Norm-regulated Transition System Situations
Magnus Hjelmblom
Faculty of Engineering and Sustainable Development, University of G
¨
avle, SE-80176, G
¨
avle, Sweden
Department of Computer and Systems Sciences, Stockholm University, Forum 100, SE-16440, Kista, Sweden
Keywords:
Transition System, Multi-Agent System, Norm-regulated, Norm-governed, Normative System.
Abstract:
Many multi-agent systems (MAS) and other kinds of dynamic systems may be modeled as transition sys-
tems, in which actions are associated with transitions between different system states. This paper presents
an approach to normative systems in this context, in which the permission or prohibition of actions is related
to the permission or prohibition of different types of state transitions with respect to some condition d on a
number of agents x
1
, ..., x
ν
in a state. It introduces the notion of a norm-regulated transition system situation,
which is intended to represent a single step in the run of a (norm-regulated) transition system. The normative
framework uses an algebraic representation of conditional norms and is based on a systematic exploration of
the possible types of state transitions with respect to d(x
1
, ..., x
ν
). A general-level Java/Prolog framework for
norm-regulated transition system situations is currently being developed.
1 INTRODUCTION
Many dynamic systems, including multi-agent sys-
tems (MAS), may be modeled as transition systems,
in which the actions of an agent are associated with
transitions between different states of the system.
There is a number of different approaches to norma-
tive systems in this context. The permission or prohi-
bition of a specific action in a transition system is nat-
urally connected to permissible or prohibited transi-
tions between states of the system, and norms (some-
times referred to as ‘social laws’) may then be formu-
lated as restrictions on states and state transitions.
This paper will introduce the notion of a norm-
regulated transition system situation, which is in-
tended to represent a single step in the run of a
(norm-regulated) transition system. The permission
or prohibition of actions in this framework is related
to the permission or prohibition of different types
of state transitions with respect to some condition
d on a number of agents x
1
, ..., x
ν
in a state. The
framework uses an algebraic representation of condi-
tional norms, based on the representation used in the
norm-regulated DALMAS architecture (see Previous
Work, Sect. 1.2). The novel feature presented here
is primarily an extension to the DALMAS’s norma-
tive framework, based on a systematic exploration of
the possible types of state transitions with respect to
d(x
1
, ..., x
ν
). A norm-regulated transition system sit-
uation is easily instrumentalized into a general-level
Prolog module that can be used to implement a wide
range of specific norm-regulated dynamic systems.
Important norm-related issues such as enforce-
ment of norms, norm change and consistency of nor-
mative systems are beyond the scope of this paper;
however, the approach presented here is general in
nature, and may be combined with many different ap-
proaches to, e.g., norm enforcement. The term ‘agent’
will be frequently used for some sort of ‘acting entity’
within a dynamic system, but no special assumptions
are made about for example autonomy, reasoning ca-
pability, architecture, and so on.
1.1 The Setting
A labelled transition system (LTS) is usually defined
(see for example (Craven and Sergot, 2008, p. 174))
as an ordered 3-tuple hS, E, Ri where S is a non-empty
set of states; E is a set of transition labels, often
called events; and R ⊆ S × E × S is a non-empty set
of labelled transitions. If (s, ε, s
0
) is a transition, s is
the initial state and s
0
is the resulting state of ε. An
event ε is executable in a state s if there is a transition
(s, ε, s
0
) ∈ R, and non-deterministic if there are tran-
sitions (s, ε, s
0
) ∈ R and (s, ε, s
∗
) ∈ R with s
0
6= s
∗
. A
path (or run) of length m (m ≥ 0) of a labelled transi-
tion system is a sequence s
0
ε
0
s
1
··· s
m−1
ε
m−1
s
m
such
that, for i ∈ 1..m, (s
i−1
, ε
i−1
, s
i
) ∈ R.
In the following, we restrict our attention to tran-
sition systems in which all events are deterministic.
This means that, for each state s, the labels associated
with the outgoing transitions from s are distinct. Fur-
thermore, we assume that a ν-ary condition d is true
or false on ν agents x
1
, ..., x
ν
∈ Ω in s, where Ω is
109
Hjelmblom M..
Norm-regulated Transition System Situations.
DOI: 10.5220/0004260801090117
In Proceedings of the 5th International Conference on Agents and Artificial Intelligence (ICAART-2013), pages 109-117
ISBN: 978-989-8565-38-9
Copyright
c
2013 SCITEPRESS (Science and Technology Publications, Lda.)
a set of agents associated with s; this will be written
d(x
1
, ..., x
ν
;s). In the special case that the sequence of
agents is empty, i.e. ν = 0, d represents a proposition
which is true or false in s.
1.2 Related Work
This section will give a brief overview of different ap-
proaches to the design of normative systems and the
formulation of norms. A common feature of many ap-
proaches is the idea to partition states and (possibly)
transitions into two categories, for example ‘permit-
ted’ and ‘non-permitted’. This may be accomplished
with the use of if-then-else rules or constraints on the
states and/or the transitions between states. The Ball-
room system in (Gaertner et al., 2007) and the an-
ticipatory system for plot development guidance in
(Laaksolahti and Boman, 2002) both serve as exam-
ples of this approach. Some approaches are purely
algebraic or based on modal logics, for example tem-
poral or deontic logic. The DALMAS architecture (see
Previous Work below) for norm-regulated MAS is
based on an algebraic approach to the representation
of normative systems. Dynamic deontic logic (Meyer,
1987) and Dynamic logic of permission (van der
Meyden, 1996) are two well-known examples of the
modal logic approach. Other examples are the combi-
nation of temporalised agency and temporalised nor-
mative positions (Governatori et al., 2005), in the set-
ting of Defeasible Logic, and Input/Output Logic by
Makinson and van der Torre (see for example (Makin-
son and van der Torre, 2007)). V
´
azquez-Salceda et
al. use a language consisting of deontic concepts
which can be conditional and can include temporal
operators. They characterize norms by whether they
refer to states (i.e., norms concerning that an agent
sees to it that some condition holds) or actions (i.e.,
norms concerning an agent performing a specific ac-
tion), whether they are conditional, whether they in-
clude a deadline, or whether they are norms concern-
ing other norms. (Vazquez-Salceda et al., 2004) nC +,
an extension of the action language C +, is employed
within the context of ‘coloured agent-stranded transi-
tion systems’ (Craven and Sergot, 2008) to formulate
two kinds of norms: state permission laws and ac-
tion permission laws. A state permission law states
that certain (types of) states are permissible or pro-
hibited, while an action permission law states that
specific (types of) transitions are permissible or pro-
hibited in certain states. By picking out the compo-
nent (‘strand’) corresponding to an individual agent’s
contribution to an event, different categories of non-
compliant behaviour (‘sub-standard’ resp. ‘unavoid-
ably non-compliant’ behaviour) can be distinguished.
Cliffe et al. use Answer Set Programming (ASP) for
representing institutional norms, as part of the repre-
sentation and analysis of specifications of agent-based
institutions. (Cliffe et al., 2006; Cliffe et al., 2007) In
Deontic Petri nets, and variants thereof such as Or-
ganizational Petri nets, varying degrees of ‘ideal’ or
‘sub-ideal’ (more or less ‘allowed’ or ‘preferred’) be-
haviour is modeled by preference orderings on exe-
cutions of Petri nets; see for example (Raskin et al.,
1996; Combettes et al., 2006).
1.2.1 Previous Work: The DALMAS
Architecture
DALMAS (Odelstad and Boman, 2004) is an abstract
architecture for a class of (norm-regulated) multi-
agent systems. A deterministic DALMAS is a simple
multi-agent system in which the actions of an agent
are connected to transitions between system states.
In a deterministic DALMAS the agents take turns to
act; only one agent at a time may perform an action.
Therefore, each individual step in a run of the system
may be represented by a transition system situation.
A DALMAS is formally described by an ordered
9-tuple, where the arguments are various sets, oper-
ators and functions which give the specific DALMAS
its unique features. Of particular interest is the de-
ontic structure-operator, which for each situation of
the system determines an agent’s deontic structure
(i.e., the set of permissible acts) on the feasible acts
in the current situation, and the preference structure-
operator, which for each situation determines the pref-
erence structure on the permissible acts. In a norm-
regulated simple deterministic DALMAS, the deontic
structure consists of all acts that are not explicitly
prohibited by a normative system; thereby employ-
ing what is often referred to as ‘negative permission’.
The preference structure consists of the most prefer-
able (according to the agent’s utility function) of the
acts in the deontic structure. In other words, a DAL-
MAS agent’s behaviour is regulated by the combina-
tion of a normative system and a utility function. The
normative system consists of conditional norms based
on the Kanger-Lindahl theory of normative positions,
expressed in an algebraic notation for norms. See for
example (Lindahl, 1977; Lindahl and Odelstad, 2004;
Odelstad, 2008) for an introduction. A general-level
Java/Prolog implementation of the DALMAS architec-
ture has been developed, to facilitate the implementa-
tion of specific systems. The Colour & Form system,
the Waste-collector system and the Forest Cleaner
system are three specific systems that have been im-
plemented using this framework. The reader is re-
ferred to (Odelstad and Boman, 2004; Hjelmblom,
2008; Hjelmblom and Odelstad, 2009; Hjelmblom,
ICAART2013-InternationalConferenceonAgentsandArtificialIntelligence
110
2011) for a description of these systems and their im-
plementations.
2 NORMATIVE SYSTEMS
AND TYPES OF STATE
TRANSITIONS
Let us consider the transition from a state s to
a following state s
+
, and focus on the condition
d(x
1
, ..., x
ν
). To facilitate reading, X
ν
will be used as
an abbreviation for the argument sequence x
1
, ..., x
ν
.
With regard to d(X
ν
), there are four possible alterna-
tives for the transition from s to s
+
, since in s as well
as in s
+
, d(X
ν
) or ¬d(X
ν
) could hold:
(I) d(X
ν
;s) ∧ d(X
ν
;s
+
)
(II) ¬d(X
ν
;s) ∧ d(X
ν
;s
+
)
(III) d(X
ν
;s) ∧ ¬d(X
ν
;s
+
)
(IV) ¬d(X
ν
;s) ∧ ¬d(X
ν
;s
+
)
Each alternative represents a basic type of transition
with regard to the state of affairs d(X
ν
). Following
the notation in (Sergot, 2008), (I) could be written 0 :
d(X
ν
)∧1 : d(X
ν
), (II) could be written 0 : ¬d(X
ν
)∧1 :
d(X
ν
), and similarly for (III) and (IV).
2.1 Prohibition of State Transition
Types
The set {(I),(II),(III),(IV)} has 16 subsets. Let us ex-
plore the idea that each subset might represent a pro-
hibited combination of basic transition types with re-
gard to d(X
ν
). We may then formulate (conditional)
norms whose normative consequents are represented
by such prohibited combinations. A specific event ε is
taken to be prohibited if, in a certain state s, the nor-
mative system contains a norm which prohibits the
type of transition represented by ε.
The subsets of {(I),(II),(III),(IV)} are summarized
in Table 1, where ‘-’ denotes a permissible transi-
tion type while ‘X’ denotes a prohibited transition
type. For each row, we form the disjunction of the
prohibited transition types to obtain conditions on
state transitions. E.g., row 5 represents the condi-
tion ¬d(X
ν
;s) ∧ d(X
ν
;s
+
); now, we make the corre-
sponding stipulation that if ¬d(X
ν
;s) and d(X
ν
;s
+
),
and (s, ε, s
+
) ∈ R, then the event ε is not permissible
in state s. Another example is row 8, in which the
disjunction of (II),(III) and (IV) yields the condition
¬d(X
ν
;s) ∧ d(X
ν
;s
+
)
∨
d(X
ν
;s) ∧ ¬d(X
ν
;s
+
)
∨
¬d(X
ν
;s) ∧ ¬d(X
ν
;s
+
)
which may be simplified to ¬d(X
ν
;s) ∨ ¬d(X
ν
;s
+
).
A closer look at Table 1 reveals, however, that
not all rows seem to represent meaningful norms.
For example, the disjunction (I) and (III) in
row 11 yields the condition (d(X
ν
;s) ∧ d(X
ν
;s
+
)) ∨
(d(X
ν
;s) ∧ ¬d(X
ν
;s
+
)), which may be simplified to
d(X
ν
;s). Let us try to make the corresponding stipu-
lation that if d(X
ν
;s), and (s, ε, s
+
) ∈ R, then ε is not
permissible in s. This would mean that, if d(X
ν
;s),
then no events are permissible in s. In other words,
if d(X
ν
;s) holds before the execution of an event,
then all events are prohibited, no matter their re-
sult. This can hardly represent a meaningful norm;
after all, an event in a state s can only affect the
truth of d(X
ν
) in a following state s
+
. Another ex-
ample is row 6, which expresses that if ¬d(X
ν
;s),
then no event is permissible, since neither of transi-
tion type (II) or (IV) is permissible. We conclude that
all rows that contain {(I),(III)} or {(II),(IV)} repre-
sent norms that are not meaningful. Table 2 contains
the rows (slightly reordered) that represent mean-
ingful normative conditions. For each of the rows
in Table 2 (except Π
1
, which expresses no restric-
tions at all) we define a ‘transition type operator’ τ
j
,
j ∈ {2, 2
0
, 4, 4
0
, 5, 6, 6
0
, 7}, based on the corresponding
row in the table:
1
For all ν-ary conditions d and for all agents
x
1
, . . . , x
ν
and all states s and s
0
such that (s, ε, s
0
) ∈ R,
1. τ
ε
2
d(X
ν
;s) iff [d(X
ν
;s) ∧ ¬d(X
ν
;s
0
)]
2. τ
ε
2
0
d(X
ν
;s) iff [¬d(X
ν
;s) ∧ ¬d(X
ν
;s
0
)]
3. τ
ε
4
d(X
ν
;s) iff [¬d(X
ν
;s) ∧ d(X
ν
;s
0
)]
4. τ
ε
4
0
d(X
ν
;s) iff [d(X
ν
;s) ∧ d(X
ν
;s
0
)]
5. τ
ε
5
d(X
ν
;s) iff ¬d(X
ν
;s
0
)
6. τ
ε
6
d(X
ν
;s) iff ¬[d(X
ν
;s) ↔ d(X
ν
;s
0
)]
7. τ
ε
6
0
d(X
ν
;s) iff [d(X
ν
;s) ↔ d(X
ν
;s
0
)]
8. τ
ε
7
d(X
ν
;s) iff d(X
ν
;s
0
)
The ‘transition type condition’ τ
ε
j
d(X
ν
;s) indicates
whether or not, in state s, the event ε has transition
type j with regard to d(X
ν
). We note in passing that
the following ‘symmetry principles’ hold (cf. the ob-
servation in (Odelstad and Boman, 2004, p. 148)):
1. τ
ε
2
d(X
ν
;s) iff τ
ε
4
¬d(X
ν
;s)
2. τ
ε
2
0
d(X
ν
;s) iff τ
ε
4
0
¬d(X
ν
;s)
3. τ
ε
5
d(X
ν
;s) iff τ
ε
7
¬d(X
ν
;s)
4. τ
ε
6
d(X
ν
;s) iff τ
ε
6
¬d(X
ν
;s)
5. τ
ε
6
0
d(X
ν
;s) iff τ
ε
6
0
¬d(X
ν
;s)
1
The numbering is based on the numbering used in (Hjelm-
blom, 2011).
Norm-regulatedTransitionSystemSituations
111
Table 1: Possible Combinations of Prohibited State Transition Types.
(I) (II) (III) (IV)
1 - - - - -
2 - - - X ¬d(X
ν
;s) ∧ ¬d(X
ν
;s
+
)
3 - - X - d(X
ν
;s) ∧ ¬d(X
ν
;s
+
)
4 - - X X ¬d(X
ν
;s
+
)
5 - X - - ¬d(X
ν
;s) ∧ d(X
ν
;s
+
)
6 - X - X ¬d(X
ν
;s)
7 - X X - ¬(d(X
ν
;s) ↔ d(X
ν
;s
+
))
8 - X X X ¬d(X
ν
;s) ∨ ¬d(X
ν
;a(x, s))
9 X - - - d(X
ν
;s) ∧ d(X
ν
;s
+
)
10 X - - X d(X
ν
;s) ↔ d(X
ν
;s
+
)
11 X - X - d(X
ν
;s)
12 X - X X d(X
ν
;s) ∨ ¬d(X
ν
;s
+
)
13 X X - - d(X
ν
;s
+
)
14 X X - X ¬d(X
ν
;s) ∨ d(X
ν
;s
+
)
15 X X X - d(X
ν
;s) ∨ d(X
ν
;s
+
)
16 X X X X >
Table 2: Meaningful Combinations of Prohibited State Transition Types.
(I) (II) (III) (IV) τ
ε
j
d(X
ν
;s)
- - - - -
- - X - d(X
ν
;s) ∧ ¬d(X
ν
;s
+
)
- - - X ¬d(X
ν
;s) ∧ ¬d(X
ν
;s
+
)
- X - - ¬d(X
ν
;s) ∧ d(X
ν
;s
+
)
X - - - d(X
ν
;s) ∧ d(X
ν
;s
+
)
- - X X ¬d(X
ν
;s
+
)
- X X - ¬(d(X
ν
;s) ↔ d(X
ν
;s
+
))
X - - X d(X
ν
;s) ↔ d(X
ν
;s
+
)
X X - - d(X
ν
;s
+
)
Next, we define a normative ‘transition type prohibi-
tion operator’ Π
j
, j ∈ {1, 2, 2
0
, 4, 4
0
, 5, 6, 6
0
, 7}, such
that for all ν-ary conditions d and for all agents
x
1
, . . . , x
ν
and for all states s, and for all events ε such
that (s, ε, s
0
) ∈ R,
Π
j
d(X
ν
;s) iff [τ
ε
j
d(X
ν
;s) is forbidden].
Now suppose that Π
j
d(X
ν
;s) holds (is ‘in effect’),
and that the corresponding transition type condition
τ
ε
j
d(X
ν
;s) also holds for some event ε. Then we forbid
the transition (s, ε, s
0
): For all states s and all events ε
such that (s, ε, s
0
) ∈ R,
Prohibited
s
(ε) if there exists a condition c,
a sequence of agents x
1
, . . . , x
ν
,
and a j ∈ {2, 2
0
, 4, 4
0
, 5, 6, 6
0
, 7} such that
Π
j
d(x
1
, . . . , x
ν
;s) & τ
ε
j
c(x
1
, . . . , x
ν
;s).
The representation of a normative system N
based on transition type conditions Π
j
d is similar to
the one used in the DALMAS architecture (see Sect.
1.2.1); a norm in N is represented by an ordered pair
hG, Ci where G is the (descriptive) ground of the norm
and C is its (normative) consequence.
2
For example,
2
This view of normative systems is further developed
and compared with other approaches in (Lindahl and Odel-
stad, 2012). It bears some resemblance to the treatment
of norms in Input/Output Logic (Makinson and van der
Torre, 2007); conditional norms are simply treated as or-
dered pairs, and are not assumed to bear truth-values. In In-
ICAART2013-InternationalConferenceonAgentsandArtificialIntelligence
112
the elementary norm hg, Π
j
ci represents the sentence
∀x
1
, x
2
, ..., x
ν
∈ Ω :
g(x
1
, x
2
, ..., x
p
;s) → Π
j
c(x
1
, x
2
, ..., x
q
;s)
where Ω is the set of agents and ν = max(p, q).
If the condition specified by the ground of a norm
is true in some situation, then the (normative) con-
sequence of the norm is in effect in that situa-
tion. In the example above, if hg, Π
j
ci is a norm in
N , and g(x
1
, x
2
, ..., x
p
;s), and τ
ε
j
c(x
1
, . . . , x
q
;s), then
Prohibited
s
(ε) according to N :
Prohibited
s
(ε) according N
if there exists a condition g and a condition c and a
j ∈ {2, 2
0
, 4, 4
0
, 5, 6, 6
0
, 7} such that hg, Π
j
ci is a norm
in N , and there exists x
1
, . . . , x
ν
such that
g(x
1
, . . . , x
ν
;s) & τ
ε
j
c(x
1
, . . . , x
ν
;s).
2.2 Prohibition of Actions
By adding the requirements that (1) each event ε is of
the form x : a, i.e., represents an action a performed
by an individual agent x, and that (2) norms apply
to an individual agent x in a state s, it is straightfor-
ward to interpret the prohibition of state transitions as
prohibition of actions. A specific action a is taken
to be prohibited for x in s if the normative system
contains a norm which prohibits the type of transition
represented by the event ε = x : a: Prohibited
x,s
(a) if
Prohibited
s
(x : a). This idea will be further elaborated
in the following section.
3 NORM-REGULATED
TRANSITION SYSTEM
SITUATIONS
In the following, we focus on an arbitrary state in a
deterministic LTS, with the added requirement that
each event ε represents an action performed by a sin-
gle agent x. The term ‘transition system situation’ will
be used for an ordered 5-tuple S = hx, s
0
, A, Ω, Si char-
acterized by a set of states S, an initial state s
0
∈ S, an
agent-set Ω = {x
1
, ..., x
n
}, an acting (‘moving’) agent
x, and an action-set A = {a
1
, ..., a
m
}. An event ε is of
the form x : a, i.e., it refers both to an agent x and an
action a. In this setting, an action a may be regarded
as a function such that a(x, s) = s
+
means that s
+
is
the resulting state when x performs act a in state s.
3
put/Output Logic however, it is the conditional as a whole
that represents, for example, a permission or prohibition,
while in Lindahl’s and Odelstad’s algebraic approach it is
the consequence of the norm that is normative.
3
Note that s
0
may also be an element of S, i.e. the action a may
lead back to s
0
.
Figure 1: A state diagram for a transition system situation
with three events.
In the following, the abbreviation s
+
will be used for
a(x, s
0
) when there is no need for an explicit reference
to the action a and the acting agent x.
As indicated by Fig. 1, a transition system situ-
ation is intended to represent, for example, a ‘snap-
shot’ of a labelled transition system in which each
transition is deterministic and represents the action of
a single agent. In this case, s
0
represents an arbitrar-
ily chosen state in the LTS, and S is the set of states
reachable from s
0
by the transitions x : a, for all a ∈ A.
At the same time, a transition system situation is de-
signed to be general enough to also represent a step
in a run of other kinds of dynamic systems, including
systems modeled by finite automata (see for example
(Laaksolahti and Boman, 2002)) or Petri nets, and de-
terministic DALMASes.
A norm-regulated transition system situation is
represented by an ordered pair hS, N i where S =
hx, s, A, Ω, Si is a transition system situation and
N is a normative system. For each τ
ε
j
, j ∈
{2, 2
0
, 4, 4
0
, 5, 6, 6
0
, 7}, we define a corresponding op-
erator C
a
j
: For all ν-ary conditions d and for all agents
x
1
, . . . , x
ν
, x
ν+1
and all actions a ∈ A,
C
a
j
d(x
1
, ..., x
ν
, x
ν+1
;x, s) iff
τ
ε
j
d(x
1
, ..., x
ν
;s) & ε = x
ν+1
: a.
The extra argument x
ν+1
denotes the agent to which
the normative condition applies.
4
Next, we define
a normative transition type prohibition operator P
j
,
j ∈ {1, 2, 2
0
, 4, 4
0
, 5, 6, 6
0
, 7}, such that for all ν-ary
conditions d and for all agents x
1
, . . . , x
ν
, x
ν+1
∈ Ω and
for all actions a ∈ A,
4
It may seem unnecessary to introduce x
ν+1
; one might
expect that the agent to which the normative condition ap-
plies is always the same as the ’moving’ agent x. This dis-
tinction is made to allow for normative systems in which,
for example, agents other than the ’moving’ agent may per-
form punishments or other ’reaction acts’.
Norm-regulatedTransitionSystemSituations
113
P
j
d(x
1
, ..., x
ν
, x
ν+1
;x, s) iff
[C
a
j
d(x
1
, ..., x
ν
, x
ν+1
;x, s) is forbidden].
Now suppose that P
j
d(X
ν
, x
ν+1
;x, s) holds (is ‘in ef-
fect’), and that the corresponding transition type con-
dition C
a
j
d(X
ν
, x
ν+1
;x, s) also holds for some action a
and some agent x
ν+1
. Then we forbid a for x
ν+1
: For
all actions a ∈ A and all agents x
ν+1
∈ Ω,
Prohibited
x,s
(x
ν+1
, a) if there exists a condition d,
a sequence of agents x
1
, . . . , x
ν
,
and a j ∈ {2, 2
0
, 4, 4
0
, 5, 6, 6
0
, 7}, such that
P
j
d(x
1
, . . . , x
ν
, x
ν+1
;x, s) & C
a
j
d(x
1
, ..., x
ν
, x
ν+1
;x, s).
To ensure that the agent x
ν+1
to which the norm ap-
plies is the same as the moving agent, we apply the
‘move operator’ M which transforms the condition
g on ν agents in a state s to a condition on ν + 1
agents in the situation hx, si, characterized by s and
the moving agent x, while at the same time identify-
ing x
ν+1
with the ‘moving’ agent x. (See (Odelstad
and Boman, 2004; Hjelmblom, 2008) for a deeper ex-
planation of the operator M.) Action a is prohibited
for x
ν+1
in the situation hx, si if the normative system
contains a norm which prohibits the type of transition
represented by the event x
ν+1
: a:
Prohibited
x,s
(x
ν+1
, a) according to N
if there exists a condition g and a condition c
and a j ∈ {2, 2
0
, 4, 4
0
, 5, 6, 6
0
, 7}
such that hMg, P
j
ci is a norm in N ,
and there exists x
1
, . . . , x
ν
such that
Mg(x
1
, . . . , x
ν
, x
ν+1
;s) & C
a
j
c(x
1
, . . . , x
ν
, x
ν+1
;s).
As in Sect. 2.1, a norm in N is represented by an
ordered pair hG, Ci, where G is a descriptive, and C a
normative, condition on a situation hx, si. For exam-
ple, hMg, P
j
ci represents the sentence
∀x
1
, x
2
, ..., x
ν
, x
ν+1
∈ Ω :
Mg(x
1
, x
2
, ..., x
p
, x
ν+1
;x, s) →
P
j
c(x
1
, x
2
, ..., x
q
, x
ν+1
;x, s)
where Ω is the set of agents, x
ν+1
is the agent to which
the norm applies, x is the ‘moving’ agent in the situa-
tion hx, si, and ν = max(p, q). If the condition speci-
fied by the ground of a norm is true in some situation,
then the (normative) consequence of the norm is in ef-
fect in that situation. Since each situation for a DAL-
MAS can be viewed as a transition system situation,
it is straightforward to evolve the existing general-
level Java/Prolog implementation of the DALMAS ar-
chitecture (see Sect. 1.2.1) into a general-level imple-
mentation of a norm-regulated transition system situ-
ation. A norm is then represented by a Prolog term
n/3 of the form n(Id/N,OpG*G,OpC*C), where Id is
an identifier of a norm-system and N is an identifier
of an individual norm. OpG*G is a compound term
representing an operator OpG applied to (the functor
of) a state condition predicate G, forming the norm’s
ground. Similarly, OpC*C represents the norm’s con-
sequence.
3.1 Applications
The existing implementations of the Colour & Form
system, the Waste-collector system and the For-
est Cleaner system are easily adapted to serve as
demonstrations of the use of norm-regulated transi-
tion system situations. However, the use of norm-
regulated transition system situations is not limited to
the DALMAS context. Many kinds of dynamic sys-
tems (including different types of transition systems
and multi-agent systems) in which state transitions are
connected to the actions of a single ‘moving’ agent,
could be modelled and implemented by (iterated) use
of a norm-regulated transition system situation. The
general-level Java/Prolog implementation is intended
to serve as a tool for the implementation of such sys-
tems, with a Prolog logic server as a backend and
a Java user interface as frontend
5
, functioning as a
lookup-service that answers questions such as ‘is act
a permissible for x in state s, according to normative
system N ’ or ‘which acts are permissible for x in state
s, according to N ’. At the system level, it could be
used to maintain a normative system for some society,
in combination with some norm enforcement strategy.
At the agent level, it could be used as a common nor-
mative framework that is shared by individual agents
that take norms into account in their reasoning cycle,
or as part of an agent’s internal architecture, either
to represent a model of society’s normative system
or to represent an agent’s ‘internal’ normative system
(‘ethics’). Naturally, the use of both Java and Prolog
as implementation languages has both advantages and
disadvantages. The primary advantage is that this ap-
proach combines the strengths of two different pro-
gramming paradigms and languages. On the other
hand, it puts high demands regarding skills in both
object-oriented and logic programming on the devel-
oper wishing to use the framework to develop a spe-
cific system.
4 CONCLUSIONS AND FUTURE
WORK
This paper has introduced the notion of a transition
system situation, which is intended to represent a sin-
5
The source code will be made available for download
and will be publicly and freely disseminated.
ICAART2013-InternationalConferenceonAgentsandArtificialIntelligence
114
gle step in the run of many kinds of transition sys-
tems. In a norm-regulated transition system situa-
tion, the permission or prohibition of actions is related
to the permission or prohibition of different types of
state transitions with respect to some condition d on a
number of agents x
1
, . . . , x
ν
in a state. The framework
uses a representation of conditional norms based on
the algebraic approach
6
to normative systems used in
(Odelstad and Boman, 2004) and a systematic explo-
ration of the possible types of state transitions with
respect to d(x
1
, . . . , x
ν
). A general-level Java/Prolog
framework for norm-regulated transition system situ-
ations is currently being developed, by adaption of the
existing implementation of the DALMAS architecture.
The set of eight transition type conditions C
a
i
is an ex-
tension of the set of six E
a
i
conditions in (Odelstad
and Boman, 2004). These conditions were intended
as an interpretation in the DALMAS context of Lin-
dahl’s set of one-agent types of normative positions.
The (potential) connection between the combination
of τ
i
and C
a
i
and the Kanger-Lindahl theory of nor-
mative positions is interesting. It has been partly in-
vestigated in (Hjelmblom, 2011), but deserves to be
further explored.
Lindahl and Odelstad argue that a normative sys-
tem should express ”... general rules where no in-
dividual names occur. If the task is to represent a
normative system this feature of generality has to be
taken into account.” (Lindahl and Odelstad, 2012,
p. 5) One of the strengths of their algebraic ap-
proach to normative systems is in fact the expres-
sive power it yields. The algebraic normative frame-
work presented in this paper allows the construction
of norms based on conditions on an arbitrary num-
ber of agents, in contrast to for example Dynamic
deontic logic (Meyer, 1987) and Dynamic logic of
permission (van der Meyden, 1996) which both have
their roots in Propositional Dynamic Logic (PDL).
Unlike in the agent-stranded coloured transition sys-
tems (Craven and Sergot, 2008; Sergot, 2008), the
framework presented in this paper does not explicitly
distinguish between state permission laws and action
permission laws. It allows, however, a state permis-
sion law to be represented implicitly as a special case,
by a norm which prohibits all transitions that lead to
an undesired state. Our framework treats all norms as
action permission laws, in the sense that actions are
prohibited in different states as a consequence of cer-
tain transition types being prohibited by the norma-
tive system. It allows the creation of norms that forbid
6
This approach was originally developed in a series of
papers; see for example (Lindahl and Odelstad, 2003; Lin-
dahl and Odelstad, 2008; Lindahl and Odelstad, 2011; Lin-
dahl and Odelstad, 2012).
specific named actions in certain situations, by choos-
ing a normative consequence that forbids the agent to
act so that it ends up in a state where the last action
performed was the prohibited action. This requires
some sort of history of actions to be part of the state
of the system.
The idea to base norms on permissible and pro-
hibited types of state transitions has, to the author’s
knowledge, not been systematically explored before.
It appears that the language for action permission laws
used by Craven and Sergot also allows the formu-
lation of norms that prohibit certain types of tran-
sitions, but an example of this feature is not given
in (Craven and Sergot, 2008). In Dynamic deon-
tic logic it is only the state resulting from a transi-
tion that determines if the transition is classified as
‘permitted/non-permitted’, while in Dynamic logic of
permission, it is executions of actions that are classi-
fied as ‘permitted/non-permitted’. van der Meyden’s
treatment of permission uses the process semantics
for actions, in which the denotation of an action ex-
pressions is a set of sequences of states. This allows
for the description of the states of affairs during the
execution of an action; the permission of an action
is not dependent only on the state resulting from the
execution of the action, but also on the intermediate
states.
The systematic treatment of the different types of
transitions ensures that the set of transition type oper-
ators C
a
j
and the corresponding prohibition operators
P
i
exhaust the space of meaningful transition type pro-
hibitions. Therefore, norm-regulated transition sys-
tem situations could be used in a given problem do-
main to systematically search for the ‘best’ normative
system for (a class of) dynamic systems, according
to some criteria for evaluation of the system’s perfor-
mance. For example, as suggested in (Hjelmblom,
2011), a genetic algorithm or some other mechanism
from machine learning could be employed to seek the
optimal normative system for a particular task.
The requirement that each event ε in a norm-
regulated transition system situation represents an ac-
tion performed by a single agent deserves further at-
tention. This bears some resemblance to the restric-
tion in the ‘rooms’ example in (Craven and Sergot,
2008, p. 178ff) that only one agent at a time can
move through a doorway in this environment. This
raises a number of questions regarding the relation-
ship between norm-regulated transition system situ-
ations and transition systems in which a single tran-
sition may correspond to the simultaneous action of
several agents, possibly including ‘actions’ by the en-
vironment itself. These issues deserve a deeper dis-
cussion, which is left for future papers.
Norm-regulatedTransitionSystemSituations
115
It is possible to combine a norm-regulated transi-
tion system situation with some mechanism for norm
change, but in the current formulation norms may not
be changed as a consequence of an action by an agent
in a state s, since the normative system N is not itself
considered a part of s. An interesting line of future
work is to explore the possibility to let the normative
system be a part of the state, thereby letting agents
choose actions that modify the normative system. An-
other interesting issue is consistency; an inconsistent
normative system may lead to a situation in which the
deontic structure is empty, i.e. all actions are prohib-
ited. How the system should behave in such a situ-
ation is heavily dependent on the nature of the spe-
cific application at hand; this is not specified by the
general-level framework.
ACKNOWLEDGEMENTS
The author wishes to thank Jan Odelstad and Magnus
Boman for valuable ideas and suggestions.
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