Late Bindings in AgentSpeak(L)

Frantisek Zboril

, Frantisek Vidensky

, Radek Koci

and Frantisek V. Zboril

Department of Intelligent Systems, Brno University of Technology, Bozetechova 2, Brno, Czech Republic

Keywords: BDI Agents, Planning, Agent Interpretation, AgentSpeak(L).

Abstract: For agents based on BDI theory, some problems remain open. These include parts of the interpretation of

these systems that are nondeterministic in the original specifications, and finding methods for their

determinism should lead to improved rationality of agent behaviour. These problems include the choice of a

plan suitable for achieving the goal, then the choice of the intention to be pursued by the agent at any given

time, and if a language based on predicate logic is used to implement such an agent, then there is also the

problem of choosing variable substitutions. One such agent-based system is systems using the AgentSpeak(L)

language, which will be the basis for this paper. We will introduce late binding into the interpretation of this

language and show that they do not make the agent lose the possibility of achieving the goal by making

unnecessary or incorrect substitutions in cases where such a decision is not necessary. We show that with late

binding substitutions the agent operates with all possible substitutions given by the chosen plan to the goals

in the plan structure, and that these substitutions are always valid with respect to the acts performed so far

within this plan.

1 INTRODUCTION

One way to create artificial agents is based on

Bratman's theory of intentions (Bratman, Intention,

Plans and Practical Reason, 1987). Intentions, as the

mental states of agents, are persistent goals (Cohen &

Levesque, 1990), which an agent decides to achieve

at the moment some of its desires have been found

attainable and are pursued until they are achieved or

the agent finds them unachievable. Systems

combining intentions with other mental states,

namely beliefs and desires, known as BDI systems

(Rao & Georgeff, Modeling Rational Agents within a

BDI-Architecture, 1991), have gained popularity.

Implementations of such systems include IRMA

(Bratman, Israel, & Pollack, Plans and resource-

bounded practical reasoning, 1988), PRS (Georgeff &

Lansky, 1987), dMars (d'Iverno, Kinny, Luck, &

Wooldridge, 1998), 2APL (Dastani, 2008), CAN

(Sardina & Padgham, 2011), as well as systems

interpreting the AgentSpeak(L) language (Rao,

AgentSpeak(L): BDI agents speak out in a logical

https://orcid.org/0000-0001-7861-8220

https://orcid.org/0000-0003-1808-441X

https://orcid.org/0000-0003-1313-6946

https://orcid.org/0000-0002-6965-4104

computable language, 1996). The principle of these

systems is based on the choice of intentions from the

desirec that the agent has, which may become

intentions at an opportune moment based on the

current state of the agent's beliefs. The plan then tries

to fulfill this intention by executing a plan, or plans,

that are hierarchically ordered. Each plan may declare

goals to be achieved, and a subplan is chosen to

achieve those goals. If the top-level plan that has been

chosen to achieve the goal of an intention is achieved,

that intention is also achieved.

The problems that are still a matter of research are

found in the non-deterministic parts of an agent's

decision making. This involves both the selection of

the appropriate means to achieve goals or subgoals in

the form of plans, but it also involves, when a

language based on predicate logic is used, the choice

of appropriate substitutions. In addition, the agent, if

pursuing multiple goals simultaneously, also has the

dilemma of which goal to pursue at any given step.

This text offers a solution for dealing with

substitutions such that if a plan is chosen, then

substitutions need not be applied immediately, but

Zboril, F., Vidensky, F., Koci, R. and Zboril, F.

Late Bindings in AgentSpeak(L).

DOI: 10.5220/0010897000003116

In Proceedings of the 14th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2022) - Volume 3, pages 715-724

ISBN: 978-989-758-547-0; ISSN: 2184-433X

715

can be deferred until a decision needs to be made,

typically before a particular action is taken. Poorly

chosen substitutions during the practical reasoning

phase may indicate the choice of the wrong resource,

which may become apparent later during the

execution of the plan thus instantiated. Several ways

of handling substitutions during agent execution are

currently proposed. The 2APL system (Dastani,

2008) offers the possibility to create rules that specify

decision processes including the handling of possible

substitutions. For the CAN system (Sardina &

Padgham, 2011), knowledge of the substitutions used

is stored and, if a new plan has to be searched if a

previous attempt to achieve the intentions failed with

some substitutions, other possible substitutions can

be chosen for the same plan.

Our approach is based on trying to preserve all

possible variable bindings and during the execution

of a plan, it works with all of them. If some step (act)

of the plan is executed, then the number of these

options may be limited and only some of them survive

on. However, if at least one of the options persists,

then the plan can continue. We first introduced the

principle that leads to late variable binding in (Zboril,

Koci, Janousek, & Mazal, 2008) and later discussed

it in (Zboril, Zboril, & Kral, Flexible Plan Handling

using Extended Environment, 2013). In this paper, we

give a more precise and modified form of the basic

operations we need to create a late-binding

AgentSpeak(L) interpreter and specify the use of each

operation here.

The individual sections are organized as follows.

In Section 2, we briefly introduce the AgentSpeak(L)

language and give the motivation for introducing late

binding for its interpreter. Section 3 defines the

operations that are necessary for such an

interpretation, and in Section 4, we introduce the

concept of weak plans and events, which we build on

by introducing the execution of a single plan or a

hierarchy of plans with a late binding approach.

2 EXECUTION OF AgentSpeak(L)

An agent that is executed according to some program

written in AgentSpeak(L), is a tuple

〈

𝑃𝐵,𝐸𝑄,𝐵𝐵,𝐴𝑆,𝐼𝑆,𝑆



,𝑆



,𝑆



〉

where PB is a plan base,

EQ is an event queue, BB is a belief base, AS is a set

of actions that are executable by the agent, IS is a set

of intention structures and 𝑆



,𝑆



,𝑆



are functions for

selections of events, options and intentions (Rao,

AgentSpeak(L): BDI agents speak out in a logical

computable language, 1996). The program itself

written in this language defines knowledge as

formulas of first-order predicate logic, for simplicity

we will consider beliefs in the form of atomic

formulas of this logic as facts are defined in PROLOG

as a predicate. Goals can be declared as achievement

or test goals. In the first case, the predicate is given

with the preposition ! and in the latter with the

preposition ?. The core part of a program in

AgentSpeak(L) is the plan of how to achieve some

goal in the form of plans. This looks like the

following 𝑡



:Ψ⟵𝑝𝑙𝑎𝑛 and consists of the

triggering even 𝑡



, context conditions Ψ and plan’s

body. Triggering events are written in the form

+event or -event, where an event can be a test goal,

an achievement or just a predicate. A plan is then

relevant to some goal in the form of an event, when

its triggering event is unifiable with that event, and it

is further applicable if the context conditions are valid

in the current state of the agent's BB. If we have a

substitution for which the plan is relevant and

applicable, then the plan is a possible means of

achieving the goal.

The interpretation of an agent programmed in

AgentSpeak(L) is presented in the original paper

(Rao, AgentSpeak(L): BDI agents speak out in a

logical computable language, 1996), and the

operational semantics is presented more formally in

(Winikoff, 2005). The problem we address and offer

a solution to is that if a plan is chosen, a substitution

is chosen that may not be appropriate for the agent's

next acts. As difficult as it is to predict which of the

possible behaviors will lead to success in a dynamic

environment, it is possible to either try to estimate

these substitutions based on previous experience, or

to defer the decision about substitutions until this

becomes necessary. We use the latter approach and

now wish to demonstrate its potential appropriateness

and usefulness.

Let us return to the aforementioned interpretation

of plan selection for some goal in the form of an event

e. Then there may exist one or more plans from

agent’s plan base such that their triggering events are

unifiable with e for some most general unifier (mgu)

σ. Thus, let us have such plans

𝑡𝑒



:Ψ



⟵𝑝𝑙𝑎𝑛



𝑡𝑒



:Ψ



⟵𝑝𝑙𝑎𝑛



… 𝑡𝑒



:Ψ



⟵𝑝𝑙𝑎𝑛



and 𝜎





𝑚𝑔𝑢



𝑒,𝑡𝑒





, 𝜎



𝑚𝑔𝑢



𝑒,𝑡𝑒





… 𝜎



𝑚𝑔𝑢



𝑒,𝑡𝑒





are

the mgu for the event and individual triggering

events. These plans are relevant but only applicable if

𝐵𝐵|  Ψ



𝜎



for the first plan, etc. Thus, again, we

look for substitutions that unify the individual context

conditions in the agent's BB, let’s denote them as

𝜌



,𝜌



…𝜌



. Then the agent selects one of these plans

to achieve the goal represented by event e, say plan j,

and the agent is going to execute the body of this plan

ICAART 2022 - 14th International Conference on Agents and Artiﬁcial Intelligence

716

with the appropriate substitutions to achieve the

event.

Thus, only one option is chosen from all that can

be found. Not only is it potentially possible to find

multiple plans for a single goal, but also it is possible

to find multiple substitutions for which these plans

are relevant and applicable.

By introducing the context as a set of possible

substitutions, we generalize the execution of one plan

to the execution of several possibilities found for that

plan during practical reasoning. We will refer to a

plan as a structure given in the program,

supplemented with context as a set of substitutions, as

a weak instance of the plan. That is, if for the plan

𝑡𝑒



:Ψ



⟵𝑝𝑙𝑎𝑛



it is possible to find multiple

substitutions that unify this plan’s 𝑡𝑒



with the event

𝑒



and for which this plan is applicable in the agent's

BB, let’s say

𝑐𝑡𝑥, then 𝑒



:Ψ



⟵𝑝𝑙𝑎𝑛



,ctx is a

weak instance of the plan 𝑒



:Ψ



⟵𝑝𝑙𝑎𝑛



. The context

𝑐𝑡𝑥 will allow to postpone the decision about

substitutions until the moment when this is necessary.

We therefore view this approach as a late binding for

the interpretation of AgentSpeak(L). The weak plan

instance is executed using the operations we present

in the next section. After introducing them, we will

also show how the acts in the weak plan instance are

interpreted and how subplans and transitions between

plan levels are handled, which will allow us to

conclude by showing that such an interpretation is

done correctly and that it expands the decision-

making options for these types of agents.

3 OPERATIONS AND

FUNCTIONS FOR LATE

BINDINGS

The basic terms of the formal description of our

approach are the sets of unifiers and substitutions.

(Substitution is a set of tuples, in this text [t/x], where

x is variable and t is term. Unifier is a substitution that

unifies some two formulas). Sets of substitutions play

an essential role in our interpretation, as they are

created and modified during agent's interpretation.

For a specification on how the corresponding

operations work, we must write some definitions.

First, we introduce broad unification as a function that

creates a set of unifiers.

Definition 1: Broad unification is denoted as 𝜌𝑈 is a

function that maps the predicate p and predicates p’

from the BB to a set of all possible mgu without

variables renaming. Using function 𝑚𝑔𝑢



𝑝,𝑝





for

finding the mgu of 𝑝 and 𝑝



without variable

renaming we define it formally as

𝝆𝑼



𝒑,𝑩𝑩



≝



𝒎𝒈𝒖



𝒑,𝒑





:𝒑



∈𝑩𝑩



(1)

In this definition we say that BB is a set of

predicates. Because beliefs are also in the form of

predicates, then any BB is also a set of predicates.

Variable renaming is important in the unification for

making the formulas’ logical equivalent. However, in

our case we need the unification just for specification

of the predicate 𝑝 in such a way that it is valid in the

BB in any interpretation. For this reason, we need not

unify the name of the variables. However, even

though such a function from Definition 1 does not

unify the predicates in the correct sense, we will use

the /term unification for this anyway.

The result of the broad unification is a set of

unifiers that we call to be a possible unifier set (in this

text we write PUS). The set of predicates consists in

this case of all beliefs that form the agent's BB.

Further, in the text we will use a symbolic

representation of the broad unification of a predicate

p and a belief base BB in the simplified form 𝜌𝑈





or when the parts are not necessary, we will write just

𝜌𝑈. Second, we use in some further definitions when

we refer to the set just as certain PUSs and we do not

need a precise determination of the predicate and

belief base.

Conversely, an instance set is a set of predicates

containing every predicate that rises, after the

application of every unifier, from a PUS to a

predicate.

Definition 2: We denoted the instance set as 𝐼𝜎,

and it is a function that maps a predicate and a PUS

to a set of predicates. We define it in the following

way:

𝐼𝜎



𝑝,𝜌𝑈



≝



𝑝𝜎:𝜎 ∈ 𝜌𝑈



Notice that an instance set is not inverse to the

broad unification in the sense that if we have 𝜌𝑈





then 𝐼𝜎𝑝,𝜌𝑈





 need not be equal to the BB and vice

versa.

Definition 3: Shorting is a function denoted by ≺ and

we define it as:

𝜌𝑈 ≺ 𝑝 ≝ (2)

𝜎: ∃𝜎



𝜎



∈𝜌𝑈,𝜎⊆𝜎



,∀



𝑡/𝑥



∈𝜎





𝑋∈𝑉𝑎𝑟



𝑝



→



𝑡/𝑥



∈𝜎





The resulting substitution set contains just those

substitutions that substitute free variables from p. The

Var function is used here in the usual way, which

means that by using Var(p) we get a set of free

variables in p. Then, only the variable substitutions

from

𝜌𝑈 unifiers that substitute any of the Var(p)

persist in

𝜌𝑈 ≺ 𝑝.

Late Bindings in AgentSpeak(L)

717

The functions and operations that we call

merging, restriction, shorting, and decision-making

are used for the transformation of PUSs and play an

essential role for late bindings. The purpose of the

first operation, which we call ‛merging’, is to create

substitution from two other substitutions. It unites the

substitutions when every variable substituted in both

of them is substituted for the same term. This means

that there must not be a conflict where the

substitutions map the same variable to two different

terms. If this occurs, then the result of the merging is

an empty set.

Definition 4: We denote the merging by ⋉ and it

is defined for a certain two unifiers as follows:

𝜎



⋉𝜎



≝ (3)

𝜎



∪𝜎



𝑖𝑓𝑓 ∀



𝑥



/𝑡



∈𝜎



∀



𝑥



/𝑡



∈𝜎





𝑥



𝑥



→𝑡



𝑡







𝑒𝑙𝑠𝑒

Let the first substitution be a unifier for a

predicate and a belief base, and the second unifies

another predicate in another belief base. If the result

of the merging of these unifiers is a nonempty set of

substitutions, then this set of substitutions unify both

predicates with some belief in the belief bases. If the

result is an empty set, then the unifiers cannot be

united. However, there can be another pair of unifiers

for which the merging produces a non-empty set. For

finding each such pair, we define the restriction

operator.

Definition 5: The restriction operator is denoted

⊓ and is defined for some two PUS 𝜌𝑈



and 𝜌𝑈



𝝆𝑼

𝟏

⊓𝝆𝑼

𝟐

≝

⋃

𝝈

𝟏

⋉𝝈

𝟐𝝈

𝟏

∈𝝆𝑼

𝟏

,𝝈

𝟐

∈𝝆𝑼

𝟐

(4)

Using the restriction, an agent obtains pairs of

unifiers from both PUSs that work as extended

unifiers for both pairs of predicate/belief bases for

which it makes the original PUSs.

The following Theorem 1 shows a key property

of this operation, which is fundamental for the

functionality of the late bindings.

Theorem 1: The restriction of two PUSs created

by some broad unifications 𝜌𝑈









and 𝜌𝑈









results

in a set of unifiers that contains all the mgu that unify

both 𝑝



in belief base 𝐵𝐵



and 𝑝



in another belief

base 𝐵𝐵



Proof: First, we prove that all such unifiers work

properly for both pairs of predicates and belief bases

(1). Moreover, we need to show that they are in their

most general form (2). Then we prove that there is no

other mgu (3).

Ad 1) Let 𝜎∈𝜌𝑈









⊓𝜌𝑈









be a unifier that

unifies p

in BB

and p

in BB

. From Definitions 4

and 5 it follows that 𝜎 is a union of the original

unifiers. If the predicates are unified to a ground

belief, then every variable in p

and p

must be

substituted in the same way as they were substituted

in the original unifiers. There cannot be a conflict,

because if both unifiers substitute the same variable,

by definition of the merging operator, it follows that

it was substituted in the same way. Even if it unifies

them with a lighted predicate, then one unifier may

cause the substitution of a variable that remained free

in the second unification; this means that, for

example, 𝜎 substitutes a variable in p

, but in the

original 𝜌𝑈









this variable remained free. However,

this is just the specification in the predicate logic and

by the rule of specification this substitution makes p

also valid in BB.

ad 2) If 𝜎 is not the mgu for both unifications,

then there must be a unifier

𝜎



and a substitution 𝛿

where 𝜎𝜎



𝛿. Definition 5 says that every mapping

of a variable in

𝛿 must also be in either 𝜌𝑈









𝜌𝑈









. If 𝜎



works for both unifications, then there

must be a unifier 𝜎



for the first or second unification

which is more general, then a unifier from

𝑈









and

𝜌𝑈









, respectively. However, because these unifiers

are mgu, this is not possible. From another point of

view, because unifications produce ground

predicates, then the number of variable substitutions

in the unifier must be equal to the number of free

variables in 𝑝



and 𝑝



that is valid for the results of

the restriction operation.

ad 3) Let us consider that there is, a unifier, 𝜎 that

is mgu for both predicates and belief sets and that this

unifier is not included in the restricted set. Then there

must be some other mgu unifiers

𝜎



⊆𝜎 and 𝜎



⊆𝜎

which are equal to or more general than 𝜎, which

means

𝜎⊆𝜎



∧𝜎⊆𝜎



and 𝑝



𝜎



∈𝐵𝐵



and 𝑝



𝜎



∈

𝐵𝐵



. However because 𝜌𝑈









𝑎𝑛𝑑 𝜌𝑈









contains

every mgu for individual predicates and belief bases,

then both 𝜎



and 𝜎



belong to these sets and because

𝜎⊆𝜎



∪𝜎



then either 𝜎 or a more general unifier 𝜎

must be included in the restriction.

From Theorem 1 it follows that if a restriction is

applied to 𝜌𝑈







𝑎𝑛𝑑 𝜌𝑈







then it results in every

possible substitution that still fulfils both queries of

𝑝



to 𝐵𝐵



and 𝑝



to 𝐵𝐵



We will add one more lemma to this definition,

which we will use later.

Lemma 1: If we obtain PUS by restriction

𝜌𝑈  𝜌𝑈



⨅𝜌𝑈



then it is valid that

∀σ∃σ





𝜎∈𝜌𝑈∧𝜎



∈𝜌𝑈



∧ 𝜎



⊆𝜎∧𝜎∈



𝜎





⨅𝜌𝑈





ICAART 2022 - 14th International Conference on Agents and Artiﬁcial Intelligence

718

Proof: We need to show that if a substitution

arises by performing a restriction operation, then for

every substitution σ in this result there is at least one

substitution in the first operand 𝜎



, that is a subset of

𝜎 such that the substitution 𝜎 would arise by

performing the restriction even if the first operand

were the set containing olny that substitution 𝜎



 .

Since such a substitution must have arisen as a result

of the merging operation 𝜎𝜎



⋉𝜎



and 𝜎



∈𝜌𝑈



From Definition 4, this operation is performed as

𝜎



∪𝜎



and it is clear that 𝜎⊆𝜎



∪𝜎



. But then

since 𝜎



∈𝜌𝑈



then 𝜎



⨅𝜌𝑈



will also perform

merging 𝜎



⋉𝜎



and its result will also be part of the

resulting set of substitutions.

An essential aspect of the BDI agents is that they

build a hierarchy of plans. If the agent is about to go

from one plan to another in the hierarchy, it must also

transfer appropriate substitutions. For this purpose,

we introduce the PUS intersection function.

Definition 6: The intersection is a function

denoted by the symbol ∽ and is defined for one PUS

𝜌𝑈 and two predicates 𝑝



a 𝑝



as follows:

𝑝



,𝜌𝑈∽𝑝



≝𝜌𝑈𝑝



,𝐼𝜎



𝑝



,𝜌𝑈



 (5)

Definition 7: The intersection function for two

PUS 𝜌𝑈



and 𝜌𝑈



and two predicates 𝑝



and 𝑝



defined as follows:

𝑝



,𝜌𝑈



∽𝑝



,𝜌𝑈



≝



𝑝



,𝜌𝑈



∽𝑝





⊓𝜌𝑈



(6)

The last function which we define is called

decision-making. Contrary to the previous operations

and functions, this one is defined abstractly. Here we

only declare its structure without any further

specification on how to realize it.

Definition 8: Decision-making is a function

denoted as Dec that for a PUS and a predicate p

provides a ground substitution which is a subset of a

substitution from the PUS.

When we write that

𝐷𝑒𝑐



𝜌𝑈,𝑝



𝜎 then it must be

valid that

∃𝜎



𝜎



∈𝜌𝑈,𝜎⊆𝜎



,𝑉𝑎𝑟



𝑝



𝑑𝑜𝑚𝜎) (7)

and 𝜎 is a ground substitution for p.

So the agent decides on some substitution from

𝜌𝑈 and from that determines the binding of the free

variables in 𝑝. If this substitution is applied to 𝑝, then

it makes

𝑝 a ground predicate. But in addition, this

substitution is based on those substitutions that the

agent keeps as possible.

4 EXECUTING PLANS USING

LATE BINDINGS

We have already mentioned that we will be working

with a set of substitutions, i.e. some PUS that will be

stored during the execution of the agent's plan. The

plan becomes part of the agent's intent when the

reasoning process chooses it for some event. At that

same moment, the plan is assigned a PUS that the

agent operates on while executing the plan. These

substitutions are made by the agent in the course of

its practical reasoning. So, unlike other

AgentSpeak(L) interpreter systems, in our case the

plan variables do not have to be replaced

immediately, but can be kept separately as PUSs,

which we will now call the plan context. This context

changes as the agent performs actions and achieves

goals from the plan body. If such a PUS is assigned

to an event, we will speak of an event context. This

will arise from the actual context of the plan that

triggered the event, and we explain this in more detail

below. Further in this text, we will talk about weak

instances of plans and events when plans and events

have contexts and other information associated with

them according to the following definitions.

Definition 9: A weak plan instance is a triple

〈

𝑡𝑒, 𝒉,𝑐𝑡𝑥

〉

containing a plan trigger event 𝑡𝑒, a plan

body

h = ℎ



;ℎ



;…ℎ



, and a plan context ctx.

Similarly, we define weak event instances as

tuples.

Definition 10: A weak event instance is a tuple

〈

𝑒𝑣𝑡, 𝑐𝑡𝑥

〉

where 𝑒𝑣𝑡 is an event, and ctx is a context.

Its purpose is to represent an event that has arisen

during the execution of a weak plan instance. It is

necessary to have weak event instances because when

an agent is pursuing a goal by a weakly instantiated

plan, it need not know the exact substitution of the

goal variables. Instead, the agent selects a single

event with a context representing all possible goals

that are sufficient to satisfy the declared goal. The

agent decides how to proceed based on which goal or

goals the agent subplan will achieve. It means, that it

is sufficient to meet only one or a subset of these

goals.

Furthermore, we will use the abbreviations WEI

for weak event instances and WPI for weak plan

instances.

4.1 Practical Reasoning for Weak

Instances

What differs from the original interpretation is the

way the applicability and relevance are recognized.

Late Bindings in AgentSpeak(L)

719

When there are WEIs and a plan, then a plan is

relevant to the WEIs when its triggering event and the

WEI event are unifiable concerning the WEI context.

At the beginning, the plan has no context that would

play a role in the unification process. Then it is

enough that the PUS intersection of the triggering

event, the goal event, and its context create a set

containing at least one substitution. This can be seen

as a query by triggering event to the base, which is

formed as a set of instantiations from the WEI. Notice

that it is valid also when the substitution is an empty

set, i.e. if the resulting set contains a single element,

namely the empty set. This is what would happen if

the plan’s triggering event were ground. Now let us

define this formally.

Definition 11: A plan

𝑡𝑒:𝑏



∧𝑏



∧…𝑏



←

ℎ



;ℎ



;…ℎ



is relevant to a WEI

〈

𝑒𝑣𝑡, 𝑐𝑡𝑥

〉

when



𝑒𝑣𝑡,𝑐𝑡𝑥 ∽ 𝑡𝑒









A plan is applicable when every of the plan's

context conditions (here 𝑏



…𝑏



) is satisfied in the

actual state of the agent's belief base. This means that

it is possible to find a PUS for every context condition

and the belief base and consequently to make

restrictions among them. From Theorem 1, it follows

that if the restrictions result in a nonempty set, then

there is at least one unifier for every context condition

of the plan and an agent's belief in its belief base.

Definition 12:

A plan 𝑡𝑒:𝑏



∧𝑏



…𝑏



←

ℎ



;ℎ



;…ℎ



is applicable in the actual agent’s belief

base BB, when 𝜌𝑈



𝑏



,𝐵𝐵



⊓…⊓𝜌𝑈



𝑏



,𝐵𝐵









Every plan which is both relevant and applicable

can be considered as a means for the goal and the

agent may choose it as its intended means. When the

plan is both relevant and applicable, then the

restriction of the PUS from Definition 11 and the PUS

from Definition 12 must be also a non-empty PUS.

Using the PUS intersection from Definition 7, we

may write

𝑐𝑡𝑥



𝑒𝑣𝑡,𝑐𝑡𝑥∽𝑡𝑒⊓ 𝜌𝑈



𝑏



,𝐵𝐵



⊓…⊓

𝜌𝑈



𝑏



,𝐵𝐵









and then 𝑐𝑡𝑥



is an context which,

together with the plan’s body ℎ



;ℎ



;…ℎ



, forms

new WPI that can be an intended means for the WEI.

We can think of a WEI as a specification of more

than one goal in the form of an event. By creating a

set of instances for a given predicate in a WEI and for

all substitutions from a given context, we obtain all

event instances that represent all reachable goals

represented by that WEI. To satisfy a given WEI, it is

sufficient for the agent to satisfy at least one of these

goals. The following Lemma will show that for a WEI

and a chosen plan, its default context contains all

substitutions for which this plan is relevant and

applicable to some WEI instance. In the remainder of

this paper, when we refer to a WEI instance, we will

mean a single instance created by applying the WEI

context to a WEI predicate. By a WEI instance, we

will then also mean one particular goal represented by

this WEI.

Lemma 2: The result of practical reasoning for

some WEI

〈

𝑒𝑣𝑡,𝑐𝑡𝑥

〉

, if a suitable plan is found, is a

PUS that contains all the most general substitutions

for which the chosen plan

𝑡



:Ψ⟵𝑝𝑙𝑎𝑛 is relevant

with respect to a given some event instance

𝑒∈

𝐼𝜎



evt,ctx



and applicable in the current state of the

agent's belief base.

Proof: First consider a plan without the context

conditions

𝑡



⟵𝑝𝑙𝑎𝑛. Then the result of practical

reasoning is Θ



𝑒𝑣𝑡,𝑐𝑡𝑥 ∽ 𝑡𝑒,



. By Definition 6,



𝑒𝑣𝑡,𝑐𝑡𝑥 ∽ 𝑡𝑒



𝜌𝑈𝑡



,𝐼𝜎



𝑒𝑣𝑡, 𝑐𝑡𝑥



 and further by

Definition 1,

Θ



𝑚𝑔𝑢



𝑡



,𝑝





:𝑝



∈𝐼𝜎



𝑒𝑣𝑡, 𝑐𝑡𝑥





Assume that there is a substitution φ∉Θ, for which

this plan is relevant with respect to some

𝑒∈

𝐼𝜎



evt,ctx



and applicable in the agent's BB. Since,

due to the absence of context, every relevant plan is

also applicable, then suppose that φ is mgu for some

𝑡



and an event instance 𝑒 , but then φ∉



𝑚𝑔𝑢



𝑡



,𝑝



:𝑝∈𝐼𝜎



𝑒𝑣𝑡,𝑐𝑡𝑥





which is a

contradiction. From a different perspective, we can

view such inference as querying 𝑡



into the BB

containing all instances of a given WEI, and the result

of this query is all possible mgu for which this query

is satisfied. Thus, in the first stage of practical

reasoning, we obtain all the substitutions that make

this plan relevant to some instance of the event. All

these substitutions are mgu for all event instances of

the WEI and the plan’s triggering event. Assume, that

the context were non-empty

Ψ𝑏



∧…∧𝑏



and we

have φ which is a substitution for which the plan is

relevant and applicable. Suppose now that φ∉Ξ

where Ξ is calculated, as we showed above, as

intersection by

𝑒𝑣𝑡, 𝑐𝑡𝑥 ∽ 𝑡𝑒 ⊓ 𝜌𝑈



𝑏



,𝐵𝐵



⊓…⊓

𝜌𝑈



𝑏



,𝐵𝐵



 and we know that Θ



evt,𝑐𝑡𝑥 ∽ 𝑡





provides all mgu for 𝑡



and the instance of WEI under

consideration, φ must also be a superset of some such

mgu of Θ . Furthermore, if we consider the first

condition 𝑏



of the context condition Ψ and perform

Θ⊓,𝜌𝑈



𝑏



,𝐵𝐵



, then by Theorem 1 we obtain all

substitutions for which they are valid both with

respect to the relevance of the plan and with respect

to the context condition 𝑏



. Similarly, for Θ⊓

,𝜌𝑈



𝑏



,𝐵𝐵



,…⊓𝜌𝑈



𝑏



,𝐵𝐵



, the result is all

substitutions that make the plan relevant and all

context conditions are satisfied, so the condition φ

must be present in Ξ.

Thanks to Lemma 2, we know that if a plan is

adopted and a WPI containing that plan and some

context is created, then that context contains all

possible substitutions for which the execution of that

ICAART 2022 - 14th International Conference on Agents and Artiﬁcial Intelligence

720

plan is applicable and, in particular, valid against

some WEI for which it was chosen. Again, this can

be looked at from a different perspective. The

selection of the appropriate plan can be seen as a

successful query in which we use the plan’s triggering

event and event instance set. The choice of the

applicable plan is then seen as a series of queries for

each part of the context conditions to the agent's BB.

The resulting substitutions according to Theorem 1

contain all substitutions that satisfy all these queries.

Now we want to find out all the substitutions that,

on the one hand, still satisfy that the plan is a relevant

means for them to achieve some event instance, but

also make all the acts of the plan, including possible

subplans, feasible. We will do this sequentially.

4.2 Plan Body Execution

We start with the simplest example where the plan

does not contain any act, and choosing it would

already result in the event being fulfilled. Then the

substitutions found during the practical reasoning

provide the goals that have been achieved.

Example 1: For some WEI, where the event

predicate is !transport_meansX,Y,M. is to

represent the location X from which someone or

something is to be moved to location Y and M is the

means of transport and the WEI context may be

𝑏𝑒𝑟𝑙𝑖𝑛/𝑋,𝑝𝑟𝑎𝑔𝑢𝑒/𝑋. The relevant plan is for

example !𝑡𝑟𝑎𝑛𝑠𝑝𝑜𝑟𝑡_𝑚𝑒𝑎𝑛𝑠X,Y,M:𝑔𝑒𝑡_𝑡𝑚𝑋,𝑀

and this contains only one context condition and if the

agent's belief base contains for example

𝑔𝑒𝑡_𝑡𝑚𝑝𝑟𝑎𝑔𝑢𝑒,𝑐𝑎𝑟.

𝑔𝑒𝑡_𝑡𝑚𝑏𝑒𝑟𝑙𝑖𝑛,𝑎𝑖𝑟𝑝𝑙𝑎𝑛𝑒.

𝑔𝑒𝑡_𝑡𝑚𝑏𝑒𝑟𝑙𝑖𝑛,𝑡𝑟𝑎𝑖𝑛.

𝑔𝑒𝑡_𝑡𝑚𝑝𝑎𝑟𝑖𝑠,𝑏𝑢𝑠.

then for

the given WEI the initial and resulting context is

𝑏𝑒𝑟𝑙𝑖𝑛/𝑋,𝑎𝑖𝑟𝑝𝑙𝑎𝑛𝑒/𝑀,𝑏𝑒𝑟𝑙𝑖𝑛/𝑋,𝑡𝑟𝑎𝑖𝑛/

𝑀,𝑝𝑟𝑎𝑔𝑢𝑒/𝑋,𝑐𝑎𝑟/𝑀.

In the following paragraphs, our focus will be on

how the WPI changes from the moment of its creation

to the moment of its successful execution. We won't

even go into how the agent responds to situations

where the WPI fails. This would occur when the

context in the WPI transforms to an empty set.

However, we will assume that each execution of an

act in the body of the WPI plan will result in the

transformation of its context into another context that

is not an empty set. We will show specifically how

this context will be transformed, one by one, by the

possible acts that the agent can execute.

First, we define the test goals and their

interpretation.

Definition 13: A test goal is executed for some

WPI

𝑡



,?𝑝



𝒕



;𝑝𝑙𝑎𝑛𝑏𝑜𝑑𝑦,𝑐𝑡𝑥 and if successfully

executed, then the new WPI is

𝑡



,𝑝𝑙𝑎𝑛𝑏𝑜𝑑𝑦,𝑐𝑡𝑥⊓𝜌𝑈



𝑝



𝒕



,𝐵𝐵





By executing one test goal ?𝑝





𝒕



the context is

changed to include all substitutions that match all

previous queries and must now include this new

query. Let us denote the result of the testing goal by

𝜌𝑈 and this contains all mgu for 𝑝





𝒕



and some

belief from the agent's current BB. If we perform

𝑐𝑡𝑥



𝑐𝑡𝑥⊓𝜌𝑈, then we get a new context 𝑐𝑡𝑥



which, by Theorem 1, contains all substitutions that

are valid for all queries performed so far (including

those performed during practical inference).

Definition 14: Performing actions either external

or internal transforms the context from WPI

𝑡



,𝑎



𝒕



;𝑝𝑙𝑎𝑛𝑏𝑜𝑑𝑦,𝑐𝑡𝑥 so that the new WPI is

𝑡



,𝑝𝑙𝑎𝑛𝑏𝑜𝑑𝑦,𝑐𝑡𝑥 ⊓ 𝐷𝑒𝑐𝑐𝑡𝑥,𝑎



𝒕



 

The agent can create a ground predicate for the

action before executing it. Thus, all free variables

must be bound to a specific atom. This is the case

when the

𝐷𝑒𝑐 function of Definition 8 is used. This is

a situation that causes non-determinism in the agent's

behavior and we do not provide a concrete

implementation of it in this text. Anyway, the agent

performs the action described as

𝑎



𝒕



where 𝒕 are

terms (in our case we consider atoms or variables).

After this action is performed, all variables from the

terms 𝒕 are uniquely bound to the same variables in

all context substitutions.

Example 2: For an action 𝑔𝑜



𝐴,𝐵



with context

𝑐𝑡𝑥  



𝑝𝑎𝑟𝑖𝑠/𝐴





𝑐𝑎𝑟/𝐵



,𝑚𝑜𝑛/𝑋





𝑟𝑖𝑜/

𝐴,𝑝𝑙𝑎𝑛𝑒/𝐵,𝑡𝑢𝑒/𝑋





𝑏𝑟𝑛𝑜/𝐴.𝑏𝑦𝑐𝑖𝑐𝑙𝑒/𝐵,𝑠𝑡𝑑/

𝑋





𝑏𝑟𝑛𝑜/𝐴.𝑏𝑦𝑐𝑖𝑐𝑙𝑒/𝐵,𝑠𝑢𝑛/𝑋



 the agent can

decide to perform one of 𝑔𝑜



𝑝𝑎𝑟𝑖𝑠,𝑐𝑎𝑟



𝑔𝑜



𝑝𝑎𝑟𝑖𝑠,𝑝𝑙𝑎𝑛𝑒



or 𝑔𝑜



𝑏𝑟𝑛𝑜, 𝑏𝑦𝑐𝑖𝑐𝑙𝑒



. The first action

is created by performing the first substitution on the

original action, the second action by performing the

second, and the third action by performing the third

or fourth. The resulting contexts are then

𝑐𝑡𝑥 





𝑝𝑎𝑟𝑖𝑠/𝐴





𝑐𝑎𝑟/𝐵



,𝑚𝑜𝑛/𝑋



for the first action,

𝑐𝑡𝑥  



𝑟𝑖𝑜/𝐴,𝑝𝑙𝑎𝑛𝑒/𝐵,𝑡𝑢𝑒/𝑋



for the second

action, and

𝑐𝑡𝑥  



𝑏𝑟𝑛𝑜/𝐴,𝑏𝑦𝑐𝑖𝑐𝑙𝑒/𝐵,𝑠𝑡𝑑/

𝑋





𝑏𝑟𝑛𝑜/𝐴,𝑏𝑦𝑐𝑖𝑐𝑙𝑒/𝐵,𝑠𝑢𝑛/𝑋



for the third

action.

The result of performing an action is to reduce the

context to only those substitutions that are consistent

with the action performed and the agent loses those

options that do not match the action performed. From

the above, we summarize the following Lemma. Its

proof follows from the above definitions and their

analysis.

Lemma 3: Executing a plan. which contains only

actions and test goals (i.e., does not invoke subplans

by the act of achievement goal) the original WPI is

transformed into another whose context contains all

Late Bindings in AgentSpeak(L)

721

the substitutions for which this one is relevant to the

original WEI that invoked it, was applicable with

respect to the specified context conditions, and

corresponds to all the executed acts of this plan.

Before we move on to the implementation of the

subplans, let's take a closer look at the relationship

between the resulting context and the WEI that

triggered the plan. We want to show that it is true that

if the execution of a WPI succeeded, then for some

event that can be instantiated from the predicate and

context in that WEI, the plan would also succeed, it

means that our approach sounds.

Lemma 4: If the execution of a plan invoked by a

WEI succeeds, then the execution of the plan would

also succeed for at least one event instance from the

WEI if used alone.

Proof: The context is changed only by using the

restriction operation in the case of both types of acts

(testing, action). According to Lemma 1, we know

that each substitution thus produced is a non-strict

superset of a substitution from the previous context.

The second operand in the restriction is given by the

act performed. Since non-strict superset is a transitive

relation, then every substitution in the resulting

context is an non-strict superset to some mgu arising

during the practical reasoning, which we denote by Θ

in the proof of Lemma 2. Thus, if the context changes

from Θ to Ξ𝑐𝑡𝑥



and then by carrying out the acts

of the plans to 𝑐𝑡𝑥



,𝑐𝑡𝑥



…𝑐𝑡𝑥



, then by Lemma 1

for each 𝜎



∈𝑐𝑡𝑥



there is a sequence 𝜎



∈Θ, 𝜎



∈

ctx



,𝜎



∈𝑐𝑡𝑥



…𝜎



∈𝑐𝑡𝑥



such that 𝜎



⊆

𝜎



⊆𝜎



⊆ …𝜎



⊆𝜎



. Furthermore, thanks to the

same Lemma, this sequence would be replayed using

the same actions, according to Definition 13 and

Definition 14. That is, if some one mgu from Θ were

taken during practical reasoning, above 𝜎



, then the

plan would be feasible.

A corollary of this Lemma is that if the plan is

successfully executed, the analysis of the resulting

substitutions can determine for which all mgu of Θ

the plan would be successfully executable. A simple

way to determine this is that if an element from Θ is

a subset of some element from the resulting context,

then this element determines the goal that will be

achieved by the plan, since it is the mgu obtained

during practical reasoning in the search for a relevant

plan for some WEI.

Example 3: For a WEI

!𝑔𝑜



𝑋,𝑌,𝑍



,𝑝𝑎𝑟𝑖𝑠/

𝑋.𝑟𝑖𝑜/𝑋 

the plan with triggering event

!𝑔𝑜



𝐴,𝐵,𝐶



is chosen. The plan is relevant and the

Θ



𝑝𝑎𝑟𝑖𝑠/𝐴,𝑟𝑖𝑜/𝐴



 contains all the mgu for

the WEI and triggering event of the plan. The

resulting context after such a plan is executed is for

example

𝑝𝑎𝑟𝑖𝑠/𝐴,𝑐𝑎𝑟/𝐵,𝑚𝑜𝑛/𝐶,𝑝𝑎𝑟𝑖𝑠/

𝐴,𝑡𝑟𝑎𝑖𝑛/𝐵,𝑡𝑢𝑒/𝐶.

We can see, that this plan is

feasible for the goal 𝑔𝑜



𝑝𝑎𝑟𝑖𝑠,𝑌,𝑍



because 𝑝𝑎𝑟𝑖𝑠/

𝐴 ⊆ 𝑝𝑎𝑟𝑖𝑠/𝐴,𝑐𝑎𝑟/𝐵,𝑚𝑜𝑛/𝐶and 𝑝𝑎𝑟𝑖𝑠/𝐴 ⊆

𝑝𝑎𝑟𝑖𝑠/𝐴,𝑡𝑟𝑎𝑖𝑛/𝐵,𝑡𝑢𝑒/𝐶,

but is not feasible for

𝑔𝑜



𝑟𝑖𝑜,𝑌,𝑍



. Moreover, we know that this plan could

have been feasible if we had driven on Monday, or

taken the train on Tuesday.

The above example shows that plan execution

could be used to find the answer as understood in

(Sardina & Padgham, 2011). The answer-seeking

principle is familiar from the execution of programs

in PROLOG, and the process of querying and

obtaining answers can be seen in the execution of the

goals specified in the agent's plans. The query is given

in the form of a predicate that need not be ground, and

the answers are possibly bindings to variables for

which such a predicate is valid in the database. Since

we have shown that the resulting context contains all

substitutions for which a plan is feasible for some

WEI, these substitutions are also all possible answers

to the queries, which can be considered as all the

entities of a given WEI. The example above shows

that these answers could be reached through the

trigger event and final context of the plan. Creating

the instance set from them creates a base, which we

query with a predicate from the WEI. Thus, in

Example 3, such a base would contain two predicates

𝑔𝑜



𝑝𝑎𝑟𝑖𝑠,𝑐𝑎𝑟,𝑚𝑜𝑛



and 𝑔𝑜



𝑝𝑎𝑟𝑖𝑠,𝑡𝑟𝑎𝑖𝑛,𝑡𝑢𝑒



which we obtained by applying substitutions

𝑝𝑎𝑟𝑖𝑠/

𝐴,𝑐𝑎𝑟/𝐵,𝑚𝑜𝑛/𝐶 and 𝑝𝑎𝑟𝑖𝑠/𝐴,𝑡𝑟𝑎𝑖𝑛/𝐵,𝑡𝑢𝑒/

𝐶

to the trigger event 𝑔𝑜



𝐴,𝐵,𝐶



. The question

would be

𝑔𝑜



𝑋,𝑌,𝑍



, which will provide answers

𝑝𝑎𝑟𝑖𝑠/𝑋,𝑐𝑎𝑟/𝑌,𝑚𝑜𝑛/𝑍,𝑝𝑎𝑟𝑖𝑠/𝑋,𝑡𝑟𝑎𝑖𝑛/

𝑌,𝑡𝑢𝑒/𝑍

We summarize the above in the following

Lemma, which we provide without proof, noting that

we have demonstrated its validity above.

Lemma 5: For some WEI and for a selected

relevant and applied plan containing only actions or

test goals, the interpreted system interpreting the

program with late bindings will provide all possible

answers with respect to the chosen actions.

We can now consider plans that contain some

achievement goals, and show that what is stated in the

previous Lemma holds for them as well. But first we

need how the achievement goal is implemented.

Definition 15: Achievemnt goal is executed for

some WPI

𝑡



.!𝑝



𝒕



;𝑝𝑙𝑎𝑛𝑏𝑜𝑑𝑦,𝑐𝑡𝑥 such that WEI

!𝑝



𝒕



,𝑐𝑡𝑥



 where 𝑐𝑡𝑥



𝑐𝑡𝑥≺𝑝



𝒕



generated, (see Definition 3). The execution of the

WPI is suspended until the sub-plan is successfully

executed, which ends up as

!𝑝



𝒕





,𝑛𝑢𝑙𝑙,𝑐𝑡𝑥



.

The original WPI then continues as

ICAART 2022 - 14th International Conference on Agents and Artiﬁcial Intelligence

722

𝑡



.𝑝𝑙𝑎𝑛𝑏𝑜𝑑𝑦,𝑝



𝒕

𝟐



,𝑐𝑡𝑥



∽𝑝



𝒕



,𝑐𝑡𝑥

The change of context here is done according to

Definitions 6 and 7, by which we can break this

operation down as

𝑝



𝒕

𝟐



,𝑐𝑡𝑥



∽𝑝



𝒕



,ctx



𝑝



𝒕

𝟐



,𝑐𝑡𝑥



∽𝑝



𝒕





⊓ctx

𝜌𝑈



𝑝



𝒕



,𝐼𝜎𝑝



𝒕

𝟐



,𝑐𝑡𝑥







⊓ctx.

Thus, the query

𝑝



𝒕



to the basis 𝐼𝜎𝑝



𝒕

𝟐



,𝑐𝑡𝑥





is performed as a broad unification using the

predicate of the original achievement goal

!𝑝



𝒕



. This

query provides all answers according to the

successfully executed chosen plan. These answers are

then restricted by the context of the original plan and

the result is the new context of the original WPI.

Example 4: Suppose that there is a WPI

!𝑣𝑖𝑠𝑖𝑡



𝑉,𝑋



,!𝑔𝑜



𝑋,𝑌,𝑍



; 𝑝𝑙𝑎𝑛𝑏𝑜𝑑𝑦,



𝑝𝑎𝑟𝑖𝑠/

𝑋,𝑢𝑛𝑐𝑙𝑒/𝑉,𝑟𝑖𝑜/𝑋,𝑓𝑟𝑖𝑒𝑛𝑑/𝑉





which was generated for this WEI. Such a plan

could be implemented for an agent who wants to go

on a trip and visit someone at the same time. The

context of this plan gives the possibilities of visiting

an uncle in Paris, or a friend in Rio. The achievement

goal

!𝑔𝑜



𝑋,𝑌,𝑍



is to find a means of getting to some

place the agent would like to be. This plan in Example

3 ended with the agent receiving the answers

𝑝𝑎𝑟𝑖𝑠/𝑋,𝑐𝑎𝑟/𝑌,𝑚𝑜𝑛/𝑍,𝑝𝑎𝑟𝑖𝑠/𝑋,𝑡𝑟𝑎𝑖𝑛/

𝑌,𝑡𝑢𝑒/𝑍

. after obtaining all possible answers

according to this plan. This is the result of the

operation

𝑝



𝑔𝑜𝐴, 𝐵,𝐶



,𝑝𝑎𝑟𝑖𝑠/𝐴,𝑐𝑎𝑟/𝐵,𝑚𝑜𝑛/

𝐶,𝑝𝑎𝑟𝑖𝑠/𝐴,𝑡𝑟𝑎𝑖𝑛/𝐵,𝑡𝑢𝑒/𝐶 ∽ 𝑔𝑜𝑋,𝑌,𝑍. By

Definition 15, the restriction with the original WPI

context which was



𝑝𝑎𝑟𝑖𝑠/𝑋,𝑢𝑛𝑐𝑙𝑒/𝑉,𝑟𝑖𝑜/

𝑋,𝑓𝑟𝑖𝑒𝑛𝑑/𝑉



is to be made and then the new

context is

𝑝𝑎𝑟𝑖𝑠/𝑋,𝑐𝑎𝑟/𝑌,𝑚𝑜𝑛/𝑍,𝑢𝑛𝑐𝑙𝑒/

𝑉,𝑝𝑎𝑟𝑖𝑠/𝑋,𝑡𝑟𝑎𝑖𝑛/𝑌,𝑡𝑢𝑒/𝑍,𝑢𝑛𝑐𝑙𝑒/𝑉

Theorem 2: If for some WEI

〈

𝑒𝑣𝑡, 𝑐𝑡𝑥

〉

a plan is

successfully executed, and the resulting WPI of this

plan is

𝑡



,𝑛𝑢𝑙𝑙,𝑐𝑡𝑥



 , then the result of executing

the achievement goals according to the predicate

𝑒𝑣𝑡

and the substitutions in the context of

𝑐𝑡𝑥 is a result

encompassing all the goals achieved by the plan with

respect to the selected actions and subplans.

Proof: Thanks to the above Lemma 5, we know

that if we have a WEI of the form

〈

𝑒𝑣𝑡,𝑐𝑡𝑥

〉

and a WPI

of some plan that does not contain an achievement

goal is successfully executed, then all possible

answers to the given goals that can be achieved by

executing this WPI are obtained. Thus, if we have an

achievement goal

!𝑝



𝒕



within some plan, then a WEI

!𝑝



𝒕



,𝑐𝑡𝑥′ wbere 𝑐𝑡𝑥′ is the actual context is

created, and if this goal was achieved by the WPI of

some plan without other achievement goals, then the

agent would receive all possible answers to

!𝑝



𝒕



According to the definition of the implementation

of this type of goals, we know that the new context is

given by these (all) answers and restrictions with the

original context and thus, according to Theorem 1,

contains all possible substitutions that agree with the

original context and the answers received. Such a

WPI, if successfully executed, will provide through

the resulting context all possible substitutions that

agree with the executed plans. By induction, if an

agent would need more immersions in subplans for

some goal, say up to level n, then we can prove that

WPIs that need at most two levels of subplans achieve

all possible substitutions, and then proceed similarly

up to level n.

5 CONCLUSIONS

In this paper, we have presented an interpretation of

the execution of a plan written in AgentSpeak(L)

using late binding of variables. What this approach

brings to the interpretation of this language is that the

agent thus has available at any point in time all the

substitutions that are valid with respect to all the acts

it has performed with a given WPI. This in itself does

not necessarily improve the rationality of the agent's

behaviour. Still, at some points the agent has to decide

for some substitutions when it has to perform an

action. However, late binding of variables can help

avoid premature erroneous choices of substitutions,

because by delaying their selection, the agent has the

opportunity to evaluate the situation when it must

make the decision. Moreover, if the agent maintains

all the ways it can still bind variables, this can be

beneficial even when the agent is pursuing multiple

goals and can look for synergies between plans for

these goals, but also for synchronizing the actions of

agents in a multi-agent group.

Systems that have addressed this so far either do

not work with variables at all, or consider variable

binding as an alternative when the initially chosen

binding fails (Sardina & Padgham, 2011) (Dastani,

2008). When researchers do work with focuses and

with the choice of plans for goals, they currently work

with goal plan trees, in which formulas are written as

propositional and thus do not consider predicates as

such (Yao & Logan, Action-Level Intention Selection

for BDI Agents, 2016) (Yao, Logan, & Thangarajah,

Robust execution of BDI agent programs by

exploiting synergies between intentions, 2016). We

believe that a system that can handle multiple options

at once by storing possible bindings until a decision

Late Bindings in AgentSpeak(L)

723

must be made will yield an increase in agent

behaviour in these respects. However, these are

suggestions for our future research, in which we

intend to use the principle of late bindings introduced

in this paper.

ACKNOWLEDGEMENTS

This work has been supported by the internal BUT

project FIT-S-20-6427.

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