Distributed Strategies and Managements based on

State Constraint Logic with Predicate for Communication

Susumu Yamasaki and Mariko Sasakura

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

Keywords:

Logical Approach, Organization of Strategy, Logic for Distributed Systems, Model Theory.

Abstract:

From the views on cognitive management, this paper deals with state constraint and distributed systems, where

communication between states is a key function of complexity. The primary purpose is concerned with logical

analysis of complex, distributed system structure which contains strategies (procedures) designed in states.

Between states, strategies may be communicative and transferrable, where the transferrability is supposed

to be given by predicates for communication between states. The strategy as a procedure is assumed to be

inductively constructed by other distributed strategies. The structure to represent the designed way of strate-

gies takes an inductively deﬁned form, on which some logical relation is characterized with respect to the

compound construction of strategies (procedures). The logical relation is in accordance with possibly inﬁnite

set of propositional formulas constrained by states. As regards procedural executions, implementation, the

undeﬁned (implementation), and non-implementation may be considered for the remarked strategy. Based

on the discussions of implementability for strategic constructions, a structural analysis of distributed strate-

gies may be settled as 3-valued model theory of logical expressions. It is related to 3-valued model theory,

where some ﬁxed point theory is now examined, with respect to the mapping (which is in general monotonic)

associated with a logical expression. The logical expression of this paper can be denoted as a propositional

logic formula with default negation. As an application to logical system, logical formulas with both strong

and default negations may be analyzed with 3-valued domain. This paper thus abstracts application of logical

expressions to structural analysis of distributed strategies. Structure of strategies is complex owing to distri-

bution of state constraint strategies, but effectiveness may be endowed with logical approach and abstraction

of communication facility.

1 INTRODUCTION

Cognitive management subjects include complexity

analysis of distributed systems containing computing

and communication facilities. As such a distributed

system, we consider an abstract state machine frma-

work, where the states are distributed and strategies

(as procedures) are integrated into each state. Be-

tween states, communications and behavioral interac-

tions are allowed, as well as state transitions which

are caused by strategic actions in states. As strategies

in abstract state machine possibly for data operations,

there are procedural, logical and algebraic views.

(1) Procedural strategy is expressed by denotational

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

ral method is in accordance with operational imple-

mentation for programs to be executed. The strate-

gies may be abstracted, with functional programs

(Bertolissi et al., 2006). These are essentially pro-

grams on data, being interpreted as data operations.

(2) Strategies are also captured in logical systems

from the viewpoints of sequential process, as in the

papers (Giordano et al., 2000; Hanks and McDermott,

1987). These are concerned with dynamism of ac-

tions, and modality in logical systems. The action

is formulated as a key role in strategic reasoning of

abstract state machine, as well as concretized actions

in dynamic logic. Acting and sensing failures are

discussed as advanced works (Spalazzi and Traverso,

2000).

(3) Primarily from the viewpoints of transitions, ab-

stract state machine is discussed, in the paper (Reps

et al., 2005). We can have speciﬁed structure of

streams caused by abstract state transitions, by the

note (Rutten, 2001). Algebraic structures are formally

discussed with respect to state transitions (Droste

et al., 2009).

With relevance to practical senses of strategies,

AI reasoning may be common interests even in

distributed system designs. With the concerns to

78

Yamasaki, S. and Sasakura, M.

Distributed Strategies and Managements based on State Constraint Logic with Predicate for Communication.

DOI: 10.5220/0010468700780085

In Proceedings of the 6th International Conference on Complexity, Future Information Systems and Risk (COMPLEXIS 2021), pages 78-85

ISBN: 978-989-758-505-0

Copyright

c

2021 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

AI programming, we have studied nonmonotonic

reasoning where ﬁxed point theory is not alway a

routine. In knowledge representations with respect

to logical approaches to reasoning, we have seen

backgrounds:

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

classical, where knowledge (data) representation

and reasoning as operation in knowledge base are

systemized in logics.

(2) “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.

(3) Regarding distributed systems, software and

knowledge engineerings like mobile ambients

(Cardelli and Gordon, 2000; Merro and Nardelli,

2005) have been formulated with environments

to make communication reasonable. As proofs in

programming and data science, the papers (Dam and

Gurov, 2002; Kozen, 1983) are classical enough to

formulate the proof systems with ﬁxed points and

their approximations.

Following the ideas of strategies in abstract state

machine, and AI reasonings, this paper aims at struc-

tural aspects of strategies as procedures which may

be basis possibly for AI programming with algebra

or logic, rather than the whole algebraic structure

of abstract state machine. This paper deals with a

framework, different from those established works,

and examines distributed strategies just with respect

to inductive structure of strategy construction, where

strategies are inductively deﬁned in various states,

and integrated into each state. Between states, strate-

gies are communicative and transferrable, where each

state contains strategies on data. For descriptions

of strategy constructions in a distributed system,

possibly inﬁnite set of propositional logic formulas

is adopted in the sense that it may express abstract

and clear structures. With universal denotations for

possibly inﬁnite set of propositional logic formulas

(which may represent ﬁrst-order logic formula with

Herbramd base), we have effectiveness (with respect

to computing) for the system construction of this

paper. As regards implementability of strategies

with communication between states in a distributed

system, 3-valued model theory of logical expressions

provides analysis with default negation (in accor-

dance with negation as failure) and the unknown

for implementability. Abstracted from the structural

implementability of strategy constructions, logical

expressions constrained by states (virtually with

intensionality sensitive to states and communication

between states) may be formulated for a distributed

system.

As a primary goal of this paper, 3-valued model

theory of logical expressions is given. The pred-

icates of implementability and default negative are

next organized. As an application, we examine logical

database with query predicates where we could have

model theory of propositional logic with both strong

and default negations, and with communication fa-

cility. The paper is organized as follows. Section 2

presents an outlook on inductive structure of strategy

constructions from the view of distributed resources.

In Section 3, logical expressions and model theory are

discussed, abstracted from implementability of strat-

egy constructions. Section 4 presents the predicates of

evaluations for expressions and derivations as query

of strategy (procedure) implementability, with refer-

ence to application to logical database, where strategy

names are regarded as organizing database. Conclud-

ing remarks are described in Section 5.

2 ORGANIZATION OF

STRATEGIES

2.1 State Constraint Objects

With respect to process theories (Hennessy and Mil-

ner, 1985; Kucera and Esparza, 2003; Milner, 1999),

we consider the case that state constraint objects are

distributed but relational, where the case conceives

complexity in managements for cognitive or comput-

ing merits, but representations by means of distributed

objects are easily available as long as communica-

tion between objects is implicitly guaranteed. This

paper treats such objects distributed and constrained

by states with manageable relations.

As an illustration, we present state constraint

propositions represented as adjective for strategic

senses:

(i) secure[h], remedied[h], sa f e[h] and risky[h] at a

state h (as regards health-affair), and

(ii) bound[s] and open[s] at a state s (as regards social-

affair),

where logical and “∧” and implication “→” are used

as well as negation “

∼

”, in addition to parentheses for

discriminations.

h-state:

secure[h]

∧(

∼

remedied[h] ∧

∼

sa f e[h] → risky[h])

∧(

∼

open[s] →

∼

sa f e[h])

Distributed Strategies and Managements based on State Constraint Logic with Predicate for Communication

79

s-state:

(secure[h] → bound[s])

∧(bound[s] ∧

∼

risky[h] → open[s])

Communication of s-state might be supposedly avail-

able to h-state such that the entailed propositions can

be acknowledged at state h. When the proposition

secure[h] is assumed, then what reasoning may be

taken into consideration, on the basis of propositional

logic constrained by states. The negation can be

thought of as default (i.e., negative as failure of rea-

soning, in 3-valued logic containing the unknown to

be reasoned or not to be reasoned.)

Assume no communication of h-state to s-state

(i.e. neglect of h-state propositions at s-state) ex-

cept secure[h]: (i) At state s, with secure[h], bound[s]

is inferred. Even with bound[s], (“without” default

∼

risky[h],) open[s] is unknown. (ii) At state h,

∼

sa f e[h] is unknown from unknown

∼

open[s], such

that even with default

∼

remedied[h], risky[h] may

be unknown, and even the default

∼

risky[h] may not

transferrable to state s.

Without implementation details of communica-

tion, we may see reasoning from declarations of log-

ical formulas of state constraint propositions, which

is regarded as coordination of computing (based on

reasoning) with communication. In this paper, we

deal with such a coordination for distributed (proposi-

tional) variables (at states), which can represent pro-

cedural or strategic objects by name.

2.2 Representation of Strategy

Constructions

As a formal system for distributed programming with

state environments, we are interested in strategic pro-

gramming where strategies as procedures are dis-

tributed such that each strategy may be compiled into

construction with other distributed strategies. Then

the structure of strategy constructions should be rep-

resented in a simpler form than verbal accounts with

reﬁned procedural words. In case of problem solv-

ing, the strategy is called by name and the reference

structure is based on recursion manners of goals, con-

structed by subgoals (such that the subgoal may be

constructed by subgoals). We thus make the exami-

nations of the expression for the ways of (i) how to

represent the inductive structure of distributed strat-

egy constructions, and (ii) how to represent the con-

structive structure of strategies.

As in object-oriented programming language, a

procedure can be designed in a class, such that a form

{pr

1

[o

1

],. .., pr

n

[o

n

]} B pr[o]

may be taken to see that the strategy pr in the class o

may be inductively constructed, to contain the strate-

gies pr

1

(in the class o

1

), ... , and pr

n

(in the class o

n

)

(as components).

This paper treats a logical approach to represent

structures of a distributed system involving strategies

as procedures at states:

(a) Propositional variables denoting strategies are dis-

tributed, depending on the state variables.

(b) Some standard form of distributed strategy (pro-

cedure) constructions is assumed, and the form would

be logically described.

(c) Virtual communications between states are deﬁn-

able by predicates containing propositional variables,

such that a strategy (procedure) may be regarded as

transferrable from a state to another.

3 STATE CONSTRAINT LOGIC

WITH COMMUNICATIONS

3.1 Formal Description of Distributed

Strategy

Following Motague grammar (by R.Montague), in-

tentionality is deﬁned in the manner of λs.p for a state

variable s within the scope of λ-notation to the propo-

sition p. We here make use of the extension in the

manner of p[s], rather than intension. We then make

a description of formal system in terms of logic in co-

ordination to communication between states. Now let

us see a sequence of the procedural constructions in-

cluding state constraints (like class-constraints as in

subsection 2.2):

{pr

1

[s

1

],. .., pr

n

[s

n

]} B pr[s],

.. .,

.. .,

{pr

0

1

[s

0

1

],. .., pr

0

m

[s

0

m

]} B pr

0

[s

0

],

with strategy (procedure) names pr

1

[s

1

], . .. , pr[s],

pr

0

1

[s

0

1

], .. . , pr

0

[s

0

].

The left hand of the composing B is referred to by

body and the right hand is referred to by head. The

body consists of (none or) ﬁnitely many expressions

of the form pr[s] (with or without negation), where

pr is a procedure name with a state s. The head con-

sists of an expression of the form pr[s]: The procedure

may be represented by proposition or its negative,

constrained by a state. Because the procedural im-

plementation is sensitive to logical values, if calling

by name for procedures is adopted where procedural

structures may be of sense. Effectiveness (for com-

puting) of such structures may be guaranteed by in-

COMPLEXIS 2021 - 6th International Conference on Complexity, Future Information Systems and Risk

80

ﬁniteness of propositional logic, denoting ﬁrst-order

logic based on Herbrand base.

With communication applied to logic as coordina-

tion of logic to communication, this paper treats the

abstract strategy constrained by the set S of states as

well as communicative predicates. From semantics

views, strategic structure might be logically deﬁned,

while from formal description viewpoints, they are

deﬁned in Backus Naur Form (BNF) as follows:

Therefore, to describe the structure “body B

head” (just with a symbol B) referred to as a rule,

we take BNF of:

(a) literal ::= p[s] |

∼

p[s]

(b) head ::= literal

(c) body ::= { } | {literal} ∪ body

(d) rule ::= body B head

where (i) the notation “

∼

” is reserved for the negative

sign, (ii) p is a propositional variable and s is a state

variable, and (iii) ∪ denotes the set union operation,

applicable to the empty body, { }.

The rule body B head is contained by the whole

strategy, which is a sequence of such structures as

rules. Instead of the sequence, the whole structure

(Strategy in BNF) may be alternatively deﬁned as a

ﬁnite or denumerably inﬁnite set of such structures.

(d) Strategy ::=

/

0 | {rule} ∪ Strategy

where (i)

/

0 stands for the empty set, and (ii) as

Condition for Strategy, at most one of any comple-

mentary pair (p[s],

∼

p[s]) occurs at heads of rules.

Note that this Condition is in accordance with the

sense that the head of a rule may denote a constructed

strategy by name such that both positive and negative

are not needed. Throughout this paper, this condition

is assumed even without mentioning.

Then a set of rules (deﬁned as Strategy) is consid-

ered as the whole structure, where its inductive struc-

ture is to be interpreted. For the interpretation, com-

munications between states must be included, since

strategies in Strategy contain state constraint proposi-

tions to represent implementations of distributed pro-

cedures.

As coordination of strategies with communica-

tions, we just make use of higher-order predicates of

the form and a set Commu:

(e) Commu ::=

/

0 | {commu(s

0

,s, p[s

0

])} ∪Commu

where commu is a predicate symbol, with state vari-

ables s

0

and s as well as state constraint propositional

variable p[s

0

], such that commu(s,s, p[s]) is suppos-

edly included in Commu for any p[s].

Finally we have a set of programs to be organized,

where their inductive structures of implementations

are expressed in terms of call by name and universal-

ities fo computing and communication may be guar-

anteed in the ﬁrst-order logic.

(f) Program ::= Strategy ∪Commu

For the structural interpretation with respect to

Program, we adopt 3-valued logic with default nega-

tion. With reference to implementability for the struc-

ture of procedural constructions, whether some pro-

cedure is implementable or non-implementable is to

be noted. This paper deals with the implementability

case of procedure to be undeﬁned, so that 3-valued

domain may be taken for implementability.

Assume in 3-valued domain (underlying set)

{0,1/2, 1} that (i) the implication is based on

(possibly inﬁnite) propositional logic, and (ii) the

evaluation of

∼

p[s] follows the way:

p[s]

∼

p[s]

1 0

1/2 1/2

0 1

Example 1. Take the logical expressions at states h

and s of subsection 2.1. Then as Strategy, we can

have a set:

Strategy

= { { } B secure[h],

{

∼

remedied[h],

∼

sa f e[h]} B risky[h],

{

∼

open[s]} B

∼

sa f e[h],

{secure[h]}B bound[s],

{bound[s],

∼

risky[h]} B open[s] }

As Commu, we assume:

Commu =

{ commu(s,s, bound[s]),commu(s, s,open[s]),

commu(s,h, open[s]),commu(h, h,secure[h]),

commu(h,h, sa f e[h]),commu(h, h,remedied[h]),

commu(h,h, risky[h]), commu(h,s,secure[h]) }

3.2 Evaluation of Strategy

Implementation

The assignment of the values in 3-valued domain

{0,1/2, 1} to the propositional variable with state

constraint causes the program to have the value 0,

1/2. or 1 for implementability. To deﬁne such an in-

terpretation of the program implementation, we con-

sider a mapping associated with a given program to

be represented in terms of logic with predicates for

communication. The mapping is for a ﬁxed point se-

mantics which can be an interpretation of the pro-

gram structure inductively constructed in subsection

3.1. Such semantics would be related to retrieval in

Distributed Strategies and Managements based on State Constraint Logic with Predicate for Communication

81

logical database expressed by the program with state

constraints and with predicates for communication.

The logical database will be shown in Section 4.

Note that the predicates of the form

commu(s

0

,s, p[s

0

])

(in Commu) are supposedly included in Program (in

subsection 3.1), with name P. Let A

P

(or A when P is

explicitly supposed) be {p[s] | p[s] ∈ literal}, a set of

state constraint variables occurring in P.

For exp ::= literal | body | rule | Program and

a pair (I, J) ∈ 2

A

× 2

A

, to have the evaluation of the

expression in 3-valued domain, we deﬁne a valuation

Val : exp → 2

A

× 2

A

→ {0, 1/2,1} in the following

manner, with respect to (possibly inﬁnite) proposi-

tional logic with state constraint.

(a) Val[[p[s]]](I,J) =

1 if p[s

0

] ∈ I with

commu(s

0

,s, p[s

0

])

0 if p[s

0

] ∈ J with

commu(s

0

,s, p[s

0

])

1/2 otherwise

(b) Val[[

∼

p[s]]](I,J) =

1 if p[s

0

] ∈ J with

commu(s

0

,s, p[s

0

])

0 if p[s

0

] ∈ I with

commu(s

0

,s, p[s

0

])

1/2 otherwise

(c) Val[[body]](I,J) =

1 if Val[[litreal]](I,J) = 1

for any literal of body

0 if Val[[literal]](I, J) = 0

for some literal of body

1/2 otherwise

(with Commu)

(d) Val[[rule]](I,J) =

1 if Val[[body]](I, J) is less than

or equal to Val[[head]](I,J)

0 otherwise

(for rule = body B head with Commu)

(e) Val[[Program]](I,J) =

1 if Val[[rule]](I,J) = 1

for any rule of Strategy

0 if Val[[rule]](I,J) = 0

for some rule of Strategy

1/2 otherwise

(for Program = Strategy ∪Commu)

If Val[[P]](I, J) = 1 for a given program P and a pair

(I, J) ∈ 2

A

×2

A

, such that I ∩J =

/

0 (i.e., the pair (I,J)

is consistent), then (I, J) is called a model of P.

With the name P (of Program), a mapping Tr

P

is deﬁned, such that its ﬁxed point may be a model

of P, that is, an evaluation of P as 1. The model is

regarded as presenting consistency of P which may

denote strategy constructions with communication.

The mapping Tr

P

(associated with a program P),

applied to a pair (I

1

,J

1

) for providing a pair (I

2

,J

2

),

is deﬁned in the manner as follows.

Tr

P

: 2

A

× 2

A

→ 2

A

× 2

A

,

Tr

P

(I

1

,J

1

) = (I

2

,J

2

).

Deﬁnition of Tr

P

:

(1) For some rule body B p[s] such that (i) for any

q[s

0

] of body, q[s

0

] is in I

1

with commu(s

0

,s, q[s

0

]),

and (ii) for any

∼

r[s

00

] of body, r[s

00

] is in J

1

with

commu(s

00

,s, r[s

00

]), p[s] is in I

2

.

(This case contains the one that body is the empty set,

where p[s] is in I

2

.)

(2)(a) For any rule of the form body B p[s] such

that (i) for some q[s

0

] of body, q[s

0

] is in J

1

with

commu(s

0

,s, q[s

0

]), or (ii) for some

∼

r[s

00

] of body,

r[s

00

] is in I

1

with commu(s

00

,s, r[s

00

]), p[s] is in J

2

.

(This case contains the one that there is no rule of the

form bodyB p[s] without any rule of the form body

0

B

∼

p[s], where p[s] is in J

2

.)

(b) For some rule body B

∼

p[s] such that (i) for any

q[s

0

] of body, q[s

0

] is in I

1

with commu(s

0

,s, q[s

0

]),

and (ii) for any

∼

r[s

00

] of body, r[s

00

] is in J

1

with

commu(s

00

,s, r[s

00

]), p[s] is in J

2

.

(This case contains the one that body is the empty set,

where p[s] is in J

2

.)

Fixed Point of Tr

P

:

If Tr

P

(I, J) = (I, J), then (I,J) is called a ﬁxed point

of Tr

P

. By “componentwise subset inclusion ⊆

c

” (a

binary relation on 2

A

× 2

A

), we mean that I

1

⊆ I

2

and

J

1

⊆ J

2

iff (I

1

,J

1

) ⊆

c

(I

2

,J

2

). When Tr

P

(I, J) ⊆

c

(I, J), (I,J) is a preﬁxpoint of Tr

P

. A ﬁxed point of

Tr

P

is a preﬁxpoint. We will see that for a ﬁxed point

(I, J), if I ∩ J =

/

0 (i.e., (I,J) is consistent), then (I, J)

can be a model of P, that is, P is evaluated as 1 by the

pair (I, J).

Fixed Point Models:

The mapping Tr

P

is monotonic, that is, if (I

1

,J

1

) ⊆

c

(I

2

.J

2

), then Tr

P

(I

1

,J

1

) ⊆

c

Tr

P

(I

2

,J

2

). The method

by ﬁxed point of Tr

P

is always available as a mod-

elling of the given program P.

Example 2. Assume the Strategy and Commu as in

Example 1:

Strategy

= { { } B secure[h],

{

∼

remedied[h],

∼

sa f e[h]} B risky[h],

{

∼

open[s]} B

∼

sa f e[h],

{secure[h]}B bound[s],

{bound[s],

∼

risky[h]} B open[s] }

COMPLEXIS 2021 - 6th International Conference on Complexity, Future Information Systems and Risk

82

Commu =

{ commu(s,s, bound[s]),commu(s, s, open[s]),

commu(s,h, open[s]),commu(h,h,secure[h]),

commu(h,h, sa f e[h]), commu(h, h,remedied[h]),

commu(h,h, risky[h]),commu(h,s, secure[h]) }

Then a pair ({secure[h],bound[s]},{remedied[h]})

may be a ﬁxed point of Tr

P

(P = Strategy ∪Commu).

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

A

× 2

A

for a

given program P. If a pair (I,J) is a consistent ﬁxed

point of the mapping Tr

P

, then (I, J) is a model of P.

Proof. Let Tr

P

(I, J) = (I

0

,J

0

). Following the deﬁni-

tion of the mapping Tr

P

, we examine the mapping

case by case, to see why (I

0

,J

0

) = (I,J) causes (I,J)

to be a model.

(1) If p[s] ∈ I

0

= I for some rule body B p[s], then

Val[[p[s]]](I,J) = 1 such that

Val[[body

0

B p[s]]](I,J) = 1

for any rule body

0

B p[s].

(2) (i) If p[s] ∈ J

0

= J, then there may the case: for

any rule of the form body B p[s], q[s

0

] is in J with

commu(s

0

,s, q[s

0

]) for some q[s

0

] of body, or r[s

00

] is

in I with commu(s

00

,s, r[s

00

]) for some

∼

r[s

00

] of body

(i.e., Val[[body]](I,J) = 0) such that

Val[[body B p[s]]](I, J) = 1

for any rule body B p[s]. (This case contains the one

that there is no rule of the form body B p[s] for p[s]

without any rule of the form body

0

B

∼

p[s].)

(ii) If p[s] ∈ J

0

= J, then there may be the case: for

some rule of the form body B

∼

p[s], q[s

0

] is in I with

commu(s

0

,s, q[s

0

]) for any q[s

0

] of body, and r[s

00

] is in

J with commu(s

00

,s, r[s

00

]) for any

∼

r[s

00

] of body (i.e.,

Val[[body]](I,J) = 1) such that

Val[[body

0

B

∼

p[s]]](I,J) = 1

for any rule body

0

B

∼

p[s].

(3) If p[s] 6∈ I

0

∪ J

0

= I ∪ J, then for any rule body B

p[s] or any rule body B

∼

p[s], Val[[body]](I,J) =

0, or 1/2 with the pair (I,J) (and Commu). Since

Val[[p[s]]](I,J) = 1/2 and Val[[

∼

p[s]]](I,J) = 1/2,

Val[[body B p[s]]](I, J) = 1.

Thus all the rules are evaluated as 1, with respect to

the pair (a ﬁxed point of Tr

P

) (I,J). This may con-

clude the proposition.

4 ANALYSIS OF LOGIC WITH

COMMUNICATION

4.1 Predicates for Implementability of

Strategy

To make the sense of the mapping Tr

P

less complex

from implementation views, we have simple predi-

cates to relate the mapping Tr

P

with. The predicates

(of higher-order for propositions) imple

P

(p[s]) and

de f ault

P

(p[s]) are deﬁnable. Formally, the predicates

are deﬁned inductively, for a given program P with

the communicative predicates Commu (shown in the

subsection 3.1):

Predicates imple

P

(-) and de f ault

P

(-):

(1) If there is a rule bodyB p[s] such that imple

P

(q[s

0

])

with commu(s

0

,s, p[s

0

]) for any q[s

0

] of body, and

de f ault

P

(r[s

00

]) with commu(s

00

,s, r[s

00

]) for any

∼

r[s

00

] of body, then imple

P

(p[s]).

(This case contains the one that body is the empty set,

where imple

P

(p[s]).)

(2)(a) If for any rule body B p[s] such that

de f ault

P

(q[s

0

]) with commu(s

0

,s, q[s

0

]) for some q[s

0

]

of body, or imple

P

(r[s

00

]) with commu(s

00

,s, r[s

00

]) for

some

∼

r[s

00

] of body, then de f ault

P

(p[s]).

(This case contains the one that there is no rule of the

form bodyB p[s] without any rule of the form body

0

B

∼

p[s], where de f ault

P

(p[s]).)

(b) If there is a rule body B

∼

p[s] such that

imple

P

[q[s

0

]) with commu(s

0

,s, q[s

0

]) for any q[s

0

] of

body and de f ault

P

(r[s

00

]) with commu(s

00

,s, r[s

00

]) for

any

∼

r[s

00

] of body, then de f ault

P

(p[s]).

These predicates are concerned with a ﬁxed point

model of P, where they are made use of for analysis

of strategy construction with communication and are

related to strategy implementability.

Proposition 2. Assume a program P over the set

A. Let (I,J) be a pair deﬁned by the imple

P

and

de f ault

P

predicates in the manner:

I = {p[s] | imple

P

(p[s])} and

J = {p[s] | de f ault

P

(p[s])}.

Then Tr

P

(I, J) = (I,J), that is, (I, J) is a ﬁxed point

of Tr

P

.

Proof. Let Tr

P

(I, J) = (I

0

,J

0

).

(1) We prove by induction that (I

0

,J

0

) ⊆

c

(I, J), i.e.

I

0

⊆ I and J

0

⊆ J.

(i) If p[s] is in I

0

, then there is a rule body B p[s] such

that q[s

0

] is in I with commu(s

0

,s, q[s

0

]) for any q[s

0

] of

body, and r[s

00

] is in J with commu(s

00

,s, r[s

00

]) for any

∼

r[s

00

] of body. This reason is applied to the case

that body is the empty set. By deﬁnition of (I,J),

imple

P

(q[s

0

]), for q[s

0

] to be in I, and de f ault

P

(r[s

00

]),

for r[s

00

] to be in J. It follows from the inductive deﬁ-

nition that imple

P

(p[s]). Thus, p[s] is in I, i.e., I

0

⊆ I.

(ii) Assume that p[s] is in J

0

. Then, there are two

cases: • For any rule body B p[s], q[s

0

] is in J with

commu(s

0

,s, q[s

0

]) for some q[s

0

] of body, or r[s

00

]) is

in I with commu(s

00

,s, r[s

00

]) for some

∼

r[s

00

] of body.

That is, there is de f ault

P

(q[s

0

]), or imple

P

(r[s

00

]). It

follows that de f ault

P

(p[s]), i.e. p[s] is in J.

Distributed Strategies and Managements based on State Constraint Logic with Predicate for Communication

83

(This case contains the one that there is no rule of the

form bodyB p[s] without any rule of the form body

0

B

∼

p[s], where p[s] is in J.)

• For some rule body B

∼

p[s], q[s

0

] is in I with

commu(s

0

,s, q[s

0

]) for any q[s

0

] of body, and r[s

00

] is

in J with commu(s

00

,s, r[s

00

]) for any

∼

r[s

00

] of body.

That is, there are imple

P

(q[s

0

]), and de f ault

P

(r[s

00

]).

It follows that de f ault

P

(p[s]), i.e. p[s] is in J.

In both cases, if p[s] ∈ J

0

, then p[s] ∈ J. Thus

J

0

⊆ J.

(2) We prove that (I,J) ⊆

c

(I

0

,J

0

), i.e., I ⊆ I

0

and

J ⊆ J

0

.

(i) If p[s] is in I, then imple

P

(p[s]). Thus

there is a rule body B p[s] such that imple

P

(q[s

0

])

with comuu(s

0

,s, q[s

0

]) for any q[s

0

] of body, and

de f ault

P

(r[s

00

]) with commu(s

00

,s, r[s

00

]) for any

∼

r[s

00

] of body. By deﬁnition of (I,J), q[s

0

] is

in I with commu(s

0

,s, q[s

0

]), and r[s

00

] is in J with

commu(s

00

,s, r[s

00

]). By the deﬁnition of Tr

P

, p[s] is

in I

0

, obtained by applying of Tr

P

to the pair (I, J).

Thus, if p[s] ∈ I then p[s] ∈ I

0

(i.e. I ⊆ I

0

).

(ii) If p[s] is in J, then de f ault

P

(p[s]). There are two

cases:

• For any rule body B p[s] such that de f ault

P

(q[s

0

])

with commu(s

0

,s, q[s

0

]) for some q[s

0

] of body,

or imple

P

(r[s

00

]) with commu(s

00

,s, r[s

00

]) for some

∼

r[s

00

] of body. By the deﬁnition of (I,J), q[r

0

]

is in J with commu(s

0

,s, q[s

0

]), or r[s

00

] is in I with

commu(s

00

,s, r[s

00

]). By the deﬁnition of Tr

P

, p[s] is

in J

0

, obtained by applying of Tr

P

to the pair (I, J).

(This case contains the one that there is no rule of the

form bodyB p[s] without any rule of the form body

0

B

∼

p[s], where p[s] is in J

0

.)

• For some rule body B

∼

p[s] such that imple

P

(q[s

0

])

with commu(s

0

,s, q[s

0

]) for any q[s

0

] of body, and

de f ault

P

(r[s

00

]) with commu(s

00

,s, r[s

00

]) for any

∼

r[s

00

] of body. By the deﬁnition of (I,J), q[r

0

] is

in I with commu(s

0

,s, q[s

0

]), and r[s

00

] is in J with

commu(s

00

,s, r[s

00

]). By the deﬁnition of Tr

P

, p[s] is

in J

0

, obtained by applying of Tr

P

to the pair (I, J).

In both cases, if p[s] ∈ J then p[s] ∈ J

0

. Thus J ⊆

J

0

.

With Propositions 1 and 2, we can have the mean-

ing that the predicates may be related to a model of

the program.

Proposition 3. Assume that for the program P, a pair

(I, J) is deﬁned such that

I = {p[s] | imple

P

(p[s])}, and

J = {p[s] | de f ault

P

(p[s])}.

If I ∩ J =

/

0, then (I, J) is a model of P.

4.2 Application to Logical Database and

Related Works

By regarding strategies (called by name) as data, the

program P (in coordination of strategy with commu-

nication) describe as above is applicable to logical

database where rules are logically described:

(i) As regards negatives, prohibition (strong nega-

tion not) can be used, as well as default

∼

for

impermisisibility, in 3-valued domain.

(ii) The predicates imple

P

(-) and de f ault

P

(-) are

extended to the predicates for queries to logical

database.

Formally we have database Database with com-

munication Commu (of Section 3), in BNF:

Literal ::= p[s] |

∼

p[s] | not p[s]

Head ::= p[s] |

∼

p[s]

Body ::= { } | {Literal} ∪ Body

Rule ::= Body B Head

database ::=

/

0 | {Rule} ∪ database

Database ::= database ∪Commu

Note that Head does not contain strong negation

not, and that Condition for Strategy is also assumed

for database where the rules of Rule have restric-

tions such that at most one any complementary pair

(p[s],

∼

p[s]) occurs at heads.

About related works on logical frameworks pos-

sibly for cognitive managements and for defeasible

logic (Governatori et al., 2004), we have examined

concepts and ideas. On the one side, quantiﬁcations

for proposition variables are studied. On the other

hand, modal operators are invented with theoretical

basis and applicable aspects. They may be relevant to

ontology views on structural and knowledgeable anal-

yses of procedures and reasoning:

(i) Applying 3-valued models to the Heyting algebra,

we made the papers on modal mu-calculus, a lan-

guage system and reference data abstraction (S. Ya-

masaki et al. in COMPLEXIS 2020 and DATA 2020).

The algebraic expressions of those papers are evalu-

ated in 3-valued domain, to possibly represent an in-

ﬁnite conjunctive form of propositional logic, where

the form of algebraic expressions should supposedly

take conditions similar to Condition of this paper, and

the preﬁxpoint models must be a little more restricted

than those described there.

(ii) There is a paper presenting second-order propo-

sitional frameworks, with epistemic and intuitionistic

logic (P. Kremer, 2018). It may be relevant to the ex-

tension of this paper with logical expressions to be

quantiﬁed.

(iii) For an extension of propositional modal logic

without quantiﬁcation (whose transition system is

COMPLEXIS 2021 - 6th International Conference on Complexity, Future Information Systems and Risk

84

captured as abstract state machinery), the paper (Fit-

ting, 2002) introduces relations and terms with scop-

ing mechanism by lambda abstraction.

(iv) Based on beliefs and intentions, modal opera-

tions have been applied to mental states (Dragoni

et al., 1985). The paper (Beddor and Goldstein, 2018)

presents the belief predicate with the credence func-

tion of agents, concerning epistemic contradictions.

The contradictions of complexity may be avoided by

grades of such a function.

5 CONCLUSION

This paper reﬁnes 3-valued model theory of logical

expressions from structural aspects of strategic con-

structions. There is a complexity in that the state con-

straint strategies inductively form a strategy possibly

assigned to another state. Implementability of a strat-

egy is supposedly indebted to implementabilities of

the strategies as components to the primary strategy.

Non-implementability is denoted with default nega-

tion, as well as the undeﬁned for implementability

for strategies. The structure of strategy constructions

is so far examined with respect to implementability

evaluation. A communicative predicate on the set

of states is assumed in a simpler manner, such that

transferrability of distributed strategies may be vir-

tually supposed. The structure is regarded as rele-

vant to interests of cognitive management complex-

ity caused by combination of computing mechanism

with communication between distributed states. With

respect to structure of strategies for computing imple-

mentability backed by communication facilities, the

main results are listed up:

(i) A general form of logical expressions for strat-

egy constructions is formulated with distributions to

states, which may be also modeling of distributed log-

ical formulas with default negation.

(ii) 3-valued model theory of such expressions is

given in terms of ﬁxed point of the mapping (corre-

sponding to a transformation) associated with expres-

sions attached to states, with respect to predicates for

communication. The model denotes implementability

of constructed strategies.

(iii) From the predicate and derivation views, we can

have the implementable and default predicates for ex-

pression evaluations, in accordance with the success

and failure derivations for query in a framework of

logical expressions, both of which may be sound to a

ﬁxed point model. This is a kind of reasoning with

respect to a ﬁxed point model of logical expressions,

applicable to logical database.

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