Multi-modal Mu-calculus Semantics for Knowledge Construction

Susumu Yamasaki

and Mariko Sasakura

HCI Group, Okayama University, Okayama, Japan

Department of Computer Science, Okayama University, Okayama, Japan

Keywords:

Multi-modal Logic Semantics, Action Logic, Knowledge Construction.

Abstract:

This position paper aims at setting a new semantics for multi-modal mu-calculus to represent interactive states

where abstract actions may be applied to. A least ﬁxed point formula may be available to denote states allowing

interaction. A simple algebraic representation for interactive states can be deﬁnable. For communication

between human and machinery, a modality is reserved. In applicative task domains, knowledge construction is

focused on with respect to interactive action applications through communications. Panel touch behaviour on

iDevice as practice, URL references as functions and grammatical rule applications for sequential effects are

studied, as knowledge construction technologies. These views coherent with abstract state machine are ﬁnally

related to recent trends as semiring in algebraic structure and coalgebra for streams as sequential knowledge

structures. A reﬁnement of interactive techniques is positioned into a formal approach to multi-modal logic,

applicable to some practices.

1 INTRODUCTION

This positioning is motivated by an intention to

present the uniﬁed machinery framework of action in

knowledge construction and interactive communica-

tion with human ideas, for a human machine interac-

tion as illustrated below, where the environments of

human and machinery are virtually regarded as states.

Human Machinery

Communications ⇒ Reasoning

Cognition ⇐ Actions

As actions of both artistic and technological meth-

ods with respect to knowledge engineering in inter-

active artiﬁcial intelligence, this positioning supposes

working (action as reasoning) in (i) design of paper

folding to make some forms, (ii) knowledge acquisi-

tion by references to URLs, and (iii) grammatical rule

application for language learnings, as case studies.

As the book (Jackson, 1989) describes, the art of

paper folding is rich enough in terms of simple and

beautiful fascinations. Anyone can do anywhere, any-

time by means of papers which are also attractive in

practices as well as ﬁne displays. With respect to in-

teractive computing and convenience, iDevice panel

touch, as action, may be interesting for the art. Com-

pared with the paper as a medium, 2D panel touch

is simpler even for knowledge construction to the 3D

form made by paper medium. However, simplicity of

panel touch may cause difﬁculty in graphical visual-

izations. This is regarded as trade off for simplicity

and compactness automated by modeling and mecha-

nized panel touch. This positioning aims at design for

implementation methods and tools as reasoning as-

pects, in respect to machanized action and interaction

with human.

As regards URLs, it may involve knowledge con-

struction by means of location references such that

acquisition of knowledge can be implemented as ac-

tions to have insights into contexts. Observing and

enjoying knowledge construction can be interactive to

human behaviours with automated eInfrastructure.

Concerning language learning, grammatical rule

applications are respected as in case of recovery from

language incapability written in the book (Chapey,

2001). The cognitive process of clients often needs

interactions to other human, whose work may be par-

tially realized by machine intelligences. To recover

language capability or to learn more, the cognitive

process must be supported with respect to and con-

sistently by formally grammatical rules.

For a method of unifying machinery with ac-

tions and interaction to human communication, we

reﬁne multi-modal logic as representation of moni-

358

Yamasaki, S. and Sasakura, M..

Multi-modal Mu-calculus Semantics for Knowledge Construction.

In Proceedings of the 7th International Joint Conference on Knowledge Discovery, Knowledge Engineering and Knowledge Management (IC3K 2015) - Volume 2: KEOD, pages 358-363

ISBN: 978-989-758-158-8

toring states (implementable environments). Seman-

tics is newly reﬁned with ideas from Hennesy-Milner

Logic denotation to states and Keene-Kripke model

for modality as well as ﬁxed point theory. By means

of states, interaction can be made and programmed

actions are implemented. This is a primary purpose

of this positioning. As another side, state-constraint

mechanism is viewed from the concept like abstract

state machine originating from the works of Gure-

vich,Y. As reﬁned techniques, we classify an action

into a postﬁx-modality following a formula, where in-

teractions are captured with communication modality

(of preﬁx-modality preceding a formula). Thus in-

teraction is acknowledged even in the formulation of

logic. We then consider a sequence of actions. It is

related to the sequential process modeled in modal

logic. It is also relevant to semiring structure with re-

spect to state-transitions in the handbook and papers

(Droste et al., 2009; Reps et al., 2005).

The positioning is organized as follows. Section

2 is to formulate multi-modal mu-calculus with se-

mantics regarding interaction for monitoring the state

sets. It involves an interactive state-constraint imple-

mentation for actions. A ﬁxed point formula may

be made use of in representations of actions at in-

teractive states. Section 3 presents some working

in progress, practical or theoretical. This section

contains (i) realization of folding paper by iDevice,

as a state-constraint implementation, (ii) acquisition

through URL references, as functions, and (iii) gram-

matical rule sequence with state-constraints. Section

4 gives some remarks regarding formal model of this

positioning.

2 MULTI-MODAL

MU-CALCULUS FOR

STATE-CONSTRAINT

We make use of actions (Mosses, 1992), Hennessy-

Milner logic with reference to the papers (Cardelli

and Gordon, 2000; Merro and Nardelli, 2005; Hen-

nessy and Milner, 1985; Milner, 1999) and multi-

modal µ-calculus for modeling of state constraint im-

plementation. With reference to multi-process moni-

toring states regarding negations “

∼

vs. ¬”, we have

the set Φ of formulas deﬁned by:

ϕ ::= tt | p | ¬ϕ |

∼

ϕ | ϕ∨ ϕ | hciϕ | µx.ϕ | ϕiai

We here have a (standard) preﬁx modality hci (for

communication), a postﬁx one iai (for action), and

a negation (sign)

∼

as regards incapability of interac-

tion, in addition to standard propositions p, the logical

negation ¬ and a least ﬁxed point operator µ.

The semantics for formulas are deﬁnable on the

basis of a transition system, which is modiﬁed and ex-

tended for denoting “interaction” states where some

implementation and action are available. The transi-

tion system is

S = (S,C, Ac,Re,Rel,V

pos

neg

inter

where:

(i) S is a set of states.

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

(iii) Ac is a set of actions.

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

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

(vi) V

pos

neg

inter

: Prop → 2

, map to each propo-

sition (variable) a set of states, respectively.

The reason why 3 assignments of V

pos

, V

neg

and V

inter

are adopted comes from a motivation to introduce an

assignment for monitoring interaction and the exis-

tence of negation “¬”. Given a transition system S ,

the functions [[ ]]

pos

,[[ ]]

neg

,[[ ]]

inter

: Φ → 2

are de-

ﬁned 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, while [[

∼

ϕ]]

inter

to demonstrate that the formula (process) denotation,

the state set for interaction, is empty.

Meaning concerned with two modalities hci, iai:

(1) [[tt]]

pos

= S, [[tt]]

neg

0, and [[tt]]

inter

(2) [[p]]

pos

= V

pos

(p), [[p]]

neg

= V

neg

(p), and

[[p]]

inter

= V

inter

(p) = S\ ([[p]]

pos

∪ [[p]]

neg

(3) [[¬ϕ]]

pos

= [[ϕ]]

neg

, [[¬ϕ]]

neg

= [[ϕ]]

pos

and [[¬ϕ]]

inter

= [[ϕ]]

inter

(4) [[

∼

ϕ]]

pos

= [[ϕ]]

neg

, and

[[

∼

ϕ]]

neg

= [[ϕ]]

pos

∪ [[ϕ]]

inter

([[

∼

ϕ]]

inter

0).

(5) [[ϕ

∨ ϕ

]]

pos

= [[ϕ

]]

pos

∪ [[ϕ

]]

pos

[[ϕ

∨ ϕ

]]

neg

= [[ϕ

]]

neg

∩ [[ϕ

]]

neg

, and

[[ϕ

∨ ϕ

]]

inter

= S \ ([[ϕ

∨ ϕ

]]

pos

∪ [[ϕ

∨ ϕ

]]

neg

(6) [[hciϕ]]

pos

= {s ∈ S | ∃s

′

. s Re(c) s

′

∧ s

′

∈ [[ϕ]]

pos

[[hciϕ]]

neg

= {s ∈ S | ∀s

′

. s Re(c) s

′

⇒ s

′

∈ [[ϕ]]

neg

and [[hciϕ]]

inter

= S \ ([[hciϕ]]

pos

∪ [[hciϕ]]

neg

Multi-modal Mu-calculus Semantics for Knowledge Construction

359

(7) ([[µx.ϕ]]

pos

,[[µx.ϕ]]

neg

)

{(T

pos

neg

) ⊆ S × S |

([[ϕ]]

pos [x:=T

pos

]

,[[ϕ]]

neg [x:=T

neg

]

) ⊆ (T

pos

neg

)},

and [[µx.ϕ]]

inter

= S \ ([[µx.ϕ]]

pos

∪ [[µx.ϕ]]

neg

where every free occurrence of x in ϕ is positive.

(8) [[ϕiai]]

pos

= {s

′

∈ S | ∀s. s Rel(a)s

′

⇒ s ∈ [[ϕ]]

pos

[[ϕiai]]

neg

= {s

′

∈ S | ∀s. s Rel(a) s

′

⇒ s ∈ [[ϕ]]

neg

[[ϕiai)]]

inter

= S\ ([[ϕiai]]

pos

∪ [[ϕiai]]

neg

(Note) Implementation sense of modality hci is

from communication labelled by c, like the standard

modality. Modality iai possibly comes from actions

which machinery virtually causes. When it is applied

to a state s, it conceives a relation Rel(a).

By the deﬁnition for [[-]]

inter

to be concerned with

denoting admissible interaction, we can see that:

[[ϕiai]]

inter

= {s

′

∈ S|∃s. sRel(a)s

′

∧ s ∈ [[ϕ]]

inter

Some algebraic treatments are available with re-

spect to denoting interactive states.

(A1) We have Heyting algebra

H = ({0,1/2,1},≤,

,0,1,−→),

where 0 ≤ 1/2 ≤ 1, and the expression a −→ b de-

notes a greatest element c of {0,1/2, 1} such that

c ≤ b. We have a semantic function Mon, to see

whether a process (supported by a formula) on a state

is legal (by value 1/2) for interaction:

Mon : Φ → S → {0,1/2,1},

Mon[[ϕ]]s =







1 s ∈ [[ϕ]]

pos

1/2 s ∈ [[ϕ]]

inter

0 s ∈ [[ϕ]]

neg

Given a transition system S and H for monitoring,

with ϕ ∈ Φ and s ∈ S:

(i) Mon[[¬ϕ]]s = if s ∈ [[ϕ]]

inter

then Mon[[ϕ]]s

else Mon[[ϕ]]s −→ 0.

(ii) Mon[[

∼

ϕ]]s = Mon[[ϕ]]s −→ 0.

(iii) Mon[[

∼

¬ϕ]]s ≤ Mon[[¬

∼

ϕ]]s.

(iv) Mon[[ϕ

∨ ϕ

]]s = Mon[[ϕ

]]s

Mon[[ϕ

]]s.

(v) If s Re(c) s

′

, and Mon[[hciϕ]]s = 1/2, then

Mon[[ϕ]]s

′

= 1/2.

(vi) If s Rel(a) s

′

, and Mon[[ϕ]]s = 1/2, then

Mon[[ϕiai]] s

′

= 1/2.

(A2) A ﬁxed point formula is applicable to the modal-

ity denotations. The meaning of formulas ϕiai con-

tains the states, to which the states supported by the

formula (process) ϕ might transit. With the ﬁxed

point formula,

| iai |

inter

= [[µp. piai]]

inter

may be regarded as meanings with the functions a

within modalities, respectively, for interaction.

3 KNOWLEDGE

CONSTRUCTION

The formula, say ϕ, is considered as a process (gov-

erning the states), abstracted from an interaction

scheme and cognition as follows.

As suggested later, the panel touch in iDevice is

represented by action in modality iai. As regards the

cognition of concepts with references, technologies

of the internet URLs are available such that modality

iai can be applied. As to learning grammatical rules,

modality iai may be adopted. The modalities are to

be conveniently placed as followers, because they are

concerned with the roles (effects) of actions in formal

reasoning:

Interaction Scheme

Interaction with communication (C) and action (A)

between human (H) and machinery (M) as artiﬁcial

intelligence:

→ M : Communications

hciϕ ϕ : Supporting formulas

→ H : Actions

ϕ ϕiai : Presentations of actions

Cognition

Human (H) cognition of action (A):

→ H

′

(advanced H) : Cognition

ϕ ϕiai : Acquiring actions

3.1 Interactive Paper Folding

Folding Model

We assume some points for an art of folding pa-

per (origami) to be virtually mechanized or imple-

mentable by iDevice, while folding is an action in

modal operator at states:

• An origami contains a set of faces.

• A face is an area of no thicknesss, enclosed with

edges. A face is, for iDevice techniques, restricted

to a triangle of 3 edges and 3 vertexes, while the

initial sheet paper is supposedly regarded as con-

taining 2 triangles.

• A crease line is an edge adjacent to 2 faces.

• An edge is a line segment ended by 2 vertexes.

KEOD 2015 - 7th International Conference on Knowledge Engineering and Ontology Development

360

Making origami is (i) to specify a crease line, and

(ii) to fold 2 adjacent faces with an angle between 0

and 180 degrees.

Valley and Mountain Foldings

A primitive but so fundamental folding is to specify a

crease line, folding 2 faces like a valley or a mountain,

called valley folding (V-fold) or mountain folding (M-

fold), respectively.

As implementations of V-fold and M-fold on iDe-

vices (virtually with correspondence to communica-

tion and action modal operators), we have 2 alterna-

tive methods below. Each of them is interpreted as

an abstract action constrained by a state modeling the

modal logic formula in the previous section, and as

abstractly causing a transition to the next state after

V-fold or M-fold virtually results.

(I1) It is to determine the positions of 2 vertexes:

A panel touch with long-press from the ﬁrst

point and with pan (drag) to another suggests

a line segment. For the suggestive line seg-

ment, a perpendicular is provided as an effective

crease line, at already-decided position crossing

the given (line segment), so that V-fold or M-

fold might be implemented.

(I2) It is to directly by touch specify a crease line: By

an operation pinch in/out, a crease line is pro-

vided between the suggested points so that ﬂick

operation might be effective for either face con-

cerning the crease line to be folded.

They are basic tools for a more reﬁned visualisa-

tion (Sasakura et al., 2013).

Sequential Foldings

As a sequence formation for making (ﬂying) plane,

we can now have a sequence of folding by recursion

(cycle):

(i) At a state of communicating (C), machinery sees

by interaction where machinery makes folding

and how it does, and transit to the next concern-

ing state: This is regarded as monitored by a for-

mula of the form hciϕ with interaction admis-

sion, in which communication c virtually repre-

sents an interaction.

(ii) At the next state of reasoning (R), machinery

makes an implementation of iDevice for folding

determined in the previous step on the point and

by the method: This is regarded as monitored by

a formula of the form ϕ with legal interaction ad-

mission.

(iii) We then see that a folding (F) is made by the pre-

vious implementation: This is regarded as mon-

itored by a formula of the form ϕiai with inter-

action admission.

The cycle is regarded as monitoring realized by for-

mula denotations in the following manner, where we

take abbreviations for C, R and F:

Monitoring: hciϕ ϕ ϕiai

| | |

Interaction for: C R F

In a more concretized folding to an airplane, a

whole sequence, virtually causing state-transitions for

implementation, is given with an interactive commu-

nication at each state (step):

(i) V-fold, to make the rightmost and uppermostpo-

sition set into the centre.

(ii) V-fold, to make the leftmost and uppermost po-

sition set into the centre.

(iii) V-fold, to operate by a central and vertical crease

line.

(iv) V-fold, to operate for one out of 2 faces (made in

step (iii)) by a crease line parallel to the crease

line of step (iii), with an angle of 90 degrees

open to the outer side.

(v) V-fold, to operate for another out of 2 faces

(made in step (iii)) by a crease line parallel to

the crease line of step (iii), with an angle of 90

degrees open to the outer side.

A visualization of a simple plane may be de-

signed, on the basis of the above action sequence.

3.2 Reference Recursion

As URL structures, a referenced page (named x) may

contain some senses recursively linked with other ref-

erences, as a function x 7→ (y

,.. . ,y

) (n ≥ 0) with

references x, and y

, ..., y

Let X be a set of references. With respect to

modality iai, u ≡ u

is deﬁnable as a function, u :

X → 2

. The function ˆu : 2

→ 2

is deﬁned for the

function u by: ˆu(Y) = ∪

x∈Y

u(x). Note for any ˆu that

ˆu(

0) =

0. For the function 1

: X → 2

, 1

(x) = {x},

the function

(Y) = Y, the identity on 2

Let U

be a set of functions of 2

to 2

such that

0) =

0 for f ∈ U

. The composition of f and g in

, g◦ f : 2

→ 2

is (g◦ f)(Y) = g( f(Y)). Then, the

composition is associative. With respect to the iden-

tity element

, f ◦

◦ f = f.

Then hU

,◦,

i is a monoid (semigroup with

identity).

On the other hand to the operation ◦, the alterna-

tion + is considerable. f + g : 2

→ 2

is deﬁned to

be ( f + g)(X) = f (X) ∪ g(X). It is seen that the op-

eration + is commutative and associative. With the

function 0

: X → 2

such that 0

(x) =

(Y) =

Multi-modal Mu-calculus Semantics for Knowledge Construction

361

Then, as the identity element

, f +

+ f = f.

It is clear from the deﬁnition of “+” that f + f = f

holds (idempotence).

As above mentioned, hU

,+,

i is a commuta-

tive monoid. As regards the composition, we can see

for any f ∈ U

that

◦ f = f ◦

, because

(Y) =

0. By properties of the operations + and ◦,

on the set U

, we have:

Proposition 1. hU

,+, ◦,

i is an (idempotent)

semiring.

Proof. It remains to see distributive laws, which hold

with the following reasons:

(g+ h) ◦ f(Y) = g( f(Y)) ∪ h( f(Y))

= (g◦ f + h◦ f)(Y)

f ◦ (g+ h)(Y) = f(g(Y) ∪ h(Y))

= ( f ◦ g+ f ◦ h)(Y)

Q.E.D.

We now examine the property of compositional

sequences of functions of the form ˆu.

Deﬁnition 2. The composition of ˆu

, ˆu

,.. . , ˆu

n−1

and

ˆu

is successful for x ∈ X if ˆu

◦ . .. ◦ ˆu

({x}) =

Let U

(⊆ U

) be a set of functions of the form

ˆu,

. To provide a composition (sequence)

σ of the functions from U

, successful for a given

x ∈ X, we have a recursive procedure Pro(x) as fol-

lows, where ⊥ stands for a failure: ⊥+σ = σ+⊥ = σ

and ⊥ ◦ σ = σ◦ ⊥ = ⊥ for any ﬁnite sequence σ con-

structed by the functions from U

Procedure Pro:

Pro(x,U

) ⇐ if U

0 then ⊥

else +

{ ˆu∈U

}

Check(x,u)

Check(x,u) ⇐ if u(x) =

0 then ˆu

else ◦

{y∈u(x)}

Pro(y,U

− { ˆu}) ◦ ˆu

Proposition 3. On the basis of the procedure Pro for

a given x ∈ X, Pro(x) contains a non-⊥ sequence iff

some sequence from Pro(x) is successful for x.

Proof. (1) If Pro(x,U

) contains a sequence success-

ful for x, then it must be a non-⊥ sequence, by the

construction of the procedure Pro with Check.

(2) Assume that Pro(x,U

) contains a non-⊥ se-

quence. Induction is made on recursion included

in Pro with Check: (i) If some u exists such that

u(x) =

0, then Pro(x,U

) contains ˆu, successful for

(ii) If u(x) 6=

0, suppose each procedure Pro(y,U

−

{ ˆu}) for y ∈ u(x), with the preceding function ˆu, in

Check(x,u). By the procedure Pro(y,U

− { ˆu}) to

(by induction hypothesis) contain a sequence success-

ful for y, and by distributive laws of ◦ over +, there

may be a sequence from U

, beginning with ˆu (as

in Check(x,u)), successful for the given x. This con-

cludes the induction step. Q.E.D.

3.3 Rewriting Rules

Learning the rules r and r

′

, the human’s state may be

transited from s to s

′

, which can be mechanized in

rewritings with states:

state state

→ s

′

→

To intuitively see such a structure, we now have

a function sequence virtually with reference to states.

Let Nt and Σ be a set of nonterminals and a set of ter-

minals, respectively. With respect to modality iai,

a rule r regarded as an action a is deﬁned to be a

function r : Nt ∪ Σ → (Nt ∪ Σ)

∗

such that (Nt ∪ Σ)

∗

is the set of all ﬁnite sequences, formed from the set

Nt ∪Σ, containing the nil sequence nil, and r(t) = t for

any t ∈ Σ. The function r can be extended to the one

¯r : (Nt ∪ Σ)

∗

→ (Nt ∪Σ)

∗

as deﬁned to be

¯r(nil) = nil, and

¯r(z) = cons(r(head(z)), ¯r(tail(z))),

where (i) head takes the ﬁrst symbol from a given

non-nil sequence, (ii) tail is a sequence constructed

by cutting off the ﬁrst symbol for a given sequence,

and (iii) cons is an operation to get a sequence by

combining a symbol with a sequence.

The composition of ¯r

and ¯r

can be deﬁned to be

(¯r

• ¯r

)(z) = ¯r

(¯r

(z))

with the identity function

1(z) = z for any z ∈ (Nt ∪

Σ)

∗

. As regards the composition, the associative laws

holds. Now let

= { fn | fn is a function of (Nt ∪Σ)

∗

→ (Nt ∪Σ)

∗

Then hV

,•,

1i is a monoid.

Given a nonterminal m ∈ Nt, whether there is a

sequence ¯r

, ..., ¯r

such that (¯r

• ... • ¯r

)(m) ∈ Σ

∗

can be decided, if the set Nt is ﬁnite: Neglecting the

elements of Σ, at least a similar procedure like the

procedure Pro (as regards reference completion for

successful termination) can work for a given nonter-

minal.

4 CONCLUDING REMARKS

We have summaries regarding this work in progress

for formal methods in interactive techniques.

(i) This formality of multi-modal logic, to monitor

states virtually interactive for iDevice may be closely

relevant to the recent trend of game semantics (Ven-

ema, 2008).

(ii) As well as newly presented semantics for this cal-

culus based on a state set, actions regarding a modal

KEOD 2015 - 7th International Conference on Knowledge Engineering and Ontology Development

362

operator are discussed, with respect to their functional

and algebraic aspects and with reference to works

(Giordano et al., 2000; van der Hoek et al., 2005;

Kucera and Esparza, 2003). The accounts of actions

are made by applications of Heyting algebra, ﬁxed

point theory, and semiring structure. They are also

related to trend coalgebra (Kurz, 1989).

(iii) Fixed point logic (Venema, 2006) may include

the present version, since the action modality may be

denoted by a ﬁxed point. However, the mu-operator

requires some restriction that the operator may be as-

sociated with a monotone function. For a nonmono-

tonic case, we have backgrounds (Genesereth and

Nilsson, 1987; Yamasaki, 2006; Yamasaki, 2010).

(iv) As regards sequence formation in iDevice, it

is closely related to knowledge structure, where

the well-done sequence presents a beauty based on

mechanized formation of reasonable (simple) state-

transitions. Whether well mechanized formations of

a sequence for the origami crane by iDevice would

be a problem from the views of interaction techniques

with graphical designs of practical impacts. Concern-

ing URL references, a referential closeness is dis-

cussed with respect to the idea of successful sequence

of references. The sequence is related to the struc-

ture of semiring, captured by coalgebraic behaviours.

For a task as implementing heuristic cognition stages

of recovery from language, it is a methodological or

technical idea to automate grammatical rule appli-

cations, while the discussion of this positioning is

really concerned with state-constraint grammars be-

tween context-free and context-sensitive grammar hi-

erarchies (Kasai, 1970), simpler than the class of

constraint functional programming (Bertolissi et al.,

2006) and more classical than the recent studies of

(inﬁnite) streams and languages by coalgebra (Rut-

ten, 2001; Winter et al., 2013; Winter et al., 2015).

We might, however, have formal reasonings to im-

plementing cognitions or to making them graded up

as artiﬁcial (machine) intelligences, following the ad-

vanced theories from those (Genesereth and Nilsson,

1987; Reiter, 2001).

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