ALGORITHMS FOR AI LOGIC OF DECISIONS

IN MULTI-AGENT ENVIRONMENT

Vladimir Rybakov and Sergey Babenyshev

Manchester Metropolitan University, John Dalton Bld, Chester Str, Manchester, U.K.

Siberian Federal University, Svobodnyi Ave 79, Krasnoyarsk, Russia

Keywords:

Multi-agent logic, temporal logic, interacting agents, decision algorithms, satisﬁability, inference rules.

Abstract:

This paper

1

suggests a temporal multi-agent logic LD

M A

(with interacting agents), to imitate decision-making

of independent agents, supported by access to knowledge through interaction with other agents. The interac-

tion is modeled by considering all possible communication paths between agents in temporal Kripke/Hintikka

like models. The logic LD

M A

distinguishes local and global decision-making and is based on temporal

Kripke/Hintikka models with agents accessibility relations deﬁned between the states of time clusters. The

main result provides a decision algorithm for LD

M A

(so, we prove that the set of theorems of LD

M A

is

decidable), which also solves the satisﬁability problem for LD

M A

.

1 INTRODUCTION

Applications of multi-modal and temporal logic to

AI and CS is a popular area of research. In partic-

ular, they can be seen (based on the formalism of

multi-agent logics) as a part (or implementation) of

epistemic logic. Among epistemic logics to model

knowledge a range from S4 to S5 has been investi-

gated (Hintikka (1962) — logic S4, Kutschera (1976)

argued for S4.4, Lenzen (1978) suggested S4.2, van

der Hoek (1996) had proposed to strengthen knowl-

edge according to system S4.3, van Ditmarsch, van

der Hoek and Kooi together with Fagin, Halpern,

Moses and Vardi (Fagin et al., 1995) and others as-

sume knowledge to be S5 valid, see also (Halpern and

Shore, 2004)). The approach, developed to model

multi-agent environment in AI, often combines not

only modal operations for agents‘ knowledge and

Boolean logical operations, but also some other —

e.g. operations for time — temporal operations, dy-

namic logic operations (cf. (Schmidt and Tishkovsky,

2004)). Through the prism of multi-agent approach

we may view the logic of discovery, which has a

solid prehistory, starting possibly from the mono-

graph “Logic of Discovery and Logic of Discourse”

by Jaakko Hintikka and Fernand Vandamme (Hin-

tikka and Vandamme, 1986).

1

This research is supported by Engineering and Phys-

ical Sciences Research Council (EPSRC), U.K., grant

EP/F014406/1

The Decision Logics apparently have interdis-

ciplinary origin and they were inﬂuenced by ideas

coming from researchers of widely varying back-

ground (cf. (Ohsawa and McBurney, 2003)). In

particular, the modeling of environmental decision-

support systems has been undertaken (Cort´es et al.,

2000; Avouris, 1995), tools involved in sematic

web and multi-agent systems had been developed

(Harmelem and Horrocks, 2000; Hendler, 2001; Ar-

isha et al., 1999). Instruments for decision procedures

in equational causal logic were created in (Peltier,

2003). Regardingmulti-agent logics, many developed

tools were inspired by the techniques of modal and

temporal logic through mathematical semantics of

Kripke/Hintikka models (Goldblatt, 2003; van Ben-

them, 1983).

In our paper we study a temporalmulti-agent logic

LD

M A

with interacting agents with the purpose of

ﬁnding a decision algorithm for this logic. The main

idea is to study ways of passing knowledge between

agents via possible communication paths, then model

them in the temporal Kripke/Hintikka-like models by

modal-like operation D

l

(locally taken decision), and

extend the method to the global decision. We build

our logic in a language, which considers and distin-

guishes local decision D

l

and global decision D

g

op-

erators applied to formulas. An approach, which we

use, is based on the research (Rybakov, 1997; Ry-

bakov, 2005b; Rybakov, 2005a; Rybakov, 2006; Ry-

bakov, 2007) on the representations of knowledge by

125

Rybakov V. and Babenyshev S. (2008).

ALGORITHMS FOR AI LOGIC OF DECISIONS IN MULTI-AGENT ENVIRONMENT.

In Proceedings of the Tenth International Conference on Enterprise Information Systems - AIDSS, pages 125-129

DOI: 10.5220/0001672301250129

Copyright

c

SciTePress

inference rules in AI logics. The ﬁnal result of the

paper is a decision algorithm, which recognizes the-

orems of LD

M A

(i.e., we prove that LD

M A

is decid-

able). The algorithm works by reducing a formula to

a logical consecution — inference rule, and checking

validity of this rule in special Kripke-Hintikka models

of size double exponential in the size of the original

formula.

2 NOTATION, PRELIMINARIES

In the sequel we use the standard notation and well-

known facts concerning modal, multi-modal and tem-

poral logics, hence, some familiarity with the area is

assumed. To formulate the logic LD

M A

, we proceed

by introducing a semantic motivation for the choice

of its language and interpretation. The Kripe/Hintikka

models, upon which we base our logical language, are

the following tuples — linear-time frames with time

clusters to model agents‘ accessibility relations to in-

formation:

N

C

:= h

[

i∈N

C(i),R,R

1

,... ,R

m

i,

where N is the set of natural numbers, C(i) are some

nonempty sets, R,R

1

,... R

m

are binary accessibility

relations. For all elements a and b from

S

i∈N

C(i),

aRb ⇐⇒ ∃i, j ∈ N : i 6 j & a ∈ C(i) & b ∈ C( j);

any R

j

is a reﬂexive, transitive and symmetric rela-

tion, and

∀a,b ∈

[

i∈N

C(i) : aR

j

b =⇒ ∃k ∈ N : a,b ∈ C(k).

The models based on these frames are intended

to represent the reasoning (computation) in discrete

time, so each i ∈ N (any natural number i) is the time

index for a cluster of states arising at a step in compu-

tation. Any C(i) is a ﬁnite set of all possible states in

the time point i, and the relation R represents discrete

current of time. Relations R

j

are intended to model

the access to knowledge (information) for agents at

any state of the cluster of states C(i). As usual, any

R

j

is supposed to be S5-like relation — a binary sym-

metric, reﬂexive and transitive relation.

The relation R models the discrete ﬂow of time,

so, aRb means that a and b are some states at the

same time point or the state b will be achievedat some

point in the future, after the time point with the state

a. The ﬂow of time is supposed to be linear, which re-

ﬂects well the human perception. We suppose the rea-

sonong/computation to be concurrent — after a step

in computation a new cluster of possible states ap-

pears, and agents will be given new access rules to the

information in this time cluster. However, the agents

cannot predict, which access rules they will have (that

is why, in particular, we do not use nominals).

What is new in our approach is that we in-

tend to make decision-making being based on non-

deterministic agents‘ accessibility to knowledge (with

the aim of ﬁnding algorithms recognizing logical laws

of the proposed logic). To handle the problem we

use a special logical language. It combines an ex-

tended language for agents‘ knowledge logic and a

non-standard modal language for modeling time. The

foundation of the language is the set of propositional

letters P, the logical operations include usual Boolean

operations, usual unary agent knowledge operations

K

i

, 1 ≤ i ≤ m and the modal operation ♦ for time. We

also augment the language by taking weak necessity

operation

w

and decision operations D

l

and D

g

, for

local and global decision-making. Formation rules for

formulas are as usual, and

• K

i

ϕ can be read: an agent i knows ϕ at the cur-

rent state (informational node) of the current time

cluster;

• ♦ϕ means: ϕ is possible in a future time cluster

accessible from the current state;

•

w

ϕ stands for: in any future time cluster there

is a state where ϕ holds (so to say, ϕ is weakly

necessary);

• D

l

ϕ has the meaning : it is decided that ϕ holds

locally;

• D

g

ϕ has the meaning: it is decided that ϕ holds

globally.

A concrete knowledge situation is modeled by dis-

tribution of truth values of propositions from P at the

elements of the frame N

C

. More formally, we con-

sider valuations V of P, which are mappings of P into

the set of all subsets of the set

S

i∈N

C(i), so in sym-

bols,

∀p ∈ P :V(p) ⊆

[

i∈N

C(i).

If, for an element a ∈

S

i∈N

C(i), a ∈ V(p), we say that

the fact p is true at the state a.

Validness of formulasis deﬁned as follows (below,

N

C

,a

V

ϕ is meant to say that the formula ϕ is true

at the state a in the model N

C

w.r.t. the valuation V):

∀p ∈ P, ∀a ∈ N

C

: N

C

,a

V

p ⇐⇒ a ∈ V(p);

N

C

,a

V

ϕ∧ ψ ⇐⇒ [N

C

,a

V

ϕ and N

C

,a

V

ψ];

N

C

,a

V

ϕ∨ ψ ⇐⇒ [N

C

,a

V

ϕ or N

C

,a

V

ψ];

ICEIS 2008 - International Conference on Enterprise Information Systems

126

N

C

,a

V

ϕ → ψ ⇐⇒ [N

C

,a 6

V

ϕ or N

C

,a

V

ψ];

N

C

,a

V

¬ϕ ⇐⇒ N

C

,a 6

V

ϕ;

N

C

,a

V

♦ϕ ⇐⇒ ∃b ∈ N

C

: aRb and N

C

,b

V

ϕ;

N

C

,a

V

w

ϕ ⇐⇒

(a ∈ C(i) =⇒ ∀ j ≥ i ∃b ∈ C( j) : N

C

,b

V

ϕ);

N

C

,a

V

K

i

ϕ ⇐⇒ ∀b ∈ N

C

: aR

i

b =⇒ N

C

,b

V

ϕ;

N

C

,a

V

D

l

ϕ ⇐⇒

∃a

i

1

,... ,a

i

n

∈ N

C

: aR

i

1

a

i

1

...R

i

n

a

i

n

& N

C

,a

i

n

V

ϕ;

N

C

,a

V

D

g

ϕ ⇐⇒ ∀b ∈ N

C

:aRb =⇒ N

C

,b

V

D

l

ϕ.

From the rules above, we immediately see that in-

troduced logical operations are not independent from

the semantical viewpoint and that

D

g

ϕ ≡ ¬♦¬D

l

ϕ.

Therefore we omit D

g

from the further consideration.

Deﬁnition 1. The logic LD

M A

is the set of all formu-

las which are true at any state of any frame N

C

w.r.t.

any valuation.

To compare the logical laws of LD

M A

with the

laws of the standard multi-modal logic, note that the

following holds:

w

p → p,

w

p ≡ ¬♦¬p, ♦p ≡ ¬

w

¬p 6∈ LD

M A

.

This can be derived immediately from the deﬁnitions,

using only simple frames. Moreover,

w

(p → q) → (

w

p →

w

q) 6∈ LD

M A

,

therefore,

w

-fragment of LD

M A

is not a fragment

of any reﬂexive modal logic, and also

w

and ♦ are

not mutual counterparts of each other. Thus LD

M A

differs from any standard normal or not-normal multi-

modal logic. It is interesting, whether

w

and ♦ may

be mutually expressed by other non-standard ways.

At the same time (where := ¬♦¬)

Lemma 1. The following holds

• (p → q) → (

w

p →

w

q) ∈ LD

M A

,

•

w

p →

w

w

p ∈ LD

M A

,

• (

w

p →

w

q) ∨ (

w

q →

w

p) ∈ LD

M A

,

• ϕ ∈ LD

M A

=⇒

w

ϕ ∈ LD

M A

.

Thus

w

-fragment of LD

M A

has some similar-

ity with modal logics extending S4.3. The primal

questions for any logic are the decidability problem

and satisﬁability problem (for a logic L it is of vi-

tal importance to recognize correct logical law of this

logic, and solving the decidability problem means

constructing an algorithm, which could recognize the

logical laws). We address the decidability problem

for LD

M A

in the next section.

3 DECIDING ALGORITHMS

The decidability problem for LD

M A

, which we will

be dealing with in the sequel, is how, for any given

formula ϕ, to determine whether ϕ is a theorem of

LD

M A

or not, in other words whether ϕ ∈ LD

M A

or

ϕ 6∈ LD

M A

(if there is an algorithm for solving this

task for a logic L , then the logic L is said to be de-

cidable). The logic LD

M A

is an extension of a spe-

cial many-modal logic, and we could try to use some

well-known techniques to tackle the problem at hand.

However the standard ways meet the obstacle of the

operator D

l

used in our approach, mainly because it

fails such techniques as ﬁltration and dropping points.

Several ways to approach the problem are neverthe-

less possible, and we will apply a technique based on

our own approach with employing logical consecu-

tions and validity of inference rules, which was tested

for several logics earlier (Rybakov, 1997; Rybakov,

2005a; Rybakov, 2006; Rybakov, 2007). This tools

use a representation of formulas by rules, and an al-

gorithmic translations of rules into some normal re-

duced forms. Recall that a (sequential) rule is an ex-

pression of the form

r :=

ϕ

1

(x

1

,... , x

n

),... , ϕ

m

(x

1

,... , x

n

)

ψ(x

1

,... , x

n

)

,

where ϕ

1

(x

1

,... , x

n

),... , ϕ

m

(x

1

,... ,x

n

),ψ(x

1

,... , x

n

)

are some formulas constructed out of letters x

1

,... , x

n

.

Letters x

1

,... , x

n

are called variables of r, symboli-

cally

Var(r) = {x

1

,... , x

n

}.

A formula ϕ is valid in a frame N

C

(notation

N

C

ϕ) if, for any valuation V of Var(ϕ) and for any

element a of N

C

, N

C

,a

V

ϕ.

Deﬁnition 2. A rule r is said to be valid in the Kripke

model hN

C

,Vi with the valuation V (we will use no-

tation N

C

V

r) if

∀a : N

C

,a

V

^

1≤i≤m

ϕ

i

=⇒ ∀a : N

C

,a

V

ψ.

ALGORITHMS FOR AI LOGIC OF DECISIONS IN MULTI-AGENT ENVIRONMENT

127

Otherwise we say that r is refuted in N

C

, or refuted in

N

C

by V, and write N

C

6

V

r.

A rule r is valid in a frame N

C

(notation N

C

r)

if, for any valuation V of Var(r), N

C

V

r. A rule r

is said to be in the reduced normal form if r = ε

r

/x

1

where

ε

r

=

_

1≤ j≤m

^

1≤i≤n

"

x

t( j,i,0)

i

∧ (♦x

i

)

t( j,i,1)

∧ (

w

x

i

)

t( j,i,2)

∧(D

l

x

i

)

t( j,i,3)

∧

^

1≤s≤m

(¬K

s

¬x

i

)

t( j,i,4,s)

#!

,

where all x

l

are variables, t( j, i,z),t( j,i,k,z) ∈ {0,1}

and, for any formula α above, α

0

:= α, α

1

:= ¬α.

For any formula ϕ we can convert it into the rule

x → x/ϕ and employ technique of reduced normal

forms as explained below.

Deﬁnition 3. Given a rule r

nf

in the reduced normal

form, r

nf

is said to be a normal reduced form for a rule

r iff, for any frame N

C

,

N

C

r ⇐⇒ N

C

r

nf

.

Based on proofs of Lemma 3.1.3 and Theorem

3.1.11 from (Rybakov, 1997), by similar technique,

we obtain

Theorem 1. There exists an algorithm running in

(single) exponential time, which, for any given rule

r, constructs its normal reduced form r

nf

.

It is immediate to see that a formula ϕ is valid in a

frame N

C

iff the rule x → x/ϕ is valid in N

C

, so from

Theorem 1 we obtain

Lemma 2. A formula ϕ is a theorem of LD

M A

iff the

rule (x → x/ϕ)

nf

is valid in any frame N

C

.

Thus, to solve the question about decidability of

LD

M A

it is sufﬁcient to ﬁnd an algorithm recognizing

rules in reduced normal form which are valid in all

frames N

C

. To begin with, we ﬁrst will bound effec-

tively the number of states in clusters on frames refut-

ing the consecution. This step deals only with prob-

lems arising from interaction of agents. In the follow-

ing lemma the representation of formulas by normal

reduced forms of rules is essential.

Lemma 3. If a consecution r = ε

r

/x

1

where ε

r

:=

W

1≤ j≤m

(

V

1≤i≤n

[x

t( j,i,0)

i

∧(♦x

i

)

t( j,i,1)

∧(

w

x

i

)

t( j,i,2)

∧

(D

l

x

i

)

t( j,i,3)

∧

V

1≤s≤m

(¬K

s

¬x

i

)

t( j,i,4,s

]), is refuted in

a model N

C

, then r is refuted in a such model with

clusters C(i) linear in the size of r.

In the proof of this lemma we cannot just use

the standard ﬁltration technique, because paths of in-

terchanging knowledge-accessibility relations cannot

be bounded by any ﬁltration, decision formulas D

l

x

j

pose the problem. But a reﬁned technique based on

normal forms of the rules works.

To further describe our algorithm we need the

following special ﬁnite Kripke models. Take the

frame N

C

and some numbers n,m, where m > n >

1. The frame N

C

(n,m) has the following structure:

N

C

(n,m) := h

S

1≤i≤m

C(i),R, R

1

,... , R

m

i, where R is

the accessibility relation from N

C

extended by pairs

(x,y), where x,y ∈ [n,m], so xRy holds for all such

pairs, and relations R

1

,... , R

m

are inherited from N

C

.

Given a valuation V of letters from a formula ϕ in

N

C

(n,m), the truth values of ϕ can be deﬁned on ele-

ments of N

C

(n,m) by the same rules as for frames N

C

above (actually, in accordance with the standard deﬁ-

nitions of truth values for modalities). For illustration,

we describe below basic steps for modalities.

N

C

(n,m),a

V

♦ϕ ⇐⇒

∃b ∈ N

C

: aRb & N

C

(n,m),b

V

ϕ;

N

C

(n,m),a

V

w

ϕ ⇐⇒

[a ∈ C(i) & i ≤ n &

∀ j(m ≥ j ≥ i)∃b ∈ C( j) : N

C

(n,m),b

V

ϕ]∨

[a ∈ C(i) & i > n &

∀ j(n ≤ j ≤ m)∃c ∈ C( j) : N

C

(n,m),b

V

ϕ].

For K

j

ϕ and D

l

ϕ steps are exactly the same as for

the models based on frames N

C

. Using this modiﬁed

Kripke structures N

C

(n,m) we derive

Lemma 4.

A rule

r

nf

in the reduced normal form is

refuted in a certain frame

N

C

w.r.t. a valuation

V

if

and only if

r

nf

can be refuted in a special model,

based on a frame

N

C

(n,m)

, by a valuation

V

1

, where

(i) The size of any cluster

C(i)

in

N

C

(n,m)

is linear

in the size of

r

nf

;

(ii)

n

and

m

are exponential in the size of

r

nf

;

(iii) The size of the frame

N

C

(n,m)

is double expo-

nential in the size of

r

nf

.

We did not specify the details of the “special

model” and the valuation V

1

, due to restriction on

the size of the paper, but those conditions may be ef-

fectively veriﬁed. Therefore based on Theorem 1,

Lemma 2 and Lemma 4 we conclude

Theorem 2. The logic LD

M A

is decidable.

The veriﬁcation of the fact that a formula ϕ is

a theorem of LD

M A

consists in verifying of valid-

ity of the rule (x → x/ϕ)

nf

in Kripke/Hintikka frames

N

C

(n,m) of size double exponential in the size of re-

duced normal forms. The overall complexity includes

also the complexity of reducing a rule to the normal

reduced form, which is single exponential.

ICEIS 2008 - International Conference on Enterprise Information Systems

128

4 CONCLUSIONS

We propose the logic LD

M A

, which combines linear

discrete time and agents’ accessibility relations in-

side time clusters with interaction of agents via arbi-

trary paths of individual accessibility relations. This

logic seems to be new and interesting, because it is

able to model situations resistant to description by the

standard modal language. We propose an algorithm

for recognizing theorems of LD

M A

(so, we prove

that LD

M A

is decidable), the same algorithm solves

the satisﬁability problem for LD

M A

. Future research

might include, for instance, the case of non-linear

time, i.e., with the time ﬂow modeled by arbitrary re-

ﬂexive and transitive binary relation, with no further

restrictions. Another open and interesting problem is

an explicit axiomatization of LD

M A

and its above-

mentioned extensions.

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