In order to provide personalised advice, behaviour support agents need to consider the user's needs and preferences. This user model should be easily adaptable as the user's requirements will change during the long-term use of the agent. We propose a formal framework for such an agent in which the knowledge and the beliefs of the agent are represented explicitly and can be updated directly. Our framework is based on ordered default logic as defeasible reasoning allows the agent to infer additional information based on possibly incomplete knowledge about the world and the user. We also define updates on each component of the agent's framework and demonstrate how these updates can be used to resolve potential misalignments between the agent and the user. Throughout the paper we illustrate our work using a simplified example of a behaviour support agent intended to assist the user in finding a suitable form of exercise.
The rise of artificial assistants has lead to an increased interest in behaviour change support agents [1], which can support the user in establishing new routines and ifnding ways to achieve their goals consistently. In order for these agents to support each user as efectively as possible, the agents need to model the user’s desires, needs and preferences as accurately as possible [2]. Since the agent should ofer support over longer periods of time, it is likely that both the user and the surrounding context will go through changes throughout the agent’s use [3]. Based on the emerging design principles of hybrid intelligence [4],we propose that the agent and the user should be able to collaborate in order to identify and implement the updates that are necessary to adapt the agent over time. This means that the user is in control of the agent’s knowledge and beliefs [5], but the agent should be able to assist the user in determining how each change can be realised and explaining the efects that this will have.
While data-driven approaches such as machine learning, can be used to create a detailed and accurate user model [6], these models can be hard to adapt when the user’s needs change [7]. The concepts captured in these Workshop Proce dings htp:/ceur-ws.org ISN1613-073
CEUR user models are often not explicitly represented, which in turn means that they cannot be updated directly. This also makes it dificult for the user to understand exactly how their changes will afect the agent’s output [8]. By using knowledge-driven methods, we can formalise changes to the user model within the framework, similarly for example to the work in [7]. In particular, we use default logic to model an agent with both dynamic knowledge and beliefs.
In this paper, we introduce a formal framework for a behaviour support agent which includes a model of the world and the user (Section 3.1). We use this framework to represent the agent’s knowledge and beliefs explicitly within a default theory (Section 3.2). We use the defeasible nature of default logic to express beliefs about both the user and the world, which allows the agent to reason with incomplete knowledge and provide advice based on this. In order to make changes to the agent’s knowledge and beliefs possible, we define updates to our formal framework (Section 4). These updates are based on existing work on belief revision updates for default logic [9]. We then compare this to previous work on user-agent misalignment [10, 11] and showcase how the formal updates can be used to resolve potential misalignment scenarios (Section 5) . Throughout the paper we will illustrate the framework using a simple running example of a support agent intended to assist the user in ifnding a suitable exercise based on their needs.
We begin by introducing some preliminaries about the CEUR
ceur-ws.org © 2024 Copyright for this paper by its authors. Use permitted under Creative Commons License ordered default logic that we will be using for our agent Attribution 4.0 International (CC BY 4.0). framework in Section 3.2. We also present the belief revision operators that we will be using in Section 4.
(Δ)
We define (Δ) , (Δ)
and to be the set of prerequisites, justifications and theory of the form ( , , <)
in which is a set of sen- the prerequisite of otherwise inapplicable, higher ranked
Default logic was first introduced in [ 12] to formalise inference rules which are usually true but allow for exceptions. This is done using default rules of the form Prerequisite
∶ Justification Consequent .
This rule states that if the prerequisite is proven and it is consistent to assume the justification, then the consequent is inferred.
In the work of [13] there is additionally a strict partial ordering 1 < 2 on these default rules which expresses that 1 should only be applied if 2 has already been applied or is inapplicable. This results in an ordered default tences, is a set of default rules and < is an ordering on the default rules in . Intuitively, we understand the sentences in to describe our, possibly incomplete, knowledge of the world while the default rules in allow us to derive additional information based on our beliefs. The ordering < may be used to express either preferences or priorities between these beliefs. A theory of this ordered default logic can be translated into standard default logic, allowing for an implementation in theorem provers for standard default logic [13].
When working with default theories, we are interested in the complete views of the world that are consistent with the initial theory, which we refer to as extensions. For an ordered default theory = ( , , <) and any set of sentences , we define Γ() to be the smallest set satisfying the following properties: 1. ⊆ Γ() 2. ℎ(Γ()) = Γ() 3. for all default rules ∶
∈ , if ∈ Γ()
and ¬ ∉ then ∈ Γ() Here ℎ(Γ())
stands for the deductive closure of Γ() .
We call a set of sentences an extension of the theory if = Γ()
. In the following, we will discuss only the consistent extensions of a theory. If we restrict ourselves to normal default rules where the justification and the consequent are the same, [12] has shown that this ensures the existence of a consistent extension. In the following, we will only consider default rules of this form.
We define
ℰ ( ) to be the set of all consistent extensions of the default theory = ( , , <) .
The consistent extensions we have defined above do not yet take the ordering < into account. To include this we use the notion of <-preserving extensions from [13]. consequents of the default rules in Δ. We take (, ) to be the set of default rules which generate the extension and a grounded enumeration ( )∈ of (, ) order in which these rules can be applied. to be an (, ) holds that
For a theory = ( , , <) , an extension ∈ ℰ ( ) <-preserving if there is a grounded enumeration ( )∈ of so that for all , ∈ and ∈ ∖ (, ) is it 1. if < then < and 2. if
({ < then ( )
∉ 0, … , −1 }) ⊢ ¬ ( ) .
or ∪
Even if we know that ℰ ( ) is not empty, this does not ensure that a <-preserving extension of = ( , , <) exists. Intuitively, this is because lower ranked default rules may have a consequent which can be used to infer default rules. This means that a higher ranked rule may be applied after the application of a lower ranked rule. As a result, the grounded enumeration of (, ) not satisfy the first condition from Definition 2.
will
In [14] these inference relationships between the default rules of a theory are formalised using the dependency graph of the theory.
The dependency graph (, )
captures whether default rules influence the applicability of other default rules, either positively by inferring the prerequisite or negatively by inferring the negation of the justification. We take (, ) to be the set of directed edges between the default rules in .
In [13] this is used to specify conditions under which an order default theory has a <-preserving extension. For this, a default theory is considered even if all cycles of the dependency graph have an even number of negative relations. Intuitively this means that the application of a default rules does not negatively influence its own applicability. The ordering < specifies that a lower ranked rule is only applicable after all higher ranked rules have been applied. This means that for each relation ( < ′), we want to ensure that does not afect the applicability of ′.
Proposition 1. As proven in [13], an ordered default theory = ( , , <)
is guaranteed to have a <-preserving extension if the dependency graph (, ) 1. is even and 2. including the ordering < does not create new cycles, so for all cycles of (, ) ∪ {( is a cycle of (, ) .
′, ) ∣ < ′},
Since the ordering < is not necessarily total, it is possible that there are multiple <-preserving extensions.
The field of belief revision is concerned with formalising changes to knowledge and belief bases. Since the knowledge and beliefs of a behaviour support agent are subject to change over time, we want to use update operations from belief revision to reflect this.
In general, belief revision is used to update a set of sentences . We will be working with theory base revision operators [15], which do not require to be deductively closed, as opposed to AGM operators [16]. Specifically we will use the operator ∗ to add a sentence to while ensuring the resulting set remains consistent and the operator ÷ to remove sentences from until can no longer be inferred.
There is a range of work specifically concerned with integrating belief revision methods into default logic such as [17, 18, 19]. In our work we will use the operators defined in [ 9], which includes updates to the knowledge base and the default rules of a default theory.
If we use theory base revision operators, ∗ and ÷ on the knowledge base of a default theory , [9] shows that this can be used to either add to all extensions of or remove from ℎ( ) .
Additionally, [9] introduces updates on the set of default rules . We use ÷ = ∖{ } as an operator which removes the default rule from and ∗ = ∪ { } which adds a default rule to .
In the following section we introduce our framework that can be used to formalise a behaviour support agent. The agent will be able to select a suitable goal for the user to pursue based on the context that the user is currently in. The agent will then recommend actions which result in this goal being achieved, based on the user’s preferences.
We define a support agent for a set of possible actions that the agent can recommend, the set of goals the user may have and a set of contexts that may afect the user’s goals and actions.
Definition 3 (Atoms). We define the following sets of propositional atoms: • = { 1, … , } describing the possible actions, • = { 1, … , } describing the goals, • = { 1, … , } describing diferent contexts and • = ∪ ∪ .
Definition 4 (Language). Let = {⊤, ¬, ∧, ∨, →} be a standard set of logical operators. We introduce the following propositional languages, defined over the operators and sets of atoms in the usual way: • The action language ℒ over and atoms • The goal language ℒ over and atoms • The context language ℒ over and atoms • The agent language ℒ over and atoms
A plan for a goal ∈ is a tuple of the form (, ) , in which is a formula from ℒ describing the actions that must be taken or avoided to achieve the goal .
Definition 5. The set of all possible plans is defined as follows: = {(, ) ∣ ∈ , ∈ ℒ , ≢ ⊥} .
We introduce several types of rules which allow the agent to infer information based on its initial knowledge.
Each rule is represented as a tuple (, ) in which is the prerequisite and is the consequent. These rules will capture a form of defeasible reasoning in which we only infer the consequent if it is consistent with all other information. This means that if is true and nothing suggests otherwise, then is inferred. For all types of rules may be ⊤ to signify that the rule has no prerequisite.
Context assumption rules are of the form (, ) with , ∈ ℒ describing aspects of the context. We can use these rules to make assumptions about the standard context that the user is in or to represent the beliefs the agent has about the relation between diferent contexts.
Definition 6. The set of all possible context assumption rules is defined as ℛ = {(, ) ∣ , ∈ ℒ }.
Goal selection rules are of the form (, ) with ∈ ℒ describing the context and ∈ describing the goal that should be achieved in this context. These are used to describe which goal the user should strive for in a certain context, if possible.
Definition 7. The set of all possible goal selection rules is defined as ℛ = {(, ) ∣ ∈ ℒ , ∈ } .
Action selection rules are of the form (, ) with ∈ ℒ and ∈ ℒ . Here describes the circumstances in which the actions described in may be taken, if they are possible. These circumstances can include certain context factors, the selected goals and other selected actions depending on the application.
Definition 8. The set of action selection rules is defined as ℛ = {(, ) ∣ ∈ ℒ , ∈ ℒ }.
We use ℛ = ℛ ∪ ℛ ∪ ℛ to refer to all rules collectively. In order to be able to reason with these rules, we assign each rule ∈ ℛ a unique name ( ) . For this, we define an injective naming function from the set of all rules ℛ to a set of names . We use these names to define an ordering on the rules. For simplicity of notation we will use the name and the rule itself interchangeably.
We represent the current state of the agent through its configuration, a tuple which specifies the formulas, rules and orderings that the agent reasons with.
The configuration of an agent is a tuple
This results in an agent configuration = , , , < , < , < ) where ⊆ ℒ a goal this should be specified through disjunctions in , and the ordering < based on < , < and < . We take
For a given configuration
Conf of the agent, we deifne a corresponding theory of ordered default logic = ( , , <) ,
. We define the knowledge base based on and , the set of default rules based on , and = (, ) { → ∣ (, ) ∈ } the sentences in to describe the agent’s, possibly incomplete, knowledge of the world while the default rules in allow the agent to derive additional information based on its beliefs. The ordering < provides a way to prioritise between these beliefs, either based on other beliefs about the world or based on the the user’s preferences.
For this we translate every plan ∈ of the form into a formula → ∈ ℒ
. We write ( ) = for the set of all such translated plans.
We also translate each rule = (, ) ∈ in the agent’s configuration to a default rule of the form for ∈ {, , } ∶ and therefore included in the agent’s advice.
This is needed to ensure that the agent only gives one recommended action each day. The current context tains the blood pressure information of the user. In this example we will assume that the blood pressure is high, so = { } = {( , ∨ ), ( , ∨ ℎ)}
We assume that if we have no information suggesting
= {(⊤, ¬ )} out should only be selected if
is true, but the goal of a lower intensity workout can be selected in any situation so = {( , ), (⊤, )}
. For the sake of this example we assume that all the considered actions can be done in any context, which gives us the action selection rules {(⊤, ), (⊤, ), (⊤, ), (⊤, ℎ)}
Since we only have one context assumption rule, we do not specify any ordering on this type of rule. The goal of a higher intensity workout, if applicable, is more important than the lower intensity workout so we have which stand for low- ings to obtain strict partial orderings and define < as the
We take the transitive closures <+ , <+, <+ of the ordercon- and then selects actions based on this.
series composition partial order of , and . This means in addition to ordering given in Conf, we also consider all rules regarding the context to be ranked higher than goal and action selection rules and we rank all goal selection rules higher than the action selection rules. We do this to make sure that the agent first considers the context it is in, then selects a goal for the user to pursue
We define the ordered default theory . The plans corresponding to the goals are ( Conf) corresponding to the agent whose configuration consistent and all default rules in are normal. However, as discussed in Section 2, this does not yet guarantee the existence of a <-preserving extension. For this we define the notion of an efective configuration.
An agent configuration is efective if it is valid and the ordered default theory ( Conf) fulfils the requirements from Proposition 1.
We argue that an agent which is defined in an intuitively sensible way, will fulfil these conditions. If the dependency graph of the theory ( Conf) is not even, or goes against the ordering <, then this signifies an implicit
In the previous section we have defined the configuration of a behaviour support agent and detailed how this determines the advice that the agent gives. In practice, the knowledge and beliefs of the agent change over time, so we need to be able to adapt the configuration of the agent. In this section, we define update operations on the agent’s configuration which will allow us to add or remove information from each component individually.
For each of these components, we want the updates to be defined in such a way that the knowledge base ∪ ∪ ( ) remains consistent. This is necessary to inconsistency in the reasoning formalised in the agent. ensure that we obtain a valid configuration as the result of However, these conditions are dificult to formalise for the configuration of the agent, as they require us to consider the default theory. In future work we hope to deter- requirements from Definition the update. Unfortunately, we cannot always guarantee that the new configuration will also be efective due to the 12. We will formally define mine clear requirements for agent configurations which guarantee for the existence of a <-preserving extension.
We use the <-preserving extensions of the default the- 4.1. Updates to Knowledge Base ory (
Conf) based on the agent’s configuration to determine the advice that the agent should give the user. If there are multiple suitable extensions of (
Conf) then the agent requires a way to choose one of these extensions. This requires a meta-logic above the default logic that we have specified, so we will simply assume that such a selection can be made.
tion Conf = ( , , ,
For an agent with the efective configura
, , , < , < , < ) which is translated into the ordered default theory ( Conf) with the <-preserving extension , the agent’s advice consists of the set of selected goals = ∩ mended actions = () ∩ , ∶ 6, ℎ
⊤∶ ℎ 1 , 1 , , → ( ∨ ℎ), → 2, ⊤∶ 3, ⊤∶
4, 7} and
the updates and also highlight such possible problems. and the set of recom- eration Conf ∗ = ( ′, , , The knowledge base of the agent is made up of the world knowledge , the current context information and the set of plans . We want to be able to update these parts individually, but as explained above we have to consider them all to ensure the updates yield a valid configuration.
We can update the world knowledge of the agent by adding a sentence ∈ ℒ using the following update.
has four possible extensions. We is possible that is still contained in an extension of 1 = { , }, 4 = { , } 2 = { , ℎ}, 3 = { , }
and . However, only 1 is <-preserving. Therefore the agent’s advice consists of the selected goal the recommended action .
(
Conf’) due to the information in ∪ ( ) and the rules in , and . and following operators, similarly to the ones for .
In order to update the current context we use the context
Conf context = in= ining a plan = (, ) we use similar updates as for the The beliefs of the agent are made up of the context assumption rules , the goal selection rules and the action selection rules . Since all types of rules and their respective orderings are defined and translated in the same way, we will only go through the updates of the context assumption rules in detail, the rest are analogous.
When adding a new context assumption rule to the agent’s belief base, it is likely that this belief should also be integrated into the ordering < . However, this is not mandatory and can be done separately with the update operator on < that we introduce below. = =
When an existing context assumption rule needs to be removed from , then we have to remove it from the ordering < as well. This follows from the requirement that the ordering < should be defined on the set .
′ , , , < , < , < ) where are well-defined. If the default theory Lemma 2. The update operators Conf ∗ and Conf ÷ (
Conf) has a consistent extension, then the updated theory will also have a consistent extension.
Proof. This follows directly from the definition and Proposition 2, since all default rules are still normal and the knowledge base is still consistent.
Unfortunately, when adding a new rule we cannot guarantee the existence of a <-preserving extension as this rule could generate new cycles in the dependency graph that might not be even. However, when removing a rule this does result in a configuration
Conf ′ for which
Conf) has a <-preserving extension.
Proposition 3. For an efective configuration
Unfortunately we cannot make the same claim regard- 1. Removing a rule from the configuration and thereby ing <-preserving consistent extensions. This is because any update to the knowledge base of a theory will afect the dependency graph (, ) of the theory (
We can add a relation to < using the following update. to these situations as misalignment scenarios, based on Conf is an efective configuration, we know that all existing cycles are even, which means that the dependency graph of (
Conf ′) is even too. Additionally, any cycles that are removed from the dependency graph by removing are also removed from the ordering <, so the ordering cannot introduce any new cycles.
The ordering of the agent is made up of the orderings < , < and <
on , and respectively. We can update each of these orderings individually and only need to ensure that the resulting ordering is acyclic. Since all three orderings of the agent are defined in the same way, we will only go through the updates to the context ordering in detail; the others are analogous.
Conf ∗<
( 1, 2) where <′ =< ∪{( 1, 2)}.
= with 1, 2 ∈ and ( 2, 1) ∉ <+ we define the update , , , < , < , < ) and a relation ( 1, 2) = , , , <′ , < , < )
When removing a relation ( 1, 2) from < , we ideally want to remove the relation from < does not appear in (
Conf). However, this may require removing multiple relations from < in the process. Since we have multiple choices for this, we will not include this in the update. If necessary, the ordering will need to be updated multiple times to fully remove the relation + to make sure it
With the framework we introduced, the agent is able to reason about a user model and a world model in order to provide personalised support to the user. By representing this explicitly, the user can interact with and adapt the agent’s reasoning process directly using the updates that we have defined in the previous section. We chose to use default logic for this purpose because this allows the user to interact with and adapt the agent’s reasoning process directly using the updates that we have defined in the previous section. A revision of the agent’s reasoning process is necessary if the agent’s advice does not align with the needs and wants of the user because the agent’s advice contains either an action or a goal that the user does not agree with. In the following, we refer [10, 11]. In this section we will discuss the causes of misalignments that are identified in [ 10] and discuss how these can be resolved using the update operators defined in the previous section.
The three causes for these misalignments that are differentiated in [10] are the reasoning process of the agent being wrong, the agent’s user model being wrong or something having changed in such a way that the agent needs to adapt to the new situation. For our purposes, we do not need to diferentiate whether the misalignments occur due to a change or because of a mistake in the initialisation of the agent. Formally, these are handled the same way in this framework. We will discuss how each of the scenarios can be addressed by updating the configuration of the agent. We will give examples of potential misalignments with the advice provided by the agent we introduced in Example 1 and showcase how the realignment updates afect that configuration.
For the sake of this section we will assume that the agent and the user are able to identify the exact cause of the misalignment together. While this is not a trivial assumption and still a topic of active research, this is not something that can be achieved purely within the logical paper. For simplicity, we also assume that there is only one misalignment at a time.
The reasoning process of the agent is based on logical inference, which cannot be incorrect by itself. However, if the world model of the agent is incorrect, then the agent may draw the wrong conclusions even if the user model is correct. This may refer to either the knowledge or the beliefs about the world, the latter including the prioritisation of these beliefs.
Conf) has a consis- framework of an agent, making it out of the scope of this <′ is acyclic. If the default theory ( Lemma 3. The update operators Conf ∗ < ( 1, 2) and Conf ÷< ( 1, 2) are well-defined. The resulting ordering tent extension, then the updated theory will also have a consistent extension.
Proof. This follows directly from definition as the knowledge base is still consistent and the default rules are still normal.
When adding a new relation to the ordering < , this may create new cycles when combined with the dependency graph of (
Conf), which means we cannot guarantee that the resulting configuration will be efective.
On the other hand, it is obvious that removing a relation does not have this issue, meaning that a useful configuration will be updated to another useful configuration. 5.1.1. Incorrect World Knowledge The first misalignment scenario we consider is the situation in which the agent has incorrect knowledge about the world. This means that there is either a sentence ∉
that the agent does not know or the agent incorrectly accepts ∈ as known.
¬∶
specify that unless we have other knowledge, we assume it is not raining. We use the update 2 = 1 ∗ ⊤∶¬ ¬
We remove the action selection rule 6 which is concerned with running through
3 = 2 ÷ 6 . We add the modified action selection rule and obtain 4 = 3 ∗ }. We now restore the ordering by including 7 < date the configuration
Conf of the agent using Conf ∗ . 5 ∗< ( 7, 9) If the agent is missing the information , we can up- 9. This gives us the final updated configuration ′ = By definition, this update requires to be consistent with ∪ ( )
. If we assume that is the only cause of misalignment then this requirement also makes sense intuitively. In order for the agent to also be able to give advice, there are the additional requirements of the effective configuration. While we have explained above that these requirements are reasonable, they might be hard for the user to understand, especially if the agent becomes more complex. In future work we hope to look into ways to identify problematic cycles in the agents configuration and assist the user in resolving them.
Example 3. In Example 2, we have identified that the agent’s advice would be to pursue higher intensity exercising and specifically to go for a run. However, the user may be unable to go for a run because their regular running route is under construction. While this is related to a certain context in a way, which we discuss later on, we can treat this as direct information about the world. This means we update the agent’s configuration with As a result, { , }
is no longer an extension of the agent’s advice will instead be based on the extension ¬ ′, and ′ = ∗
′ = { , ℎ}
However, if the user thinks that a new context factor should be considered which is not yet in the language of the agent, then simply adding it to the current context
is not enough. We likely need to add a context assumption rule which specifies whether this context factor is normally assumed to be true or false. Additionally, we probably want to include this context factor in the relevant goal and action selection rules. Since we do not have an update that can modify individual rules, this needs to be achieved by deleting the original rule, then including the modified rule and lastly reinstating the relevant orderings.
Example 4. We consider that the user does not want to go for a run because it is raining. The original configuration of the agent did not account for the context of rain, so we need to perform a series of updates to include this. We begin by adding 1 = ∗ to the description of the current context using . We add a context assumption rule to
The resulting default theory ( preserving extension ′ = { , ℎ} ′) only has one <the current context and the knowledge about the world and the dependency graph to fulfil the requirements from Proposition 1. The agent will need to collaborate with the user to ensure that the user model is as accurate as possible while still meeting the formal requirements. 5.2.1. Incorrect Goals The goals of the user are what motivates the advice that the agent gives, but they are also subject to change as the needs and desires of the user develop. Each goal ∈
has to correspond to a plan ∈ , so that the agent knows how each goal can be achieved. Additionally, a goal using the updates Conf∗< and Conf÷< . updates Conf∗ and Conf÷ . If the preferences of the user regarding the actions need to be changed, this can be handled analogous to the change of the ordering < , as in Example 3, ′ = { , ℎ} Example 7. In Example 3, the user was not able to go for a run and we added ¬
to the agent’s world knowledge.
This time we will remove the action selection rule for instead. By performing the update tion selection rule and its corresponding ordering in < are removed. This leads to the same < −preserving extension ′ = ÷
6, the ac. While these updates should occur in the consequent of a goal selection rule, formally have the same result, they intuitively mean diferotherwise it cannot be considered in the agent’s advice. ent things. In Example 3 the user is not able to run due to Any changes to the goals of the user therefore have to be outside circumstances from the world that may be resolved captured in the set of plans and the goal selection rules.
If a new goal is added, this goal needs a correspondUsually we will also add a goal selection rule ( ∶ /) ing plan , which can be added with the update Conf ∗ . for a sentence ∈ ℒ describing the context in which , the goal can be selected, by Conf ∗ . The goal selec tion rule will then likely need to prioritised adequately, by adding relations to the ordering < and Conf÷< . Each of these updates will afect the dependency graph, which means there is a risk that the using Conf∗<
resulting agent is not efective.
If the user no longer wants to pursue a goal , then the relevant goal selection rules as well as their orderings need to be removed using the appropriate update.
If a plan or a goal selection rule needs to be changed, the original rule has to be deleted and the new version needs to be added in separate updates as for updates to the world model. ′ = 2 configuration {, ¬ , ¬ , ¬,
¬ ℎ} Example 6. We want the agent to consider the additional goal Rest when giving advice. This goal is achieved if no exercise is done, so the plan is = ( Rest, ¬ ∧
¬ ∧ ¬ ∧ ¬ ℎ)
. For now we do not have any context requirements for this goal to be selected but we prioritise it above the other goal selection rules. We begin by including the plan in the set of plans using 1 = ∗ . We then include the goal selection rule ⊤ ∶ /
with the update 2 = 1 ∗
. Lastly, we include the relations ( 2, ) and ( 3, ) in the ordering < through the update ∗< {( 2, ), ( 3, )}. This results in an efective ′ with the <-preserving extension ′ = 5.2.2. Incorrect Actions at some point. We can remove ¬
from the knowledge base and are still able to use the previous user model. The update in this example on the other hand removes running as a possible action from the user model, indicating that the user no longer views this as an option.
We have introduced a formal framework which can be used to specify the configuration of a behaviour support agent. The configuration can be translated into a theory of ordered default logic and the <-preserving expansions of this theory determine the advice that the agent presents to the user. We have also defined updates on the configuration of the agent which can add or remove information from each of the components. These updates can be used to resolve misalignments between the user and the agent.
In order to use the updates for realignment, it is necessary for the agent and the user to accurately identify the precise cause of the misalignment. While this problem needs to be addressed through communication between the agent and the user [10], we want to facilitate this process using the formal framework of the agent. In future work we hope to study whether the structure of the framework is understandable to users, how we can formally identify potential causes of misalignment and how we can explain problematic cycles in the dependency graph to assist the user in resolving these.
So far we have only included the basic updates which add or remove information from each component of the agent’s configuration. However, in [ 9] there are also other updates on default theories, such as introducing a possibility by ensuring that there is at least one consisThe actions that are recommended by the agent are deter- tent extension which contains a sentence . It may be mined by the plans for the selected goals and the action interesting to see whether these updates can be adapted selection rules. The user may either want to change the for ordered default logic and what they would mean for context prerequisites for selecting certain actions, add a the agent’s configuration. The updates we have used so new action selection rule or remove an existing action far have also each been permanent changes of the agent’s selection rule. These can each be achieved using the configuration. In practice there may be situations which
This research was partly funded by the Hybrid Intelligence Center, a 10-year programme funded by the Dutch Ministry of Education, Culture and Science through the Netherlands Organisation for Scientific Research, https:// hybrid-intelligence-centre.nl, grant number 024.004.022. require diferent advice in the moment but should not be considered in the future. These might require different types of updates to optimise the computational complexity of the agent.
In order to further demonstrate the potential of our framework, we also hope to implement the example agent we have presented in this paper. Since ordered default logic can be translated into regular default logic using the process described in [13], we can use existing solvers for default logic to implement the reasoning of the agent. By combining this with implementations of belief revision operators, we can study how our framework behaves in a real application. For this, we will likely also need to optimise the agent to reduce the computational complexity. A first step for this is to consider the work of [ 14] which discusses specific types of default theories for which an extension can be found in polynomial time. 0004-3702(94)90087-6. [15] M.-A. Williams, Iterated theory base change: A computational model., Proceedings of the Fourteenth International Joint Conference on Artificial Intelligence (1995) 1541–1549. [16] C. E. Alchourrón, P. Gärdenfors, D. Makinson, On the logic of theory change: Partial meet contraction and revision functions, The Journal of Symbolic Logic 50 (1985) 510–530. [17] T. I. Aravanis, P. Peppas, Belief revision in answer set programming, in: Proceedings of the 21st PanHellenic Conference on Informatics, PCI ’17, Association for Computing Machinery, New York, NY, USA, 2017. doi:10.1145/3139367.3139387. [18] P. Krümpelmann, G. Kern-Isberner, Belief base change operations for answer set programming, in: L. F. del Cerro, A. Herzig, J. Mengin (Eds.), Logics in Artificial Intelligence, Springer Berlin Heidelberg, Berlin, Heidelberg, 2012, pp. 294–306. [19] S. Pandžić, On the dynamics of structured argumentation: Modeling changes in default justiifcation logic, Lecture Notes in Computer Science (including subseries Lecture Notes in Artiifcial Intelligence and Lecture Notes in Bioinformatics) 12012 LNCS (2020) 222 – 241. doi:10.1007/ 978-3-030-39951-1_14.