Semantics and Algebra for Action Logic Monitoring State Transitions

Susumu Yamasaki

HCI Group, Okayama University, Okayama, Japan

Keywords:

State Constraint Agent, Interactive Monitoring, Modal Logic and Algebra.

Abstract:

This position paper is concerned with aspects of an interactive state transition system (based on abstract state

machine) by means of the state monitoring in action logic (as multi-modal logic), towards a step to the design

for complex systems. Logical models are here presented as theories for implementation design on iDevice,

with respect to the algebraic structure caused by state transitions. As a simpler design of complex AI, the envi-

ronmental constraint is captured as a state, where the function applications are available at each state with the

transition to the next states. For communication to the state, and function applications at the state, multi-modal

logic model may be of use, where the formula or the condition monitors the state. Then interaction availability

is signiﬁcant, expressed in some algebra on the basis of the meaning deﬁnitions for formulas (conditions). By

the state transition system, URL searching operations are now formally considered as in algebraic structure.

The application of predicates to the states is regarded as applications of functions (transforming conditions)

such that its algebraic structure may be given.

1 INTRODUCTION

As methods of monitorng and analysis for systems,

this positioning is concerned with the following con-

cepts, for extensions of action logic (as multi-modal

logic): (i) interaction modeling for abstract state ma-

chine as a state transition system, (ii) monitoring

states in action logic with modal operators, (iii) logic

of action, with respect to multi-modal logic and appli-

cations of functions, and (iv) denotational semantics

for action logic.

Abstract state machine (by Y. Gurevich) can be

a basis for the framework of state-transition systems

applicable to complex AI systems. Containing state-

transitions, action logic is needed for design meth-

ods even on iDevice, as in the paper (Yamasaki and

Sasakura, 2015).

For the required design methods, interactive stage

would be here formulated as being monitored, to re-

ﬂect behaviours on popular iDevices (for interactive

AI-tools), with relevance to usage of ideas in func-

tional programming (Thompson, 1991) or process al-

gebra (Cardelli and Gordon, 2000; Milner, 1999) to

AI-tools. Combined with abstract state machine, the

function applications can be made with state transi-

tions. However, the state can be nowadays realized

by a panel display to be touched on iDevices. The

iDevice is, on one hand, to present the state transi-

tion where the function may be applied. On the other

hand, it is interpreted to support an interactive pro-

cess, in which function applications are executed in

a programming system so that the evaluation may be

obtained.

The state transition system is involved in action

logic (Hennessy and Milner, 1985; Kucera and Es-

parza, 2003) and modal logic (Venema, 2008). Based

on the new aspect of the meaning deﬁnition for for-

mulas (or conditions), an interaction state (where in-

teractive actions like communications and/or function

applications are available) can be presented.

As regards knowledge-based systems, URL

searching forward and backward would be exam-

ined with respect to the operations (composition –

multiplication– and alternation – addition –) of func-

tion applications in a state transition system. The

view on the operations is closely related to automata

theory as in the book (Droste et al., 2009) where al-

gebraic structures like semiring are compiled. In this

positioning, a new technique on the reduction of mul-

tiplicative inverse is to be presented, after the URL

searching is well organized as an algebraic structure,

caused by a state-constraint system. For knowledge-

based systems (which involve technologies in state

transition systems as well as in action logic), actions

by predicates or logical formulas may be regarded as

function applications.

The position paper is organized as follows. In Sec-

tion 2 we observe Yale Shooting Problem as an intro-

110

Yamasaki, S.

Semantics and Algebra for Action Logic Monitoring State Transitions.

DOI: 10.5220/0006345901100115

In Proceedings of the 2nd International Conference on Complexity, Future Information Systems and Risk (COMPLEXIS 2017), pages 110-115

ISBN: 978-989-758-244-8

ductory motivation to this positioning. It is in Section

3 followed by a revised formulation of multi-modal

mu-calculus to monitor the state transition system. In

Sections 4 and 5, some function applications of state

transition systems are discussed from algebraic views.

A concluding remark, with related topics, is brieﬂy

given, as well.

2 A KNOWLEDGE-BASED

SYSTEM FOR ACTIONS

Solution Display of Yale Shooting Problem

The solution of the AI problem is often expressed

by a sequence of state-transitions. The state is tempo-

rally prepared for and the operation is equipped with,

where the formula or condition is attached to the state.

AI solution and display can thus be made in terms of

(interactive) state-constraint system. Let us have an

outlook on the display of solutions in the Yale Shoot-

ing Problem (Hanks and McDermott, 1987).

The Yale Shooting Problem is a scenario in logic:

(i) A turkey is initially alive and a gun is initially un-

loaded.

(ii) Loading the gun, the shooter waits for a moment.

(iii) Then shooting at the turkey is expected to kill it.

The scenario is captured with the condition chang-

ing truth values over time, like alive and loaded. As-

suming four time points 0, 1, 2, and 3, alive(t) and

loaded(t) supposedly denote the conditions alive and

loaded to be true (i.e. to hold) at time t, respectively.

Then the scenario must satisfy:

alive(0)

¬loaded(0)

true → loaded(1)

loaded(2) → ¬alive(3)

where “¬” stands for the (classical) negation and “→”

does for the entailment. As an implicit assumption

that alive(0) ≡ alive(1), loading the gun only changes

the value of “loaded”: The conditions do not change

unless an action changes them (which is the frame

problem).

By the concept of ﬂuent (which is a condition to

change truth values) (Hanks and McDermott, 1987),

we can have one evaluation of the conditions whose

changes are minimized:

alive(0), alive(1), alive(2),¬alive(3);

¬loaded(0),loaded(1),loaded(2),loaded(3)

where another evaluation is not satisfactory, though

the changes of the conditions are minimized:

alive(0), alive(1), alive(2),¬alive(3);

¬loaded(0),loaded(1),¬loaded(2),¬loaded(3)

The former sequence of evaluations can be repre-

sented in a state-constraint system as follows, where

Fn is a function, and C1, C2 are conditions:

state 0 1 2 3

Fn gun-load null shooting null

C1 alive alive alive ¬alive

C2 ¬loaded loaded loaded loaded

The solution with the state transitions suggests a

basic structure of:

state s Fn

state s

... ...

state s

where: (i) Fn

, Fn

, . . . , Fn

are terms or functions

applicable (with or without conditions), at the state s

(conditioned or monitored by some logical formula).

(ii) The states s

, s

, . . . , s

are regarded as the con-

straints, respectively, transited from the state s after

the application of each of those functions.

3 COMMUNICATION AND

FUNCTION APPLICATION

Hennessy-Milner Logic (HML, for short) is concep-

tually relevant to the state-constraint system. With an

action (or a communication) hci as below, formulae

as in Hennessy-Milner Logic (HML) are described by

the form:

ϕ ::= tt | ϕ ∨ ϕ | ¬ϕ | hciϕ.

Following the denotations of formulas, we have

extended Hennessy-Milner logic to a multi-modal

logic version, based on the papers (Cardelli and Gor-

don, 2000; Merro and Nardelli, 2005; Hennessy and

Milner, 1985; Milner, 1999), for modeling of states

monitored in implementation. Possibly with moni-

torable states as meaning for each formula, we now

have the set Φ of formulas, modiﬁed from the ﬁrst

version (Yamasaki and Sasakura, 2015):

ϕ ::= tt | p | ¬ϕ |

∼

ϕ | ϕ ∨ ϕ | hciϕ | µ.ϕ | ϕiti

We here have a preﬁx modality hci (for communica-

tion), a postﬁx one iti (for function application), a

negation (sign)

∼

for incapability of interaction, stan-

dard propositions p, the logical negation ¬ and a least

ﬁxed operator µ.

The semantics for formulas are deﬁnable on the

basis of a transition system. The transition system

(for semantics of logic) is

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

pos

neg

inter

Semantics and Algebra for Action Logic Monitoring State Transitions

111

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 t ∈ A a relation Rel(t) on S.

(vi) V

pos

neg

inter

: Prop → 2

, map to each

proposition (variable) a set of states, respec-

tively.

The reason why 3 assignments of V

pos

, V

neg

and V

inter

are adopted comes from introduction to monitoring

interaction. Given a transition system S , the func-

tions [[ ]]

pos

,[[ ]]

neg

,[[ ]]

inter

: Φ → 2

are deﬁned 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.

The Meaning is concerned with two modalities

hci, iti:

(1) [[tt]]

pos

= S, [[tt]]

neg

0, and [[tt]]

inter

(2) [[p]]

pos

= V

pos

(p), [[p]]

neg

= V

neg

(p), and

[[p]]

inter

= S\ ([[p]]

pos

∪ [[p]]

neg

) = V

inter

(p).

(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

(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

and s

∈ [[ϕ]]

pos

[[hciϕ]]

neg

= {s ∈ S | ∀s

. s Re(c) s

entails s

∈ [[ϕ]]

neg

and [[hciϕ]]

inter

= S \ ([[hciϕ]]

pos

∪ [[hciϕ]]

neg

(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,

and both the intersection “∩” and the subset “⊆”

are componentwise, with assignments of T

pos

and

neg

to x.

(8) [[ϕiti]]

pos

= {s

∈ S | ∀s. s Rel(t)s

entails s ∈ [[ϕ]]

pos

[[ϕiti]]

neg

= {s

∈ S | ∀s. s Rel(t) s

entails s ∈ [[ϕ]]

neg

[[ϕiti]]

inter

= S \ ([[ϕiti]]

pos

∪ [[ϕiti]]

neg

Modality hci is from communication labelled by c,

Modality iti possibly comes from function applica-

tions. When the latter modality is applied to a state s,

it may hold a relation Rel(t). It follows that:

[[ϕiti]]

inter

= {s

∈ S|∃s. sRel(t)s

,s ∈ [[ϕ]]

inter

4 FUNCTIONS REGARDING URL

SEARCHING

ϕiti with a function t is interpreted as monitoring the

state at which the function (a kind of term) is (in in-

teraction mode) applied and possibly transited to the

next state.

Upon URL searching, the operations contain un-

folding to reﬁne the next references included in the

present reference, and folding to return back to the

former reference. In this context, we here have some

algebraic structure, as a state-constraint system be-

haviour.

Assume a simple structure of references in URL:

(Home page name) A

(Contents) (Reference names) B

. . .

We then examine two functions of:

(i) unfolding to open the HP (home page) named A to

see reference names B

, . . . , and B

, and (ii) folding

to close reference names B

, . . . , and B

to have the

(HP) name A, as illustrated below.

(Home page name) A →

un f olding

; ←

f olding

(Contents) (Reference names) B

. . .

The sequence of operations by folding and unfold-

ing can be described in an algebraic structure. To see

it, let X be a set of reference (including HP) names,

where its power set 2

contains all the subsets of X ,

including the emptyset

0 and X.

For a function (like unfolding) f : X → 2

, we

have

f (x) = ∪

x∈X

f (x), such that the function f is

extended to the function

f : 2

→ 2

. For a function

COMPLEXIS 2017 - 2nd International Conference on Complexity, Future Information Systems and Risk

112

(like folding) g : 2

→ X, we have ˆg(Y ) = {g(Y )},

such that the function g is extended to the function

ˆg : 2

→ 2

We therefore assume the set Ψ of functions of the

power set 2

to the power set 2

, that is, 2

→ 2

where each function should supposedly assign the

emptyset (

0 ∈ 2

) to the emptyset. The set includes

the identity Id : 2

→ 2

, Id(Y ) = Y . The function

φ is assumed to assign the empty set (

0 ∈ 2

) to any

Y ∈ 2

The composition (G ◦F) of the functions F,G ∈ Ψ

is deﬁned to be

(G ◦ F)(Y ) = G(F(Y )) for Y ∈ 2

where (G ◦ F) may be represented by G ◦ F. The al-

ternation (F + G) : 2

→ 2

of F,G ∈ Ψ is deﬁned to

(F + G)(Y ) = F(Y ) ∪ G(Y ),

where (F + G) may be represented by F + G.

The relation ≡ on the set Ψ is deﬁned:

F ≡ G iff F(Y ) = G(Y ) for any Y ∈ 2

It follows that the relation ≡ is an equivalence rela-

tion.

We can see the following properties, which show

that (Ψ,+,◦,φ,Id) is a semiring:

Proposition 1.

(i) F + G ≡ G + F.

(ii) F + (G + H) ≡ (F + G) + H.

(iii) F + φ ≡ φ + F ≡ F.

(iv) F ◦ (G ◦ H) ≡ (F ◦G) ◦ H.

(v) F ◦ Id ≡ Id ◦ F ≡ F.

(vi) F ◦ (G + H) ≡ (F ◦ G) + (F ◦ H).

(vii) (F + G) ◦ H ≡ (F ◦ H) + (G ◦ H).

(viii) F ◦ φ ≡ φ ◦ F ≡ φ.

Proof. (i) (F +G)(Y ) = F(Y )∪G(Y ) = G(Y )∪F(Y )

= (G + F)(Y ) for any Y ∈ 2

. Thus it holds.

(ii) It can be seen for the same reason as in (i), owing

to the associative law in taking the union ∪.

(iii) Because φ(Y ) =

0 by the deﬁnition, it holds.

(iv) (F ◦ (G ◦ H))(Y ) = F(G(H(Y ))) = (F ◦

G)(H(Y )) = ((F ◦ G) ◦ H)(Y ) for any F ∈ 2

, as the

associative law of function compositions. It therefore

hold.

(v) Because Id is an identity function on composition,

it holds.

(vi) (F ◦ (G +H))(Y ) = F(G(Y )∪ H(Y )) = F(G(Y ))

∪ F(H(Y )) = ((F ◦ G) + (F ◦ H))(Y ). It so holds.

(vii) It is seen by the similar reason of (vi) for this dis-

tributive law to hold.

(viii) Because φ(Y ) =

0 and F(

0) is deﬁned to be

this annihilation holds.

That is, the properties (i), (ii) and (iii) show that

(Ψ,+,φ) is a commutative monoid (a commutative

semigroup with the identity φ for +). The properties

(iv) and (v) show that (Ψ,◦,Id) is a monoid (a semi-

group with the identity for ◦). (vi) and (vii) are dis-

tributive laws, while (viii) is annihilation (in a semir-

ing).

For F ∈ Ψ, let F

∗

= ∪

n∈ω

, where:



Id (n = 0)

n−1

◦ F (n > 0)

It follows that F

∗

∈ Ψ such that:

∗

= Id + F ◦F

∗

≡ Id + F

∗

◦ F,

that is, (Ψ,+,◦,φ,Id) is a star semiring.

Semiring with Multiplicative Inverse

The structure may contain the case that

∀F ∈ Ψ, ∃F

∈ Ψ. F ◦ F

= F

◦ F = Id

(where F

is a multiplicative inverse represented by

−1

(Note) We may take the reduction to have the identity

Id from G ◦ G

−1

or G

−1

◦ G, until no more reduction

could be made, to have an equivalent expression (with

respect to “≡”) for a given expression.

Proposition 2. Given an expression with the opera-

tions +, ◦ and ∗, there is a procedure to reduce it to

the equivalent expression (with respect to “≡”) until

no more reduction of right or left inverse can be made.

Proof. We can have a mapping h : EX P → EXP re-

cursively deﬁned as follows, where for an expression

Ex ∈ EX P (the set of expressions with the operations

+, ◦ and ∗) to denote a member in the set Ψ:

h(Ex) =











φ (Ex = φ)

Id (Ex = Id)

F (Ex = F)

−1

(Ex = F

−1

)

h(Ex

) + h(Ex

) (Ex = Ex

+ Ex

)

, G

−1

h(Ex

) ◦ h(G

−1

\Ex

)

+ +

−1

, H

h(Ex

−1

) ◦ h(H

\Ex

)

+h(Ex

) ◦ h(Ex

) (E = Ex

◦ Ex

)

, G

−1

h(Ex

∗

) ◦ h(G

−1

\Ex

)

+ +

−1

, H

h(Ex

∗

−1

) ◦ h(H

\Ex

)

+Id + h(Ex

∗

) ◦ h(Ex

) (Ex = Ex

∗

)

with the expressions:

(1) Ex

and G

−1

\Ex

are the right residual

and the left residual, respectively.

Semantics and Algebra for Action Logic Monitoring State Transitions

113

(2) Ex

−1

and H

\Ex

denote the right residual

and the left residual, respectively.

Note that h(E

∗

), and h(E

∗

−1

) are included

in h(E

∗

), such that:

h(E

∗

) =

, G

−1

h(Ex

∗

) ◦ h(G

−1

\Ex

)

+ +

−1

, H

h(Ex

∗

−1

) ◦ h(H

\Ex

)

+h(Ex

∗

) ◦ h(Ex

), and

h(E

∗

−1

) =

, G

−1

h(Ex

∗

) ◦ h(G

−1

\Ex

−1

)

+ +

−1

, H

h(Ex

∗

−1

) ◦ h(H

\Ex

−1

)

+h(Ex

∗

) · h(Ex

−1

A well-known ﬁxed point technique (following

S.C. Kleene) is available, for the expression Ex as be-

low to be deﬁned:

Ex = Ex

+ Ex ◦ Ex

⇒ Ex = Ex

◦ Ex

∗

such that h(Ex) may be the required expression.

5 PREDICATES AS APPLIED

FUNCTIONS

Bothe positive and negative predicates are thought of

as function applications, with reference to formula

conditions monitoring states. They may induce re-

lations between states, where the relations are con-

cerned with the operations, union ∪ and composition

◦. As in interaction mode, we here formulate a postﬁx

modality consisting of the pair as below:

With an assumed predicate set P-Set, a par of

(i) a set of sequences of predicates for a collection of

sequences of positive conditions, and

(ii) a set of predicates for negative conditions

is to be considered.

Deﬁnition 3. Given a set P-Set, P-Set

∗

is the set of

all ﬁnite sequences of elements from P-Set, including

the empty sequence ε such that εl = lε = l for any

l ∈ P-Set

∗

Then a postﬁx modality iti, where t takes the form

(pseq,neg) for pseq ⊆ P-Set

∗

and neg ⊆ P-Set, can

be built into the formulas of this paper. Then a rela-

tion Rel(pseq, neg) may be deﬁned in the transition

system S .

With correspondences and modiﬁcations of addi-

tion + to union ∪ for the relation Rel(pseq, neg) and

of multiplication • to concatenation ◦ for the relation

Rel(pseq,neg), an algebraic structure of some set of

pairs (pseq,neg) is below discussed.

For a “consistent” pair of the form (pseq, neg), we

make use of:

Deﬁnition 4. l ∈ pseq is consis to the set neg if any

element of l is not in neg. The pair (pseq,neg) is Con-

sis if any l in pseq is consis to neg.

We now have the algebraic structure: For Pred2

= {(pseq,neg) | pseq ⊆ P-Set

∗

and neg ⊆ P-Set,

(pseq,neg) is Consis }, the structure

hPred2,+,•i

is to be a semiring, where the operations + as addition

and • as multiplication are deﬁned, as well as the star:

(a) (pseq

,neg

) + (pseq

,neg

)

de f

(pseq

∪ pseq

,neg

∩ neg

(b)

(pseq

,neg

) • (pseq

,neg

)

de f

(pseq

· pseq

− {l ∈ pseq

|l is not consis to neg

} · pseq

− pseq

· {l ∈ pseq

|l is not consis to neg

neg

∪ neg

where “·” means a concatenation of two (se-

quence) sets, which provides a new set of se-

quences obtained by arranging two sequences

taken from the two given sets in order.

de f

(pseq

∗

0) for a star “?”, where

pseq

∗

= ∪

n∈ω

pseq

(for pseq

= {ε} and

pseq

= pseq

n−1

· pseq, n > 0, with “·” as con-

catenation of two sets of sequences).

It can be easily observed that:

(i) hPred2,+,(

0,P-Set)i is a commutative semi-

group with the identity (

0,P-Set), that is , a com-

mutative monoid.

(ii) hPred2,•,({ε},

0)i is a semigroup with the iden-

tity ({ε},

0), that is , a monoid.

The we have:

Proposition 5. The algebraic structure hPred2, +,•i

is a (star) semiring.

Proof. (Outline) With observations of two monoids

hPred2,+i (+: commutative addition) and hPred2,•i

(•: multiplication), we can see that the multiplication

distributes over addition from the left and from the

right. Then we have the annihilation that (

0,P-Set) •

(pseq,neg) = (pseq,neg) • (

0,P-Set) = (

0,P-Set). It

is ﬁnally seen for the star operation that:

(pseq,neg)

= ({ε},

0) + (pseq,neg)

• (pseq,neg)

= ({ε},

0) + (pseq,neg) • (pseq, neg)

= (pseq

∗

0).

COMPLEXIS 2017 - 2nd International Conference on Complexity, Future Information Systems and Risk

114

6 CONCLUDING REMARKS

As a conclusion of this positioning, the interactive,

functional applications (as in complex AI) may be

settled with state transition system concepts, which

is also viewed by multi-modal logic with meanings

of formulas (conditions). This view has been from

a display of AI system solution like Yale Shoot-

ing Problem. Then (1) URL searching structures in

knowledge-based systems and (2) predicates applica-

ble at states can have been interpreted in algebraic

structures. As theories, the view may be relevant to

the algebraic informatics as in the classical category

theory. The view may be practical in an interactive

design for origami (Yamasaki and Sasakura, 2015).

As related topics on abstract state machine or

state transition system, we have already principles

and backgrounds as follows, such that this position-

ing may be regarded as a reﬁned and original work.

(i) Communication models are well established in

relation to action logic with reference to algebraic

aspects (Cardelli and Gordon, 2000; Merro and

Nardelli, 2005; Milner, 1999) where behavioural se-

quences may be considered as essential. The modal

operator of this positioning may be regarded as rele-

vant to such backgrounds.

(ii) As regards functional programming and actions,

semantics are examined (Bertolissi et al., 2006;

Mosses, 1992; Reiter, 2001) from operational and

declarative methods. This positioning follows an al-

gebraic way different from these semantic views.

(iii) The state-constraint system is relevant to coalge-

bra (Rutten, 2001; Venema, 2006; Winter et al., 2013;

Winter et al., 2015). As regards pushdown store man-

agements, weighted automaton concept (Reps et al.,

2005) is known, which is related to a semiring struc-

ture with multiplicative (right) inverse of this paper.

(iv) The problem solving methods (Genesereth and

Nilsson, 1987; Osorio et al., 2004) are established

such that it may have conceived the step by step man-

agements even for action logic (Giordano et al., 2000;

van der Hoek et al., 2005). This positioning might

present reﬁnements in those directions.

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