Modal Mu-calculus Extension with Description of Autonomy and Its

Algebraic Structure

Susumu Yamasaki and Mariko Sasakura

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

Keywords:

Modal Logic, Fixed Point, Abstract State Machinery, Human Computer Interaction, Application to Autonomy.

Abstract:

This paper deals with complex abstract state machinery, clearly represented by modal logic with ﬁxed point

operator. The logic is well known as modal mu-calculus, which is extended to the version involving human

computer interaction as well as involving awareness, communication and behavioral predicates of propo-

sitional variables as in autonomy systems. The extended version contains complexity for human machine

interaction, whose meaning is represented by Heyting algebra but not by Boolean algebra. In the sense of

Heyting algebra, human computer interaction of complexity can be described such that related predicates of

communication and behavior may be simpliﬁed. Then the extended version can be applied to some process by

means of awareness to an expertise, communication and behavior processes, and repetitions represented with

ﬁxed point operator (that is, mu-operator). This version is also concerned with model theory caused by postﬁx

modal operator, where composition and alternation of modal operators may be organized into an algebraic

structure.

1 INTRODUCTION

The complex information system may be more intu-

itively expressed if abstract state machinery would be

taken into consideration on the assumption of the state

concept:

(1) In the literatures (Droste et al., 2009; Reps et al.,

2005), algebraic systems of abstract state machin-

ery are compiled, which are related to more ab-

stract structure of streams in the note (Rutten,

2001).

(2) On the state concept, the action causing state tran-

sitions are also signiﬁcant, whether or not it is ab-

stract or not. In the papers (Giordano et al., 2000;

Hanks and McDermott, 1987), actions are cap-

tured in logical systems, while the action is a key

issue in strategic reasoning.

(3) Relative to functional programs (Bertolissi et al.,

2006; Thompson, 1991), the actions are to be

viewed. In the book (Mosses, 1992), the proce-

dural action is expressed by its denotation.

(4) With composed actions and programs in dynamic

logic, we may see an application system of acting

and sensing failures, and actions to generate and

execute plans (Spalazzi and Traverso, 2000).

If we aim at the human computer interaction

(HCI) of complexity to the programming systems,

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

Merro and Nardelli, 2005) may be effective in the

sense that communication environments are well

described, and

(b) From views of AI system developments, we need

the bases of logics (Genesereth and Nilsson,

1987) as well as of knowledge (Reiter, 2001).

Based on the classics for beliefs and intentions,

modal operations concerning mental states are cap-

tured (Dragoni et al., 1985), which are applied to form

state sequences. The modal mu-calculus contains a

ﬁxed point notation to reﬂect some goal where actions

and communications are satisfactory for given condi-

tions. In the papers (Dam and Gurov, 2002; Kozen,

1983; Park, 1970), the proof systems with ﬁxed point

approximations are in details formulated.

In the context of AI programming system formu-

lations, we have in this paper an extension of modal

logic with ﬁxed point operator (Venema, 2006; Ven-

ema, 2008), where modal logic with ﬁxed point oper-

ator is settled with its transition system:

• To reﬂect abstract state machinery, and

• To condition the states statically containing pro-

grams and their implementations.

Yamasaki, S. and Sasakura, M.

Modal Mu-calculus Extension with Description of Autonomy and Its Algebraic Structure.

DOI: 10.5220/0009322300630071

In Proceedings of the 5th International Conference on Complexity, Future Information Systems and Risk (COMPLEXIS 2020), pages 63-71

ISBN: 978-989-758-427-5

Copyright

c

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

63

We here have the extended version of multi-modal

mu-calculus (Yamasaki and Sasakura, 2015) for hu-

man computer interaction, with respect to algebraic

meanings. It may contain a model of autonomy as re-

covery process, in terms of the meanings. The mean-

ing of extended logical formulas may realize HCI

such that its algebraic basis is given by Heyting al-

gebra (corresponding to intuitionistic logic).

In this paper, it should be applied to a design of

autonomy with HCI for some process, so that the de-

signed autonomy needs the predicates of awareness

(to expertise) as well as communication and behav-

ior implementing HCI. The autonomy system can be

applied to a practice for recovery from the worse con-

dition of patients. Before making the application in

details, this paper gives an outlook on the construc-

tion of the autonomy system which may provide a

sequence of behavioral and communicative actions.

The constructed system is initiated by awareness to

the expert knowledge, by which the system speciﬁ-

cation is made. Then the system does work, as hu-

man computer interaction, with behavior (displaying

such a sequence) and communication (responded by

the person). The system construction is regarded as

practise of the autonomy described by a formula of

the modal mu-calculus extension of this paper.

The paper is organized as follows. Section 2 gives

the full syntax of the modal mu-calculus extension

the primary part of which we originally present. The

meanings of logical formulas are deﬁned in Section

3, where HCI may be allowable. In Section 4, the au-

tonomy is considered from theoretical views with a

practise. In Section 5, the whole logic is viewed from

algebraic structures. Section 5 also deals with model

theories for the logical or algebraic expression (re-

garded as a program), represented for (a term) form-

ing a postﬁx modal operator. Section 6 concludes this

paper and refers to advanced theories.

2 MODAL MU-CALCULUS

EXTENSION

Not only for human computer interaction but also for

the autonomy system components of predicates as re-

gards awareness, communication and behavior, we

have more forms extended from modal mu-calculus

than our former version. That is, some predicates are

newly made use of:

(i) Aw(ϕ) as awareness to condition (the logical for-

mula) ϕ,

(ii) Be(ψ,ϕ,t) as behavior with a term t, for a rela-

tion between conditions ψ and ϕ to hold, and

(iii) Cm(ψ,ϕ, c) as communication with a communi-

cation c, for a relation between conditions ψ and

ϕ to hold.

The set Φ of (logical) formulas is deﬁned induc-

tively as follows.

ϕ ::= tt | p | ¬ϕ |

∼

ϕ | ϕ ∨ ϕ | µx.ϕ | hciϕ | ϕiti

| Aw(ϕ) | Be(ϕ,ϕ,t) | Cm(ϕ, ϕ,c)

Note that the intuitive meanings of symbols are

described as below, where the formal meanings are

given, in the next section, with the transition system

as below.

(i) tt is the truth, and p denotes propositions.

(ii) ∨ stands for the disjunction, and ¬ is the logical

negation.

(iii)

∼

is another negation as interactive incapabil-

ity.

(iv) µ is a least ﬁxed point operator.

(v) hci is a preﬁx modality with communication c.

(vi) iti is a postﬁx modality with term t.

(vii) Aw is an awareness operator.

(viii) Be is a behavior operator with respect to term

t.

(ix) Cm is a communicative operator with respect to

communication c.

A Transition System S :

For the set Φ of formulas, a transition system S is

deﬁned to be:

(S,C,U,Re,Rel,V

pos

,V

neg

,V

inter

,V

Aw

,r

Be

,r

Cm

)

where:

(i) S is a set of states.

(ii) C is a set of labels for communications.

(iii) U is a set of labels for terms.

(iv) Re maps to each c ∈ C a relation Re(c) on S.

(v) Rel maps to each t ∈ U a relation Rel(t) on S.

(vi) V

pos

,V

neg

,V

inter

: Prop → 2

S

map to each propo-

sition (variable) a set of states, respectively.

(vii) V

Aw

is a mapping of S to 2

Φ

.

(viii) r

Be

is a subset of Φ × Φ.

(ix) r

Cm

is a subset of Φ × Φ.

COMPLEXIS 2020 - 5th International Conference on Complexity, Future Information Systems and Risk

64

3 MEANING OF FORMULAS

FITTING HUMAN COMPUTER

INTERACTION

The meaning of a formula may be a subset of the state

set (in the transition system), as in Hennessy-Milner

Logic (HM-Logic). However, to represent the states

where human interaction may be made in computer

working process, we have classiﬁed the state set into

3 subsets (parts): The ﬁrst is to express the states

positive to computing, the second is to designate the

states for possible interaction with human, and the

third is to contain the states negative to computing.

In such a way, complexity is more than in HM-Logic,

Given a transition system S , the functions

[[ ]]

pos

,[[ ]]

neg

,[[ ]]

inter

: Φ → 2

S

are deﬁned as meanings of formulas such that:

(i) [[ϕ]]

pos

∪ [[ϕ]]

neg

∪ [[ϕ]]

inter

= S, and

(ii) [[ϕ]]

pos

, [[ϕ]]

neg

and [[ϕ]]

inter

are mutually disjoint,

for ϕ ∈ Φ.

Note that V

pos

(p) ∪ V

inter

(p) ∪ V

neg

(p) = S for

each proposition (variable) p.

(1) [[tt]]

pos

= S, [[tt]]

neg

=

/

0, and [[tt]]

inter

=

/

0.

(2) [[p]]

pos

= V

pos

(p), [[p]]

neg

= V

neg

(p),

and [[p]]

inter

= S\ ([[p]]

pos

∪ [[p]]

neg

)

(p ∈ Prop).

(3) [[¬ϕ]]

pos

= [[ϕ]]

neg

, [[¬ϕ]]

neg

= [[ϕ]]

pos

,

and [[¬ϕ]]

inter

= [[ϕ]]

inter

.

(4) [[

∼

ϕ]]

pos

= [[ϕ]]

neg

, [[

∼

ϕ]]

neg

= [[ϕ]]

pos

∪ [[ϕ]]

inter

,

and [[

∼

ϕ]]

inter

=

/

0.

(5) [[ϕ

1

∨ ϕ

2

]]

pos

= [[ϕ

1

]]

pos

∪ [[ϕ

2

]]

pos

,

[[ϕ

1

∨ ϕ

2

]]

neg

= [[ϕ

1

]]

neg

∩ [[ϕ

2

]]

neg

, and

[[ϕ

1

∨ ϕ

2

]]

inter

= S \ ([[ϕ

1

∨ ϕ

2

]]

pos

∪ [[ϕ

1

∨ ϕ

2

]]

neg

).

(6) [[hciϕ]]

pos

= {s ∈ S | ∃s

0

. s Re(c) s

0

and s

0

∈ [[ϕ]]

pos

},

[[hciϕ]]

neg

= {s ∈ S | ∀s

0

. s Re(c) s

0

entails s

0

∈ [[ϕ]]

neg

}, and

[[hciϕ]]

inter

= S \ ([[hciϕ]]

pos

∪ [[hciϕ]]

neg

).

(7) ([[µx.ϕ]]

pos

,[[µx.ϕ]]

neg

)

=

T

{(T

pos

,T

neg

) ⊆ S × S |

([[ϕ]]

pos [x:=T

pos

]

,[[ϕ]]

neg [x:=T

neg

]

) ⊆ (T

pos

,T

neg

)},

and [[µx.ϕ]]

inter

= S \ ([[µx.ϕ]]

pos

∪ [[µx.ϕ]]

neg

),

where every free occurrence of x in ϕ is positive

such that the occurence x is replaced by T

pos

and

T

neg

, respectively, and the operations

T

and ⊆ are

componentwise.

(8) [[ϕiti]]

pos

= {s

0

∈ S | ∀s. s Rel(t) s

0

entails s ∈ [[ϕ]]

pos

},

[[ϕiti]]

neg

= {s

0

∈ S | ∀s. s Rel(t) s

0

entails s ∈ [[ϕ]]

neg

},

and [[ϕiti]]

inter

= S \ ([[ϕiti]]

pos

∪ [[ϕiti]]

neg

).

(9) [[Aw(ϕ)]]

pos

= {s ∈ S | ϕ ∈ V

Aw

(s)},

[[Aw(ϕ)]]

neg

= S \ [[Aw(ϕ)]]

pos

, and

[[Aw(ϕ)]]

inter

=

/

0.

(10) [[Be(ψ,ϕ,t)]]

pos

= [[ϕiti]]

pos

if (ψ,ϕ) ∈ r

Be

, and

[[ϕ]]

pos

otherwise. [[Be(ψ, ϕ,t)]]

neg

= [[ϕiti)]]

neg

if (ψ,ϕ) ∈ r

Be

, and [[ϕ]]

neg

otherwise.

[[Be(ψ, ϕ,t)]]

inter

= S \ ([[Be(ψ,ϕ,t]]

pos

∪ [[(ψ, ϕ,t)]]

neg

).

(11) [[Cm(ψ,ϕ,c)]]

pos

= [[hciϕ]]

pos

if (ψ, ϕ) ∈ r

Cm

,

and [[ϕ]]

pos

otherwise. [[Cm(ψ, ϕ,t)]]

neg

=

[[ϕiti)]]

neg

if (ψ, ϕ) ∈ r

Cm

, and [[ϕ]]

neg

other-

seise.

[[Cm(ψ, ϕ,t)]]

inter

= S \ ([[Cm(ψ,ϕ,t]]

pos

∪ [[Cm(ψ,ϕ,t)]]

neg

).

Although the deﬁnitions are clear later in terms of

Heyting algebra (in Section 5), the conjunction ∧ and

the (Heyting algebra) implication −→ are intuitively

given as follows.

(i) ϕ

1

∧ ϕ

2

is deﬁned by ¬(¬ϕ

1

∨ ¬ϕ

2

),

(ii) ϕ

1

−→ ϕ

2

is regarded as “ϕ

1

implies ϕ

2

”.

We have seen term use for admissible interaction

by the deﬁnition for [[-]]

inter

, in terms of:

[[ϕiti]]

inter

= {s

0

∈ S|∃s. sRel(t)s

0

and s ∈ [[ϕ]]

inter

}.

Note the usage of mu-operator for description of

term use meaning, | iti |

inter

, with the ﬁxed point op-

erator. That is, we might have got:

| iti |

inter

= [[µp. piti]]

inter

.

4 AUTONOMY AND

APPLICATION TO RECOVERY

With the modal mu-calculus extension, we can have

a representation of an autonomy to be aware of ex-

pertise for behavior and communication with human,

as well as to implement human computer interaction

scheme designated by logical formulas.

Modal Mu-calculus Extension with Description of Autonomy and Its Algebraic Structure

65

4.1 Description of Autonomy

With awareness to an expertized condition for behav-

iors and communications with human (for HCI), the

logical conditions might be given by means of the fol-

lowing formula (with paraentheses and with a propo-

sitional variable q to the mu-operator).

µq.(((q −→ Aw(ψ)) −→ (Be(ψ,q,t) ∧Cm(ψ, q, c)))

∧(∨

i

(hc

i

iq ∧ qit

i

i)))

where

(1) “−→” is Heyting algebra implication, whose def-

inition is given in the next section but intuitively

considered as an implication, and

(2) the terms and communications are to be imple-

mented:

(a) t

i

and c

i

are terms and communications within

postﬁx and preﬁx modal operators, respec-

tively, virtually in human computer interaction.

(b) t and c are term (with respect to Be-formula)

and communication (with respect to Cm-

formula), respectively, with reference to an ex-

pertized condition by the formula ψ.

The mu-operator can be regarded as inductively

extending the states to which conditions are kept to

be satisﬁed, where the conditions (within the operator

scope) are speciﬁed to describe awareness to exper-

tise and repetitions of communication and behavior

implementations.

4.2 Application to Recovery Process

As an example of recovery process which autonomy

is applied to, the recovery of an aphasic patient is ex-

amined. It may be regarded as the one led by speech

therapist (ST), where ST presents a picture to a pa-

tient and encourages him or her to explain the content

of picture orally. Most of patients cannot explain the

picture because they do hardly ﬁnd proper words. We

may design such a system with views of autonomy as

follows, to cope with such difﬁculty: (i) (Expertise in-

put) ST speaks a sound sentence at ﬁrst with display

of a picture, and then a patient is encouraged to re-

peat the sentence. (ii) (Interaction model) While they

repeat ST’s machinery messages and responses of the

patient, the patient might possibly speak a satisfactory

sentence.

With respect to the formula of the previous sub-

section as seen for autonomy, and the implementation

need of the design by T. Kojima (2019), we may think

of ST’s speech as awareness of the autonomy system

to ST’s expertise, followed by (or implicating) behav-

ior predicate (by interaction of ST with machinery

message) and receipt of the patient with communica-

tive predicate (including no need of receipt response),

in terms of:

((q −→ Aw(ψ)) −→ (Be(ψ,q,t) ∧Cm(ψ,q, c))).

The repeated process of messages and responses

is represented, in accordance to the (sub)formula of

the whole formula:

∨

i

(hc

i

iq ∧ qit

i

i).

The mu-operator can be seen as denoting some

stable state set, with which the recovery process may

supposedly continue.

Speciﬁcation of Autonomy System:

The system (which we have implemented) pro-

vides a recovery process: The system shows pictures

according to ST’s interaction, with machinery mes-

sages. For instance, we can present a sentence as an

exercise: “a girl eats pasta with a fork”. The sentence

has three nouns and one verb. Since it may be difﬁ-

cult for a patient to speak the whole sentence at once,

we divide the sentence into three phrases of “a girl”,

“eats pasta”, “with a fork”. They are associated with

pictures, as well.

The autonomy system contains three parts: “pic-

ture part”, “sentence part” and “instruction part”.

• In the picture part, a picture is displayed.

• In the sentence part, the sentence phrase which

corresponds to a picture sequence is displayed.

• In the instruction part, the interactive instruction

of ST is displayed.

As below, there is a situation in which ST interac-

tive with a patient to speak the sentence “a girl eats

pasta”. The speciﬁcation (which is above given) may

be implemented as an autonomy system for ST by

the object-oriented programming. Each part is un-

der control of an object in a programming language,

where each object has a state, such that ST may tap

the screen for each object to transfer to the next state.

Picture Part Sentence Part

A picture of the girl (1) “A girl”

(2) “eats pasta”

Instruction Part

Please look at the picture. Make a sentence.

5 ALGEBRAIC STRUCTURE

We here have an algebraic structure of the domain

which can be composed with respect to the meanings

COMPLEXIS 2020 - 5th International Conference on Complexity, Future Information Systems and Risk

66

in the set of formulas. As a part of autonomy descrip-

tion, we examine the model theory of algebraic or log-

ical expressions as a “program”, which is described

for the term in postﬁx modality.

5.1 Meaning and Algebra

Complexity is caused by double negation, one

of which is classical and another of which is for

interaction incapability. With respect to interaction

capability, Heyting algebra (a bounded lattice with

a speciﬁed implication: See the book (Crole, 1993)

for it and category theories to semantics) is made use

of, for the whole set of formulas to be represented in

terms of algebraic structure.

The set Φ of formulas is related to a bounded lat-

tice:

([Φ],

W

,

V

,[ff],[tt]),

where:

(i) [Φ] = {[ϕ] | ϕ ∈ Φ} for [ϕ] = ([[ϕ]]

pos

,[[ϕ]]

neg

).

(ii)

W

and

V

denotes a join and a meet, respectively.

for which the partial order ≤ may be deﬁned on

the set [Φ] by means of:

[ϕ

1

] ≤ [ϕ

2

] iff

[[ϕ

1

]]

pos

⊆ [[ϕ

2

]]

pos

and [[ϕ

2

]]

neg

⊆ [[ϕ

1

]]

neg

.

(iii) [ff] = [¬ tt] = [

∼

tt] = (

/

0,S), and [tt] = (S,

/

0).

The implication ⇒ is equipped with, on the set

[Φ]:

[ϕ] ≤ [ϕ

1

] ⇒ [ϕ

2

] iff [ϕ

1

]

V

[ϕ] ≤ [ϕ

2

]

so that a Heyting algebra [Φ] is associated with the

set Φ of formulas.

We then have some properties derived to denote a

relation between the set Φ of formulas and the set [Φ]

as model base.

Proposition 1. (1) [ϕ

1

∨ ϕ

2

] = [ϕ

1

]

W

[ϕ

2

].

(2) [ϕ

1

∧ ϕ

2

] = [ϕ

1

]

V

[ϕ

2

], where

ϕ

1

∧ ϕ

2

is deﬁned as ¬ (¬ ϕ

1

∨ ¬ ϕ

2

).

(3) [

∼

ϕ] = [ϕ] ⇒ [ff], where

[

∼

ϕ] = ([[ϕ]]

neg

,S \ [[ϕ]]

neg

).

Proof. (1)

[ϕ

1

∨ ϕ

2

]

= ([[ϕ

1

∨ ϕ

2

]]

pos

,[[ϕ

1

∨ ϕ

2

]]

neg

)

= ([[ϕ

1

]]

pos

∪ [[ϕ

2

]]

pos

,[[ϕ

1

]]

neg

∩ [[ϕ

2

]]

neg

)

= [ϕ

1

]

W

[ϕ

2

].

(2) With respect to the negation ¬, we can see that:

[ϕ

1

∧ ϕ

2

]

= ([[ϕ

1

]] ∧ [[ϕ

2

]]

pos

,[[ϕ

1

∧ ϕ

2

]]

neg

)

= ([[ϕ

1

]]

pos

∩ [[ϕ

2

]]

pos

,[[ϕ

1

]]

neg

∪ [[ϕ

2

]]

neg

)

= [ϕ

1

]

V

[ϕ

2

].

(3) Since [ϕ] ⇒ [ff] is the greatest element [ψ] such

that [ϕ]

V

[ψ] ≤ [ff], and [ff] = (

/

0,S) over the state set

S, we have:

[

∼

ϕ] = ([[ϕ]]

neg

,S \ [[ϕ]]

pos

)

= [ϕ] ⇒ [ff].

We ﬁnally have a Heyting algebra implication on

the set Φ of formulas, with a correspondence to the

implication ⇒ on the set [Φ].

Deﬁnition 2. A binary operation “−→” (on the set Φ)

may be deﬁned such that:

[ϕ

1

−→ ϕ

2

] = [ϕ

1

] ⇒ [ϕ

2

].

5.2 Model of Term in Postﬁx Modality

To the term t in the postﬁx modality iti of such a

formula ϕiti, the relation Rel(t) is assigned. In this

subsection, we have a case that the term t might be

described by the propositional expression F over a

set A

F

(A, for short) of proposition letters, as below,

with a correspondence to the relation Rel(t) in the

transition system S :

Syntax Semantics

term t relation on state sets: Rel(t)

expression F model of expression

With reference to logical and algebraic expres-

sions as programs, we pay attention to the proposi-

tional expression F (over the set A of proposition let-

ters) is of the form:

∧

j∈ω

(l

j

1

∧ .. . ∧ l

j

n

j

→ l

j

)

where l

j

i

denote literals, that is, propositions or their

negations with not, and both the implication “→” and

the conjunction ∧ are interpreted with respect to Heyt-

ing algebra. Its 3-valued models are to be examined.

Note that the expression “1/2 → 1/2” is evaluated as

1. The expression F as a program is descriptive, con-

taining logical and algebraic properties.

Modal Mu-calculus Extension with Description of Autonomy and Its Algebraic Structure

67

5.2.1 Logic Program

The logic program with its Herbrand base is associ-

ated with the expression F as above, containining the

predicate pr (with or without “not” as a procedure)

followed by

pr

1

,. . . , pr

m

(as a procedural body)

for pr, pr

1

,. . . , pr

m

(predicates or their negations).

This is a different view on logic programs from

the one on answer set programming (Osorio et al.,

2004). The model of the expression F of the above

form is now discussed over the 3-valued domain.

3-Valued Model of Propositional Expression:

For the negation “not” in 3-valued domain

{0,1/2, 1} as de f ault negation, we assume, for the

proposition (letter) p, that:

p not p

1 0

1/2 1/2

0 1

Applicable Fixed Point as Model:

Extending the methods of M. Fitting (1985) and

A. van Gelder (1991), we may have models of least

ﬁxed points of mappings as follows, where the pair

(I, J) denotes the set I of propositions assigned to 1,

and the set J of propositions assigned to 0.

For the set A of proposition letters, a monotonic

mapping is extended to this case, on the basis of the

originally proposed mapping:

Φ

F

: 2

A

× 2

A

→ 2

A

× 2

A

,

Φ

F

(I

1

,J

1

) = (I

2

,J

2

),

is deﬁned.

I

2

=

{p | ∃(p

1

∧ .. . ∧ p

i

∧ not p

i+1

∧ .. . ∧ not p

j

→ p)

in F. ∀p

k

(1 ≤ k ≤ i). p

k

∈ I

1

, and

∀p

k

0

(i + 1 ≤ k

0

≤ j)}. p

k

0

∈ J

1

},

J

2

=

{q | ∀(q

1

∧ . . . ∧ q

i

∧ not q

i+1

∧ . . . ∧ not q

j

→ q)

in F. ∃q

k

(1 ≤ k ≤ i). q

k

∈ J

1

, or

∃q

k

0

(i + 1 ≤ k

0

≤ j). q

k

0

∈ I

1

, or

∃(q

1

∧ .. . ∧ q

i

∧ not q

i+1

∧ .. . ∧ not q

j

→ not q)

in F. ∀q

k

(1 ≤ k ≤ i). q

k

∈ I

1

, and

∀q

k

0

(i + 1 ≤ k

0

≤ j). q

k

0

∈ J

1

}.

(Extended version of the method by M. Fitting)

The least ﬁxed point of Φ

F

, that is, the pair (I,J)

is obtained such that, with componentwise subset in-

clusion order ⊆

c

,

Φ

F

(I, J) = (I, J),and

if Φ

F

(I

0

,J

0

) = (I

0

,J

0

) then (I,J) ⊆

c

(I

0

,J

0

).

The least ﬁxed point (I,J) of Φ

F

may be a model of

F, if it is consistent.

(Note: The pair (I,J) is said to be consistent, if I ∩ J

=

/

0.)

For the set A, a monotonic mapping is extended on

the basis of the original mapping:

Π

F

: 2

A

× 2

A

→ 2

A

× 2

A

,

Π

F

(I

1

,J

1

) = (I

2

,J

2

),

is deﬁned:

I

2

=

{p | ∃(p

1

∧ .. . ∧ p

i

∧ not p

i+1

∧ .. . ∧ not p

j

→ p)

in F. ∀p

k

(1 ≤ k ≤ i). p

k

∈ I

1

, and

∀p

k

0

(i + 1 ≤ k

0

≤ j). p

k

0

∈ J

1

},

J

2

= GU

F

(I

1

,J

1

)

(With (I,J), GU

F

(I, J) is the greatest unfounded set

un f ounded

F

(I, J) inductively deﬁned as follows).

q ∈ un f ounded

F

(I, J)

↔ ∀(q

1

∧ . . . ∧ q

i

∧ not q

i+1

∧ .. . ∧ not q

j

→ q) in F.

∃q

k

(1 ≤ k ≤ i). q

k

∈ J ∪ un f ounded

F

(I, J), or

∃q

k

0

(i + 1 ≤ k

0

≤ j). q

k

0

∈ I, or

∃(q

1

∧ .. . ∧ q

i

∧ not q

i+1

∧ .. . ∧ not q

j

→ not q)

in F. ∀q

k

(1 ≤ k ≤ i). q

k

∈ I, and

∀q

k

0

(i + 1 ≤ k

0

≤ j). q

k

0

∈ J ∪ un f ounded

F

(I, J).

(Extension of the method by A.van Gelder et al.)

The least ﬁxed point of Π

F

is obtained as a model

of F, if it is consistent.

5.2.2 Algebraic Expression

In a Heyting algebra (HA) (A,

W

,

V

,⊥, >) equipped

with the partial order v and an implication ⇒:

Any expression E derives some expression F of the

“form”

V

j

(x

j

1

V

.. .

V

x

j

n

j

⇒ y

j

),

where x

j

i

and y

j

are an expression a or not a (denoting

a ⇒ ⊥), for a ∈ A, such that

F v E.

We here have a procedure to get models of a given

Heyting algebra expression F. For the negation “not”

in 3-valued domain {0,1/2, 1}, we assume, for the

algebraic element a, that:

a not a

1 0

1/2 0

0 1

COMPLEXIS 2020 - 5th International Conference on Complexity, Future Information Systems and Risk

68

As a procedural way, a model construction (for the

expression F of the form) is presented.

Procedure of 3-Valued Model Construction:

(a) Assume a pair (I,J) ∈ 2

A

× 2

A

as an input.

(b) If the part x

j

1

V

.. .

V

x

j

n

j

⇒ y

j

contains not a for a

∈ J in the left side of the implication, then remove

it from the part.

(c) If the part x

j

1

V

.. .

V

x

j

n

j

⇒ y

j

contains not a for

a 6∈ J in the left side of the implication, then

this part is replaced by 1 (the greatest element of

{0,1/2, 1}).

(d) Find a model (I

0

,J

0

) for the expression obtained

by repeating procedure applications of (b) and (c)

(until no more procedure can be applied).

(e) If (I,J) ⊆ (I

0

,J

0

) by pointwise (componentwise),

get (I

0

,J

0

) successfully as a return. Otherwise, go

back to the ﬁrst item (a), or halt in failure.

Proposition 3. If the pair (I

0

,J

0

) is successfully got

in the Procedure of Model Construction and I

0

∩ J

0

=

/

0, then it is a 3-valued model of the given expression.

Proof. Observing the Procedure with a pair (I,J), we

may see that:

(i) By the routine (b), the expression not a (for a ∈ J

may be evaluated as 1, such that the part may be

reduced to the (sub)expression without not a.

(ii) By the routine (c), the part may be evaluated as

1, such that the part may be removed from the

scope of the whole meet

V

j

.

(iii) By the routine (d), a model (I

0

,J

0

) may be ob-

tained without any element not a in the left side

of any part (with an implication), because the re-

peated routines with (b) or (c) may remove the

forms not a.

(iv) If (I,J) ⊆ (I

0

,J

0

) componetwise, any part (with

an implication) within the scope of the whole

meet

V

j

may be settled as 1 by the pair (I

0

,J

0

),

since the right side of any such part contains y

j

of the form b or not b (for b ∈ A).

6 CONCLUSION

With respect to abstract state machinery, human com-

puter interaction (HCI) is included in the modal mu-

calculus extension of the paper such that

(i) The meanings of formulas as conditions to the

state set may be more complex, but

(ii) The meaning may be clearer on the basis of

Heyting algebra.

This version has got reﬁnements of postﬁx modal op-

erator from algebraic senses. Model theories are orig-

inally constructed, in case that the postﬁx modal oper-

ator contains programs based on logical or algebraic

expressions. A semiring structure is viewed from the

point that the models of programs may cause state

transitions in abstract state machine, which is given

by an explicit nondeterminism description expanded

from the way (Yamasaki, 2017).

Some remarks are summarized:

(i) With the version of this paper, which contains

new predicates of awareness, communication

and behavior, an autonomy may be designed in

addition to mu-operator (least ﬁxed point opera-

tor).

(ii) With the autonomy design, we may have a re-

covery system for human to practice.

(iii) A complexity of HCI is now relaxed by the de-

scription of meanings of formulas conditioning

HCI, such that the description of meanings may

be related to Heyting algebra.

(iv) Model theories for programming within postﬁx

modal operator may be described on the basis of

logical and algebraic methods.

As regards further reﬁnements of this extended

version of modal mu-calculus as a logical framework,

• More complex human computer interaction with

some concept more cognitive, and

• More sophisticated “awareness”

should be examined.

From views of a logical framework, it should be

noted that the modal mu-calculus extension of the pa-

per contains

• the second-order propositions with ﬁxed point op-

erator, and

• the predicates of awareness, communication and

behavior to include formulas as arguments.

As advanced works, we should have references to:

(i) Regarding the second-order (quantiﬁed) propo-

sitions, the paper (Goranko and A., 2018) treats

concepts of (in)dependence functions.

(ii) The paper concerned with epistemic and intu-

tionistic logic is to be viewed.

(iii) With respect to quantiﬁed variables by means of

quantiﬁes ranging over the set of agents, “dis-

tributed knowledge” is discussed (Naumov and

Tao, 2019).

Modal Mu-calculus Extension with Description of Autonomy and Its Algebraic Structure

69

(iv) For an extension of propositional modal logic

without quantiﬁcation, the paper (Fitting, 2002)

introduces relations and terms with scoping

mechanism by lambda abstraction.

(v) Concerning the second-order predicates, the pa-

per (Kooi, 2016) treats the concept of knowing,

which is more complex than the autonomy with

awareness to be designed in this paper.

(vi) As regards epistemic contradictions, the pa-

per (Beddor and Goldstein, 2018) presents the

belief predicate with the credence function of

agents, which is, from the epistemic view, much

more complex than the awareness predicate for

autonomy system of this paper.

With respect to communication technology

(Kowalski and Toni, 1996),

(i) Argumentation was, in terms of non-classical

negation, formulated for lawful affairs, and

(ii) Abstract attack and defense are the argumenta-

tion concepts to have been used rather than com-

munications for recovery processes,

while HCI may be captured in argumentation and de-

bate theories for us to design recovery process of suc-

cess and failure examinations.

From model theoretic views, it is notable that the

argumentation model may be expressed by means of

3-valued logic. The 3-valued model of Heyting alge-

bra expressions discussed in this paper is related to

the semantics for defeasible reasonings able to imple-

ment argumentation (Governatori et al., 2004):

(i) Defeasibility is beforehand assumed in the given

rules, to be more complex, and

(ii) The plain program consisting of rules or Heyting

algebra expressions is simpler in the sense that

propagation of ambiguity (caused by contradic-

tory predicates) must be well reasoned or ruled

out for its blocking.

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