=Paper=
{{Paper
|id=Vol-3835/paper8
|storemode=property
|title=Defining an Adaptable Framework for Behaviour Support Agents in Default Logic
|pdfUrl=https://ceur-ws.org/Vol-3835/paper8.pdf
|volume=Vol-3835
|authors=Johanna Wolff,Victor de Boer,Dirk Heylen,M. Birna van Riemsdijk
|dblpUrl=https://dblp.org/rec/conf/nmr/WolffBHR24
}}
==Defining an Adaptable Framework for Behaviour Support Agents in Default Logic==
https://ceur-ws.org/Vol-3835/paper8.pdf
Defining an Adaptable Framework for Behaviour Support
Agents in Default Logic
Johanna Wolff1,∗ , Victor de Boer2 , Dirk Heylen1 and M. Birna van Riemsdijk1
1
University of Twente, Drienerlolaan 5, 7522 NB Enschede, The Netherlands
2
Vrije Universiteit Amsterdam, De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands
Abstract
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.
Keywords
Default Logic, Belief Revision, Behaviour Support Agent
1. Introduction user models are often not explicitly represented, which
in turn means that they cannot be updated directly. This
The rise of artificial assistants has lead to an increased also makes it difficult for the user to understand ex-
interest in behaviour change support agents [1], which actly how their changes will affect the agent’s output
can support the user in establishing new routines and [8]. By using knowledge-driven methods, we can for-
finding ways to achieve their goals consistently. In order malise changes to the user model within the framework,
for these agents to support each user as effectively as pos- similarly for example to the work in [7]. In particular, we
sible, the agents need to model the user’s desires, needs use default logic to model an agent with both dynamic
and preferences as accurately as possible [2]. Since the knowledge and beliefs.
agent should offer support over longer periods of time, it In this paper, we introduce a formal framework for a
is likely that both the user and the surrounding context behaviour support agent which includes a model of the
will go through changes throughout the agent’s use [3]. world and the user (Section 3.1). We use this framework
Based on the emerging design principles of hybrid intelli- to represent the agent’s knowledge and beliefs explic-
gence [4],we propose that the agent and the user should itly within a default theory (Section 3.2). We use the
be able to collaborate in order to identify and implement defeasible nature of default logic to express beliefs about
the updates that are necessary to adapt the agent over both the user and the world, which allows the agent to
time. This means that the user is in control of the agent’s reason with incomplete knowledge and provide advice
knowledge and beliefs [5], but the agent should be able based on this. In order to make changes to the agent’s
to assist the user in determining how each change can knowledge and beliefs possible, we define updates to our
be realised and explaining the effects that this will have. formal framework (Section 4). These updates are based
While data-driven approaches such as machine learn- on existing work on belief revision updates for default
ing, can be used to create a detailed and accurate user logic [9]. We then compare this to previous work on
model [6], these models can be hard to adapt when the user-agent misalignment [10, 11] and showcase how the
user’s needs change [7]. The concepts captured in these formal updates can be used to resolve potential misalign-
ment scenarios (Section 5) . Throughout the paper we
22nd International Workshop on Nonmonotonic Reasoning, November will illustrate the framework using a simple running ex-
2-4, 2024, Hanoi, Vietnam
∗
Corresponding author.
ample of a support agent intended to assist the user in
Envelope-Open j.d.wolff@utwente.nl (J. Wolff); v.de.boer@vu.nl (V. d. Boer); finding a suitable exercise based on their needs.
d.k.j.heylen@utwente.nl (D. Heylen);
m.b.vanriemsdijk@utwente.nl (M. B. v. Riemsdijk)
Orcid 0009-0005-0178-9570 (J. Wolff); 0000-0001-9079-039X 2. Preliminaries
(V. d. Boer); 0000-0003-4288-3334 (D. Heylen); 0000-0001-9089-5271
(M. B. v. Riemsdijk) We begin by introducing some preliminaries about the
© 2024 Copyright for this paper by its authors. Use permitted under Creative Commons License
Attribution 4.0 International (CC BY 4.0). ordered default logic that we will be using for our agent
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framework in Section 3.2. We also present the belief Definition 2. We define 𝑃𝑟𝑒𝑟𝑒𝑞(Δ),𝐽 𝑢𝑠𝑡𝑖𝑓 (Δ) and
revision operators that we will be using in Section 4. 𝐶𝑜𝑛𝑠𝑒𝑞(Δ) to be the set of prerequisites, justifications and
consequents of the default rules 𝛿 in Δ. We take 𝐺𝐷(𝐷, 𝐸)
2.1. Ordered Default Logic to be the set of default rules which generate the extension
𝐸 and a grounded enumeration (𝛿𝑖 )𝑖∈𝐼 of 𝐺𝐷(𝐷, 𝐸) to be an
Default logic was first introduced in [12] to formalise order in which these rules can be applied.
inference rules which are usually true but allow for ex- For a theory 𝑇 = (𝐾 , 𝐷, <), an extension 𝐸 ∈ ℰ (𝑇 ) is
ceptions. This is done using default rules of the form <-preserving if there is a grounded enumeration (𝛿𝑖 )𝑖∈𝐼 of
𝐺𝐷(𝐷, 𝐸) so that for all 𝑖, 𝑗 ∈ 𝐼 and 𝛿 ∈ 𝐷 ∖ 𝐺𝐷(𝐷, 𝐸) it
Prerequisite ∶ Justification
𝛿 holds that
Consequent .
1. if 𝛿𝑖 < 𝛿𝑗 then 𝑗 < 𝑖 and
This rule states that if the prerequisite is proven and it 2. if 𝛿𝑖 < 𝛿 then 𝑃𝑟𝑒𝑟𝑒𝑞(𝛿) ∉ 𝐸 or 𝐾 ∪
is consistent to assume the justification, then the conse- 𝐶𝑜𝑛𝑠𝑒𝑞({𝛿0 , … , 𝛿𝑖−1 }) ⊢ ¬𝐽 𝑢𝑠𝑡𝑖𝑓 (𝛿).
quent is inferred.
In the work of [13] there is additionally a strict partial Even if we know that ℰ (𝑇 ) is not empty, this does not
ordering 𝛿1 < 𝛿2 on these default rules which expresses ensure that a <-preserving extension of 𝑇 = (𝐾 , 𝐷, <)
that 𝛿1 should only be applied if 𝛿2 has already been ap- exists. Intuitively, this is because lower ranked default
plied or is inapplicable. This results in an ordered default rules may have a consequent which can be used to infer
theory of the form (𝐾 , 𝐷, <) in which 𝐾 is a set of sen- the prerequisite of otherwise inapplicable, higher ranked
tences, 𝐷 is a set of default rules and < is an ordering default rules. This means that a higher ranked rule may
on the default rules in 𝐷. Intuitively, we understand be applied after the application of a lower ranked rule.
the sentences in 𝐾 to describe our, possibly incomplete, As a result, the grounded enumeration of 𝐺𝐷(𝐷, 𝐸) will
knowledge of the world while the default rules in 𝐷 allow not satisfy the first condition from Definition 2.
us to derive additional information based on our beliefs. In [14] these inference relationships between the de-
The ordering < may be used to express either preferences fault rules of a theory are formalised using the depen-
or priorities between these beliefs. A theory of this or- dency graph of the theory. The dependency graph
dered default logic can be translated into standard default 𝒢 (𝐷, 𝐾 ) captures whether default rules influence the
logic, allowing for an implementation in theorem provers applicability of other default rules, either positively by
for standard default logic [13]. inferring the prerequisite or negatively by inferring the
When working with default theories, we are interested negation of the justification. We take 𝒢 (𝐷, 𝐾 ) to be the
in the complete views of the world that are consistent set of directed edges between the default rules in 𝐷.
with the initial theory, which we refer to as extensions. In [13] this is used to specify conditions under which
For an ordered default theory 𝑇 = (𝐾 , 𝐷, <) and any an order default theory has a <-preserving extension. For
set of sentences 𝑆, we define Γ(𝑆) to be the smallest set this, a default theory is considered even if all cycles of
satisfying the following properties: the dependency graph have an even number of negative
relations. Intuitively this means that the application of
1. 𝐾 ⊆ Γ(𝑆) a default rules does not negatively influence its own ap-
2. 𝑇 ℎ(Γ(𝑆)) = Γ(𝑆) plicability. The ordering < specifies that a lower ranked
𝛼∶𝛽
3. for all default rules 𝛾 ∈ 𝐷, rule is only applicable after all higher ranked rules have
if 𝛼 ∈ Γ(𝑆) and ¬𝛽 ∉ 𝑆 then 𝛾 ∈ Γ(𝑆) been applied. This means that for each relation (𝛿 < 𝛿 ′ ),
we want to ensure that 𝛿 does not affect the applicability
Here 𝑇 ℎ(Γ(𝑆)) stands for the deductive closure of Γ(𝑆). of 𝛿 ′ .
We call a set of sentences 𝐸 an extension of the theory
𝑇 if 𝐸 = Γ(𝐸). In the following, we will discuss only the Proposition 1. As proven in [13], an ordered default
consistent extensions of a theory. If we restrict ourselves theory 𝑇 = (𝐾 , 𝐷, <) is guaranteed to have a <-preserving
to normal default rules where the justification and the extension if the dependency graph 𝒢 (𝐷, 𝐾 )
consequent are the same, [12] has shown that this ensures
1. is even and
the existence of a consistent extension. In the following,
we will only consider default rules of this form. 2. including the ordering < does not create new cycles,
so for all cycles 𝒞 of 𝒢 (𝐷, 𝐾 ) ∪ {(𝛿 ′ , 𝛿) ∣ 𝛿 < 𝛿 ′ },
Definition 1. We define ℰ (𝑇 ) to be the set of all consistent 𝒞 is a cycle of 𝒢 (𝐷, 𝐾 ).
extensions of the default theory 𝑇 = (𝐾 , 𝐷, <).
Since the ordering < is not necessarily total, it is pos-
The consistent extensions we have defined above do sible that there are multiple <-preserving extensions.
not yet take the ordering < into account. To include this
we use the notion of <-preserving extensions from [13].
2.2. Belief Revision • The action language ℒ𝐴 over 𝑂 and atoms 𝐴
• The goal language ℒ𝐺 over 𝑂 and atoms 𝐺
The field of belief revision is concerned with formalising
• The context language ℒ𝐶 over 𝑂 and atoms 𝐶
changes to knowledge and belief bases. Since the knowl-
edge and beliefs of a behaviour support agent are subject • The agent language ℒ over 𝑂 and atoms 𝐴𝑡𝑜𝑚𝑠
to change over time, we want to use update operations A plan for a goal 𝑔 ∈ 𝐺 is a tuple of the form (𝑔, 𝜑), in
from belief revision to reflect this. which 𝜑 is a formula from ℒ𝐴 describing the actions that
In general, belief revision is used to update a set of sen- must be taken or avoided to achieve the goal 𝑔.
tences 𝑆. We will be working with theory base revision
operators [15], which do not require 𝑆 to be deductively Definition 5. The set of all possible plans 𝐿𝑃 is defined
closed, as opposed to AGM operators [16]. Specifically as follows: 𝐿𝑃 = {(𝑔, 𝜑) ∣ 𝑔 ∈ 𝐺, 𝜑 ∈ ℒ𝐴 , 𝜑 ≢ ⊥}.
we will use the operator 𝑆 ∗ 𝜑 to add a sentence 𝜑 to 𝑆 We introduce several types of rules which allow the
while ensuring the resulting set remains consistent and agent to infer information based on its initial knowledge.
the operator 𝑆 ÷ 𝜑 to remove sentences from 𝑆 until 𝜑 can Each rule is represented as a tuple (𝜑, 𝜓 ) in which 𝜑 is
no longer be inferred. the prerequisite and 𝜓 is the consequent. These rules will
There is a range of work specifically concerned with capture a form of defeasible reasoning in which we only
integrating belief revision methods into default logic such infer the consequent if it is consistent with all other infor-
as [17, 18, 19]. In our work we will use the operators mation. This means that if 𝜑 is true and nothing suggests
defined in [9], which includes updates to the knowledge otherwise, then 𝜓 is inferred. For all types of rules 𝜑 may
base and the default rules of a default theory. be ⊤ to signify that the rule has no prerequisite.
If we use theory base revision operators, 𝐾 ∗𝜑 and 𝐾 ÷𝜑 Context assumption rules are of the form (𝜑, 𝜓 ) with
on the knowledge base 𝐾 of a default theory 𝑇, [9] shows 𝜑, 𝜓 ∈ ℒ𝐶 describing aspects of the context. We can
that this can be used to either add 𝜑 to all extensions of 𝑇 use these rules to make assumptions about the standard
or remove 𝜑 from 𝑇 ℎ(𝐾 ). context that the user is in or to represent the beliefs the
Additionally, [9] introduces updates on the set of de- agent has about the relation between different contexts.
fault rules 𝐷. We use 𝐷 ÷𝛿 = 𝐷 ∖{𝛿} as an operator which
removes the default rule 𝛿 from 𝐷 and 𝐷 ∗ 𝛿 = 𝐷 ∪ {𝛿} Definition 6. The set of all possible context assumption
which adds a default rule 𝛿 to 𝐷. rules is defined as ℛ𝐶 = {(𝜑, 𝜓 ) ∣ 𝜑, 𝜓 ∈ ℒ𝐶 }.
Goal selection rules are of the form (𝜑, 𝑔) with 𝜑 ∈ ℒ𝐶
3. Behaviour Support Agent describing the context and 𝑔 ∈ 𝐺 describing the goal that
should be achieved in this context. These are used to
In the following section we introduce our framework that describe which goal the user should strive for in a certain
can be used to formalise a behaviour support agent. The context, if possible.
agent will be able to select a suitable goal for the user to
Definition 7. The set of all possible goal selection rules is
pursue based on the context that the user is currently in.
defined as ℛ𝐺 = {(𝜑, 𝑔) ∣ 𝜑 ∈ ℒ𝑐 , 𝑔 ∈ 𝐺}.
The agent will then recommend actions which result in
this goal being achieved, based on the user’s preferences. Action selection rules are of the form (𝜑, 𝜓 ) with 𝜑 ∈ ℒ
and 𝜓 ∈ ℒ𝐴 . Here 𝜑 describes the circumstances in
3.1. Syntax which the actions described in 𝜓 may be taken, if they
are possible. These circumstances can include certain
We define a support agent for a set of possible actions context factors, the selected goals and other selected
𝐴 that the agent can recommend, the set of goals 𝐺 the actions depending on the application.
user may have and a set of contexts 𝐶 that may affect the
user’s goals and actions. Definition 8. The set of action selection rules is defined
as ℛ𝐴 = {(𝜙, 𝜓 ) ∣ 𝜙 ∈ ℒ , 𝜓 ∈ ℒ𝐴 }.
Definition 3 (Atoms). We define the following sets of
propositional atoms: We use ℛ = ℛ𝐶 ∪ ℛ𝐺 ∪ ℛ𝐴 to refer to all rules col-
lectively. In order to be able to reason with these rules,
• 𝐴 = {𝑎1 , … , 𝑎𝑛 } describing the possible actions,
we assign each rule 𝑟 ∈ ℛ a unique name 𝑛(𝑟). For this,
• 𝐺 = {𝑔1 , … , 𝑔𝑚 } describing the goals,
we define an injective naming function 𝑛 from the set of
• 𝐶 = {𝑐1 , … , 𝑐𝑙 } describing different contexts and
all rules ℛ to a set of names 𝑁. We use these names to
• 𝐴𝑡𝑜𝑚𝑠 = 𝐴 ∪ 𝐺 ∪ 𝐶.
define an ordering on the rules. For simplicity of notation
Definition 4 (Language). Let 𝑂 = {⊤, ¬, ∧, ∨, →} be a we will use the name and the rule itself interchangeably.
standard set of logical operators. We introduce the following We represent the current state of the agent through its
propositional languages, defined over the operators 𝑂 and configuration, a tuple which specifies the formulas, rules
sets of atoms in the usual way: and orderings that the agent reasons with.
Definition 9. The configuration of an agent is a tuple This results in an agent configuration 𝐸𝑥 =
Conf = (𝑊 , 𝐶𝐶, 𝑃, 𝐷𝐶 , 𝐷𝐺 , 𝐷𝐴 , <𝐶 , <𝐺 , <𝐴 ) where 𝑊 ⊆ ℒ (𝑊 , 𝐶𝐶, 𝑃, 𝐷𝐶 , 𝐷𝐺 , 𝐷𝐴 , <𝐶 , <𝐺 , <𝐴 )
is the world knowledge, 𝐶𝐶 ⊆ 𝐿𝐶 are literals describing the
current context, 𝑃 ⊆ 𝐿𝑃 is a set of plans, 𝐷𝐶 ⊆ ℛ𝐶 is a 3.2. Determining the Agent’s Advice
set of context assumption rules, 𝐷𝐺 ⊆ ℛ𝐺 is a set of goal
selection rules, 𝐷𝐴 ⊆ ℛ𝐴 is a set of action selection rules For a given configuration Conf of the agent, we de-
and <𝐶 , <𝐺 , <𝐴 are acyclic orderings on 𝐷𝐶 , 𝐷𝐺 and 𝐷𝐴 . fine a corresponding theory of ordered default logic
We also specify that for each goal 𝑔 ∈ 𝐺, there is only one 𝑇 = (𝐾 , 𝐷, <). We define the knowledge base 𝐾 based on
plan 𝑝 = (𝑔, 𝜑) ∈ 𝑃. If there are multiple ways to achieve 𝑊, 𝐶𝐶 and 𝑃, the set of default rules based on 𝐷𝐶 , 𝐷𝐺 and
a goal this should be specified through disjunctions in 𝜑, 𝐷𝐴 and the ordering < based on <𝐶 , <𝐺 and <𝐴 . We take
rather than separate plans. the sentences in 𝐾 to describe the agent’s, possibly incom-
plete, knowledge of the world while the default rules in
To illustrate the use of each component of the agent’s 𝐷 allow the agent to derive additional information based
configuration we introduce a simplified example. We on its beliefs. The ordering < provides a way to prioritise
consider an agent which can give the user advice on how between these beliefs, either based on other beliefs about
to lead a healthier lifestyle based on the user’s medical the world or based on the the user’s preferences.
data. For our purposes we assume that the agent should For this we translate every plan 𝑝 ∈ 𝑃 of the form
recommend one exercise for the user each day, but if 𝑝 = (𝑔, 𝜑) into a formula 𝑔 → 𝜑 ∈ ℒ. We write 𝑇 𝑟(𝑃) =
the user’s blood pressure is elevated, this should be a {𝑔 → 𝜑 ∣ (𝑔, 𝜑) ∈ 𝑃} for the set of all such translated plans.
higher-intensity workout. The agent knows of two types We also translate each rule 𝑟 = (𝜑, 𝜓 ) ∈ 𝐷𝑖 for 𝑖 ∈ {𝐶, 𝐺, 𝐴}
of low-intensity exercises, namely walking and yoga, and in the agent’s configuration to a default rule of the form
two types of higher-intensity exercises, namely going for
a run and weight training. 𝜑 ∶ 𝜓
𝑟
𝜓 .
Example 1. The agent is defined for the context factor
𝐶 = {𝐵𝑃} which indicates that the user’s blood pressure is We take the transitive closures <+ + +
𝐴 , <𝐺 , <𝐶 of the order-
elevated, the set of goals 𝐺 = {𝐿𝐼 , 𝐻 𝐼 } which stand for low- ings to obtain strict partial orderings and define < as the
intensity or higher-intensity workouts and the set of actions series composition partial order of 𝐷 , 𝐷 and 𝐷 . This
𝐶 𝐺 𝐴
𝐴 = {Walk, Yoga, Run, Weights} which are available. means in addition to ordering given in Conf, we also con-
The world knowledge is given by 𝑊 = {𝜑1𝑔 , 𝜑1𝑎 }, in sider all rules regarding the context to be ranked higher
which the formulas 𝜑1𝑔 , 𝜑1𝑎 express that at most one goal than goal and action selection rules and we rank all goal
and one action proposition can be true at the same time selection rules higher than the action selection rules. We
and therefore included in the agent’s advice. This is do this to make sure that the agent first considers the
needed to ensure that the agent only gives one recom- context it is in, then selects a goal for the user to pursue
mended action each day. The current context 𝐶𝐶 con- and then selects actions based on this.
tains the blood pressure information of the user. In this
example we will assume that the blood pressure is high, Definition 10. We define the ordered default theory
so 𝐶𝐶 = {𝐵𝑃}. The plans corresponding to the goals are 𝐷𝐿(Conf) corresponding to the agent whose configuration
𝑃 = {(𝐿𝐼 , 𝑊 𝑎𝑙𝑘 ∨ 𝑌 𝑜𝑔𝑎), (𝐻 𝐼 , 𝑅𝑢𝑛 ∨ 𝑊 𝑒𝑖𝑔ℎ𝑡𝑠)}. is given by Conf = (𝑊 , 𝐶𝐶, 𝑃, 𝐷𝐶 , 𝐷𝐺 , 𝐷𝐴 , <𝐶 , <𝐺 , <𝐴 ) to
We assume that if we have no information suggesting be 𝐷𝐿(Conf) = (𝐾 , 𝐷, <) where 𝐾 = 𝑊 ∪ 𝐶𝐶 ∪ 𝑇 𝑟(𝑃) is the
otherwise, the user’s blood pressure is normal. Therefore knowledge base, 𝐷 = {𝜑 ∶ 𝜓 /𝜓 ∣ (𝜑, 𝜓 ) ∈ 𝐷𝐶 ∪ 𝐷𝐺 ∪ 𝐷𝐴 } is
𝐷𝐶 = {(⊤, ¬𝐵𝑃)}. The goal of a higher intensity work- the set of default rules that make up the belief base and <
out should only be selected if 𝐵𝑃 is true, but the goal of a is the series partial order of 𝐷𝐶 , 𝐷𝐺 and 𝐷𝐴 .
lower intensity workout can be selected in any situation
so 𝐷𝐺 = {(𝐵𝑃, 𝐻 𝐼 ), (⊤, 𝐿𝐼 )}. For the sake of this exam- Based on our definition of a configuration, it is not
ple we assume that all the considered actions can be done yet guaranteed that this ordered default theory has a
in any context, which gives us the action selection rules consistent extension. We define valid configurations to
{(⊤, 𝑊 𝑎𝑙𝑘), (⊤, 𝑌 𝑜𝑔𝑎), (⊤, 𝑅𝑢𝑛), (⊤, 𝑊 𝑒𝑖𝑔ℎ𝑡𝑠)}. be those which do.
Since we only have one context assumption rule, we Definition 11. A configuration Conf is valid if 𝑊 ∪ 𝐶𝐶 ∪
do not specify any ordering on this type of rule. The 𝑇 𝑟(𝑃) is consistent.
goal of a higher intensity workout, if applicable, is more
important than the lower intensity workout so we have Proposition 2. For an ordered default theory 𝐷𝐿(Conf) =
(⊤, 𝐿𝐼 ) <𝐺 (𝐵𝑃, 𝐻 𝐼 ) The user has expressed that they prefer (𝐾 , 𝐷, <) based on a valid configuration Conf of an agent,
Yoga over Walking and Running over Weights so we specify the unordered default theory (𝐾 , 𝐷) has at least one con-
(⊤, 𝑊 𝑎𝑙𝑘) <𝐴 (⊤, 𝑌 𝑜𝑔𝑎) and (⊤, 𝑊 𝑒𝑖𝑔ℎ𝑡𝑠) <𝐴 (⊤, 𝑅𝑢𝑛). sistent extension.
This follows directly from [12] as, by definition, 𝐾 is 4. Agent Updates
consistent and all default rules in 𝐷 are normal. However,
as discussed in Section 2, this does not yet guarantee the In the previous section we have defined the configura-
existence of a <-preserving extension. For this we define tion of a behaviour support agent and detailed how this
the notion of an effective configuration. determines the advice that the agent gives. In practice,
the knowledge and beliefs of the agent change over time,
Definition 12. An agent configuration is effective if it is so we need to be able to adapt the configuration of the
valid and the ordered default theory 𝐷𝐿(Conf) fulfils the agent. In this section, we define update operations on
requirements from Proposition 1. the agent’s configuration which will allow us to add or
We argue that an agent which is defined in an intu- remove information from each component individually.
itively sensible way, will fulfil these conditions. If the For each of these components, we want the updates
dependency graph of the theory 𝐷𝐿(Conf) is not even, or to be defined in such a way that the knowledge base
goes against the ordering <, then this signifies an implicit 𝑊 ∪ 𝐶𝐶 ∪ 𝑇 𝑟(𝑃) 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 difficult to formalise for the update. Unfortunately, we cannot always guarantee
the configuration of the agent, as they require us to con- that the new configuration will also be effective due to the
sider the default theory. In future work we hope to deter- requirements from Definition 12. We will formally define
mine clear requirements for agent configurations which the updates and also highlight such possible problems.
guarantee for the existence of a <-preserving extension.
We use the <-preserving extensions of the default the- 4.1. Updates to Knowledge Base
ory 𝐷𝐿(Conf) based on the agent’s configuration to de-
The knowledge base of the agent is made up of the world
termine the advice that the agent should give the user. If
knowledge 𝑊, the current context information 𝐶𝐶 and the
there are multiple suitable extensions of 𝐷𝐿(Conf) then
set of plans 𝑃. We want to be able to update these parts
the agent requires a way to choose one of these exten-
individually, but as explained above we have to consider
sions. This requires a meta-logic above the default logic
them all to ensure the updates yield a valid configuration.
that we have specified, so we will simply assume that
We can update the world knowledge 𝑊 of the agent
such a selection can be made.
by adding a sentence 𝜑 ∈ ℒ using the following update.
Definition 13. For an agent with the effective configura-
tion Conf = (𝑊 , 𝐶𝐶, 𝑃, 𝐷𝐶 , 𝐷𝐺 , 𝐷𝐴 , <𝐶 , <𝐺 , <𝐴 ) which is Definition 14. For a configuration Conf =
translated into the ordered default theory 𝐷𝐿(Conf) with (𝑊 , 𝐶𝐶, 𝑃, 𝐷𝐶 , 𝐷𝐺 , 𝐷𝐴 , <𝐶 , <𝐺 , <𝐴 ) and a formula 𝜑 ∈ ℒ
the <-preserving extension 𝐸, the agent’s advice consists of with {𝜑} ∪ 𝐶𝐶 ∪ 𝑇 𝑟(𝑃) consistent, we define the update op-
the set of selected goals 𝒜𝐺 = 𝐺 ∩ 𝐸 and the set of recom- eration Conf ∗𝑊 𝜑 = (𝑊 ′ , 𝐶𝐶, 𝑃, 𝐷𝐶 , 𝐷𝐺 , 𝐷𝐴 , <𝐶 , <𝐺 , <𝐴 )
mended actions 𝒜𝐴 = 𝐿(𝐴) ∩ 𝐸. with 𝑊 ′ = (𝑊 ∗ ({𝜑} ∪ 𝐶𝐶 ∪ 𝑇 𝑟(𝑃))) ∖ (𝐶𝐶 ∪ 𝑇 𝑟(𝑃)).
We showcase how the advice is obtained from the con- This means we use the theory base revision opera-
figuration of an agent by going through the configuration tor on 𝑊 and update it with 𝜑 but also with 𝑇 𝑟(𝑃) and
from Example 1. 𝐶𝐶. While we remove 𝑇 𝑟(𝑃) and 𝐶𝐶 again afterwards,
this approach guarantees that 𝑊 ′ ∪ 𝐶𝐶 ∪ 𝑇 𝑟(𝑃) will be
Example 2. The configuration 𝐸𝑥 as defined in Example 1 consistent.
is translated into the ordered default theory 𝑇 = (𝐾 , 𝐷, <) If we want to remove a formula from the world knowl-
with edge 𝑊, this is unproblematic for the consistency of the
knowledge base.
• 𝐾 = {𝜑1𝑔 , 𝜑1𝑎 , 𝐵𝑃, 𝐻 𝐼 → (𝑅𝑢𝑛 ∨ 𝑊 𝑒𝑖𝑔ℎ𝑡𝑠), 𝐿𝐼 →
(𝑊 𝑎𝑙𝑘 ∨ 𝑌 𝑜𝑔𝑎)}, Definition 15. For a configuration Conf =
• 𝐷 = { ⊤∶¬𝐵𝑃
¬𝐵𝑃 1
𝛿 , 𝐵𝑃∶𝐻 𝐼
𝐻𝐼
𝛿 , ⊤∶𝐿𝐼
𝛿 , ⊤∶𝑊 𝑎𝑙𝑘
2 𝐿𝐼 3 𝑊 𝑎𝑙𝑘 4 𝛿 , (𝑊 , 𝐶𝐶, 𝑃, 𝐷 𝐶 , 𝐷 𝐺 , 𝐷 𝐴 , < 𝐶 , < 𝐺 , < 𝐴 ) and a for-
⊤∶𝑌 𝑜𝑔𝑎 ⊤∶𝑅𝑢𝑛 ⊤∶𝑊 𝑒𝑖𝑔ℎ𝑡𝑠 mula 𝜑 ∈ ℒ, we define the update operation
𝛿5 , 𝛿 6 , 𝑊 𝑒𝑖𝑔ℎ𝑡𝑠 7
𝛿 } and
𝑌 𝑜𝑔𝑎 𝑅𝑢𝑛 Conf ÷𝑊 𝜑 = ((𝑊 ÷ 𝜑), 𝐶𝐶, 𝑃, 𝐷𝐶 , 𝐷𝐺 , 𝐷𝐴 , <𝐶 , <𝐺 , <𝐴 ) to
• < = 𝐷𝐶 ; 𝐷𝐺 ; 𝐷𝐴 = {(𝛿𝑖 , 𝛿1 ) ∣ 𝑖 = 2, … , 7} ∪ {(𝛿𝑖 , 𝛿𝑗 ) ∣ remove 𝜑 from 𝑊 and its deductive closure 𝑇 ℎ(𝑊 ).
𝑖 = 4 … , 7; 𝑗 = 2, 3} ∪ {(𝛿3 , 𝛿2 )} ∪ {(𝛿4 , 𝛿5 ), (𝛿7 , 𝛿6 )}
We note that by defining the operator in this way, it
The default theory (𝐾 , 𝐷) has four possible extensions. We is possible that 𝜑 is still contained in an extension 𝐸 of
write only the relevant parts of each extension. These are 𝐷𝐿(Conf’) due to the information in 𝐶𝐶 ∪ 𝑇 𝑟(𝑃) and the
𝐸1 = {𝐻 𝐼 , 𝑅𝑢𝑛}, 𝐸2 = {𝐻 𝐼 , 𝑊 𝑒𝑖𝑔ℎ𝑡𝑠}, 𝐸3 = {𝐿𝐼 , 𝑌 𝑜𝑔𝑎} and rules in 𝐷𝐶 , 𝐷𝐺 and 𝐷𝐴 .
𝐸4 = {𝐿𝐼 , 𝑊 𝑎𝑙𝑘}. However, only 𝐸1 is <-preserving. There- In order to update the current context 𝐶𝐶 we use the
fore the agent’s advice consists of the selected goal 𝐻 𝐼 and following operators, similarly to the ones for 𝑊.
the recommended action 𝑅𝑢𝑛.
Definition 16. For a configuration Conf = 4.2. Updates to the Beliefs
(𝑊 , 𝐶𝐶, 𝑃, 𝐷𝐶 , 𝐷𝐺 , 𝐷𝐴 , <𝐶 , <𝐺 , <𝐴 ) and context in-
The beliefs of the agent are made up of the context as-
formation 𝜑 ∈ 𝐿(𝐶) so that {𝜑} ∪ 𝑊 ∪ 𝑇 𝑟(𝑃)
sumption rules 𝐷𝐶 , the goal selection rules 𝐷𝐺 and the
consistent, we define the update operation
action selection rules 𝐷𝐴 . Since all types of rules and
Conf ∗𝐶𝐶 𝜑 = (𝑊 , 𝐶𝐶 ′ , 𝑃, 𝐷𝐶 , 𝐷𝐺 , 𝐷𝐴 , <𝐶 , <𝐺 , <𝐴 )
their respective orderings are defined and translated in
with 𝐶𝐶 ′ = (𝐶𝐶 ∗ (𝜑 ∪ 𝐶𝐶 ∪ 𝑇 𝑟(𝑃))) ∖ (𝑊 ∪ 𝑇 𝑟(𝑃))
the same way, we will only go through the updates of the
Definition 17. For a configuration Conf = context assumption rules in detail, the rest are analogous.
(𝑊 , 𝐶𝐶, 𝑃, 𝐷𝐶 , 𝐷𝐺 , 𝐷𝐴 , <𝐶 , <𝐺 , <𝐴 ) and context in- When adding a new context assumption rule to the
formation 𝜑 ∈ 𝐶 ∪ ¬𝐶, we define the update operations agent’s belief base, it is likely that this belief should also
Conf ÷𝐶𝐶 𝜑 = (𝑊 , (𝐶𝐶 ÷ 𝜑), 𝑃, 𝐷𝐶 , 𝐷𝐺 , 𝐷𝐴 , <𝐶 , <𝐺 , <𝐴 ) to be integrated into the ordering <𝐶 . However, this is not
remove 𝜑 from 𝐶𝐶. mandatory and can be done separately with the update
operator on <𝐶 that we introduce below.
In order to update the plans 𝑃 by adding or remov-
ing a plan 𝜋 = (𝑔, 𝜑) we use similar updates as for the Definition 20. For a configuration Conf =
knowledge base. When adding a new plan to 𝑃 we need (𝑊 , 𝐶𝐶, 𝑃, 𝐷𝐶 , 𝐷𝐺 , 𝐷𝐴 , <𝐶 , <𝐺 , <𝐴 ) and a context as-
to ensure that the resulting set of plans only contains at sumption rule 𝑟 = (𝜑, 𝜓 ) ∈ ℛ𝐶 we define the update
most one plan per goal. This means we have to remove Conf ∗𝐷𝐶 𝑟 = (𝑊 , 𝐶𝐶, 𝑃, 𝐷𝐶′ , 𝐷𝐺 , 𝐷𝐴 , <𝐶 , <𝐺 , <𝐴 ) where
any previous plan (𝑔, 𝜓 ) for the goal 𝑔 before adding 𝐷𝐶′ = 𝐷𝐶 ∪ {𝑟}.
(𝑔, 𝜑) to 𝑃. We also make sure that the new information
is consistent with the other components of the knowl- When an existing context assumption rule 𝑟 needs to
edge base. be removed from 𝐷𝐶 , then we have to remove it from the
ordering <𝐶 as well. This follows from the requirement
Definition 18. For a configuration Conf = that the ordering <𝐶 should be defined on the set 𝐷𝐶 .
(𝑊 , 𝐶𝐶, 𝑃, 𝐷𝐶 , 𝐷𝐺 , 𝐷𝐴 , <𝐶 , <𝐺 , <𝐴 ) and a plan
𝜋 = (𝑔, 𝜑) ∈ 𝐿𝑃 so that {𝑇 𝑟(𝜋)} ∪ 𝑊 ∪ 𝐶𝐶 Definition 21. For a configuration Conf =
consistent, we define the update operation (𝑊 , 𝐶𝐶, 𝑃, 𝐷𝐶 , 𝐷𝐺 , 𝐷𝐴 , <𝐶 , <𝐺 , <𝐴 ) and a context as-
Conf ∗𝑃 𝜋 = (𝑊 , 𝐶𝐶, 𝑃 ′ , 𝐷𝐶 , 𝐷𝐺 , 𝐷𝐴 , <𝐶 , <𝐺 , <𝐴 ) sumption rule 𝑟 = (𝜑, 𝜓 ) ∈′ ℛ𝐶 we define the update
with 𝑃 ′ = (𝑃 ∖ {(𝑔, 𝜓 )} ∪ {𝜋}) to add 𝜋 to 𝑃. Conf ÷𝐷𝐶 𝑟 = (𝑊 , 𝐶𝐶, 𝑃, 𝐷𝐶 , 𝐷𝐺 , 𝐷𝐴 , <′𝐶 , <𝐺 , <𝐴 ) where
𝐷𝐶′ = 𝐷𝐶 ∖ {𝑟} and <′𝐶 =<𝐶 ∣𝐷𝐶′ .
If we remove a plan 𝜋 = (𝑔, 𝜑) from 𝑃, this will result
in a valid configuration. However, it is possible that the Lemma 2. The update operators Conf∗𝐷𝐶 𝑟 and Conf÷𝐷𝐶 𝑟
goal 𝑔 is still the result of a goal selection rule and may are well-defined. If the default theory 𝐷𝐿(Conf) has a
be contained in an extension of 𝐷𝐿(Conf). This means consistent extension, then the updated theory will also have
the agent may advise the user to pursue the goal, despite a consistent extension.
there not being any action recommendation which corre-
Proof. This follows directly from the definition and
sponds to this. For this reason, this update should usually
Proposition 2, since all default rules are still normal and
not be performed in isolation in practice.
the knowledge base is still consistent.
Definition 19. For a configuration Conf =
(𝑊 , 𝐶𝐶, 𝑃, 𝐷𝐶 , 𝐷𝐺 , 𝐷𝐴 , <𝐶 , <𝐺 , <𝐴 ) and a plan Unfortunately, when adding a new rule we cannot
𝜋 = (𝑔, 𝜑) ∈ 𝐿𝑃, we define the update operations guarantee the existence of a <-preserving extension as
Conf ÷𝑃 𝜋 = (𝑊 , 𝐶𝐶, 𝑃 ′ , 𝐷𝐶 , 𝐷𝐺 , 𝐷𝐴 , <𝐶 , <𝐺 , <𝐴 ) with this rule could generate new cycles in the dependency
𝑃 ′ = 𝑃 ∖ 𝜋 to remove 𝜋 from 𝑃. graph that might not be even. However, when removing
a rule this does result in a configuration Conf ′ for which
Lemma 1. The updates Conf ∗𝐾 𝜑,Conf ∗𝑊 𝜑, Conf ÷𝑊 𝜑, 𝐷𝐿(Conf) has a <-preserving extension.
Conf ∗𝐶𝐶 𝜑, Conf ÷𝐶𝐶 𝜑 Conf ∗𝑃 𝜋 and Conf ÷𝑃 𝜋, are well-
defined. Additionally, if the default theory 𝐷𝐿(Conf) has Proposition 3. For an effective configuration Conf =
a consistent extension, then the updated default theory (𝑊 , 𝐶𝐶, 𝑃, 𝐷𝐶 , 𝐷𝐺 , 𝐷𝐴 , <𝐶 , <𝐺 , <𝐴 ) where 𝐷𝐿(Conf) has a
𝐷𝐿(Conf ′ ) will also have a consistent extension. <-preserving extension and a rule 𝑟 ∈ ℛ𝐶 , the ordered
default theory 𝐷𝐿(Conf ′ ) with Conf ′ = Conf ÷𝐷𝐶 𝑟, also
Proof. This follows directly from the definitions of ∗𝜑 has a <-preserving extension.
and ÷𝜑 and Proposition 2.
Proof. To see this we check the conditions of Proposition
Unfortunately we cannot make the same claim regard- 1. Removing a rule from the configuration and thereby
ing <-preserving consistent extensions. This is because removing a default rule from the default theory, cannot
any update to the knowledge base of a theory will affect create any new cycles in the dependency graph. Since
the dependency graph 𝒢 (𝐷, 𝐾 ) of the theory 𝐷𝐿(Conf).
Conf is an effective configuration, we know that all exist- 5. Resolving Misalignments
ing cycles are even, which means that the dependency
graph of 𝐷𝐿(Conf ′ ) is even too. Additionally, any cy- With the framework we introduced, the agent is able to
cles that are removed from the dependency graph by reason about a user model and a world model in order to
removing 𝑟 are also removed from the ordering <, so the provide personalised support to the user. By representing
ordering cannot introduce any new cycles. 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
4.3. Updates to the Ordering default logic for this purpose because this allows the user
The ordering of the agent is made up of the orderings to interact with and adapt the agent’s reasoning process
<𝐶 , <𝐺 and <𝐴 on 𝐷𝐶 , 𝐷𝐺 and 𝐷𝐴 respectively. We can directly using the updates that we have defined in the
update each of these orderings individually and only need previous section. A revision of the agent’s reasoning
to ensure that the resulting ordering is acyclic. Since process is necessary if the agent’s advice does not align
all three orderings of the agent are defined in the same with the needs and wants of the user because the agent’s
way, we will only go through the updates to the context advice contains either an action 𝑎 or a goal 𝑔 that the
ordering in detail; the others are analogous. user does not agree with. In the following, we refer
We can add a relation to <𝐶 using the following update. to these situations as misalignment scenarios, based on
[10, 11]. In this section we will discuss the causes of
Definition 22. For a configuration Conf = misalignments that are identified in [10] and discuss how
(𝑊 , 𝐶𝐶, 𝑃, 𝐷𝐶 , 𝐷𝐺 , 𝐷𝐴 , <𝐶 , <𝐺 , <𝐴 ) and a relation (𝑟1 , 𝑟2 ) these can be resolved using the update operators defined
with 𝑟1 , 𝑟2 ∈ 𝐷𝐶 and (𝑟2 , 𝑟1 ) ∉ <+ 𝐶 we define the update in the previous section.
Conf ∗<𝐶 (𝑟1 , 𝑟2 ) = (𝑊 , 𝐶𝐶, 𝑃, 𝐷𝐶 , 𝐷𝐺 , 𝐷𝐴 , <′𝐶 , <𝐺 , <𝐴 ) The three causes for these misalignments that are dif-
where <′𝐶 =<𝐶 ∪{(𝑟1 , 𝑟2 )}. ferentiated in [10] are the reasoning process of the agent
When removing a relation (𝑟1 , 𝑟2 ) from <𝐶 , we ideally being wrong, the agent’s user model being wrong or
want to remove the relation from <+ something having changed in such a way that the agent
𝐶 to make sure it
does not appear in 𝐷𝐿(Conf). However, this may require needs to adapt to the new situation. For our purposes, we
removing multiple relations from <𝐶 in the process. Since do not need to differentiate whether the misalignments
we have multiple choices for this, we will not include occur due to a change or because of a mistake in the
this in the update. If necessary, the ordering will need to initialisation of the agent. Formally, these are handled
be updated multiple times to fully remove the relation the same way in this framework. We will discuss how
from <+ each of the scenarios can be addressed by updating the
𝐶.
configuration of the agent. We will give examples of
Definition 23. For a configuration Conf = potential misalignments with the advice provided by the
(𝑊 , 𝐶𝐶, 𝑃, 𝐷𝐶 , 𝐷𝐺 , 𝐷𝐴 , <𝐶 , <𝐺 , <𝐴 ) and a relation agent we introduced in Example 1 and showcase how the
(𝑟1 , 𝑟2 ) with 𝑟1 , 𝑟2 ∈ 𝐷𝐶 we define the update realignment updates affect that configuration.
Conf ÷<𝐶 (𝑟1 , 𝑟2 ) = (𝑊 , 𝐶𝐶, 𝑃, 𝐷𝐶 , 𝐷𝐺 , 𝐷𝐴 , <′𝐶 , <𝐺 , <𝐴 ) For the sake of this section we will assume that the
where <′𝐶 =<𝐶 ∖{(𝑟1 , 𝑟2 )}. agent and the user are able to identify the exact cause
of the misalignment together. While this is not a trivial
Lemma 3. The update operators Conf ∗ <𝐶 (𝑟1 , 𝑟2 ) and assumption and still a topic of active research, this is not
Conf ÷<𝐶 (𝑟1 , 𝑟2 ) are well-defined. The resulting ordering something that can be achieved purely within the logical
<′𝐶 is acyclic. If the default theory 𝐷𝐿(Conf) has a consis- framework of an agent, making it out of the scope of this
tent extension, then the updated theory will also have a paper. For simplicity, we also assume that there is only
consistent extension. one misalignment at a time.
Proof. This follows directly from definition as the knowl-
edge base is still consistent and the default rules are still 5.1. Incorrect Reasoning
normal. The reasoning process of the agent is based on logical
When adding a new relation to the ordering <𝐶 , this inference, which cannot be incorrect by itself. However,
may create new cycles when combined with the depen- if the world model of the agent is incorrect, then the
dency graph of 𝐷𝐿(Conf), which means we cannot guar- agent may draw the wrong conclusions even if the user
antee that the resulting configuration will be effective. model is correct. This may refer to either the knowledge
On the other hand, it is obvious that removing a relation or the beliefs about the world, the latter including the
does not have this issue, meaning that a useful configu- prioritisation of these beliefs.
ration will be updated to another useful configuration.
5.1.1. Incorrect World Knowledge specify that unless we have other knowledge, we assume it
The first misalignment scenario we consider is the situa- is not raining. We use the update 𝐸𝑥 2 = 𝐸𝑥 1 ∗𝐷𝐶 ⊤∶¬𝑅𝑎𝑖𝑛
¬𝑅𝑎𝑖𝑛 8
𝛿 .
tion in which the agent has incorrect knowledge about We remove the action selection rule 𝛿6 which is concerned
the world. This means that there is either a sentence with running through 𝐸𝑥 3 = 𝐸𝑥 2 ÷𝐷𝐴 𝛿6. We add the
𝜑 ∉ 𝑊 that the agent does not know or the agent incor- modified action selection rule and obtain 𝐸𝑥 4 = 𝐸𝑥 3 ∗𝐷𝐴
¬𝑅𝑎𝑖𝑛∶𝑅𝑢𝑛
rectly accepts 𝜑 ∈ 𝑊 as known. 𝑅𝑢𝑛
}. We now restore the ordering by including 𝛿7 <
If the agent is missing the information 𝜑, we can up- 𝛿9 . This gives us the final updated configuration 𝐸𝑥 ′ =
date the configuration Conf of the agent using Conf ∗𝑊 𝜑. 𝐸𝑥 5 ∗<𝐴 (𝛿7 , 𝛿9 )
By definition, this update requires 𝜑 to be consistent with The resulting default theory 𝐷𝐿(𝐸𝑥 ′ ) only has one <-
𝐶𝐶 ∪ 𝑇 𝑟(𝑃). If we assume that 𝜑 is the only cause of preserving extension 𝐸 ′ = {𝐻 𝐼 , 𝑊 𝑒𝑖𝑔ℎ𝑡𝑠}.
misalignment then this requirement also makes sense
intuitively. In order for the agent to also be able to give 5.1.2. Incorrect Beliefs about the world
advice, there are the additional requirements of the ef-
fective configuration. While we have explained above The agent’s beliefs about the world, modelled as context
that these requirements are reasonable, they might be assumption rules in 𝐷𝐶 , may also be incorrect.
hard for the user to understand, especially if the agent If a new belief 𝛿 needs to be adopted, then this can be
becomes more complex. In future work we hope to look done using the update Conf ∗𝐷𝐶 𝛿. This will cause the
into ways to identify problematic cycles in the agents dependency graph of the agent to change, which may
configuration and assist the user in resolving them. mean that the resulting configuration is not effective.
This brings similar problems as a change in the world
Example 3. In Example 2, we have identified that the knowledge.
agent’s advice would be to pursue higher intensity exercis- A belief 𝛿 can be removed from the agent’s configura-
ing and specifically to go for a run. However, the user may tion with the update Conf ÷𝐷 𝛿. While this will produce
𝐶
be unable to go for a run because their regular running an effective configuration, this may lead to the agent’s
route is under construction. While this is related to a certain advice being less specific towards the user’s context.
context in a way, which we discuss later on, we can treat The beliefs about the world may also need to be priori-
this as direct information about the world. This means we tised differently, by updating the ordering <𝐶 . While we
update the agent’s configuration with 𝐸𝑥 ′ = 𝐸𝑥 ∗𝑊 ¬𝑅𝑢𝑛. have introduced updates Conf ∗< (𝛿𝑖 , 𝛿𝑗 ) and Conf ÷<
As a result, {𝐻 𝐼 , 𝑅𝑢𝑛} is no longer an extension of 𝐸𝑥 ′ , and (𝛿𝑖 , 𝛿𝑗 ) to add or remove a relation to the ordering, in prac-
𝐶 𝐶
the agent’s advice will instead be based on the extension tice we will likely want to make more complex changes.
𝐸 ′ = {𝐻 𝐼 , 𝑊 𝑒𝑖𝑔ℎ𝑡𝑠}. While these can all be broken down into multiple appli-
Next we consider that the agent has wrongly identified cations of the two updates we have defined, this may
the current context 𝐶𝐶. If the mistake concerns a context be too complicated for the user to oversee themselves.
factor 𝑐 that is already in 𝐶, this can easily be resolved Additionally, we also need to make sure that the relation
using the updates Conf ∗𝐶𝐶 𝑐 and Conf ÷𝐶𝐶 𝑐, similarly to remains acyclic and does not contradict the implicit or-
the updates to the knowledge base we described above. dering of the default theory modelled in the dependency
However, if the user thinks that a new context factor graph.
should be considered which is not yet in the language Example 5. So far the agent’s configuration has included
of the agent, then simply adding it to the current con- the context assumption rule 𝛿 that unless other information
1
text 𝐶𝐶 is not enough. We likely need to add a context is available, the user’s blood pressure is normal. For this we
assumption rule which specifies whether this context remove the original rule 𝛿 using 𝐸𝑥 1 = 𝐸𝑥 ÷ 𝛿 and then
1 𝐷𝐶 1
factor is normally assumed to be true or false. Addition- ′ = 𝐸𝑥 1 ∗ ⊤∶𝐵𝑃 . This means
ally, we probably want to include this context factor in add the new rule through 𝐸𝑥 𝐷𝐶 𝐵𝑃
the relevant goal and action selection rules. Since we that even if the agent does not know the blood pressure
do not have an update that can modify individual rules, levels of the user, so 𝐵𝑃 ∉ 𝐶𝐶, it will still recommend
this needs to be achieved by deleting the original rule, higher intensity exercises.
then including the modified rule and lastly reinstating
the relevant orderings. 5.2. Incorrect User Model
Example 4. We consider that the user does not want to go The user model of our agent contains information about
for a run because it is raining. The original configuration of the user’s goals, the user’s possible actions and the prefer-
the agent did not account for the context of rain, so we need ences regarding these. While humans may have goals and
to perform a series of updates to include this. We begin by preferences that are not strictly logical, the formal frame-
adding 𝑅𝑎𝑖𝑛 to the description of the current context using work of the agent requires the goals to be consistent with
𝐸𝑥 1 = 𝐸𝑥 ∗𝐶𝐶 𝑅𝑎𝑖𝑛. We add a context assumption rule to the current context and the knowledge about the world
and the dependency graph to fulfil the requirements from updates Conf∗𝐷𝐴 and Conf÷𝐷𝐴 . If the preferences of the
Proposition 1. The agent will need to collaborate with user regarding the actions need to be changed, this can
the user to ensure that the user model is as accurate as be handled analogous to the change of the ordering <𝐶 ,
possible while still meeting the formal requirements. using the updates Conf∗<𝐴 and Conf÷<𝐶 .
Example 7. In Example 3, the user was not able to go for
5.2.1. Incorrect Goals
a run and we added ¬𝑅𝑢𝑛 to the agent’s world knowledge.
The goals of the user are what motivates the advice that This time we will remove the action selection rule for 𝑅𝑢𝑛
the agent gives, but they are also subject to change as the instead. By performing the update 𝐸𝑥 ′ = 𝐸𝑥 ÷𝐷𝐴 𝛿6 , the ac-
needs and desires of the user develop. Each goal 𝑔 ∈ 𝐺 has tion selection rule and its corresponding ordering in <𝐴 are
to correspond to a plan 𝜋 ∈ 𝑃, so that the agent knows removed. This leads to the same < −preserving extension
how each goal can be achieved. Additionally, a goal as in Example 3, 𝐸 ′ = {𝐻 𝐼 , 𝑊 𝑒𝑖𝑔ℎ𝑡𝑠}. While these updates
should occur in the consequent of a goal selection rule, formally have the same result, they intuitively mean differ-
otherwise 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. at some point. We can remove ¬𝑅𝑢𝑛 from the knowledge
If a new goal 𝑔 is added, this goal needs a correspond- base 𝑊 and are still able to use the previous user model. The
ing plan 𝜋, which can be added with the update Conf ∗𝑃 𝜋. update in this example on the other hand removes running
Usually we will also add a goal selection rule (𝜑 ∶ 𝑔/𝑔) 𝛿𝑔 , as a possible action from the user model, indicating that
for a sentence 𝜑 ∈ ℒ𝐶 describing the context in which the user no longer views this as an option.
the goal can be selected, by Conf ∗𝐷𝐺 𝛿𝑔 . The goal selec-
tion rule will then likely need to prioritised adequately,
by adding relations to the ordering <𝐺 using Conf∗<𝐺 6. Discussion
and Conf÷<𝐺 . Each of these updates will affect the de-
We have introduced a formal framework which can be
pendency graph, which means there is a risk that the
used to specify the configuration of a behaviour sup-
resulting agent is not effective.
port agent. The configuration can be translated into a
If the user no longer wants to pursue a goal 𝑔, then the
theory of ordered default logic and the <-preserving ex-
relevant goal selection rules as well as their orderings
pansions of this theory determine the advice that the
need to be removed using the appropriate update.
agent presents to the user. We have also defined updates
If a plan or a goal selection rule needs to be changed,
on the configuration of the agent which can add or re-
the original rule has to be deleted and the new version
move information from each of the components. These
needs to be added in separate updates as for updates to
updates can be used to resolve misalignments between
the world model.
the user and the agent.
Example 6. We want the agent to consider the additional In order to use the updates for realignment, it is neces-
goal Rest when giving advice. This goal is achieved if no sary for the agent and the user to accurately identify the
exercise is done, so the plan is 𝜋 = (Rest, ¬𝑊 𝑎𝑙𝑘 ∧ ¬𝑌 𝑜𝑔𝑎 ∧ precise cause of the misalignment. While this problem
¬𝑅𝑢𝑛 ∧ ¬𝑊 𝑒𝑖𝑔ℎ𝑡𝑠). For now we do not have any context needs to be addressed through communication between
requirements for this goal to be selected but we prioritise it the agent and the user [10], we want to facilitate this
above the other goal selection rules. We begin by including process using the formal framework of the agent. In fu-
the plan 𝑝 in the set of plans using 𝐸𝑥 1 = 𝐸𝑥 ∗𝑃 𝑝. We then ture work we hope to study whether the structure of the
include the goal selection rule ⊤ ∶ 𝑅𝑒𝑠𝑡/𝑅𝑒𝑠𝑡 𝛿𝑔 with the framework is understandable to users, how we can for-
update 𝐸𝑥 2 = 𝐸𝑥 1 ∗𝐷𝐺 𝛿𝑔 . Lastly, we include the relations mally identify potential causes of misalignment and how
(𝛿2 , 𝛿𝑔 ) and (𝛿3 , 𝛿𝑔 ) in the ordering <𝐺 through the update we can explain problematic cycles in the dependency
𝐸𝑥 ′ = 𝐸𝑥 2 ∗<𝐺 {(𝛿2 , 𝛿𝑔 ), (𝛿3 , 𝛿𝑔 )}. This results in an effective graph to assist the user in resolving these.
configuration 𝐸𝑥 ′ with the <-preserving extension 𝐸 ′ = So far we have only included the basic updates which
{𝑅𝑒𝑠𝑡, ¬𝑊 𝑎𝑙𝑘, ¬𝑌 𝑜𝑔𝑎, ¬𝑅𝑢𝑛, ¬𝑊 𝑒𝑖𝑔ℎ𝑡𝑠}. 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
5.2.2. Incorrect Actions
possibility by ensuring that there is at least one consis-
The actions that are recommended by the agent are deter- tent extension which contains a sentence 𝜑. 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
require different advice in the moment but should not L. van der Gaag, F. van Harmelen, H. van Hoof,
be considered in the future. These might require dif- B. van Riemsdijk, A. van Wynsberghe, R. Verbrugge,
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Ministry of Education, Culture and Science through the
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