Simulating collective behaviour is a challenging topic of the recent past. There is a rising demand that emerges from a variety of contexts, including the production of legislation at broad, for the simulation paradigm can be useful for determining the actual effects of introducing a new norm on a given society. Therefore, the drafters, namely those people who are in charge of producing a norm, either when designing a new norm for the general population, or when providing a norm background on the actual domain of a restricted society (a company, an association) or again to govern direct multiparty relationships (as in contracts) consider helpful to apply the norms into a simulated society for evaluating the impact the new norm has.

We then need a method to describe a society, in a way that generates a cycle of multiple agent system evolution steps able to support the impact evaluation we mentioned above.

This concept is part of a series of investigation that are conducted with the purpose of developing a complex system for law evaluation in diverse point of the production process: by design (as in the current investigation), after the enforcement, in the application phase. The schema of the application system is illustrated in Figure 1. There are essentially three phases of the application:

1. MAS Simulation -construction: a society is generated, possibly based on real-world data such as sociological evidence.

2. Normative background -specification: the existing normative background of the society simulated in Phase 1 is defined within an engine for Legal Reasoning, in particular, for the implementation of the aforementioned projects, we use the DDL reasoner Houdini [1,2].

3. MAS Simulation -execution: the MAS system defined in Phase 1 is run, and it generates, technically, bunches of facts for a Deontic Defeasible Theory, implemented to devise the normative background, as described in Phase 2.

Consequence generation -state of affairs: the DDL system generates the simulated effects of the applications of the Normative Background towards the MAS system. This changes the current state of affairs of the MAS.

Consequence generation -MAS configuration: the MAS system employs a negative feedback á la Rosenblueth, Wiener, Bigelow to resettle the MAS while configuring again the parameters in two levels of feedback layers:

• As a consequence of the normative changes, by modifying the behavioural parameters of the agents (for instance when an action which is permitted becomes forbidden the probability of individuals willing to do that action may decrease);

• As a consequence of the application of the law because some agents could have been punished, and therefore they may have been limited in their permits, or superimposed obligations and prohibitions they did not have before.

The above mentioned sequence is repeated in cycles with occasional events of actions performed in the MAS and rules changed in the normative background. Also we leave the evaluation part for further investigation.

A key aspect in modeling human behavior with respect to law compliance is given by the nondeterministic component of actions. The stochastic nature of reality shows up in at least two moments: the actual violation of a norm by a person and the chance that this violation is discovered and prosecuted by the authorities. The former phenomenon is formalized in the model we discuss here through an equation ( 1) pairing together the expected utility from the violation of a norm, compared with the one relative to the respect of that norm, and the tendency to be compliance with the law, encoded in the model discussed in this paper by a non-negative real number. The latter would call for a probabilistic application of DDL rules and, given its non trivial formalization, is postponed to a further and more specific paper.

The research plan is therefore as follows:

• Build the theoretical model that is documented in the current paper.

• Implement it within an actual simulation system. In this case we have chosen to employ GAMA [3,4,5] that has also been showed to adapt to the context of simulating ethically relevant behaviours [3].

• Develop an extension to the markup language GAML++ that expresses the aspects we discuss in this paper, and correspondingly extend GAMA to etGAMA in order to allow simulation of the law.

• Extend Houdini technology to allow law change specific management, as well as the utility functions we shall discuss in this paper.

• Build the whole system, where the communication components are managed via json/jquery components in order to align the current java implementation of both Gama and Houdini.

With this research plan in agenda, we now deal with the theoretical model, in the rest of this paper.

For what concerns the structure: Sec. 2 exposes the population model, in particular how it changes during time; Sec. 3 presents the salient characteristics which we focus on, such as age, gender and job; Sec. 4 is dedicated to model law compliance in this in vitro society; finally, Sec. 5 summarizes the whole paper.

In this section we provide the population model. First of all, for what concerns time, we adopt a discrete time 𝑇 starting with 𝑡 = 0 and indexed by natural numbers which may be interpreted as years. Let P be the countably infinite set of all the people who may, at some point, be part of the population model. At any moment, which in our discrete time means a specific year, the population of the model is given by a finite subset of P; the function 𝒫 : 𝑇 → 𝒫(P)1 associates to every year 𝑡 a finite subset of P, the current population 𝒫(𝑡) in that year denoted by 𝒫 𝑡 . The starting population 𝒫 0 , as well as its characteristics, see Sec. 3, is given; whereas the population in 𝑡 + 1, i.e. 𝒫 𝑡+1 , depends on four factors: births, deaths, immigration and emigration. Births are given by a birth function 𝐵 : 𝑇 ∖ {0} → 𝒫(P) which, for every year 𝑡 except the first one, selects the finite subset of the new born denoted by 𝒫 0 𝑡 ; the function 𝐵 satisfies reasonable constraints related to births, in particular the following two: Deaths are modeled simply by the process that every person lives exactly 80 years. Thus, if 𝒫 80 𝑡 denotes the subset of 𝒫 𝑡 of people in their eighties, then moving from 𝒫 𝑡 to 𝒫 𝑡+1 we simply remove 𝒫 80 𝑡 ; see later regarding how age is encoded in the model. Immigration and emigration are treated similarly, namely we have two functions 𝐼𝑚 : 𝑇 → 𝒫(P) and 𝐸𝑚 : 𝑇 → 𝒫(P) selecting for every year 𝑡 who is immigrating, 𝐼𝑚 𝑡 , and who is emigrating, 𝐸𝑚 𝑡 . As before we have some constraints for these functions, in particular:

• 𝐸𝑚 𝑡+1 ⊆ 𝒫 𝑡 , only people in the current population can emigrate the next year;

• 𝐼𝑚 𝑡 ∩ 𝐸𝑚 𝑡 = ∅, a person can not immigrate and emigrate the same year.

Moreover, the immigrate function 𝐼𝑚 establishes also the age and the gender of the immigrates; namely, there are two functions 𝐼𝑚𝐴𝑔𝑒 : 𝐼𝑚 𝑡 → {1, . . . , 80} and 𝐼𝑚𝐺𝑒𝑛 : 𝐼𝑚 𝑡 → {𝑚𝑎𝑙𝑒, 𝑓 𝑒𝑚𝑎𝑙𝑒} assigning to each person in 𝐼𝑚 𝑡 its age and its gender, see below for more details.

All in all, with respect to 𝑇 population satisfies the following equation:

Every person 𝑝 ∈ P has some individual characteristics, such as age, gender and so on. In this paper the following are considered:

1. Age: this is a natural number between 1 and 80.

2. Age status: there are three age status, young, adult, elderly depending on the age.

Gender: this preliminary model has two genders male and female which are fixed from birth.

4. Marital status: there are three marital status, bachelor, married and divorced. 25. Job status: there are four job status, unemployed, public job, private job and retired.

6. Ethical tendency (also called in places inclination): there are three tendencies, legalist, neutral and opportunist; related to tendency there is also a parameter 𝛼, see later for more detail.

We see these characteristics as sets, e.g. Age= {1, 2, . . . , 80} and Age status= {𝑦𝑜𝑢𝑛𝑔, 𝑎𝑑𝑢𝑙𝑡, 𝑒𝑙𝑑𝑒𝑟𝑙𝑦}, presenting them one by one after some general considerations.

Excluding the ethical tendency, devoted to modeling law compliance, the other status have been chosen since they represent three of the main social categories producing common and widespread rights and duties, e.g. age for penal responsibility, as well as three of the main characteristics used in demographic and social studies.

Except from gender, all the status may change during time, thus each status depend on both the person 𝑝 ∈ 𝒫 𝑡 and the current year 𝑡 ∈ 𝑇 . Let ℒ = 𝐴𝑔𝑒 × 𝐴𝑔𝑒 𝑠𝑡𝑎𝑡𝑢𝑠 × 𝐺𝑒𝑛𝑑𝑒𝑟 × 𝑀 𝑎𝑟𝑖𝑡𝑎𝑙 𝑠𝑡𝑎𝑡𝑢𝑠 × 𝐽𝑜𝑏 𝑠𝑡𝑎𝑡𝑢𝑠 × 𝐸𝑡ℎ𝑖𝑐𝑎𝑙 𝑡𝑒𝑛𝑑𝑒𝑛𝑐𝑦 be the set of all possible status array, then for every year 𝑡 ∈ 𝑇 there is a function ℓ 𝑡 : 𝒫 𝑡 → ℒ which associates to each person their status. Moreover, ℓ has, as function, different components one for each status and, for sake of readability, the function determining each status is denoted with an abbreviation of the status itself; thus for a person 𝑝 and a given year 𝑡, ℓ 𝑡 (𝑝) = (𝐴𝑔𝑒 𝑡 (𝑝), 𝐴𝑔𝑒𝑆𝑡𝑎𝑡 𝑡 (𝑝), 𝐺𝑒𝑛𝑑𝑒𝑟(𝑝), 𝑀 𝑎𝑟𝑆𝑡𝑎𝑡 𝑡 (𝑝), 𝐽𝑜𝑏 𝑡 (𝑝), 𝐿𝑒𝑔𝑎𝑙 𝑡 (𝑝)). We consider now each status to present it showing how it may change during time.

The age function 𝐴𝑔𝑒 𝑡 : 𝒫 𝑡 → {1, . . . , 80} assigns to every person 𝑝 in the current population 𝒫 𝑡 its age 𝐴𝑔𝑒 𝑡 (𝑝). Obviously there is a strict connection between 𝐴𝑔𝑒 𝑡 and 𝐴𝑔𝑒 𝑡+1 , more precisely the latter as the following definition:

The age function 𝐴𝑔𝑒 0 : 𝒫 0 → {1, . . . , 80} for the starting population 𝒫 0 is given. For what concerns notation, given 1 ⩽ 𝑛 ⩽ 80, we denote

The age status for a person 𝑝 ∈ 𝒫 𝑡 depends only on the current age of 𝑝, more precisely:

We

Since gender is fixed, it depends only on the initial conditions for the people in 𝒫 0 , on the birth gender function 𝐵𝑔𝑒𝑛 𝑡 for new born and on the immigrate gender function 𝐼𝑚𝐺𝑒𝑛 𝑡 for who immigrates.

There are three marital status: bachelor, married, divorced. Young people, i.e. people for which the current age is less than 21, have only one marital status available, namely bachelor; thus 𝑀 𝑎𝑟𝑆𝑡𝑎𝑡 𝑡 (𝑝) = 𝑏𝑎𝑐ℎ𝑒𝑙𝑜𝑟 for all 𝑝 ∈ 𝒫 𝑦𝑜𝑢𝑛𝑔 𝑡 . In order to model during years the marital status, as well as the job status, for adults and elderly people (who may have other status beside bachelor) we use Markov chains; more precisely different Markov chains depending on the age status. Excepted for people who have just became adult, i.e. 21 year old people, who are automatically bachelor and people who have just became elderly, i.e. 61 year old people, who preserve the previous status, 3 the transitions between these status are given by a stochastic process with different probabilities for adults and elderly people. Denoting with 𝐵, 𝑀, 𝐷 being respectively bachelor, married and divorced and setting to 99%, 95%, 5%, 1% the probabilities involved (which have been arbitrary chosen for this paper but, being editable parameters, could be instantiated with real values coming from statistical investigations), the transition schemes for adults and elderly people are the following:

This means that if at time 𝑡 a 35-year adult is bachelor there is 95% chance that he is still bachelor at 𝑡 + 1 and a 5% chance that he got married. If the current population is sufficiently large these probabilities can be seen as the fractions of the current population changing their status; namely every year the 95% of married adult remain married whereas the 5% divorced. Moreover, this setting allows for an easy introduction of further constraints. For example, if there is a waiting period of at least 𝑛 years between the declaration of divorce and a subsequent marriage, this can be encoded in the marital status in the following way: if 𝑀 𝑎𝑟𝑆𝑡𝑎𝑡 𝑡 (𝑝) = 𝑚𝑎𝑟𝑟𝑖𝑒𝑑 and 𝑀 𝑎𝑟𝑆𝑡𝑎𝑡 𝑡+1 = 𝑑𝑖𝑣𝑜𝑟𝑐𝑒𝑑, then 𝑀 𝑎𝑟𝑆𝑡𝑎𝑡 𝑡+1+𝑖 (𝑝) = 𝑑𝑖𝑣𝑜𝑟𝑐𝑒𝑑 for every 𝑖 ∈ {1, . . . , 𝑛}.

As before, the marital function 𝑀 𝑎𝑟𝑆𝑡𝑎𝑡 0 for the initial population 𝒫 0 as well as the marital status of immigrants are given.

There are four job status: unemployed, public job, private job and retired. Overall the treatment of the job status is similar to the marital status, for example young people and 21 year old people have only one status available, namely unemployed; the only difference, excepts for probabilities of the transition functions, is that adults and elderly people have two different sets of available status. More precisely adults can have one among: unemployed, public job and private job; whereas elderly people one among: retired, public job and private job. During the transition between adulthood and old age the status remains unchanged except for unemployed people who become retired; formally, if 𝐴𝑔𝑒 𝑡+1 (𝑝) = 61 then:

Moreover, as it can be seen from the transition function for elderly people, retirement is irreversible; namely if a person retires than from that year their status will always be retired. Abbreviating with 𝑈 𝑛, 𝑅𝑒𝑡, 𝑃 𝑢𝑏, 𝑃 𝑟𝑣 respectively being unemployed, retired, public job and private job, the transition functions are as follows:

As before, the job function 𝐽𝑜𝑏 0 for the initial population 𝒫 0 as well as the job status of immigrants are given.

Ethical tendency aims to formalize the behavior of people with respect to the violation of law. The main idea is that the violation of a norm by a person depends on the expected utility of that

𝛼 𝑝 is a parameter representing the tendency of 𝑝 to violate the law, there are three cases corresponding to the three Ethical tendency status:

• 𝛼 𝑝 = 0, in this case 𝑝 does not violate the law, no matter how high is the expected utility in the violation; in this case 𝑝 is a legalist.

• 0 < 𝛼 𝑝 < +∞, in this case 𝑝 may violate 𝜂 if 𝜇 𝑝 (𝜂 − ) > 𝜇 𝑝 (𝜂 + ) and the probability increases as 𝑘 = 𝜇 𝑝 (𝜂 − ) − 𝜇 𝑝 (𝜂 + ) increases; in this case 𝑝 is legally neutral.

• 𝛼 𝑝 = +∞, in this case 𝑝 violates 𝜂 as soon as 𝜇 𝑝 (𝜂 − ) > 𝜇 𝑝 (𝜂 + ); in this case 𝑝 is an opportunist.

For a quantitative estimation of the role of 𝛼 𝑝 , if 𝛼 𝑝 = 1 and 𝑘 = 𝜇 𝑝 (𝜂 − ) − 𝜇 𝑝 (𝜂 + ) = 1 then there is a probability of 1 − 𝑒 −1 ≃ 63% that 𝑝 violate 𝜂.

The Ethical tendency of a person 𝑝 may change during time according a transition function, in this case we assign the same transition process to the three age status. Denoting with 𝐿, 𝑁, 𝑂 being respectively legalist, neutral and opportunist we adopt the following status transitions (again the probabilities are editable parameters):

When the ethical tendency of a person 𝑝 becomes neutral, the transition function also assigns to 𝛼 𝑝 a strictly positive real value. As before the legal function 𝐿𝑒𝑔𝑎𝑙 0 of the starting population 𝒫 0 , as well as the ones for new born and immigrants, are given. 4

The conceptualisation of law compliance is taken from the current literature on DDL and derived models. In particular we presuppose that an agent enforces freely literals that are taken as feasible actions. On the other hand, when an agent performs a particular action, for which, as we specified before, she has a specific tendency (a probability of doing that action), the legal system takes care of that. In particular, we assume that the normative background is actually constructed as a DDT, and feasible actions are literals of that theory. Defeasible Logic [6,7] is a simple, flexible, and efficient rule-based non-monotonic formalism. Its strength lies in its constructive proof theory, allowing it to draw meaningful conclusions from (potentially) conflicting and incomplete knowledge base. In non-monotonic systems, more accurate conclusions can be obtained when more pieces of information become available.

Many variants of Defeasible Logic have been proposed for the logical modelling of different application areas, specifically agents [8,9,10], legal reasoning [11,12] and workflows from a business process compliance perspective [13,14].

In this research we focus on the Defeasible Deontic Logic (henceforth DDL) framework [15] that allows us to determine what prescriptive behaviours are in force in a given situation. For detailed descriptions of how to adopt DDL for legal reasoning we refer the reader to [16].

We start by defining the language of a Defeasible Deontic Theory (henceforth a DDT) Let PROP be a set of propositional atoms, and Lab be a set of arbitrary labels (the names of the rules). We use lower-case Roman letters to denote literals and lower-case Greek letters to denote rules.

Accordingly, PLit = PROP ∪ {¬𝑙 | 𝑙 ∈ PROP} is the set of plain literals; the set of deontic literals is ModLit = {□𝑙, ¬□𝑙 | 𝑙 ∈ PLit ∧ □ ∈ {O, P}} and, finally, the set of literals is Lit = PLit ∪ ModLit. The complement of a literal 𝑙 is denoted by ∼𝑙: if 𝑙 is a positive literal 𝑝 then ∼𝑙 is ¬𝑝, and if 𝑙 is a negative literal ¬𝑝 then ∼𝑙 is 𝑝. We will not have specific rules nor modality for prohibitions, as we will treat them according to the standard duality that something is forbidden iff the opposite is obligatory (i.e., O¬𝑝).

A DDT 𝐷 is a triple (𝐹, 𝑅, >), where 𝐹 is the set of facts, 𝑅 is the set of rules, and > is a binary relation over 𝑅 (called superiority relation).

Specifically, the set of facts 𝐹 ⊆ PLit denotes simple pieces of information that are always considered to be true, like "Sylvester is a cat", formally 𝑐𝑎𝑡(𝑆𝑦𝑙𝑣𝑒𝑠𝑡𝑒𝑟). In this paper, we subscribe to the distinction between the notions of obligations and permissions, and that of norms, where the norms in the system determine the obligations and permissions in force in a normative system. A DDT is meant to represent a normative system, where the rules encode the norms of the system, and the set of facts corresponds to a case. As we will see below, the rules are used to conclude the institutional facts, obligations and permissions that hold in a case. Accordingly, we do not admit obligations and permissions as facts of the theory.

The set of rules 𝑅 contains three types of rules: strict rules, defeasible rules, and defeaters. Rules are also of two kinds:

• Constitutive rules (non-deontic rules) 𝑅 C model constitutive statements (count-as rules);

• Deontic rules to model prescriptive behaviours, which are either obligation rules 𝑅 O that determine when and which obligations are in force, or permission rules which represent strong (or explicit) permissions 𝑅 P .

Lastly, > ⊆ 𝑅 × 𝑅 is the superiority (or preference) relation, which is used to solve conflicts in case of potentially conflicting information.

A theory is finite if the set of facts and rules are so. We only focus on finite theories. A strict (constitutive) rule is a rule in the classical sense: whenever the premises are indisputable, so is the conclusion.

On the other hand, defeasible rules are to conclude statements that can be defeated by contrary evidence. In contrast, defeaters are special rules whose only purpose is to prevent the derivation of the opposite conclusion.

A prescriptive behaviour like "Passing on zebra crossing is not permitted when the traffic light for pedestrian is red" can be formalised via the general permissive rule Following the ideas of [17], obligation rules gain more expressiveness with the compensation operator ⊗ for obligation rules, which is to model reparative chains of obligations. Intuitively, 𝑎 ⊗ 𝑏 means that 𝑎 is the primary obligation, but if for some reason we fail to obtain, to comply with, 𝑎 (by either not being able to prove 𝑎, or by proving ∼𝑎) then 𝑏 becomes the new obligation in force. This operator is used to build chains of preferences, called ⊗-expressions.

The formation rules for ⊗-expressions are as follows (i) every plain literal is an ⊗-expression and (ii) if 𝐴 is an ⊗-expression and 𝑏 is a plain literal then 𝐴 ⊗ 𝑏 is an ⊗-expression [15].

In general an ⊗-expression has the form '𝑐 1 ⊗ 𝑐 2 ⊗ • • • ⊗ 𝑐 𝑚 ', and it appears as consequent of a rule '𝐴(𝛼) ˓→ O 𝐶(𝛼)' where 𝐶(𝛼) = 𝑐 1 ⊗ 𝑐 2 ⊗ • • • ⊗ 𝑐 𝑚 ; the meaning of the ⊗-expression is: if the rule is allowed to draw its conclusion, then 𝑐 1 is the obligation in force, and only when 𝑐 1 is violated then 𝑐 2 becomes the new in force obligation, and so on for the rest of the elements in the chain. In this setting, 𝑐 𝑚 stands for the last chance to comply with the prescriptive behaviour enforced by 𝛼, and in case 𝑐 𝑚 is violated as well, then we will result in a non-compliant situation.

For instance, the previous prohibition to pass on pedestrian cross in case of red can foresee a compensatory fine, like Pedestrian_traffic_light_red ⇒ O ¬Pass ⊗ PayFine that has to be paid in case someone passes the pedestrian cross when the light is red.

It is worth noticing that we admit ⊗-expressions with only one element. The intuition, in this case, is that the obligatory condition does not admit compensatory measures or, in other words, that it is impossible to recover from its violation.

In this paper, we focus exclusively on the defeasible part of the logic ignoring the monotonic component given by the strict rules; consequently, we limit the language to the cases where the rules are either defeasible or defeaters. From a practical point of view, the restriction does not effectively limit the expressive power of the logic: a defeasible rule where there are no rules for the opposite conclusion, or where all rules for the opposite conclusion are weaker than the given defeasible rules, effectively behaves like a strict rule. Formally a rule is defined as below. If □ = C then the rule is used to derive non-deontic literals (constitutive statements), whilst if □ is O or P then the rule is used to derive deontic conclusions (prescriptive statements). The conclusion 𝐶(𝛼) is, as before, a single literal in case □ = {C, P}; in case □ = O, then the conclusion is an ⊗-expression. ⊗-expressions can only occur in prescriptive rules though we do not admit them on defeaters (Condition 5.(a).i), see [15] for a detailed explanation.

We use some abbreviations on sets of rules. The set of defeasible rules in 𝑅 is 𝑅 ⇒ , the set of defeaters is 𝑅 dft . 𝑅 □ [𝑙] is the rule set appearing in 𝑅 with head 𝑙 and modality □, while 𝑅 O [𝑙, 𝑖] denotes the set of obligation rules where 𝑙 is the 𝑖-th element in the ⊗-expression. Given that the consequent of a rule is either a single literal or an ⊗-expression (that can be understood as a sequence of elements, and then as an ordered set), in what follows we are going to abuse the notation and use 𝑙 ∈ 𝐶(𝛼). 𝑅 □ is the set of rules 𝛼 : 𝐴(𝛼) ˓→ □ 𝐶(𝛼) such that 𝛼 appears in 𝑅. For a theory as determined by Definitions 1 and 2, 𝛼 appears in 𝑅 means that 𝛼 ∈ 𝑅; thus 𝑅 P is the set of permissive rules appearing in 𝑅. We use 𝑅 ◇ and 𝑅 ◇ [𝑙] as shorthands for 𝑅 O ∪ 𝑅 P and 𝑅 O [𝑙] ∪ 𝑅 P [𝑙], respectively. The abbreviations can be combined. Finally, a literal 𝑙 appears in a theory 𝐷, if there is a rule 𝛼 ∈ 𝑅 such that 𝑙 ∈ 𝐴(𝛼) ∪ 𝐶(𝛼).