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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