Normative positions in multi-agent systems

Abstract

The Kanger-Lindahl theory of normative positions has great potential of serving as a logical foundation for normative systems for MAS, and its generality allows for great freedom when interpreting the theory. As a first step towards a typology of interpretations of the theory, the application of normative positions is studied in the context of a class of transition systems in which transitions are deterministic and associated with a single agent performing an act. By an interpretation of different types of normative positions in terms of permitting or prohibiting different state transition types in this context, lexicons for two different systems of types of normative positions are suggested and discussed. It is demonstrated that both interpretations are useful foundations for normative systems semantics in a MAS context.

Keywords

Normative MAS norm-regulated norm-governed normative positions transition system

1. Introduction

The study of norm-regulated multi-agent systems, often referred to as normative MAS, covers the formal representation and implementation of normative systems as well as applications. The ‘normative MAS roadmap’ [1] is a comprehensive introduction to and overview of the field. In many systems, the actions of an individual agent are naturally associated with transitions between different states of the system. As a consequence, the permission or prohibition of a specific action in such a system is connected to permissible or prohibited transitions between states of the system, and norms may then be formulated as restrictions on states and state transitions. Many approaches to normative systems are algebraic or based on modal logics, like temporal or deontic logic [5,6,8,27,29,31,33,34].

The Kanger-Lindahl theory of normative positions is well suited as the logical foundation for normative systems in a MAS context, since the types of normative positions are mutually exclusive and jointly exhaustive in the logical sense. From a theoretical point of view, the weak underlying assumptions makes the theory elegant and general, something which allows for great freedom when interpreting the theory. On the other hand, this may become a challenge for practical applications. The notion of agency, for example, can be understood in many quite different ways within the theory. Therefore, a kind of typology of interpretations of the theory of normative positions might be useful. This work may serve as a first step towards such a typology, by investigating how to apply the theory of one-agent normative positions in the context of a class of transition systems, in which transitions are deterministic and associated with a single agent performing an act. To study this class of systems, the notion of norm-regulated transition system situations [13,14] will be employed. Here, the permission or prohibition of actions is connected to the permission or prohibition of different types (with respect to some condition on a number of agents in a state) of state transitions. By interpreting two different extended systems of one-agent types of normative positions in terms of permitting or prohibiting different transition types, a semantics for each of these extended systems in the context of norm-regulated transition system situations will be obtained.

The paper is structured as follows. Section 1.1 gives a brief introduction to the theory of normative positions, Section 1.2 presents an algebraic approach to normative systems, and Section 1.3 presents related work, in particular an extended system of normative positions suggested by Jones and Sergot. In Section 1.4, the notion of norm-regulated transition system situations is presented, and an application to be used as a running example is introduced. Section 2, the main contribution, discusses how to map Jones and Sergot’s extended system of normative positions, as well as another extension based on an observation by Odelstad, to the set of transition type prohibition operators within norm-regulated transition system situations. Section 3 briefly discusses possible applications, and Section 5 concludes and gives some ideas for future work.

1.1. One-agent types of normative positions

Table 1
One-agent types of normative positions

$T_{1}$ $May Do (x, F) \land May [\neg Do (x, F) \land \neg Do (x, \neg F)] \land May Do (x, \neg F)$

$T_{2}$ $May Do (x, F) \land May [\neg Do (x, F) \land \neg Do (x, \neg F)] \land \neg May Do (x, \neg F)$

$T_{3}$ $May Do (x, F) \land \neg May [\neg Do (x, F) \land \neg Do (x, \neg F)] \land May Do (x, \neg F)$

$T_{4}$ $\neg May Do (x, F) \land May [\neg Do (x, F) \land \neg Do (x, \neg F)] \land May Do (x, \neg F)$

$T_{5}$ $May Do (x, F) \land \neg May [\neg Do (x, F) \land \neg Do (x, \neg F)] \land \neg May Do (x, \neg F)$

$T_{6}$ $\neg May Do (x, F) \land May [\neg Do (x, F) \land \neg Do (x, \neg F)] \land \neg May Do (x, \neg F)$

$T_{7}$ $\neg May Do (x, F) \land \neg May [\neg Do (x, F) \land \neg Do (x, \neg F)] \land May Do (x, \neg F)$

The Kanger-Lindahl theory of normative positions is based on Kanger’s ‘deontic action-logic’; see for example [19]. The theory, further developed by Lindahl in [21], contains three systems of types of normative positions. The simplest of these systems is a system of seven ‘one-agent types’ of normative positions, based on the logic of the action operator $Do$ and the deontic operator $Shall$ . $Do (x, F)$ is read as ‘x sees to it that F’ or ‘x brings it about that F’, where F is a proposition regarding some state of affairs. The logical properties assumed for $Do$ is that it is the smallest system containing propositional logic, closed under logical equivalence and containing the axiom schema $Do (x, F) \to F$ . The latter schema tries to capture the notion of successful action; if x ‘sees to it’ or ‘brings about’ that F, then F is indeed the case. The usual definitions are taken for ¬, ∧, ∨ and all other truth-functional connectives.

Each of the three statements (i) $Do (x, F)$ , (ii) $Do (x, \neg F)$ and (iii) $\neg Do (x, F) \land \neg Do (x, \neg F)$ implies the negation of each of the others, and the disjunction of all three is a tautology. Each of (i)–(iii) can be prefixed with either $May$ or $\neg May$ , where $May F$ is defined as $\neg Shall \neg F$ , and basic conjunctions containing one statement from each such pair are formed. By iterated construction of basic conjunctions, a set of eight conjunctions is obtained. One such ‘maxi-conjunction’ is self-contradictory, the others are listed in Table 1. Seven types $T_{i}$ are defined as sets of tuples $⟨ x, F ⟩$ fulfilling the different conjunctions. The type names will often be used as syntactic elements, representing the expressions in the right column. Note that Lindahl uses italic T in the type names, whereas boldface roman T is used here.

Some further extensions of the systems of normative positions have been suggested by Jones and Sergot; see Section 1.3.1. They have explored some applications of the theory within computer science, and discussed some of its limitations in this setting.

1.2. An algebraic approach to norms

In a series of papers, Lindahl and Odelstad have combined the theory of normative positions with an algebraic approach to normative systems. The reader is referred to [24] for a comprehensive summary. Their idea is to use the one-agent types of normative positions as operators on descriptive conditions to get deontic conditions. A ν-ary condition d can be true or false of ν agents $x_{1}, \dots, x_{ν}$ . Thus, $d (x_{1}, \dots, x_{ν})$ is a state of affairs which may be true or false. To facilitate the presentation, $X_{ν}$ will often be used as an abbreviation for the argument sequence $x_{1}, \dots, x_{ν}$ . In the special case when the sequence of agents is empty, i.e. $ν = 0$ , d represents a proposition which may be true or false. Note that negations $d^{'}$ , conjunctions $c \cap d$ , and disjunctions $c \cup d$ can be formed in the following way: $\begin{matrix} \begin{matrix} d^{'} (X_{ν}) iff \neg d (X_{ν}), \\ (c \cap d) (X_{ν}) iff c (X_{p}) \land d (X_{q}), and \\ (c \cup d) (X_{ν}) iff c (X_{p}) \lor d (X_{q}) \end{matrix} \end{matrix}$ where $ν = max (p, q)$ .1

¹
The free variables in $c (x_{1}, \dots, x_{p})$ must be the same, and in the same order, as the free variables in $d (x_{1}, \dots, x_{q})$ , but it is not necessary that p and q have the same arity. Cf. [29, p. 146].

Therefore, it is possible to construct Boolean algebras of conditions. A Boolean algebra together with an implicative relation R forms a so-called Boolean quasiordering (Bqo). As an application of their Theory of Joining-Systems (TJS), Lindahl and Odelstad define the notion of a normative position condition-implication structure, abbreviated np-cis, which is based on Bqo’s on descriptive and deontic conditions, so-called cis-Bqo’s. For details on Boolean quasiorderings, condition implication structures and np-cis’es, see for example [24] or [29]. An abstract architecture, the Dalmas architecture, for a class of norm-regulated multi-agent systems based on this algebraic approach to norms was developed in [29]. A general-level Java/Prolog instrumentalization of the Dalmas architecture has been developed, to facilitate the implementation of specific systems [11,16].

1.3. Related work

A number of different ways of classifying norms and normative systems are discussed in [2]. It is argued that when characterizing norms in MAS one should take into account the rule structure of norms, i.e., that norms usually have a conditional structure, as well as different types of norms, and other basic features of normative systems such as norm recognition and hierarchies, norm application, and norm change. Furthermore, three different definitions of normative MAS are given, as well as a number of guidelines for developing normative MAS. The first guideline, for example, is to motivate which definition is used and explain the choices made regarding the representation of norms in the system. It is noted in [2] that an attempt to address this guideline requires the clarification of the types of norms that may occur in normative systems.

A common simple classification of norms is into two broad categories, constitutive norms and behavioral norms. Norms in the former class specify, for example, the meaning of different kinds of communicative acts within a given institution, or the definitions of what kind of acts are meaningful in a certain context. Norms of the latter kind regulate behavior, by somehow specifying which actions are permitted, prohibited or even obligatory. Such norms may be further classified according to the ‘level’ on which they are effective; whether the system is viewed from the point of view of a system designer or an individual agent. See for example [6, p. 217f] or [31, Section 4]. As Sergot puts it, system norms

…express a system designer’s point of view of what system states and transitions are legal, permitted, desirable, and so on. There is a separate category of individual agent-specific norms that are intended to guide an individual agent’s behaviors and are supposed to be taken into account in the agent’s implementation, or reasoning processes, in one way or another. These have a different character. [31, p. 16]

A common feature of many approaches to the representation of norms and normative systems is the idea to partition states and (possibly) transitions into two categories, for example ‘permitted’ 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 ‘agent-stranded transition systems’ framework by Craven and Sergot [5,31], the Ballroom system in [6] and the anticipatory system for plot development guidance in [20] serve as examples of this approach. Some approaches are purely algebraic or based on modal logics, for example temporal or deontic logic. Dynamic deontic logic [27] and Dynamic logic of permission [33] are two well-known examples of the modal logic approach. Other examples are the combination of temporalized agency and temporalized normative positions [8], in the setting of Defeasible logic, and Input/Output logic (see for example [26]). A more extensive account of related work on norm representation can be found in [14].

The only assumption made by Kanger regarding the logic for the action operator $Do$ was that $Do (x, F)$ implies F. Besides this rather simple approach to the notion of action and agency, there are many other approaches, of which stit theory by Belnap and Perloff [3] is perhaps the most well known. Variants of the stit operator have been suggested by (among others) Horty and Belnap; a brief overview is given in [4]. The characterization of action by Governatori et al. in [8] is based on ${Brings}_{i} A$ and ${Does}_{i} A$ . The former is read as ‘agent i brings it about that A’, and is concerned with the result achieved by the agent, while the latter is used to specify an agent’s behavior. It might be fruitful to develop a theory of normative positions based on some variant of stit theory or the operators suggested by Governatori et al., but these lines of thought are not further developed here; for the purposes of this paper, Kanger’s simplistic approach will be a sufficient basis.

1.3.1. An extended set of types of normative positions

Table 2
Table 2 in [32]

$T_{1}$ $\{\begin{matrix} {P E}_{x} F \land {P E}_{x} \neg F \land P (F \land \neg E_{x} F) \land P (\neg F \land \neg E_{x} \neg F) \\ {P E}_{x} F \land {P E}_{x} \neg F \land \neg P (F \land \neg E_{x} F) \land P (\neg F \land \neg E_{x} \neg F) \\ {P E}_{x} F \land {P E}_{x} \neg F \land P (F \land \neg E_{x} F) \land \neg P (\neg F \land \neg E_{x} \neg F) \end{matrix}\}$

$T_{2}$ $\{\begin{matrix} {P E}_{x} F \land \neg {P E}_{x} \neg F \land P (F \land \neg E_{x} F) \land P (\neg F \land \neg E_{x} \neg F) \\ {P E}_{x} F \land \neg {P E}_{x} \neg F \land \neg P (F \land \neg E_{x} F) \land P (\neg F \land \neg E_{x} \neg F) \\ {P E}_{x} F \land \neg {P E}_{x} \neg F \land P (F \land \neg E_{x} F) \land \neg P (\neg F \land \neg E_{x} \neg F) \end{matrix}\}$

$T_{3}$ $\{{P E}_{x} F \land {P E}_{x} \neg F \land \neg P {Pass}_{x} F\}$

$T_{4}$ $\{\begin{matrix} \neg {P E}_{x} F \land {P E}_{x} \neg F \land P (F \land \neg E_{x} F) \land P (\neg F \land \neg E_{x} \neg F) \\ \neg {P E}_{x} F \land {P E}_{x} \neg F \land \neg P (F \land \neg E_{x} F) \land P (\neg F \land \neg E_{x} \neg F) \\ \neg {P E}_{x} F \land {P E}_{x} \neg F \land P (F \land \neg E_{x} F) \land \neg P (\neg F \land \neg E_{x} \neg F) \end{matrix}\}$

$T_{5}$ $\{O E_{x} F\}$

$T_{6}$ $\{\begin{matrix} O {Pass}_{x} F \land O F \\ O {Pass}_{x} F \land O \neg F \\ O {Pass}_{x} F \land P F \land P \neg F \end{matrix}\}$

$T_{7}$ $\{{O E}_{x} \neg F\}$

Jones and Sergot [17,30,32] have generalized and further developed the Kanger-Lindahl theory of normative positions. Using a method suitable for automation, they perform a similar analysis as Lindahl. First, a set of ‘act positions’

$E_{x} F$

$E_{x} \neg F$

$\neg E_{x} F \land \neg E_{x} \neg F$

is generated from the scheme

⟦ \pm E_{x} \pm F ⟧

, where the operator

E_{x}

corresponds to the Kanger-Lindahl

Do

operator and F is some proposition.2

$⟦ \pm E_{x} \pm F ⟧$ stands for the set of maxi-conjunctions of $\pm E_{x} \pm F$ , i.e., the maximal consistent conjunctions of expressions of the form $\pm E_{x} \pm F$ , where ± stands for the two possibilities of affirmation and negation [25].

Using O (‘obligatory’) for

Shall

and P (‘permissible’) for

May

, each of the three logically consistent act positions is then prefixed with

\pm O \pm

, which yields a set of 64 maxi-conjunctions.3

Note that italic P with indices is used in Section 1.4 to denote transition type prohibition operators on conditions.

Assuming the same logical properties of the O and

E_{x}

operators as in Lindahl’s analysis, viz., the axiom of successful action, E.T., and closure under logical equivalence, E.RE., 57 of the 64 conjunctions are internally inconsistent. The remaining seven are precisely Lindahl’s seven one-agent types of normative positions:

\begin{matrix} (1) & ⟦ \pm O \pm ⟦ \pm E_{x} \pm F ⟧ ⟧ = ⟦ \pm P ⟦ \pm E_{x} \pm F ⟧ ⟧ \end{matrix}

For weaker logics this equality does not hold. [30, p. 14] The consequences of alternative logics of O or

E_{x}

will not be explored here. Instead, consider the idea by Jones and Sergot to perform a refined analysis, based on the set of four ‘cumulative fact/act positions’

$E_{x} F$

$E_{x} \neg F$

$F \land \neg E_{x} F$

$\neg F \land \neg E_{x} \neg F$

which is generated from the scheme

⟦ \pm E_{x} \pm F ⟧ \cdot ⟦ \pm F ⟧

⁴

If Q and R represent sets of expressions, $Q \cdot R$ denotes the set of all the consistent conjunctions that can be formed by conjoining an expression from Q with an expression from R. Cf. [30, p. 11].

As before, these conjunctions are then prefixed by

\pm O \pm

\begin{matrix} (2) & ⟦ \pm O \pm ⟦ \pm E_{x} \pm F ⟧ \cdot ⟦ \pm F ⟧ ⟧ \end{matrix}

With the same basic assumptions as before, this analysis yields a set of 15 logically consistent conjunctions, shown in Table 2.5

⁵

This is a reiteration of Table 2 in [32, p. 375], using x instead of a to denote an agent. Note that ${Pass}_{x} F$ is used as an abbreviation for $\neg E_{x} F \land \neg E_{x} \neg F$ .

According to [30], Lindahl’s $T_{3}$ , $T_{5}$ and $T_{7}$ are identical to three of Jones-Sergot’s ‘normative act positions’, while each of the other four types are logically equivalent to a disjunction of three conjunctions, as shown in the table.

1.4. Previous work: Norm-regulated transition system situations

The notion of a norm-regulated transition system situation was originally presented in [13]. A transition system situation is intended to represent, for example, a ‘snapshot’ of a labelled transition system (LTS) in which each transition fulfills these criteria.

The algebraic representation of conditional norms is based on a systematic exploration of the possible types, with respect to some state of affairs $d (x_{1}, \dots, x_{n})$ , of state transitions. A transition system situation is an ordered 5-tuple $S = ⟨ x, s, A, Ω, S ⟩$ characterized by a set of states S, a state s, an agent-set $Ω = {x_{1}, \dots, x_{n}}$ , the acting (‘moving’) agent x, and an action-set $A = {a_{1}, \dots, a_{m}}$ . In this setting, the act 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. In the following, the abbreviation $s^{+}$ will be used for $a (x, s)$ when there is no need for an explicit reference to the action a and the acting agent x. It is assumed that the action by the acting agent is deterministic and that there is no simultaneous action by other agents (including the ‘environment’, which may be regarded as a special kind of agent). Furthermore, it is assumed that the ν-ary condition d is true or false of ν agents $x_{1}, \dots, x_{ν} \in Ω$ in s, where Ω is a set of agents associated with s. Thus, $d (x_{1}, \dots, x_{ν})$ is a state of affairs which may be true or false in s; this will often be written $d (X_{ν}; s)$ . Note that s may represent an arbitrary state in an LTS, and S is the set of states reachable from s by all transitions ε such that ε represents an act $a \in A$ performed by x.

Now consider the transition from a state s to the following state $s^{+}$ , and focus on the state of affairs $d (X_{ν})$ . With regard to $d (X_{ν})$ , there are four possible alternatives (shown in Table 3) for the transition from s to $s^{+}$ , since in s as well as in $s^{+}$ , $d (X_{ν})$ or $\neg d (X_{ν})$ could hold [28, p. 41]. Each alternative represents a basic type of transition with regard to the state of affairs $d (X_{ν})$ ; ${I, II, III, IV}$ is said to be the set of basic transition types with regard to $d (X_{ν})$ .

Table 3
Basic transition types

I. $d (X_{ν}; s)$ and $d (X_{ν}; s^{+})$

II. $\neg d (X_{ν}; s)$ and $d (X_{ν}; s^{+})$

III. $d (X_{ν}; s)$ and $\neg d (X_{ν}; s^{+})$

IV. $\neg d (X_{ν}; s)$ and $\neg d (X_{ν}; s^{+})$

The situation part $⟨ x, s ⟩$ of a transition system situation S is characterized by the moving agent x and the state s. A ‘basic transition type operator’ $B_{j}^{a}$ , $j \in {I, II, III, IV}$ , is defined such that the $ν + 1$ -ary ‘transition type condition’ $B_{j}^{a} d (X_{ν}, x_{ν + 1}; x, s)$ indicates whether or not, in the situation $⟨ x, s ⟩$ , the event $x_{ν + 1} : a$ (representing a being performed by agent $x_{ν + 1}$ ) has basic transition type j with regard to the state of affairs $d (X_{ν})$ :6

⁶

The extra argument $x_{ν + 1}$ denotes the agent to which the normative condition applies. Note that, in most systems, $x_{ν + 1} = x$ . The idea behind this extra argument is to allow for systems in which, for example, agents other than the ‘moving’ agent may perform punishments or other ‘reaction acts’; cf., e.g., footnote 4 in [13].

For all ν-ary conditions d and for all agents

X_{ν}, x_{ν + 1}

, all acts a and all situations

⟨ x, s ⟩

$B_{I}^{a} d (X_{ν}, x_{ν + 1}; x, s)$ iff $d (X_{ν}; s) \land d (X_{ν}; a (x_{ν + 1}, s))$

$B_{II}^{a} d (X_{ν}, x_{ν + 1}; x, s)$ iff $\neg d (X_{ν}; s) \land d (X_{ν}; a (x_{ν + 1}, s))$

$B_{III}^{a} d (X_{ν}, x_{ν + 1}; x, s)$ iff $d (X_{ν}; s) \land \neg d (X_{ν}; a (x_{ν + 1}, s))$

$B_{IV}^{a} d (X_{ν}, x_{ν + 1}; x, s)$ iff $\neg d (X_{ν}; s) \land \neg d (X_{ν}; a (x_{ν + 1}, s))$

The idea of assigning to state transitions different types (with regard to some state of affairs) lies behind the $E_{i}^{a}$ operators in [29, pp. 160ff], and closely resembles Sergot’s ideas in [31]. The main contribution here is that the transition types are systematically explored, to then be employed as the basis for a semantics for an extended set of types of normative positions. Furthermore, the state of affairs is a condition which is true or false of a number of agents, not merely a proposition. Another notable difference is that, in Sergot’s framework, the agent is said to bring it about that a transition has a certain type, rather than bringing about a certain state of affairs. For example, an agent can bring it about that a transition has type $0 : d (X_{ν}) \land 1 : d (X_{ν})$ ; in this transition formula, 0 refers to the start state of the transition and 1 to the end state, and the intended meaning is that $d (X_{ν})$ holds in both states. Thus, in the vein of [31], I could be written $0 : d (X_{ν}) \land 1 : d (X_{ν})$ , $II$ could be written $0 : \neg d (X_{ν}) \land 1 : d (X_{ν})$ , and similarly for $III$ and $IV$ .

A norm-regulated transition system situation is represented by an ordered pair $⟨ S, N ⟩$ where $S = ⟨ x, s, A, Ω, S ⟩$ is a transition system situation and $N$ is a normative system. It is assumed that norms apply to an individual agent $x_{ν + 1}$ in a state s. A norm in $N$ is represented by an ordered pair $⟨ c, P_{k} d ⟩$ , where the (descriptive) condition c on the situation $⟨ x, s ⟩$ is the ground of the norm and the (normative) condition $P_{k} d$ on $⟨ x, s ⟩$ is its consequence. (See, e.g., [29].) Define a set of ‘transition type prohibition operators’ $P_{k}$ , where k is a subset of ${I, II, III, IV}$ , such that $P_{k} d (X_{ν}, x_{ν + 1}; x, s)$ indicates that the basic transition types (with respect to the state of affairs $d (X_{ν})$ ) in k are prohibited, and a set of corresponding ‘transition type operators’ $C_{k}^{a}$ , such that $C_{k}^{a} d (X_{ν}, x_{ν + 1}; x, s)$ iff the transition from s to $a (x_{ν + 1}, s)$ has any of the basic transition types in k with respect to $d (X_{ν})$ .7

⁷

That is, $P_{k}$ and $C_{k}^{a}$ transform ν-ary conditions d on a state s to $ν + 1$ -ary conditions $P_{k} d$ and $C_{k}^{a} d$ on a situation $⟨ x, s ⟩$ . To avoid confusion, note that roman P is used in Section 1.3.1 and 2.1 for the deontic operator $May$ .

Then, for example,

⟨ c, P_{k} d ⟩

represents the sentence

\begin{matrix} \begin{matrix} \forall x_{1}, x_{2}, \dots, x_{ν}, x_{ν + 1} \in Ω : \\ c (x_{1}, x_{2}, \dots, x_{p}, x_{ν + 1}; x, s) \to \\ P_{k} d (x_{1}, x_{2}, \dots, x_{q}, x_{ν + 1}; x, s) \end{matrix} \end{matrix}

where Ω is the set of agents,

x_{ν + 1}

is the agent to which the norm applies, x is the acting agent in the situation

⟨ x, s ⟩

, and

ν = max (p, q)

. If the condition specified by the ground of a norm is true in some situation, then the (normative) consequence of the norm is in effect in that situation. If the normative system contains a norm whose ground holds in the situation

⟨ x, s ⟩

and whose consequence prohibits the type of transition represented by the event

x_{ν + 1} : a

, then action a is prohibited for

x_{ν + 1}

⟨ x, s ⟩

\begin{matrix} \begin{matrix} {Prohibited}_{x, s} (x_{ν + 1}, a) according to N \\ if there are conditions c and d \\ and a k \subseteq {I, II, III, IV}, \\ such that ⟨ c, P_{k} d ⟩ is a norm in N, \\ and there are x_{1}, \dots, x_{ν}, x_{ν + 1} such that \\ c (x_{1}, \dots, x_{p}, x_{ν + 1}; x, s) & \\ C_{k}^{a} d (x_{1}, \dots, x_{q}, x_{ν + 1}; x, s), \end{matrix} \end{matrix}

where

ν = max (p, q)

. Hence, if

c (x_{1}, \dots, x_{p}, x_{ν + 1}; x, s)

for some sequence of agents

x_{1}, \dots, x_{q}, x_{ν + 1}

, then the normative condition

P_{k} d (x_{1}, \dots, x_{q}, x_{ν + 1}; x, s)

is ‘in effect’. Thus, if

C_{k}^{a} d (x_{1}, \dots, x_{q}, x_{ν + 1}; x, s)

holds, then a is prohibited for

x_{ν + 1}

in s. For example,

\begin{matrix} \begin{matrix} if P_{{I, IV}} d, then [a is prohibited for x_{ν + 1} in s \\ if C_{{I, IV}}^{a} d (x_{1}, \dots, x_{q}, x_{ν + 1}; x, s)], \end{matrix} \end{matrix}

where

\begin{matrix} \begin{matrix} C_{{I, IV}}^{a} d (x_{1}, \dots, x_{q}, x_{ν + 1}; x, s) \\ iff [B_{I}^{a} d (X_{q}, x_{ν + 1}; x, s) or B_{IV}^{a} d (X_{q}, x_{ν + 1}; x, s)] . \end{matrix} \end{matrix}

The set of operators $C_{k}^{a}$ is derived from the freely generated Boolean algebra over $d (x_{1}, \dots, x_{ν}; s)$ and $d (x_{1}, \dots, x_{ν}; a (x_{ν + 1}, s))$ ; see Table 4.8

⁸

Table 1 in [13].

Note that the condition

P_{{I, II, III, IV}} d

would always prohibit all possible actions, since the corresponding transition type condition

C_{{I, II, II, IV}}^{a} d

is equivalent to ⊤ and thus will be true of all agents in all situations. To match the condition

P_{{}} d

, which prohibits none of the basic transition types, the operator

C_{{}}^{a}

, corresponding to the first row in the table, could be introduced. Since

C_{{}}^{a} d

would be false of all agents in all situations, it is simply omitted.

Table 4

Table 1 in [13]: Possible combinations of basic transition types

	I	II	III	IV
–	–	–	–	–	⊥
$C_{{IV}}^{a}$	–	–	–	X	$\neg d (X_{ν}; s) \land \neg d (X_{ν}; a (x_{ν + 1}, s))$
$C_{{III}}^{a}$	–	–	X	–	$d (X_{ν}; s) \land \neg d (X_{ν}; a (x_{ν + 1}, s))$
$C_{{III, IV}}^{a}$	–	–	X	X	$\neg d (X_{ν}; a (x_{ν + 1}, s))$
$C_{{II}}^{a}$	–	X	–	–	$\neg d (X_{ν}; s) \land d (X_{ν}; a (x_{ν + 1}, s))$
$C_{{II, IV}}^{a}$	–	X	–	X	$\neg d (X_{ν}; s)$
$C_{{II, III}}^{a}$	–	X	X	–	$\neg (d (X_{ν}; s) \leftrightarrow d (X_{ν}; a (x_{ν + 1}, s)))$
$C_{{II, III, IV}}^{a}$	–	X	X	X	$\neg d (X_{ν}; s) \lor \neg d (X_{ν}; a (x_{ν + 1}, s))$
$C_{{I}}^{a}$	X	–	–	–	$d (X_{ν}; s) \land d (X_{ν}; a (x_{ν + 1}, s))$
$C_{{I, IV}}^{a}$	X	–	–	X	$d (X_{ν}; s) \leftrightarrow d (X_{ν}; a (x_{ν + 1}, s))$
$C_{{I, III}}^{a}$	X	–	X	–	$d (X_{ν}; s)$
$C_{{I, III, IV}}^{a}$	X	–	X	X	$d (X_{ν}; s) \lor \neg d (X_{ν}; a (x_{ν + 1}, s))$
$C_{{I, II}}^{a}$	X	X	–	–	$d (X_{ν}; a (x_{ν + 1}, s))$
$C_{{I, II, IV}}^{a}$	X	X	–	X	$\neg d (X_{ν}; s) \lor d (X_{ν}; a (x_{ν + 1}, s))$
$C_{{I, II, III}}^{a}$	X	X	X	–	$d (X_{ν}; s) \lor d (X_{ν}; a (x_{ν + 1}, s))$
$C_{{I, II, III, IV}}^{a}$	X	X	X	X	⊤

1.4.1. An example: Ownership of an estate

A number of example systems, intended to illustrate how to use the notion of norm-regulated transition system situations for developing norm-regulated MAS, will be briefly presented in Section 3. To be able to focus on a single situation, a somewhat simpler example, without the dynamics of a full-fledged multi-agent system, will be constructed here. Although, admittedly, an oversimplification in many ways, the example is intended to capture some of the flavour of the ‘Ownership of an estate’ example employed in [24, Section 4.4.1]. Imagine a tiny world consisting of two neighboring estates, Whiteacre (numbered 1) and Blackacre (2). The world is populated by the three agents Alice, Bob and Charlie, and each estate is owned by one of these agents. Furthermore, each estate contains a main building that can be painted white (W) or black (B), and can be surrounded by a fence.9

⁹
In the original example, each estate may also contain cows belonging to the estate itself, and possibly also stray cows from the neighbouring estate. This aspect of the example is not modelled here.

To keep things simple, there are only four actions available for the acting agent: ‘do nothing’, ‘paint the main building on Whiteacre white’, ‘paint the main building on Whiteacre black’, and ‘erect a fence that goes around both Whiteacre and Blackacre’. A state of this world can, e.g., be represented by a pair consisting of two 4-tuples

⟨ E, O, M, F ⟩

where

E \in {1, 2}

O \in {Alice, Bob, Charlie}

M \in {W, B}

, and

F \in {Yes, No}

. The intended meaning of, say,

⟨ 1, Alice, W, No ⟩

is that Whiteacre is owned by Alice, its main building is painted white, and it is not surrounded by a fence.

Let $Ω = {Alice, Bob, Charlie}$ and let $A = {n, p_{1, W}, p_{1, B}, e_{1, 2}}$ where n stands for ‘do nothing’, $p_{1, c}$ stands for ‘paint the main building on Whiteacre in the color c’, and $e_{1, 2}$ stands for ‘erect a fence that goes around Whiteacre and Blackacre’. Define n such that $n (x, s) = s$ , meaning that the result of x’s doing nothing results in no change of state. Define p such that $\begin{matrix} \begin{matrix} p_{1, c} (x, ⟨ ⟨1, o_{1}, c_{1}, f_{1}⟩, ⟨2, o_{2}, c_{2}, f_{2}⟩ ⟩) = \\ = ⟨ ⟨1, o_{1}, c, f_{1}⟩, ⟨2, o_{2}, c_{2}, f_{2}⟩ ⟩, \end{matrix} \end{matrix}$ i.e., no matter the current color ( $c_{1}$ ) of the main building on Whiteacre, the new color will be c. All other parameters will be unaffected. Finally, define e such that $\begin{matrix} \begin{matrix} e_{1, 2} (x, ⟨ ⟨1, o_{1}, c_{1}, f_{1}⟩, ⟨2, o_{2}, c_{2}, f_{2}⟩ ⟩) = \\ = ⟨ ⟨1, o_{1}, c_{1}, Yes⟩, ⟨2, o_{2}, c_{2}, Yes⟩ ⟩ . \end{matrix} \end{matrix}$ In other words, no matter whether or not there is a fence around one or both estates, the effect when x performs $e_{1, 2}$ is that both estates become surrounded by a fence. Note that these simple definitions fail to capture many real-world details; for example, the effect of erecting a fence around an already fenced estate will be the same as erecting a fence around a non-fenced estate. Similarly, painting an already white house white will have no effect; the state will remain unchanged, i.e., the house will remain white.

One of the agents may now be selected as the acting agent, to form a transition system situation S. For example, let $\begin{array}{l} s = ⟨ ⟨1, Alice, W, No⟩, ⟨2, Bob, B, No⟩ ⟩, \end{array}$ and let $S = ⟨ Alice, s, A, Ω, S ⟩$ . That is, Whiteacre is owned by Alice and is not surrounded by a fence, Blackacre is owned by Bob and is not surrounded by a fence, and Alice is the acting agent. Thus, S is implicitly defined by Alice’s performing each of the actions in A in state s.

An adapted version of the normative system suggested in [24, Section 4.4.1], which is based on the types of normative positions in Section 1.1, will now be created:

$⟨ o_{1} \cap o_{2}, T_{1} w_{1} \cap T_{1} w_{2} \cap T_{1} f_{1, 2} ⟩$

$⟨ o_{1} \cap o_{2}^{'}, T_{1} w_{1} \cap T_{6} w_{2} \cap T_{4} f_{1, 2} ⟩$

$⟨ o_{1}^{'} \cap o_{2}, T_{6} w_{1} \cap T_{1} w_{2} \cap T_{4} f_{1, 2} ⟩$

$⟨ o_{1}^{'} \cap o_{2}^{'}, T_{6} w_{1} \cap T_{6} w_{2} \cap T_{6} f_{1, 2} ⟩$

Here, $o_{1}$ and $o_{2}$ denote the unary conditions of being owner to Whiteacre and Blackacre, respectively. These conditions belong to the set of potential (descriptive) grounds of norms in the normative system. Furthermore, $w_{1}$ and $w_{2}$ denote the 0-ary conditions of the main building on Whiteacre (resp., Blackacre) being painted white, and $f_{1, 2}$ denotes the 0-ary condition of there being a fence going around both estates. These are the descriptive conditions that, by combining these with type-operators $T_{i}$ and forming boolean combinations of these expressions, make up the set of potential (normative) consequences of norms.

To exemplify, $o_{1} \cap o_{2}$ means being the owner of both estates. This condition is a ground for $T_{1} w_{1} \cap T_{1} w_{2} \cap T_{1} f_{1, 2}$ , which is the normative position-condition denoting full freedom with regard to painting the two buildings, and erecting a surrounding fence. In contrast, $o_{1} \cap o_{2}^{'}$ means owning Whiteacre but not Blackacre; this condition is ground for $T_{1} w_{1} \cap T_{6} w_{2} \cap T_{4} f_{1, 2}$ . This is intended to grant full freedom regarding the painting of building on Whiteacre, and no freedom to bring about or prevent painting of building on Blackacre. Furthermore, it is intended to grant freedom to prevent erection of a fence surrounding the estates and freedom to be ‘passive’ about the matter, but no freedom to bring about the fence’s being erected. In Section 2.2.2, returning to this example, an interpretation in terms of prohibition of state transition types will be suggested.

Note the subtle difference between the formulation of the condition $f_{1, 2}$ (‘there being a fence…’) and the corresponding formulation in [24, Section 4.4.1]: ‘erecting a fence…’. Intuitively, their intended meaning is quite different, but both of these kinds of conditions are represented by the same kind of formalism in the theory of normative positions. A detailed discussion of these matters is, however, beyond the scope of this paper.

2. Mapping normative positions to transition type prohibition operators

2.1. A first approach

There are 15 conjunctions in Jones-Sergot’s set of ‘normative act positions’ (see Section 1.3.1), and also 15 rows in Table 4 (disregarding the last row that prohibits all state transitions). Therefore, a natural interpretation of $E_{x}$ in a norm-regulated transition system situation context is one that makes it possible to identify each of the ‘normative act positions’ with one of the rows in the Table 4. One way of reading $E_{x} F$ is ‘x brings it about that F’, or even ‘it is x that brings it about that F’; if F would be (or become) the case anyway, without intervention from the agent x, then it is not the case that it is x that brings it about that F.10

¹⁰
See, e.g., [18]. For a further discussion of different ways of understanding the agency operator ${Do / E}_{x}$ , see Section 3.1 in [12].

Under this interpretation, and the assumption that there is no simultaneous action by other agents or the environment, it is reasonable to identify

E_{x} F

with a state transition of type

II

. (Clearly, this is consistent with the assumption that

E_{x} F \to F

.) Consequently, one natural understanding of

{P E}_{x} F

in a norm-regulated transition system situation is that it allows transitions of type

II

, while

\neg {P E}_{x} F

can be understood as not allowing type

II

transitions.11

¹¹

Again, $E_{x}$ is used instead of $Do$ and P is used instead of $May$ , to adhere to the notation used in Section 1.3.1.

On the other hand, if F holds both before and after the agent’s action, which corresponds to a transition of type I, then it is not x that ‘brings it about’ that F; hence

P (\neg E_{x} F \land F)

can be understood to allow type I transitions, while

\neg P (\neg E_{x} F \land F)

does not allow type I transitions. Similar arguments may be given for

\neg {P E}_{x} \neg F

and

\neg P (\neg E_{x} \neg F \land \neg F)

. To summarize the discussion in [12, pp. 17f], the following principles can be assumed:

${P E}_{x} F$ means that $II$ is allowed, $\neg {P E}_{x} F$ means that it is not.

${P E}_{x} \neg F$ means that $III$ is allowed, $\neg {P E}_{x} \neg F$ means that it is not.

$P (\neg E_{x} F \land F)$ means that I is allowed, $\neg P (\neg E_{x} F \land F)$ means that it is not.

$P (\neg E_{x} \neg F \land \neg F)$ means that $IV$ is allowed, $\neg P (\neg E_{x} \neg F \land \neg F)$ means that it is not.

Using these principles, Jones-Sergot’s ‘normative act positions’ are identified with a corresponding row in Table 4. The result is shown in Table 6, with the rows ordered as in Table 2.12

¹²

In order to adhere to the notation in [13], F is represented by $d (x_{1}, \dots, x_{ν})$ . With explicit reference to states, we write $d (X_{ν}; s)$ or $d (X_{ν}; a (x, s))$ , and so on.

The table shows a straightforward interpretation of the normative act positions in terms of permissible and prohibited state transitions in the context of norm-regulated transition system situations. Note that, for example, with this interpretation,

T_{5} (x_{ν + 1}, d (X_{ν}; s))

would prohibit action a for

x_{ν + 1}

d (X_{ν})

holds in s or

\neg d (X_{ν})

holds in

a (x_{ν + 1}, s)

It can be argued that the interpretation suggested here is both simple and fairly intuitive. One way of understanding the theory of normative positions in the context of norm-regulated transition system situations has thus been found. So why not settle for this?

Table 5

Table 2 in [13]: Meaningful combinations of prohibited state transition types

I	II	III	IV	$C_{k}^{a} d (X_{ν}, x_{ν + 1}; x, s)$
–	–	–	–	–
–	–	X	–	$d (X_{ν}; s) \land \neg d (X_{ν}; a (x_{ν + 1}, s))$
–	–	–	X	$\neg d (X_{ν}; s) \land \neg d (X_{ν}; a (x_{ν + 1}, s))$
–	X	–	–	$\neg d (X_{ν}; s) \land d (X_{ν}; a (x_{ν + 1}, s))$
X	–	–	–	$d (X_{ν}; s) \land d (X_{ν}; a (x_{ν + 1}, s))$
–	–	X	X	$\neg d (X_{ν}; a (x_{ν + 1}, s))$
–	X	X	–	$\neg (d (X_{ν}; s) \leftrightarrow d (X_{ν}; a (x_{ν + 1}, s)))$
X	–	–	X	$d (X_{ν}; s) \leftrightarrow d (X_{ν}; a (x_{ν + 1}, s))$
X	X	–	–	$d (X_{ν}; a (x_{ν + 1}, s))$

The answer may be a matter of different perspectives of behavioral norms and normative systems. If, for example, one is interested in formulating system norms (i.e., norms which express a system designer’s point of view of legal/permitted/desirable states and transitions; see Section 1.3), then one could argue that the suggested understanding of normative act positions within the context of norm-regulated transition system situations is reasonable. Again considering $T_{5}$ , it might make perfect sense from a ‘system norms’ perspective to say that a is prohibited for $x_{ν + 1}$ if $d (X_{ν})$ holds in s, even though it is impossible for the agent to change the fact that $d (X_{ν})$ holds before it is about to act. This could for example indicate that, from the designer’s point of view, the system as a whole is entering into some kind of illegal system state. If we, on the other hand, are interested in agent-specific norms, i.e., norms that apply to an individual agent in a specific situation to regulate which actions the agent may or may not perform, then it makes little sense to prohibit an action a based solely on what holds before a is performed; the agent cannot possibly act so that it affects what holds before acting. It is argued in [13] that only nine of the 16 rows of Table 4 are meaningful as the basis for agent-specific norms. They are shown in Table 5.13

¹³

Table 2 in [13].

On the basis of Table 5, prohibitions for an agent

x_{ν + 1}

can be stated in the following way:

\begin{matrix} \begin{matrix} {Prohibited}_{x, s} (x_{ν + 1}, a) according to N \\ if there are conditions c and d \\ and a k \in {{I II}, {IV}, {II}, {I}, \\ {III, IV}, {II, III}, {I, IV}, {I, II}} \\ such that ⟨ c, P_{k} d ⟩ is a norm in N, \\ and there are x_{1}, \dots, x_{ν} such that \\ c (x_{1}, \dots, x_{p}, x_{ν + 1}; x, s) & \\ C_{k}^{a} d (x_{1}, \dots, x_{q}, x_{ν + 1}; x, s), \end{matrix} \end{matrix}

where

ν = max (p, q)

2.2. A second approach

Table 6
Possible interpretation of Jones-Sergot’s normative act positions (my indexation of type names)

I: $\neg E_{x} F \land F$ II: $E_{x} F$ III: $E_{x} \neg F$ IV: $\neg E_{x} \neg F \land \neg F$ ${Prohibited}_{a} (x, s)$ if

$T_{1 a}$ – – – – –

$T_{1 b}$ X – – – $d (X_{ν}; s) \land d (X_{ν}; a (x_{ν + 1}, s))$

$T_{1 c}$ – – – X $\neg d (X_{ν}; s) \land \neg d (X_{ν}; a (x_{ν + 1}, s))$

$T_{2 a}$ – – X – $d (X_{ν}; s) \land \neg d (X_{ν}; a (x_{ν + 1}, s))$

$T_{2 b}$ X – X – $d (X_{ν}; s)$

$T_{2 c}$ – – X X $\neg d (X_{ν}; a (x_{ν + 1}, s))$

$T_{3}$ X – – X $d (X_{ν}; s) \leftrightarrow d (X_{ν}; a (x_{ν + 1}, s))$

$T_{4 a}$ – X – – $\neg d (X_{ν}; s) \land d (X_{ν}; a (x_{ν + 1}, s))$

$T_{4 b}$ X X – – $d (X_{ν}; a (x_{ν + 1}, s))$

$T_{4 c}$ – X – X $\neg d (X_{ν}; s)$

$T_{5}$ X – X X $d (X_{ν}; s) \lor \neg d (X_{ν}; a (x_{ν + 1}, s))$

$T_{6 a}$ – X X X $\neg d (X_{ν}; s) \lor \neg d (X_{ν}; a (x_{ν + 1}, s))$

$T_{6 b}$ X X X – $d (X_{ν}; s) \lor d (X_{ν}; a (x_{ν + 1}, s))$

$T_{6 c}$ – X X – $\neg (d (X_{ν}; s) \leftrightarrow d (X_{ν}; a (x_{ν + 1}, s)))$

$T_{7}$ X X – X $\neg d (X_{ν}; s) \lor d (X_{ν}; a (x_{ν + 1}, s))$

	I: $\neg E_{x} F \land F$	II: $E_{x} F$	III: $E_{x} \neg F$	IV: $\neg E_{x} \neg F \land \neg F$	${Prohibited}_{a} (x, s)$ if
$T_{1 a}$	–	–	–	–	–
$T_{1 b}$	X	–	–	–	$d (X_{ν}; s) \land d (X_{ν}; a (x_{ν + 1}, s))$
$T_{1 c}$	–	–	–	X	$\neg d (X_{ν}; s) \land \neg d (X_{ν}; a (x_{ν + 1}, s))$
$T_{2 a}$	–	–	X	–	$d (X_{ν}; s) \land \neg d (X_{ν}; a (x_{ν + 1}, s))$
$T_{2 b}$	X	–	X	–	$d (X_{ν}; s)$
$T_{2 c}$	–	–	X	X	$\neg d (X_{ν}; a (x_{ν + 1}, s))$
$T_{3}$	X	–	–	X	$d (X_{ν}; s) \leftrightarrow d (X_{ν}; a (x_{ν + 1}, s))$
$T_{4 a}$	–	X	–	–	$\neg d (X_{ν}; s) \land d (X_{ν}; a (x_{ν + 1}, s))$
$T_{4 b}$	X	X	–	–	$d (X_{ν}; a (x_{ν + 1}, s))$
$T_{4 c}$	–	X	–	X	$\neg d (X_{ν}; s)$
$T_{5}$	X	–	X	X	$d (X_{ν}; s) \lor \neg d (X_{ν}; a (x_{ν + 1}, s))$
$T_{6 a}$	–	X	X	X	$\neg d (X_{ν}; s) \lor \neg d (X_{ν}; a (x_{ν + 1}, s))$
$T_{6 b}$	X	X	X	–	$d (X_{ν}; s) \lor d (X_{ν}; a (x_{ν + 1}, s))$
$T_{6 c}$	–	X	X	–	$\neg (d (X_{ν}; s) \leftrightarrow d (X_{ν}; a (x_{ν + 1}, s)))$
$T_{7}$	X	X	–	X	$\neg d (X_{ν}; s) \lor d (X_{ν}; a (x_{ν + 1}, s))$

In the previous section, it was argued that under the interpretation of $E_{x}$ that yields Table 6, the extended type $T_{5}$ is not useful when formulating agent-specific norms for norm-regulated transition system situations; cf. the discussion in [12, p. 17f]. The same can be said about, for example, $T_{7}$ . With another understanding of $Do$ that is consistent with Lindahl’s example in [21, p. 69f], a reasonable interpretation of, e.g., $T_{5}$ as an agent-specific norm is that the agent may act in such a way that F remains true or becomes true, but not in such a way that $\neg F$ remains or becomes true. In other words, transitions of type $III$ and $IV$ are to be prohibited while type $I$ and $II$ transitions are not. However, in the interpretation represented by Table 6, the prohibition of type $III$ and $IV$ transitions corresponds to $T_{2 c}$ , not to $T_{5}$ .

So, the question remains whether or not it is possible to find a mapping between the set of nine transition type prohibition operators from Section 1.4 and Lindahl’s set of seven types of one-agent normative positions, which is both suitable as the basis for formulating agent-specific norms and consistent with Lindahl’s example. One attempt in this direction follows from an observation in [28], where Odelstad defines three operators $Do$ , $Pass$ and $Act$ in terms of state transition types:14

¹⁴

Here, $d (x_{1}, \dots, x_{ν})$ is shortened as d.

\begin{array}{l} Do (ω, d; s) = & \{\begin{matrix} d (s^{+}) if d (s) & (I) \\ d (s^{+}) if \neg d (s) & (II) \end{matrix}\} \\ Pass (ω, d; s) = & \{\begin{matrix} d (s^{+}) if d (s) & (I) \\ \neg d (s^{+}) if \neg d (s) & (IV) \end{matrix}\} \\ Act (ω, d; s) = & \{\begin{matrix} \neg d (s^{+}) if d (s) & (III) \\ d (s^{+}) if \neg d (s) & (II) \end{matrix}\} \end{array}

From this follows that

\begin{matrix} Do (ω, \neg d; s) = \{\begin{matrix} \neg d (s^{+}) if d (s) & (III) \\ \neg d (s^{+}) if \neg d (s) & (IV) \end{matrix}\} \end{matrix}

and furthermore that

\begin{matrix} Pass (ω, d; s) = Pass (ω, \neg d; s) \end{matrix}

and

\begin{matrix} Act (ω, d; s) = Act (ω, \neg d; s) . \end{matrix}

A natural understanding of the statement ‘x does not see to it that F and does not see to it that not F’ is that it expresses x’s passivity with regard to a state of affairs F, in the sense that the presence or absence of the agent does not affect the truth of F. In the norm-regulated transition system situation context, this corresponds to a behavior such that x leaves F as it is, no matter if F is true or false; in other words to a behavior characterized by the transition types I and $IV$ . But, as pointed out by Odelstad, it seems that ‘x does not see to it that F and does not see to it that not F’ also covers another case, namely a ‘stubbornly active’ (‘opposive’) behavior such that x changes the truth of F, no matter if F is true or false; i.e. a behavior characterized by the transition types $II$ and $III$ :

The consistent pairs of (I)–(IV) correspond to $Do (ω, d; s)$ , $Do (ω, \neg d; s)$ , $Pass (ω, d; s)$ or $Act (ω, d; s)$ . From this follows that not $Do (ω, d; s)$ and not $Do (ω, \neg d; s)$ implies $Pass (ω, d; s)$ or $Act (ω, d; s)$ . In [29, p. 147] it is said that x’s passivity with regard to q is expressed by the formula $\begin{matrix} \neg Do (x, q) \land \neg Do (x, \neg q) (2.3) \end{matrix}$ and this is abbreviated as $Pass (x, q)$ . But this seems to disregard the possibility that x is with regard to q always active. [28, pp. 42f]

Intuitively, ‘opposivity’ is not the same as ‘passivity’ with regard to the state of affairs F, but neither is it the same as ‘seeing to it that F’ nor to ‘seeing to it that not F’. Thus, it can be argued that the ‘opposive’ kind of behavior is also covered by

\neg Do (x, F) \land \neg Do (x, \neg F)

. (See [12, p. 21f] for an illustrating and motivating example.) This is the idea that Odelstad’s definitions of

Do

Pass

and

Act

try to capture. To save space in the following, the logical operator

Do

will be abbreviated Δ.

Table 7

An extended set of types of normative positions

$P_{1}^{Λ Ω}$	$May Δ (x, F) & May Λ (x, F) & May Ω (x, F) & May Δ (x, \neg F)$
$P_{1}^{Λ}$	$May Δ (x, F) & May Λ (x, F) & \neg May Ω (x, F) & May Δ (x, \neg F)$
$P_{1}^{Ω}$	$May Δ (x, F) & \neg May Λ (x, F) & May Ω (x, F) & May Δ (x, \neg F)$
$P_{2}^{Λ Ω}$	$May Δ (x, F) & May Λ (x, F) & May Ω (x, F) & \neg May Δ (x, \neg F)$
$P_{2}^{Λ}$	$May Δ (x, F) & May Λ (x, F) & \neg May Ω (x, F) & \neg May Δ (x, \neg F)$
$P_{2}^{Ω}$	$May Δ (x, F) & \neg May Λ (x, F) & May Ω (x, F) & \neg May Δ (x, \neg F)$
$P_{3}^{_}$	$May Δ (x, F) & \neg May Λ (x, F) & \neg May Ω (x, F) & May Δ (x, \neg F)$
$P_{4}^{Λ Ω}$	$\neg May Δ (x, F) & May Λ (x, F) & May Ω (x, F) & May Δ (x, \neg F)$
$P_{4}^{Λ}$	$\neg May Δ (x, F) & May Λ (x, F) & \neg May Ω (x, F) & May Δ (x, \neg F)$
$P_{4}^{Ω}$	$\neg May Δ (x, F) & \neg May Λ (x, F) & May Ω (x, F) & May Δ (x, \neg F)$
$P_{5}^{_}$	$May Δ (x, F) & \neg May Λ (x, F) & \neg May Ω (x, F) & \neg May Δ (x, \neg F)$
$P_{6}^{Λ Ω}$	$\neg May Δ (x, F) & May Λ (x, F) & May Ω (x, F) & \neg May Δ (x, \neg F)$
$P_{6}^{Λ}$	$\neg May Δ (x, F) & May Λ (x, F) & \neg May Ω (x, F) & \neg May Δ (x, \neg F)$
$P_{6}^{Ω}$	$\neg May Δ (x, F) & \neg May Λ (x, F) & May Ω (x, F) & \neg May Δ (x, \neg F)$
$P_{7}^{_}$	$\neg May Δ (x, F) & \neg May Λ (x, F) & \neg May Ω (x, F) & May Δ (x, \neg F)$

An extended set of types of normative positions is now constructed. It is based on the idea that $\begin{matrix} \neg Δ (x, F) & \neg Δ (x, \neg F) \end{matrix}$ is logically equivalent to the disjunction of $Λ (x, F)$ (Λ for ‘Leave’) and $Ω (x, F)$ (Ω for ‘Oppose’): $\begin{matrix} \neg Δ (x, F) & \neg Δ (x, \neg F) \Leftrightarrow Λ (x, F) or Ω (x, F) \end{matrix}$ In the logic employed by Kanger, it can easily be shown that the following equivalences hold:

$May [\neg Δ (x, F) \land \neg Δ (x, \neg F)] \Leftrightarrow May Λ (x, F) \lor May Ω (x, F)$

$\neg May [\neg Δ (x, F) \land \neg Δ (x, \neg F)] \Leftrightarrow \neg May Λ (x, F) \land \neg May Ω (x, F)$

Now, relying on the tautologies

$q \land r \Leftrightarrow (q \land r) \lor (q \land \neg r) \lor (\neg q \land r)$ and

$p \land (q \lor r) \land s \Leftrightarrow (p \land q \land r \land s) \lor (p \land q \land \neg r \land s) \lor (p \land \neg q \land r \land s)$ ,

the set of extended types shown in Table 7 is obtained. As an example, consider

T_{2}

\begin{matrix} \begin{matrix} May Δ (x, F) & \\ May [\neg Δ (x, F) & \neg Δ (x, \neg F)] & \\ \neg May Δ (x, \neg F) \\ iff \\ [May Δ (x, F) & May Λ (x, F) & \\ May Ω (x, F) & \neg May Δ (x, \neg F)] or \\ [May Δ (x, F) & May Λ (x, F) & \\ \neg May Ω (x, F) & \neg May Δ (x, \neg F)] or \\ [May Δ (x, F) & \neg May Λ (x, F) & \\ May Ω (x, F) & \neg May Δ (x, \neg F)] . \end{matrix} \end{matrix}

The details can be found in [12, pp. 20ff]. An extended type containing

May Λ (x, F) \land May Ω (x, F)

is denoted

P_{i}^{Λ Ω}

, where i is the number of the corresponding ‘Lindahl-type’ in Table 1. Similarly, a type containing

May Λ (x, F) \land \neg May Ω (x, F)

is denoted

P_{i}^{Λ}

, and a type containing

\neg May Λ (x, F) \land May Ω (x, F)

is denoted

P_{i}^{Ω}

. Finally, a type containing

\neg May Λ (x, F) \land \neg May Ω (x, F)

is denoted

P_{i}^{_}

. Note the following equivalences:

\begin{matrix} \begin{matrix} T_{1} (x, F) \Leftrightarrow P_{1}^{Λ Ω} (x, F) \lor P_{1}^{Λ} (x, F) \lor P_{1}^{Ω} (x, F) \\ T_{2} (x, F) \Leftrightarrow P_{2}^{Λ Ω} (x, F) \lor P_{2}^{Λ} (x, F) \lor P_{2}^{Ω} (x, F) \\ T_{3} (x, F) \Leftrightarrow P_{3}^{-} (x, F) \\ T_{4} (x, F) \Leftrightarrow P_{4}^{Λ Ω} (x, F) \lor P_{4}^{Λ} (x, F) \lor P_{4}^{Ω} (x, F) \\ T_{5} (x, F) \Leftrightarrow P_{5}^{-} (x, F) \\ T_{6} (x, F) \Leftrightarrow P_{6}^{Λ Ω} (x, F) \lor P_{6}^{Λ} (x, F) \lor P_{6}^{Ω} (x, F) \\ T_{7} (x, F) \Leftrightarrow P_{7}^{-} (x, F) \end{matrix} \end{matrix}

With Odelstad’s operators

Do

Pass

and

Act

in mind, the additional assumptions are made that

Λ (x, F)

implies

Λ (x, \neg F)

, and that

Ω (x, F)

implies

Ω (x, \neg F)

. Then the following symmetries hold:

$P_{1}^{Λ Ω} (x, \neg F)$ iff $P_{1}^{Λ Ω} (x, F)$ , $P_{1}^{Λ} (x, \neg F)$ iff $P_{1}^{Λ} (x, F)$ , $P_{1}^{Ω} (x, \neg F)$ iff $P_{1}^{Ω} (x, F)$ ;

$P_{2}^{Λ Ω} (x, \neg F)$ iff $P_{4}^{Λ Ω} (x, F)$ , $P_{2}^{Λ} (x, \neg F)$ iff $P_{4}^{Λ} (x, F)$ , $P_{2}^{Ω} (x, \neg F)$ iff $P_{4}^{Ω} (x, F)$ ;

$P_{3}^{_} (x, \neg F)$ iff $P_{3}^{_} (x, F)$ ;

$P_{5}^{_} (x, \neg F)$ iff $P_{7}^{_} (x, F)$ ;

$P_{6}^{Λ Ω} (x, \neg F)$ iff $P_{6}^{Λ Ω} (x, F)$ , $P_{6}^{Λ} (x, \neg F)$ iff $P_{6}^{Λ} (x, F)$ , $P_{6}^{Ω} (x, \neg F)$ iff $P_{6}^{Ω} (x, F)$ .

In the terminology of [21, p. 92],

P_{2}^{*}

is the converse of the corresponding

P_{4}^{*}

, and

P_{7}^{_}

is the converse of

P_{5}^{_}

, while

P_{1}^{*}

P_{3}^{_}

and

P_{6}^{*}

are neutral types. It is straightforward to generalize Lindahl’s notions of liberty types and liberty spaces; cf. [21, pp. 106ff]. There are four basic types of one-agent liberties:

$L_{1} = {⟨ x, F ⟩ : May Δ (x, F)} = P_{1}^{Λ Ω} \cup P_{1}^{Λ} \cup P_{1}^{Ω} \cup P_{2}^{Λ Ω} \cup P_{2}^{Λ} \cup P_{2}^{Ω} \cup P_{3}^{_} \cup P_{5}^{_}$

$L_{2}^{Λ} = {⟨ x, F ⟩ : May Λ (x, F)} = P_{1}^{Λ Ω} \cup P_{1}^{Λ} \cup P_{2}^{Λ Ω} \cup P_{2}^{Λ} \cup P_{4}^{Λ Ω} \cup P_{4}^{Λ} \cup P_{6}^{Λ Ω} \cup P_{6}^{Λ}$

$L_{2}^{Ω} = {⟨ x, F ⟩ : May Ω (x, F)} = P_{1}^{Λ Ω} \cup P_{1}^{Ω} \cup P_{2}^{Λ Ω} \cup P_{2}^{Ω} \cup P_{4}^{Λ Ω} \cup P_{4}^{Ω} \cup P_{6}^{Λ Ω} \cup P_{6}^{Ω}$

$L_{3} = {⟨ x, F ⟩ : May Δ (x, \neg F)} = P_{1}^{Λ Ω} \cup P_{1}^{Λ} \cup P_{1}^{Ω} \cup P_{3}^{_} \cup P_{4}^{Λ Ω} \cup P_{4}^{Λ} \cup P_{4}^{Ω} \cup P_{7}^{_}$

Note that

\begin{matrix} \begin{matrix} L_{2}^{Λ} \cup L_{2}^{Ω} = L_{2} = \\ = {⟨ x, F ⟩ : May [\neg Δ (x, F) \land \neg Δ (x, \neg F)]} . \end{matrix} \end{matrix}

The definition of a ‘less free than’ relation for the extended system is analogous to the definition in [21]; a Hasse diagram for this relation is easily constructed (but not shown here).

Table 8

A reduced extended set of types of normative positions

$P_{1}^{Λ Ω}$	$May Δ (x, F) & May Λ (x, F) & May Ω (x, F) & May Δ (x, \neg F)$
$P_{2}^{Λ}$	$May Δ (x, F) & May Λ (x, F) & \neg May Ω (x, F) & \neg May Δ (x, \neg F)$
$P_{2}^{Ω}$	$May Δ (x, F) & \neg May Λ (x, F) & May Ω (x, F) & \neg May Δ (x, \neg F)$
$P_{4}^{Λ}$	$\neg May Δ (x, F) & May Λ (x, F) & \neg May Ω (x, F) & May Δ (x, \neg F)$
$P_{4}^{Ω}$	$\neg May Δ (x, F) & \neg May Λ (x, F) & May Ω (x, F) & May Δ (x, \neg F)$
$P_{5}^{_}$	$May Δ (x, F) & \neg May Λ (x, F) & \neg May Ω (x, F) & \neg May Δ (x, \neg F)$
$P_{6}^{Λ}$	$\neg May Δ (x, F) & May Λ (x, F) & \neg May Ω (x, F) & \neg May Δ (x, \neg F)$
$P_{6}^{Ω}$	$\neg May Δ (x, F) & \neg May Λ (x, F) & May Ω (x, F) & \neg May Δ (x, \neg F)$
$P_{7}^{_}$	$\neg May Δ (x, F) & \neg May Λ (x, F) & \neg May Ω (x, F) & May Δ (x, \neg F)$

The motivation for defining the extended set of types of normative positions was to investigate the possibility of finding a mapping between the nine transition type prohibition operators from Section 1.4 and the types of one-agent normative positions. Lindahl’s original set of types of normative positions contains seven types, which is ‘too few’ in relation to the nine transition type prohibition operators. The extended set of types, on the other hand, contains ‘too many’ types, viz. 15. Therefore the following additional assumptions are made, again with the $Do$ , $Pass$ and $Act$ operators in mind: $\begin{matrix} \begin{matrix} May Δ (x, F) & May Δ (x, \neg F) \\ iff \\ May Λ (x, F) & May Ω (x, F) \end{matrix} \end{matrix}$ Eliminating self-contradictory types, such as $P_{2}^{Λ Ω} (x, F)$ , the (reduced) set of extended types shown in Table 8 is obtained. It is interesting to note that, under the additional assumption above, type $T_{3}$ is self-contradictory and thus eliminated; this assumption does not allow for the fine-grained distinction between $T_{1}$ and $T_{3}$ . This is perhaps not so surprising, considering the fact that $T_{3}$ is a rather special type that requires special attention by Lindahl when developing the theory; see [21, pp. 94ff].

With inconsistent types removed, the equivalences and symmetries stated in the previous subsection for the extended set of types of normative positions still hold. Again, there are four basic one-agent liberty types: $\begin{matrix} \begin{matrix} L_{1} = P_{1}^{Λ Ω} \cup P_{2}^{Λ} \cup P_{2}^{Ω} \cup P_{5}^{_} \\ L_{2}^{Λ} = P_{1}^{Λ Ω} \cup P_{2}^{Λ} \cup P_{4}^{Λ} \cup P_{6}^{Λ} \\ L_{2}^{Ω} = P_{1}^{Λ Ω} \cup P_{2}^{Ω} \cup P_{4}^{Ω} \cup P_{6}^{Ω} \\ L_{3} = P_{1}^{Λ Ω} \cup P_{4}^{Λ} \cup P_{4}^{Ω} \cup P_{7}^{_} \end{matrix} \end{matrix}$

Fig. 1.

Hasse diagram for the ‘less free than’ relation.

Table 9

Transition type conditions for reduced extended types

	Reduced Extended Type of One-Agent Normative Position	Corr. C-operator	${Prohibited}_{x, s} (x, a)$ if
$P_{1}$ :	$May Δ (x, F) & May Λ (x, F) & May Ω (x, F) & May Δ (x, \neg F)$	–	–
$P_{2 Λ}$ :	$May Δ (x, F) & May Λ (x, F) & \neg May Ω (x, F) & \neg May Δ (x, \neg F)$	$C_{2 Λ}^{a}$	$d (X_{ν}; s) \land \neg d (X_{ν}; a (x, s))$
$P_{2 Ω}$ :	$May Δ (x, F) & \neg May Λ (x, F) & May Ω (x, F) & \neg May Δ (x, \neg F)$	$C_{2 Ω}^{a}$	$\neg d (X_{ν}; s) \land \neg d (X_{ν}; a (x, s))$
$P_{4 Λ}$ :	$\neg May Δ (x, F) & May Λ (x, F) & \neg May Ω (x, F) & May Δ (x, \neg F)$	$C_{4 Λ}^{a}$	$\neg d (X_{ν}; s) \land d (X_{ν}; a (x, s))$
$P_{4 Ω}$ :	$\neg May Δ (x, F) & \neg May Λ (x, F) & May Ω (x, F) & May Δ (x, \neg F)$	$C_{4 Ω}^{a}$	$d (X_{ν}; s) \land d (X_{ν}; a (x, s))$
$P_{5}$ :	$Shall Δ (x, F)$	$C_{5}^{a}$	$\neg d (X_{ν}; a (x, s))$
$P_{6 Λ}$ :	$Shall Λ (x, F)$	$C_{6 Λ}^{a}$	$\neg (d (X_{ν}; s) \leftrightarrow d (X_{ν}; a (x, s)))$
$P_{6 Ω}$ :	$Shall Ω (x, F)$	$C_{6 Ω}^{a}$	$d (X_{ν}; s) \leftrightarrow d (X_{ν}; a (x, s))$
$P_{7}$ :	$Shall Δ (x, \neg F)$	$C_{7}^{a}$	$d (X_{ν}; a (x, s))$

A Hasse diagram for the relation ‘less free than’ on this system of types is shown in Fig. 1. To investigate how the set of transition type operators (see Section 1.4) can form a semantics for this system, the following principles are first assumed: For all $X_{ν}$ , $x \in Ω : a \in A : s \in S$ ,

$May Δ (x, d (X_{ν}))$ implies that both $B_{I}^{a} d (X_{ν}, x_{ν + 1}; x, s)$ and $B_{II}^{a} d (X_{ν}, x_{ν + 1}; x, s)$ are allowed.

$May Δ (x, \neg d (X_{ν}))$ implies that both $B_{III}^{a} d (X_{ν}, x_{ν + 1}; x, s)$ and $B_{IV}^{a} d (X_{ν}, x_{ν + 1}; x, s)$ are allowed.

$May Λ (x, d (X_{ν}))$ implies that both $B_{I}^{a} d (X_{ν}, x_{ν + 1}; x, s)$ and $B_{IV}^{a} d (X_{ν}, x_{ν + 1}; x, s)$ are allowed.

$May Ω (x, d (X_{ν}))$ implies that both $B_{II}^{a} d (X_{ν}, x_{ν + 1}; x, s)$ and $B_{III}^{a} d (X_{ν}, x_{ν + 1}; x, s)$ are allowed.

$\neg May Δ (x, d (X_{ν}))$ implies that not both $B_{I}^{a} d (X_{ν}, x_{ν + 1}; x, s)$ and $B_{II}^{a} d (X_{ν}, x_{ν + 1}; x, s)$ are allowed; i.e., at least one of them must be disallowed.

$\neg May Δ (x, \neg d (X_{ν}))$ implies that not both $B_{III}^{a} d (X_{ν}, x_{ν + 1}; x, s)$ and $B_{IV}^{a} d (X_{ν}, x_{ν + 1}; x, s)$ are allowed.

$\neg May Λ (x, d (X_{ν}))$ implies that not both $B_{I}^{a} d (X_{ν}, x_{ν + 1}; x, s)$ and $B_{IV}^{a} d (X_{ν}, x_{ν + 1}; x, s)$ are allowed.

$\neg May Ω (x, d (X_{ν}))$ implies that not both $B_{II}^{a} d (X_{ν}, x_{ν + 1}; x, s)$ and $B_{III}^{a} d (X_{ν}, x_{ν + 1}; x, s)$ are allowed.

These principles are now applied to each of the types in Table 8. Let

X_{ν}, x \in Ω

a \in A

, and

s \in S

[ $P_{1}^{Λ Ω}$ ] According to principles (1)–(4), all combinations of $B_{I}^{a}$ – $B_{IV}^{a}$ are allowed.

[ $P_{2}^{Λ}$ ] The combination of $B_{I}^{a}$ and $B_{II}^{a}$ is allowed according to (1), and $B_{I}^{a} & B_{IV}^{a}$ is allowed according to (3). According to (8) the combination of both $B_{II}^{a}$ and $B_{III}^{a}$ is not allowed. Since $B_{II}^{a}$ was to be allowed, it is $B_{III}^{a}$ that must be disallowed. Similarly, according to (6), the combination $B_{III}^{a} & B_{IV}^{a}$ is not allowed, and since $B_{IV}^{a}$ was to be allowed, it is again $B_{III}^{a}$ that must be disallowed. That is, the principles are fulfilled by, for all $a \in A$ , allowing $B_{I}^{a}$ , $B_{II}^{a}$ , and $B_{IV}^{a}$ , but disallowing $B_{III}^{a}$ .

[ $P_{5}^{Λ}$ ] The combination $B_{I}^{a} & B_{II}^{a}$ is allowed according to (1). According to (6) the combination $B_{III}^{a} & B_{IV}^{a}$ is not allowed, according to (7) the combination $B_{I}^{a} & B_{IV}^{a}$ is not allowed, and according to (8) the combination $B_{II}^{a} & B_{III}^{a}$ is not allowed. The principles are fulfilled by, for all $a \in A$ , allowing $B_{I}^{a}$ and $B_{II}^{a}$ , but disallowing $B_{III}^{a}$ and $B_{IV}^{a}$ .

Similar arguments can be made for the remaining types. (See [12] for the details.) With appropriate renaming of the $C_{i}^{a}$ operators in Table 4, Table 9 is obtained. A comparison with Table 5 reveals that there is a reasonable understanding of the above extension of the Kanger-Lindahl theory of types of one-agent normative positions in terms of the set of nine transition type prohibition operators. That is, in the context of norm-regulated transition system situations, the Kanger-Lindahl type $T_{1}$ can be represented by the extended type $P_{1}$ , and is given a semantics by the transition type prohibition operator $P_{1}$ . Similarly, $T_{5}$ and $T_{7}$ can be represented by $P_{5}$ and $P_{7}$ , respectively, and are given a semantics by $P_{5}$ and $P_{7}$ . For $T_{2}$ there are two alternative representations, viz., $P_{2 Λ}$ or $P_{2 Ω}$ , and thus two alternative semantics, $P_{2 Λ}$ or $P_{2 Ω}$ , and analogously for $T_{4}$ and $T_{6}$ . For example, if $T_{6}$ is represented by the extended type $P_{6 Λ}$ , then act a is prohibited for $x_{ν + 1}$ if $P_{6 Λ} d (x_{1}, \dots, x_{ν}, x_{ν + 1}; x, s)$ , and $C_{6 Λ}^{a} d (x_{1}, \dots, x_{ν}, x_{ν + 1}; x, s)$ holds for some agents $x_{1}, \dots, x_{ν}$ .

2.2.1. Extended normative position cis’es

The use of the algebraic representation of norms by Lindahl and Odelstad, briefly introduced in Section 1.2, requires the definition of condition-implication structures (cis’es) which are based on Boolean quasiorderings on descriptive and deontic conditions. The first step is to use the extended types of normative positions as operators on descriptive conditions to obtain deontic conditions. Suppose, for example, that d is a binary condition. Then $P_{i} d$ is the ternary condition such that $P_{i} d (y, z, x)$ iff $P_{i} (x, d (y, z))$ ; cf. [29, p. 148].

In a procedure similar to the one described in [24, Section 4.4], an ‘extended normative position cis’, np15-cis for short, is constructed such that it fulfils the requirements in the extended theory of one-agent normative positions. Let $B = ⟨ B, \cap,^{'}, R ⟩$ be a cis-Bqo (see Section 1.2) with a domain B of descriptive conditions $d_{1}, d_{2}, \dots,$ let $\begin{matrix} \begin{matrix} H = {1, 1 Λ, 1 Ω, 2, 2 Λ, 2 Ω, 3, 4, 4 Λ, \\ 4 Ω, 5, 6, 6 Λ, 6 Ω, 7}, \end{matrix} \end{matrix}$ and let $P_{B}$ be the extended set of normative positions, shown in Table 7, with regard to the conditions in B: $\begin{matrix} P_{B} = {P_{i} d | d \in B - {⊤, ⊥}, i \in H} . \end{matrix}$ Next, let $P_{B}^{*}$ be the closure of $P_{B}$ under $\cap,^{'}$ . From the ‘Boolean np15-algebra’ $P = ⟨ P_{B}^{*}, \cap,^{'} ⟩$ a cis-Bqo $⟨ P_{B}^{*}, \cap,^{'}, R ⟩$ is constructed:

Definition 1.
An np15-cis over a cis $⟨ B, \cap,^{'}, R ⟩$ is a Boolean quasiordering $⟨ B_{P}^{}, \cap_{P},_{P}^{'}, R_{P} ⟩$ with $⊤_{P}$ as unit element, $⊥_{P}$ as zero element and where $Q_{P}$ is the indifference part of $R_{P}$ such that the following requirements are satisfied. For any $c, d \in B$ it holds that:
if $i \neq j$ , then $(P_{i} d \cap_{P} P_{j} d) R_{P} ⊥_{P}$ (for $i, j \in H$ ),

$⊤_{P} R_{P} (P_{1} d \cup_{P} \dots \cup_{P} P_{7} d)$ ,

$P_{1} d Q_{P} P_{1} (d^{'})$ , $P_{1 Λ} d Q_{P} P_{1 Λ} (d^{'})$ , $P_{1 Ω} d Q_{P} P_{1 Ω} (d^{'})$ , $P_{2} d Q_{P} P_{4} (d^{'})$ , $P_{2 Λ} d Q_{P} P_{4 Λ} (d^{'})$ , $P_{2 Ω} d Q_{P} P_{4 Ω} (d^{'})$ , $P_{3} d Q_{P} P_{3} (d^{'})$ , $P_{5} d Q_{P} P_{7} (d^{'})$ , $P_{6} d Q_{P} P_{6} (d^{'})$ , $P_{6 Λ} d Q_{P} P_{6 Λ} (d^{'})$ , $P_{6 Ω} d Q_{P} P_{6 Ω} (d^{'})$ ,

if $c Q d$ then $P_{i} c Q_{P} P_{i} d$ (for $i \in H$ ),

$P_{i} (⊤_{P}) Q_{P} ⊥_{P}$ if $i = 1, 1 Λ, 1 Ω, 3, 4, 4 Λ, 4 Ω, 7$ ,

$P_{i} (⊥_{P}) Q_{P} ⊥_{P}$ if $i = 1, 1 Λ, 1 Ω, 2, 2 Λ, 2 Ω, 3, 5$ .

The first three requirements in the definition express restrictions on the relation R in an np15-algebra. They correspond to three features of extended one-agent types, viz.* that the types are mutually incompatible, jointly exhaustive, and that some types are the converses of others and some are neutral. Requirements 4–6 follow from the logic of $Shall$ and $Do$ ; for example, 5 and 6 follow from the theorem $\neg May Do (x, ⊥)$ . Cf. [24, p. 610].

A ‘reduced extended normative position cis’, or np9-cis, is then constructed. Let B be a set of descriptive conditions $d_{1}, d_{2}, \dots,$ let $\begin{matrix} H_{red} = {1, 2 Λ, 2 Ω, 4 Λ, 4 Ω, 5, 6 Λ, 6 Ω, 7}, \end{matrix}$ and let $P_{B}^{red}$ be the ‘reduced extended’ set of normative positions, shown in Table 8: $\begin{matrix} P_{B}^{red} = {P_{i} d | d \in B - {⊤, ⊥}, i \in H_{red}} . \end{matrix}$ Finally, let $P_{B}^{red }$ be the closure of $P_{B}^{red}$ under $\cap,^{'}$ . From the ‘Boolean np9-algebra’ $P = ⟨ P_{B}^{red }, \cap,^{'} ⟩$ a cis-Bqo $⟨ P_{B}^{red }, \cap,^{'}, R ⟩$ is constructed:
Definition 2.
An np9-cis* over a cis $⟨ B, \cap,^{'}, R ⟩$ is a Boolean quasiordering $⟨ B_{P}^{*}, \cap_{P},_{P}^{'}, R_{P} ⟩$ with $⊤_{P}$ as unit element, $⊥_{P}$ as zero element and where $Q_{P}$ is the indifference part of $R_{P}$ such that the following requirements are satisfied. For any $c, d \in B$ it holds that:
if $i \neq j$ , then $(P_{i} d \cap_{P} P_{j} d) R_{P} ⊥_{P}$ (for $i, j \in H_{red}$ ),

$⊤_{P} R_{P} (P_{1} d \cup_{P} \dots \cup_{P} P_{7} d)$ ,

$P_{1} d Q_{P} P_{1} (d^{'})$ , $P_{2 Λ} d Q_{P} P_{4 Λ} (d^{'})$ , $P_{2 Ω} d Q_{P} P_{4 Ω} (d^{'})$ , $P_{5} d Q_{P} P_{7} (d^{'})$ , $P_{6 Λ} d Q_{P} P_{6 Λ} (d^{'})$ , $P_{6 Ω} d Q_{P} P_{6 Ω} (d^{'})$ ,

if $c Q d$ then $P_{i} c Q_{P} P_{i} d$ (for $i, j \in H_{red}$ ),

$P_{i} (⊤_{P}) Q_{P} ⊥_{P}$ if $i = 1, 4 Λ, 4 Ω, 7$ ,

$P_{i} (⊥_{P}) Q_{P} ⊥_{P}$ if $i = 1, 2 Λ, 2 Ω, 5$ .

The algebraic representation of normative systems discussed here, together with the interpretation in terms of prohibited transition types, is intended as a generic and useful tool for the development of normative MAS. Clearly, the rule structure of norms is deeply embedded in this framework, but many of the other design choices mentioned in the ‘normative MAS roadmap’ (see Section 1.3) cannot be addressed at the level of the framework itself; it is up to the developer who uses the framework to address the different guidelines, motivate which definition of normative MAS is used, etc.
2.2.2. Ownership of an estate (cont’d)

A normative system for the agents in the Estates world in Section 1.4.1 may now be formulated. Taking the ‘agent-specific norms’ perspective, the approach in Section 2.2 is adopted. For two of the types, $T_{4}$ and $T_{6}$ , there is a choice of representing these types with $P_{4 Λ}$ or $P_{4 Ω}$ (resp., $P_{6 Λ}$ or $P_{6 Ω}$ ). It is quite clear that, in this example, ‘passivity’ with regard to a condition d is intended to be read as ‘leaving things as they are’ as regards d. Hence, $P_{4 Λ}$ and $P_{6 Λ}$ are chosen, not $P_{4 Ω}$ and $P_{6 Ω}$ . For the grounds, the operator $M_{1}$ (see [13] for an explanation) is used to identify the acting agent with the argument to the unary conditions $o_{1}$ and $o_{2}$ and their boolean combinations. The following (np9-based) normative system is obtained: $\begin{matrix} \begin{matrix} ⟨M_{1} (o_{1} \cap o_{2}), P_{1} w_{1} \cap P_{1} w_{2} \cap P_{1} f_{1, 2}⟩ \\ ⟨M_{1} (o_{1} \cap o_{2}^{'}), P_{1} w_{1} \cap P_{6 Λ} w_{2} \cap P_{4 Λ} f_{1, 2}⟩ \\ ⟨M_{1} (o_{1}^{'} \cap o_{2}), P_{6} w_{1} \cap P_{1} w_{2} \cap P_{4 Λ} f_{1, 2}⟩ \\ ⟨M_{1} (o_{1}^{'} \cap o_{2}^{'}), P_{6} w_{1} \cap P_{6 Λ} w_{2} \cap P_{6 Λ} f_{1, 2}⟩ \end{matrix} \end{matrix}$

Fig. 2.

Screenshot from the Estates application.

Now recall the specific situation S, in which $s = ⟨ ⟨1, Alice, W, No⟩, ⟨2, Bob, B, No⟩ ⟩$ , from Section 1.4.1. What restrictions on Alice’s choice of action does this normative system give? The only ground condition that is fulfilled in this situation is $M_{1} (o_{1} \cap o_{2}^{'})$ , since for $x = Alice$ , it holds that $o_{1} (x) \land \neg o_{2} (x)$ and x is the acting agent. Hence, the normative condition that is ‘in effect’ is $P_{1} w_{1} \cap P_{6 Λ} w_{2} \cap P_{4 Λ} f_{1, 2}$ . As can be seen in Table 9, $P_{1}$ is the most permissible type; $P_{1} w_{1}$ gives no restriction on Alice’s actions.15

¹⁵

The notation ‘ $May$ $Δ (x, w_{1}) \land May$ $Λ (x, w_{1}) \land May$ $Ω (x, w_{1}) \land May$ $Δ (x, w_{1}^{'})$ ’ will be used, even though $May$ technically operates on propositions, not on conditions.

What about

P_{6 Λ} w_{2}

, which represents ‘

Shall

Λ (x, w_{2})

’, i.e., that matters should be left as they are regarding the color of the main building on Blackacre? If

P_{6 Λ} w_{2} (Alice, s)

, then an action a is prohibited if

C_{6 Λ}^{a} w_{2} (Alice, s)

. But none of the four available actions (i.e., n,

p_{1, W}

p_{1, B}

, and

e_{1, 2}

) can change the color of Blackacre’s main building, so for each a it is not the case that

w_{2} (s) \land \neg w_{2} (s^{+})

\neg w_{2} (s) \land w_{2} (s^{+})

, where

s^{+} = a (Alice, s)

. Hence,

C_{6 Λ}^{a} w_{2} (Alice, s)

holds for no

a \in A

, which means that

P_{6 Λ} w_{2}

does not prohibit any of the acts in A. Finally,

P_{4 Λ} f_{1, 2}

, ‘

\neg May

Δ (x, f_{1, 2}) \land May

Λ (x, f_{1, 2}) \land \neg May

Ω (x, f_{1, 2}) \land May

Δ (x, f_{1, 2}^{'})

’, prohibits an action a if

C_{4 Λ}^{a} f_{1, 2} (Alice, s)

. Now,

C_{4 Λ}^{a} f_{1, 2} (Alice, s)

holds for one action, viz.

e_{1, 2}

, since

\neg f_{1, 2} (s) \land f_{1, 2} (s^{+})

, where

s^{+} = e_{1, 2} (Alice, s)

. Hence it follows that

e_{1, 2}

is prohibited for Alice in the situation

⟨Alice, s⟩

. To summarize, in the given situation the normative system prohibits one of the four available acts in a way that seems to be in line with the intentions in [24, Section 4.4.1].

3. Applications

An instrumentalization of the Estates example in Section 1.4.1 and 2.2.2 has been developed, using the general-level Java/Prolog-framework from [16] with later extensions. The Estates application demonstrates how the framework handles non-elementary norms, i.e., norms whose grounds and/or consequences are boolean combinations of simpler conditions. Furthermore, by varying the different parameters, one can use the application to experiment with a large number of specific situations for the Estates world, to get an idea of how well the instrumentalized normative system captures the intention behind the system described by Lindahl and Odelstad in [24]. The screenshot in Fig. 2 shows the result of querying the system about the permissible acts (according to the normative system described in the previous section) for the acting agent Alice in the state $\begin{matrix} s = ⟨ ⟨1, Alice, W, No⟩, ⟨2, Bob, B, No⟩ ⟩, \end{matrix}$ i.e., Whiteacre being owned by Alice and its main building being white, and Blackacre being owned by Bob and its main building being painted black, and neither of the estates being surrounded by a fence. As expected, the actions n, $p_{1, W}$ , and $p_{1, B}$ are permitted for Alice while $e_{1, 2}$ is not. Since Alice is the owner of Whiteacre she has full freedom regarding letting the house on Whiteacre remain white or being painted black, but not to erect a fence that goes around both Whiteacre and Blackacre, since Blackacre is owned by Bob.

Assuming instead that Bob is the acting agent in the same system state, which actions are permitted for him? Selecting Bob as the acting agent and querying the system for permissible actions, gives the somewhat surprising answer that n and $p_{1, W}$ are permitted while $p_{1, B}$ and $e_{1, 2}$ are not. As expected, Bob is not allowed to erect a fence around both estates, and he is not allowed to paint the main building on Whiteacre black, since he is not the owner of Whiteacre. But apparently he may, according to the system, paint the main building on Whiteacre white. Howcome? The normative condition that is in effect regarding the color of the main building on Whiteacre is $P_{6 Λ} w_{1}$ , which states that the acting agent (i.e., Bob) should leave things as they are as regards the color of the building. That is, Bob may not act so that the building changes color. But since it is already white, and the effect of Bob’s performing act $p_{1, W}$ is that it remains white, this action is not prohibited. This is probably not in accord with the intention behind the normative system in [24, Section 4.4.1], but the problem here is the coarseness of this simple model for the Estates world, not the normative framework. This is an example of a situation in which the distinction between ‘being white’ and ‘being painted white’ may be of importance; cf. the remark in Section 1.4.1. This issue will be further discussed in Section 4.

A number of other applications that demonstrate the iterated use of norm-regulated transition system situations are presented in [14, Section 2.3]. The Color & Form Dalmas is a simple system that illustrates how two ‘agents’ can change their states (represented by their color and their form) in the presence of a simple normative system. The Wastecollector Dalmas is an instrumentalization of a variant of the example employed by Odelstad and Boman [29] when developing the abstract Dalmas architecture. A third example is the Rooms system, an instrumentalization of a variant of an example employed in [5, pp. 178ff]. The example consists of a world in which two categories of agents move around in a world of rooms that can contain any number of agents, and where the movement of the agents is restricted by norms based on the categories of the agents. A special kind of norm-regulated agent system is the Forest Cleaner system, consisting of a single agent performing forest cleaning in a rectangular forest grid. An instrumentalization of this system is presented in [12, Section 4].

Another application is shown in [15], where an evolutionary algorithm is used to evolve the ‘best possible’ normative systems for a class of problem-solving Dalmases. The approach to normative systems employed in the Dalmas framework is ideally suited for evolution of normative systems, since the set of $P_{i}$ operators exhausts the set of logical possibilities regarding prohibition of transition types. Therefore each conceivable normative system could potentially be a candidate for evaluation in the execution of the evolutionary algorithm. It is demonstrated in [15] that an evolutionary algorithm may be a useful tool when designing a normative system for a problem-solving Dalmas.

The source code for the example systems mentioned here, as well as for the general-level Java/Prolog implementation of norm-regulated transition system situations, is available for download16

¹⁶
http://drpa.se/norms/nrtssit/.

and is publicly and freely disseminated.

4. Discussion

In the Estates application discussed in the previous section, the agent Bob was allowed to paint the main building on Whiteacre white, even though he was not the owner of this estate. This was due to the coarseness of the model: painting the building white did not change its color (since it was already white) and the normative condition in effect in the specific situation only stated that the acting agent should leave the color of the building unchanged. One may assume that the intention behind the normative system goes beyond just making sure that none other than the owner may change the color of a building; it is also to prohibit an agent from entering an other agent’s property and raising scaffolds for painting, and so on. The simple model employed here is, however, not able to capture these details. In this and many other similar domains, an obvious solution is to increase the level of granularity of the model, which in turn probably requires a more fine-grained normative system. This is probably the best solution in most cases. In some scenarios, another possibility could simply be to exclude the action ‘paint the main building on Whiteacre white’ from the set of feasible actions in all situations where the building is already white. Yet another approach could be to reformulate the normative system to explicitly prohibit individual named actions in certain situations. The normative framework discussed here does not have special treatment of such norms, with explicit support for normative consequences prohibiting named actions, but as suggested in [14, p. 92] this effect may be accomplished by incorporating into the state of the world a history of performed actions, and stating norms that prohibit the last action performed to be the undesired action. A drawback of this approach is that adding more actions (e.g., ‘paint the main building on Blackacre white’, and ‘paint the main building on Blackacre black’) to the model, requires adding more norms to the normative system. One of the features of the Lindahl-Odelstad algebraic approach to norms is precisely that it makes possible the formulation of norms which do not explicitly refer to named actions. A clear separation between norms and actions allows for the specification of the normative system to be, at least to some extent, logically analyzed and modified independently of the set of available actions.

The discussion above raises more general questions regarding the expressive power of this approach to norms. It is certainly possible to construct real-world examples that are difficult or impossible to model within this approach, but one should keep in mind that the aim of the cis model (which is the basis of the Lindahl-Odelstad algebraic approach to normative systems) was not intended as the one and only treatment of normative systems. Despite the expressive power of this approach, it still has its limitations. It is argued in [14] that one of its features is that it takes into account the idea that normative systems should express general rules where no individual names occur. The normative system correlates generic antecedents with generic normative consequents, and is combined with a mechanism to instantiate those generic antecedents and consequents. In a complex normative system, however, norms are general, with cross-references between variables in antecedents and consequents. It is noted by [23, p. 230] that the cis model has limitations, compared with predicate logic, if the primary aim is to provide an overall representation of such a complex normative system. For example, the cis model has trouble representing norms which explicitly permit certain actions, even though performing the action would lead to an ‘illegal’ system state. However, in many (if not most) cases this could probably be handled by a more careful specification of the antecedents of the norms in the system.

Another kind of norms that may occur in real-world applications is norms with deontic operators in the antecedent of norms; see, e.g., [7]. The version of the normative framework discussed here does not deal with such norms, but it should be noted that the cis model approach to norms, being an application of the general theory of joining-systems (TJS; see Section 1.2), is well suited for extension in this direction. The development of TJS was motivated by, among other things, the interest by Lindahl and Odelstad for so-called intermediate legal concepts,17

¹⁷
In TJS, the technical notion representing an intermediate concept is called an intervenient. Typical examples of intermediates are ownership (“being the owner of”) and citizenship (“being a citizen”).

conceptually in between purely descriptive and purely normative concepts [24, p. 625]. Figure 5 in [24] illustrates their idea of constructing stratified normative systems with intermediate structures whose elements belong at the same time to both the set of consequences of one Bjs and the set of grounds of another Bjs. Thus, normative systems with intermediate strata such that normative conditions can occur both as consequents in one stratum and as antecedents in another stratum, would allow for norms with deontic operators in the antecedent. This aspect of the theory has yet to be exploited in the MAS context.

It is sometimes argued that standard deontic logic operators used in the analysis of the normative positions are useful for static systems, but not for dynamic (transition) systems. For example, the distinction between achievement and maintenance obligations (see, e.g., [7] with references) may be challenging. The former kind of norms prescribes that its content must hold at least once during a specified time interval, e.g., that an action is to be performed before a deadline, while the latter is an obligation that its content hold for every instant during the time interval in which the normative consequence is in force. Two other (interrelated) kinds of norms that seem to occur in real-world scenarios are so-called contrary-to-duty obligations and compensatory obligations. A contrary-to-duty obligation, for example, states that an obligation (or prohibition) is in force when the opposite of an obligation (resp., prohibition) holds [7, p. 66]. How the cis model approach to normative systems can handle these specific kinds of norms remains to be thoroughly investigated, but in many settings careful specification of the antecedents of norms should be sufficient for expressing such norms in this model. It can also be remarked that the Lindahl-Odelstad ‘anti-nivelistic’18

¹⁸

The term ‘nivelism’, coined by Odelstad, refers to the idea to treat different kinds of information in a system on the same (logical) ‘level’; see [28, p. viii].

idea to represent normative systems as networks of subsystems seems to be able to more robustly address, or altogether avoid, at least some of the problems and paradoxes associated with, for example, deontic logic or other purely logical approaches to norms. A detailed discussion of these matters is, however, beyond the scope of this paper.

5. Conclusion and future work

An investigation into how to apply the theory of one-agent normative positions in the context of a class of transition systems, in which transitions are deterministic and associated with a single agent performing an act, was performed. By interpreting two different extended systems of one-agent types of normative positions in terms of permitting or prohibiting different transition types, two lexicons were obtained for these extended systems in the context of the selected class of transition systems. It was demonstrated that both interpretations can be used as foundations for defining semantics for normative systems for MAS, depending on how the notion of agency (in the theory of normative positions represented by the action operator $Do$ ) is understood and whether a ‘system norms’ or an ‘agent-specific norms’ perspective is taken. A contribution to a typology of interpretations of the theory of normative positions was thus made.

A ‘system norms’ interpretation of the 15 normative act positions was suggested, as well as an ‘agent-specific norms’ interpretation of the ‘reduced extended’ system of normative positions. Is there a system norms interpretation of the extended 15-type system of normative positions? Is there a reduction of the set of normative act positions that can be given an agent-specific norms interpretation? It seems likely that the answer is yes to both questions, but a further investigation is left for future work.

Other ideas for the future are to generalize the concept of transition system situations, and study how Lindahl’s system of two-agent types of normative positions could be used to deal with simultaneous actions by two agents (including ‘actions’ by the environment) in this context. Furthermore, a deeper analysis of experiments with the Estates application might shed some more light on the benefits and limits of the approach to normative systems presented here.

As mentioned in Section 1.3, normative systems can besides regulative norms also contain constitutive norms, for example in the form of counts-as conditionals. A logical analysis of sentences like “x counts-as y in s”, where s can be a normative system, was proposed by Jones and Sergot [18]. The theory of normative positions does not deal with constitutive norms, but the general TJS framework developed by Lindahl and Odelstad is well suited to represent counts-as conditionals; see the discussion in [24, Section 6.2.1]. One line of future work is to instrumentalize the example employed in their discussion, and develop the norm-regulated transition system situation architecture as well as the general-level Java/Prolog instrumentalization, to allow for the representation of normative systems as networks of subsystems and relations between them, i.e., stratified normative systems consisting of both behavioral and constitutive norms [24, p. 631]. A similar line of work would be to accommodate for normative systems with intermediate strata such that normative conditions may occur both as the consequents of norms in some strata and as the antecedents of norms in other strata; cf. the discussion in Section 4.

Another idea is to try to instrumentalize the ideas regarding norm addition and subtraction in [24, Section 4.3], as a means of dealing with norm change, one of the basic features of normative systems mentioned in [2]. Another potential area of future research is the development of a full algebraic treatment of conditions, with the same expressive power as predicate logic. According to [22], this requires mathematical concepts like Tarski’s cylindric algebras [9,10].

Footnotes

Acknowledgements

The author would like to thank Jan Odelstad and Magnus Boman and two anonymous reviewers for very valuable ideas and suggestions, participants of IAT 2014 for discussions in relation to this paper, and the organizers of the conference for the opportunity to expand my conference paper.

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$T_{1}$	$May Do (x, F) \land May [\neg Do (x, F) \land \neg Do (x, \neg F)] \land May Do (x, \neg F)$
$T_{2}$	$May Do (x, F) \land May [\neg Do (x, F) \land \neg Do (x, \neg F)] \land \neg May Do (x, \neg F)$
$T_{3}$	$May Do (x, F) \land \neg May [\neg Do (x, F) \land \neg Do (x, \neg F)] \land May Do (x, \neg F)$
$T_{4}$	$\neg May Do (x, F) \land May [\neg Do (x, F) \land \neg Do (x, \neg F)] \land May Do (x, \neg F)$
$T_{5}$	$May Do (x, F) \land \neg May [\neg Do (x, F) \land \neg Do (x, \neg F)] \land \neg May Do (x, \neg F)$
$T_{6}$	$\neg May Do (x, F) \land May [\neg Do (x, F) \land \neg Do (x, \neg F)] \land \neg May Do (x, \neg F)$
$T_{7}$	$\neg May Do (x, F) \land \neg May [\neg Do (x, F) \land \neg Do (x, \neg F)] \land May Do (x, \neg F)$

$T_{1}$	$\{\begin{matrix} {P E}_{x} F \land {P E}_{x} \neg F \land P (F \land \neg E_{x} F) \land P (\neg F \land \neg E_{x} \neg F) \\ {P E}_{x} F \land {P E}_{x} \neg F \land \neg P (F \land \neg E_{x} F) \land P (\neg F \land \neg E_{x} \neg F) \\ {P E}_{x} F \land {P E}_{x} \neg F \land P (F \land \neg E_{x} F) \land \neg P (\neg F \land \neg E_{x} \neg F) \end{matrix}\}$
$T_{2}$	$\{\begin{matrix} {P E}_{x} F \land \neg {P E}_{x} \neg F \land P (F \land \neg E_{x} F) \land P (\neg F \land \neg E_{x} \neg F) \\ {P E}_{x} F \land \neg {P E}_{x} \neg F \land \neg P (F \land \neg E_{x} F) \land P (\neg F \land \neg E_{x} \neg F) \\ {P E}_{x} F \land \neg {P E}_{x} \neg F \land P (F \land \neg E_{x} F) \land \neg P (\neg F \land \neg E_{x} \neg F) \end{matrix}\}$
$T_{3}$	$\{{P E}_{x} F \land {P E}_{x} \neg F \land \neg P {Pass}_{x} F\}$
$T_{4}$	$\{\begin{matrix} \neg {P E}_{x} F \land {P E}_{x} \neg F \land P (F \land \neg E_{x} F) \land P (\neg F \land \neg E_{x} \neg F) \\ \neg {P E}_{x} F \land {P E}_{x} \neg F \land \neg P (F \land \neg E_{x} F) \land P (\neg F \land \neg E_{x} \neg F) \\ \neg {P E}_{x} F \land {P E}_{x} \neg F \land P (F \land \neg E_{x} F) \land \neg P (\neg F \land \neg E_{x} \neg F) \end{matrix}\}$
$T_{5}$	$\{O E_{x} F\}$
$T_{6}$	$\{\begin{matrix} O {Pass}_{x} F \land O F \\ O {Pass}_{x} F \land O \neg F \\ O {Pass}_{x} F \land P F \land P \neg F \end{matrix}\}$
$T_{7}$	$\{{O E}_{x} \neg F\}$

I.	$d (X_{ν}; s)$ and $d (X_{ν}; s^{+})$
II.	$\neg d (X_{ν}; s)$ and $d (X_{ν}; s^{+})$
III.	$d (X_{ν}; s)$ and $\neg d (X_{ν}; s^{+})$
IV.	$\neg d (X_{ν}; s)$ and $\neg d (X_{ν}; s^{+})$

Normative positions in multi-agent systems

Abstract

Keywords

1. Introduction

1.1. One-agent types of normative positions

1 The free variables in c ( x 1 , … , x p ) must be the same, and in the same order, as the free variables in d ( x 1 , … , x q ) , but it is not necessary that p and q have the same arity. Cf. [29, p. 146].

1.3.1. An extended set of types of normative positions

Table 3 Basic transition types I. d ( X ν ; s ) and d ( X ν ; s + ) II. ¬ d ( X ν ; s ) and d ( X ν ; s + ) III. d ( X ν ; s ) and ¬ d ( X ν ; s + ) IV. ¬ d ( X ν ; s ) and ¬ d ( X ν ; s + )

9 In the original example, each estate may also contain cows belonging to the estate itself, and possibly also stray cows from the neighbouring estate. This aspect of the example is not modelled here.

2.1. A first approach

10 See, e.g., [18]. For a further discussion of different ways of understanding the agency operator Do / E x , see Section 3.1 in [12].

16 http://drpa.se/norms/nrtssit/.

17 In TJS, the technical notion representing an intermediate concept is called an intervenient. Typical examples of intermediates are ownership (“being the owner of”) and citizenship (“being a citizen”).

Footnotes

Acknowledgements

References

¹
The free variables in $c (x_{1}, \dots, x_{p})$ must be the same, and in the same order, as the free variables in $d (x_{1}, \dots, x_{q})$ , but it is not necessary that p and q have the same arity. Cf. [29, p. 146].

Table 3
Basic transition types

I. $d (X_{ν}; s)$ and $d (X_{ν}; s^{+})$

II. $\neg d (X_{ν}; s)$ and $d (X_{ν}; s^{+})$

III. $d (X_{ν}; s)$ and $\neg d (X_{ν}; s^{+})$

IV. $\neg d (X_{ν}; s)$ and $\neg d (X_{ν}; s^{+})$

⁹
In the original example, each estate may also contain cows belonging to the estate itself, and possibly also stray cows from the neighbouring estate. This aspect of the example is not modelled here.

¹⁰
See, e.g., [18]. For a further discussion of different ways of understanding the agency operator ${Do / E}_{x}$ , see Section 3.1 in [12].

¹⁶
http://drpa.se/norms/nrtssit/.

¹⁷
In TJS, the technical notion representing an intermediate concept is called an intervenient. Typical examples of intermediates are ownership (“being the owner of”) and citizenship (“being a citizen”).