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
[
iN
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
iN
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
[
iN
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
iN
C(i), so in sym-
bols,
p P :V(p)
[
iN
C(i).
If, for an element a
S
iN
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
^
1im
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 jm
^
1in
"
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)
^
1sm
(¬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 jm
(
V
1in
[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
1sm
(¬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
1im
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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