraic ex-

pressions as terms or programs in the previous papers.

(b) The causal relations are abstractions of the static

links between data with references to each other

in distributed systems, where practices are not con-

cretized but abstracted with model theories. The

model theories are based on 3-valued domain, where

a nonmonotonic mapping is virtually associated with

Reference Data Abstraction and Causal Relation based on Algebraic Expressions

213

expressions with Heyting implication and negatives.

For a preﬁxpoint of the mapping, some inductive con-

struction is presented. We then have models for a

given expression conditioned to some representation

forms. This model theory is relevant to those in logic

programming (Yamasaki, 2006), but more general

than, with respect to strict negation. As a software

technology to analyze algebraic expression queries,

negation as failure rule is applied as sound procedure.

As another result, a semiring structure is formally

constructed with respect to state transitions virtu-

ally caused by dynamic traverses through reference

links, which is related to automata theory (Droste

et al., 2009) rather than context-free language aspects

(Winter et al., 2013). The semiring involves non-

determinism by direct sum of objects derived from

models, which require human interaction to selec-

tion of suitable objects. The abstract representation

involves nondeterministic alternation of transitions

from a state, to which human interaction may be im-

plemented which transition to select.

As related works on logical frameworks possibly

for AI, we should learn concepts and ideas as follows.

They may be hints on advancements to be considered,

as regards practical aspect of this paper:

(a) The paper (Beddor and Goldstein, 2018) presents

the belief predicate with the credence function of

agents, concerning epistemic contradictions. The

contradictions of complexity may be avoided by

grades of such a function.

(b) There is a paper (P. Kremer, 2018) presenting

second-order propositional frameworks, with epis-

temic and intuitionistic logic. It may be relevant to

the extension of this paper with HA expressions to

more facility of complex expressiveness.

(c) With the second-order (quantiﬁed) propositions,

the paper (Goranko and Kuusisto, 2018) involves

dependence and independence concepts, which may

control implementations of programs or queries if

data base is designed with such concepts of represen-

tation complexity.

(d) “Distributed knowledge” is discussed (Naumov

and Tao, 2019), with quantiﬁed variables of quantiﬁes

ranging over the set of agents. Concerning applica-

tions of the second-order predicates to knowledge, the

paper (Kooi, 2016) contains the concept of knowing.

Distributive knowledge processing is of more com-

plexity even for the state constrained programs.

(e) For an extension of propositional modal logic

without quantiﬁcation, the paper (Fitting, 2002) in-

troduces relations and terms with scoping mechanism

by lambda abstraction. It is considered as presenting

functional programming included in modal logic.

REFERENCES

Beddor, B. and Goldstein, S. (2018). Believing epistemic

contradictions. Rev.Symb.Log., 11(1):87–114.

Bertolissi, C., Cirstea, H., and Kirchner, C. (2006).

Expressing combinatory reduction systems

derivations in the rewriting calculus. Higher-

Order.Symbolic.Comput., 19(4):345–376.

Cardelli, L. and Gordon, A. (2000). Mobile ambients. The-

oret.Comput.Sci., 240(1):177–213.

Dam, M. and Gurov, D. (2002). Mu-calculus with ex-

plicit points and approximations. J.Log.Comput.,

12(1):119–136.

Dragoni, A., Giorgini, P., and Seraﬁni, L. (1985).

Mental states recognition from communication.

J.Log.Program., 2(4):295–312.

Droste, M., Kuich, W., and Vogler, H. (2009). Handbook of

Weighted Automata. Springer.

Fitting, M. (2002). Modal logics between propositional and

ﬁrst-order. J.Log.Comput., 12(6):1017–1026.

Giordano, L., Martelli, A., and Schwind, C. (2000).

Ramiﬁcation and causality in a modal action logic.

J.Log.Comput., 10(5):625–662.

Goranko, V. and Kuusisto, A. (2018). Logics for proposi-

tional determinacy and independence. Rev.Symb.Log.,

11(3):470–506.

Governatori, G., Maher, M., Autoniou, G., and Billington,

D. (2004). Argumentation semantics for defeasible

logic. J.Log.Comput., 14(5):675–702.

Hanks, S. and McDermott, D. (1987). Nonmonotonic logic

and temporal projection. Artiﬁ.Intelli., 33(3):379–

412.

Kooi, B. (2016). The ambiguity of knowability.

Rev.Symb.Log., 9(3):421–428.

Kozen, D. (1983). Results on the propositional mu-calculus.

Theoret.Comput.Sci., 27(3):333–354.

Merro, M. and Nardelli, F. (2005). Behavioral theory for

mobile ambients. J.ACM., 52(6):961–1023.

Mosses, P. (1992). Action Semantics. Cambridge University

Press.

Naumov, P. and Tao, J. (2019). Everyone knows that

some knows: Quantiﬁers over epistemic agents.

Rev.Symb.Log., 12(2):255–270.

Reiter, R. (2001). Knowledge in Action. MIT Press.

Reps, T., Schwoon, S., and Somesh, J. (2005). Weighted

pushdown systems and their application to interproce-

dural data ﬂow analysis. Sci.Comput.Program., 58(1-

2):206–263.

Rutten, J. (2001). On Streams and Coinduction. CWI.

Spalazzi, L. and Traverso, P. (2000). A dynamic logic

for acting, sensing and planning. J.Log.Comput.,

10(6):787–821.

Winter, J., Marcello, B., Bonsangue, M., and Rutten, J.

(2013). Coalgebraic characterizations of context-free

languages. Formal Methods in Computer Science,

9(3):1–39.

Yamasaki, S. (2006). Logic programming with default,

weak and strict negations. Theory Prac.Log.Program.,

6(6):737–749.

DATA 2020 - 9th International Conference on Data Science, Technology and Applications

214