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A coherence maximisation process for solving normative inconsistencies

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Abstract

Norms can be used in multi-agent systems for defining patterns of behaviour in terms of permissions, prohibitions and obligations that are addressed to agents playing a specific role. Agents may play different roles during their execution and they may even play different roles simultaneously. As a consequence, agents may be affected by inconsistent norms; e.g., an agent may be simultaneously obliged and forbidden to reach a given state of affairs. Dealing with this type of inconsistency is one of the main challenges of normative reasoning. Existing approaches tackle this problem by using a static and predefined order that determines which norm should prevail in the case where two norms are inconsistent. One main drawback of these proposals is that they allow only pairwise comparison of norms; it is not clear how agents may use the predefined order to select a subset of norms to abide by from a set of norms containing multiple inconsistencies. Furthermore, in dynamic and non-deterministic environments it can be difficult or even impossible to specify an order that resolves inconsistencies satisfactorily in all potential situations. In response to these two problems, we propose a mechanism with which an agent can dynamically compute a preference order over subsets of its competing norms by considering the coherence of its cognitive and normative elements. Our approach allows flexible resolution of normative inconsistencies, tailored to the current circumstances of the agent. Moreover, our solution can be used to determine norm prevalence among a set of norms containing multiple inconsistencies.

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Notes

  1. In this paper we will use the terms inconsistency and normative inconsistency as synonyms.

  2. The existence of normative agents that mediate between external agents or humans and multi-agent systems is not new. For example, in Electronic Institutions [19] there are governor agents that guarantee that external agents comply with the norms of the institution.

  3. In this paper, we regard norms as conditional expressions that specify under which general circumstances they must be instantiated.

  4. As defined in [6], the desirability of a formula \(\gamma \) represents to what extent an agent wants to achieve a situation in which \(\gamma \) holds.

  5. According to [6] intentions are not considered as a basic attitude. Thus, the intentions of n-BDI agents are generated on-line from the agents’ beliefs and desires. The intentionality degree of a formula \(\gamma \) is the consequence of finding a best feasible plan that permits a state of the world where \(\gamma \) holds to be achieved.

  6. See [10] for the pseudocode of the algorithm executed by n-BDI agents.

  7. In this case the static order will determine that just the instance created out of the most salient norm should prevail. In Sect. 5 we demonstrate that approaches that only rely on salience to solve normative inconsistencies can lead to undesired results, even if only two instances are considered.

    Fig. 1
    figure 1

    Instances and norms affecting the webManager agent. Ellipses represent the normative elements. Coherence relationships among these elements are represented by continuous lines, whereas dashed lines represent incoherence relationships

    Full size image

  8. Recall that deductive coherence is a symmetric relationship and, as a consequence, constraints in the coherence graph are defined over the set of all subsets of two elements of \(\mathcal {V}\).

  9. Recall that the expressions in N contain a norm and the salience of this norm.

  10. Recall that n-BDI agents translate instances into desires that will be considered for deriving intentions. Thus, intentions are not a basic attitude and there is not a direct link between instances and intentions. As a consequence, the set of intentions is not considered for resolving inconsistencies.

  11. Notice that we assume that the agent performs a reasoning process, such as the one described in [6], for inferring mental formulas (e.g., \( belief (a\wedge b, min\{\rho _{a},\rho _{b}\})\)) that are a conjunction of separate mental formulas (e.g., \( belief (a, \rho _{a})\) and \( belief (b, \rho _{b})\)).

  12. Note that agents are still under the influence of any instance even if they stop enacting the target role of this instance. Because of this, we have not defined an incoherence relationship between instances and beliefs that represent the fact that the agent is no longer playing the target role of instances.

  13. In particular, the conditional order prefers the prohibition instance to the permission instance iff

    $$\begin{aligned} \frac{\rho _{s}^{F}+\rho _{c}^{F}}{2} > \frac{\rho _{s}^{P}+\rho _{c}^{P}}{2} \end{aligned}$$

    otherwise the permission instance prevails.

  14. In each run we generate a random number for each instance. We define that the instance can be fulfilled when this number is greater than 1 minus its ease of compliance.

  15. Note that we only consider the runs in which coherence selects one instance (i.e., the inconsistency does not remain unresolved) and this instance cannot be fulfilled.

  16. Note that we only consider the runs in which both instances can be fulfilled and coherence selects the least salient instance (i.e., the inconsistency does not remain unresolved).

  17. It may be the case that the two norms were instantiated at two points in the past, when the webManager knew that its user was an academic and a university member. However, the webManager cannot determine in the current situation whether its user is still the target of the two instances.

  18. In particular, the conditional order prefers the prohibition instance to the permission instance iff

    $$\begin{aligned} \frac{\rho _{s}^{F}+\rho _{c}^{F}+\rho _{\textit{universityMember}}}{3} > \frac{\rho _{s}^{P}+\rho _{c}^{P}+\rho _{ academicStaff }}{3} \end{aligned}$$

    otherwise the permission instance prevails.

  19. Notice that the two addressing beliefs have a certainty of 1.

  20. Such semantics have been widely used in previous research on agents and norms, such as [31] and [27].

  21. The conditional order prefers the prohibition instance to the permission instance iff

    $$\begin{aligned} \frac{\rho _{s}^{F}+\rho _{c}^{F}}{2}> \frac{\rho _{s}^{P}+\rho _{c}^{P}+\rho _{ highTraffic (\textit{slow})}-\rho _{ lowTraffic (\textit{slow})}}{4} \end{aligned}$$

    otherwise the permission instance prevails.

  22. In particular, the conditional order prefers the prohibition instance to the permission instance iff

    $$\begin{aligned} \frac{\rho _{s}^{F}+\rho _{s}^{F}-\rho _{\textit{use}( fast )}}{3}> \frac{\rho _{s}^{P}+\rho _{c}^{P}}{2} \end{aligned}$$

    otherwise the permission instance prevails.

  23. We have considered alternative methods for calculating the conditional order (e.g., in the previous experiment about the activation and expiration beliefs we have also tried to calculate the conditional order as \(\frac{\rho _{s}^{F}+\rho _{c}^{F}+\rho _{ highTraffic (w)}-(1-\rho _{ lowTraffic (w)})}{4}\)), but these methods have also produced undesirable results.

  24. Note that since we are assuming that the norm, activation and expiration conditions are literals, the substitution for creating an instance from a norm is empty.

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Authors and Affiliations

  1. Liverpool John Moores University, Liverpool, UK

    Natalia Criado

  2. King’s College London, London, UK

    Elizabeth Black & Michael Luck

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  1. Natalia Criado
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Correspondence to Natalia Criado.

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Criado, N., Black, E. & Luck, M. A coherence maximisation process for solving normative inconsistencies. Auton Agent Multi-Agent Syst 30, 640–680 (2016). https://doi.org/10.1007/s10458-015-9300-x

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