$D_{\mathcal{AC}}=\left\\{\begin{array}[]{lll}\mbox{\% normal }&&\mbox{\%
abnormal}\\\ \hbox{\bf
causes}(close(sw_{1}),closed(sw_{1}),\emptyset)&&\hbox{\bf
causes}(brk,ab(b),\emptyset)\\\ \hbox{\bf caused}(\\{closed(sw_{1}),\neg
ab(r)\\},active(r))&&\hbox{\bf causes}(srg,ab(r),\emptyset)\\\ \hbox{\bf
caused}(\\{active(r)\\},closed(sw_{2}))&&\hbox{\bf causes}(srg,ab(b),\\{\neg
prot(b)\\})\\\ \hbox{\bf caused}(\\{closed(sw_{2}),\neg
ab(b)\\},on(b))&&\hbox{\bf caused}(\\{ab(b)\\},\neg on(b))\\\ \hbox{\bf
caused}(\\{\neg closed(sw_{2})\\},\neg on(b))&&\hbox{\bf
caused}(\\{ab(r)\\},\neg active(r))\\\ \hbox{\bf
executable}(close(sw_{1}),\\{\neg closed(sw_{1})\\})\end{array}\right.$

Now consider a recorded history:

$H_{1}=\left\\{\begin{array}[]{l}hpd(close(sw_{1}),0)\\\ obs(\neg
closed(sw_{1}),0)\\\ obs(\neg closed(sw_{2}),0)\\\ obs(\neg ab(b),0)\\\
obs(\neg ab(r),0)\\\ obs(prot(b),0)\end{array}\right.$

One can check that $s_{0}\ close(sw_{1})\ s_{1}$ is the only model of $H_{1}$,
where $s_{0}$ is the state depicted in Figure 4. Additionally, $H_{1}\models
holds(on(b),1)$.

Next, we formalize the key notions of diagnosis. Let $\delta=\langle
D,W\rangle$ be a diagnostic domain. A _configuration_ is a pair

${\cal S}=\langle H_{n},O^{m}_{n}\rangle$ (61)

where $H_{n}$ is the recorded history up to step $n$ and $O^{m}_{n}$ is a set
of observations between steps $n$ and $m\geq n$. Leveraging this notion, we
can now define a _symptom_ as a configuration $\langle H_{n},O^{m}_{n}\rangle$
such that $H_{n}$ is consistent and $H_{n}\cup O^{m}_{n}$ is not.

Once a symptom has been identified, the next step of the diagnostic process
aims at finding its possible reasons. Specifically, a _diagnostic explanation_
$E$ of symptom ${\cal S}=\langle H_{n},O^{m}_{n}\rangle$ is defined as a set

$E\subseteq\\{hpd(a_{i},t):0\leq t<n\mbox{ and }a_{i}\in\mbox{\bf A}_{e}\\},$
(62)

such that $H_{n}\cup O^{m}_{n}\cup E$ is consistent.

###  7.1 Diagnostic Reasoning as Answer Set Planning

As we said earlier, answer set planning can be used to determine whether a
diagnosis is needed and for computing diagnostic explanations. Next, we
introduce an ASP-based program for these purposes. Consider a domain $D$ whose
behavior up to step $n$ is described by recorded history $H_{n}$. Reasoning
about $D$ and $H_{n}$ can be accomplished by a translation to a logic program
$\Pi(D,H_{n})$ that follows the approach outlined in Section 3. Focusing for
simplicity on the direct encoding, $\Pi(D,H_{n})$ consists of:

  * •

the domain dependent rules from Section 3.1.1;

  * •

the rules related to inertia and consistency of states (11)-(13);

  * •

domain independent rule establishing the relationship between observations and
the basic relations of $\Pi$:

$\displaystyle occ(A,T)\leftarrow hpd(A,T)$ (63) $\displaystyle
holds(L,0)\leftarrow obs(L,0)$ (64)

  * •

the _reality check axiom_ , i.e. a rule ensuring that in any answer set the
agent’s expectations match the available observations (variable $L$ ranges
over fluent literals):

$\displaystyle\leftarrow obs(L,T),{not\>}holds(L,T)$ (65)

The following theorem establishes an important relationship between models of
a recorded history and answer sets of the corresponding logic program.

###### Theorem 2

If the initial situation of $H_{n}$ is complete, i.e. for any fluent $f$,
$H_{n}$ contains $obs(f,0)$ or $obs(\neg f,0)$, then $M$ is a model of $H_{n}$
iff $M$ is defined by some answer set of $\Pi(D,H_{n})$.

The proof of the theorem is in two steps. First, one shows that the theorem
holds for $n=1$, i.e., that for a history $H_{1}$ there is a one-to-one
correspondence between the transitions of the form $s_{0}a_{0}s_{1}$ and the
answer sets of $\Pi(D,H_{1})$. Then, induction is leveraged to extend the
correspondence to histories of arbitrary length. The complete proof can be
found in the paper by Balduccini and Gelfond (2003).

Next, we focus on identifying the need for diagnosis. Given a domain $D$ and
${\cal S}=\langle H_{n},O^{m}_{n}\rangle$, we introduce:

$\mathit{TEST}({\cal S})=\Pi(D,H_{n})\cup O^{m}_{n}.$ (66)

The following corollary forms the basis of this approach to diagnosis.

###### Corollary 1

Let ${\cal S}=\langle H_{n},O^{m}_{n}\rangle$ where $H_{n}$ is consistent. A
configuration ${\cal S}$ is a symptom iff $\mathit{TEST}({\cal S})$ has no
answer set.

Once a symptom has been identified, diagnostic explanations can be found by
means of the answer sets of the _diagnostic program_

$\Pi_{d}({\cal S})=\mathit{TEST}({\cal S})\cup\\{\ \ \
1\\{occ(A,T):x\\_act(A)\\}1\leftarrow time(T),T<n.\ \ \ \\}$ (67)

Specifically, every answer set $X$ of $\Pi_{d}({\cal S})$ encodes a diagnostic
explanation

$E=\\{hpd(a,t)\,\,|\,\,occ(a,t)\in X\land a\in\mbox{{\bf A}${}_{e}$}\\}.$

Note that the choice rule shown in (67) is simply a restriction of (10) to
exogenous actions. As a result, $\Pi_{d}({\cal S})$ can be viewed as a variant
of the translation $\Pi(\mathcal{P},n)$ of a planning problem, where planning
occurs over the past $0..n\\!-\\!1$ time steps and over exogenous actions
only, and the goal states are described by the observations from ${\cal S}$.

###### Example 9

Consider the domain from Example 8. According to $H_{1}$ initially switches
$sw_{1}$ and $sw_{2}$ are open, all circuit components are ok, $sw_{1}$ is
closed by the agent, and b is protected. The expectation is that $b$ will be
on at 1. Suppose that, instead, the agent observes that at time 1 bulb b is
off, i.e. $O_{1}=\\{obs(\neg on(b),1)\\}$. $\mathit{TEST}({\cal S}_{0})$,
where ${\cal S}_{0}=\langle H_{1},O_{1}\rangle$, has no answer sets and thus,
by Corollary 1, ${\cal S}_{0}$ is indeed a symptom. The diagnostic
explanations of ${\cal S}_{0}$ can be found by computing the answer sets of
$\Pi_{d}(\mathcal{S})$. Specifically, there are three diagnostic explanations:

$\begin{array}[]{l}E_{1}=\\{occ(brk,0)\\}\\\ E_{2}=\\{occ(srg,0)\\}\\\
E_{3}=\\{occ(brk,0),occ(srg,0)\\}\end{array}$

###### Remark 6

  1. 1.

Other interpretations of the relationship between agent and environment are
possible, yielding substantial differences in the overall approach to
diagnosis. The interested reader is referred to the paper by Baral et al.
(2000).

  2. 2.

In contrast to the approach by Baral et al. (2000), the approach presented in
this survey assumes that a recorded history is consistent only if observations
about fluents can be explained without assuming the occurrence of actions not
recorded in $H_{n}$.

  3. 3.

In the paper by Balduccini and Gelfond (2003), the formalization of diagnostic
reasoning presented here is extended to incorporate an account of the agent’s
interaction with the domain in order to collect physical evidence that
confirms or refutes the diagnostic explanations computed. This is accomplished
by introducing the notions of candidate diagnosis and of diagnosis.

  4. 4.

Theorem 2 is similar to the result from the paper by Turner (1997), which
deals with a different language and uses the definitions by McCain and Turner
(1995). If the initial situation of $H_{n}$ is incomplete, one can adopt
techniques discussed elsewhere in this paper or the _awareness axioms_ by
Balduccini and Gelfond (2003).

  5. 5.

As discussed in the paper by Balduccini and Gelfond (2003), the diagnostic
process may not always lead to a unique solution. In those cases, the agent
may need to perform further actions, such as repairing or replacing
components, and observe their outcomes. Balduccini and Gelfond (2003) provided
a specialized algorithm to achieve this. An alternative, and potentially more
general, option consists in leveraging conditional planning techniques (see
Section 5.2), i.e., by creating a conditional plan that determines the true
diagnosis as proposed by Baral et al. (2000).

###  7.2 Context: Planning and Diagnosis

Classical diagnosis such as the foundational work by Reiter (1987) aimed at
providing a formal answer to the question of “ _what is wrong with a system
given its (current failed) state_.” Central to classical diagnosis is the use
of the model of the system to reason about failures. Earlier formalizations
considered a single state of the system and are often referred to as _model
based diagnosis_ , which is summarized by De Kleer and Kurien (2003). Later
formalization such as the proposal by Feldman et al. (2020) considers a finite
trace of states, taking into consideration the transitions at different time
points in need of a diagnosis. This is closedly related to _dynamic diagnosis_
, as described in this paper, which has been considered in the literature
Baral et al. (2000); Baroni et al. (1999); Cordier and Thiébaux (1994);
McIlraith (1997); Thielscher (1997); Thiébaux et al. (1996); Williams and
Nayak (1996).

The close relationship between model based diagnosis and satisfiability led to
several methods for computing diagnosis using satisfiability such as the
method proposed by Grastien and Anbulagan (2013), Metodi et al. (2014), or
described in several publications on diagnosing sequential circuits (e.g., by
Feldman et al. (2020)). In a recent work by Wotawa (2020), answer set
programming has been used in the context of model-based diagnosis.

##  8 Planning in Multi-Agent Environment

The formalizations presented in the previous sections can be also extended to
deal with various problems in multi-agent environments (MAE). In these
problems, the planning (or reasoning) activity can be carried out either by
one system (a.k.a. _centralized planning_) or multiple systems (a.k.a.
_distributed planning_). In the following subsections, we discuss the use of
ASP in these settings.

###  8.1 Centralized Multi-Agent Planning

We will start with the _Multi-Agent Path Finding (MAPF)_ problem which appears
in a variety of application domains, such as autonomous aircraft towing
vehicles Morris et al. (2016), autonomous warehouse systems Wurman et al.
(2008), office robots Veloso et al. (2015), and video games Silver (2005).

A MAPF problem is defined by a tuple $\mathcal{M}=(R,(V,E),s,d)$ where

  * •

$R$ is a set of robots,

  * •

$(V,E)$ is an undirected graph, with the set of vertices $V$ and the set of
edges $E$,

  * •

$s$ is an injective function from $R$ to $V$, $s(r)$ denotes the starting
location of robot $r$, and

  * •

$d$ is an injective function from $R$ to $V$, $d(r)$ denotes the destination
of robot $r$.

Robots can move from one vertex to one of its neighbors in one step. A
collision occurs when two robots move to the same location (vertex collision)
or traveling on the same edge (edge collision). The goal is to find a
collision-free plan for all robots to reach their destinations. Optimal plans,
with the minimal number of steps, are often preferred.

A simple MAPF problem is depicted in Figure 5. In this problem, we have two
robots $r_{1}$ and $r_{2}$ on a graph with five vertices $p_{1},\ldots,p_{5}$.
Initially, $r_{1}$ is at $p_{2}$ and $r_{2}$ is at $p_{4}$. The goal consists
of moving robot $r_{1}$ to location $p_{5}$ and robot $r_{2}$ to location
$p_{3}$.

Figure 5: A Multi-Agent Path Finding Problem

It is easy to see that a MAPF $\mathcal{M}=(R,(V,E),s,d)$ can be represented
by

  * •

a set $\\{\mathcal{P}_{r}\mid r\in R\\}$ of path-planning problems for the
robots in $R$, where for each $r\in R$, $\mathcal{P}_{r}=\langle
D_{r},at(r,s(r)),at(r,d(r))\rangle$ is a planning problem with

$D_{r}=\left\\{\begin{array}[]{lll}\hbox{\bf
causes}(move(r,l,l^{\prime}),at(r,l^{\prime}),\\{at(r,l)\\})&&\mbox{ for
}(l,l^{\prime})\in E\\\ \hbox{\bf caused}(\\{at(r,l)\\},\neg
at(r,l^{\prime}))\\}&&\mbox{ for }l\neq l^{\prime},l,l^{\prime}\in V\\\
\end{array}\right.$

  * •

the set of constraints representing the collisions.

ASP-based solutions of MAPF problems have been proposed for both action-based
as well as state-based encodings. In the context of this survey, we will focus
on an action-based encoding. Consider a MAPF problem
$\mathcal{M}=(R,(V,E),s,d)$. The program developed in Section 3 for a single-
agent planning problem can be used to develop a program solving $\mathcal{M}$,
denoted by $\Pi(\mathcal{M},n)$, as follows. $\Pi(\mathcal{M},n)$ contains the
following groups of rules:

  1. 1.

the set of atoms $\\{agent(r)\mid r\in R\\}$ encoding the robots;

  2. 2.

the collection of the rules from $\Pi(\mathcal{P}_{r},n)$; and

  3. 3.

the constraints to avoid collisions:

$\displaystyle\leftarrow$ $\displaystyle agent(R),agent(R^{\prime}),R\neq
R^{\prime},$ $\displaystyle holds(at(R,V),T),holds(at(R^{\prime},V),T)$
$\displaystyle\leftarrow$ $\displaystyle agent(R),agent(R^{\prime}),R\neq
R^{\prime},$ $\displaystyle
holds(at(R,V),T),holds(at(R^{\prime},V^{\prime}),T),$ $\displaystyle
holds(at(R,V^{\prime}),T+1),holds(at(R^{\prime},V),T+1)$

Rule (3) prevents two robots to be at the same location at the same time,
while rule (3) guarantees that edge-collisions will not occur. These two
constraints and the correctness of $\Pi(\mathcal{P}_{r},n)$ imply that
$\Pi(\mathcal{M},n)$ computes solutions of length $n$ for the MAPF problem
$\mathcal{M}$.

The proposed method for solving MAPF problems can be generalized to multi-
agent planning as considered by the multi-agent community in the setting
discussed by Durfee (1999). We assume that a multi-agent planning problem
$\mathcal{M}$ for a set of agents $R$ is specified by a pair
$(\mathcal{P}_{r\in R},\mathcal{C})$ where $\mathcal{P}_{r}=\langle
D_{r},\Delta_{r},\Gamma_{r}\rangle$ is the planning problem for agent $r$ and
$\mathcal{C}$ is a set of global constraints. For simplicity of the
presentation, we will assume that

  * •

the agents in $R$ share the same set of fluents and, wherever needed,
parameterized with the names of the agents; for example, if an agent is
carrying something then $carrying(r,o)$ will be used instead of $carrying(o)$
as in a single-agent domain;

  * •

the actions in the domain in $D_{r}$ are parameterized with the agent’s name
$r$, e.g., we will use the action $move(r,l,l^{\prime})$ instead of the
traditional encoding $move(l,l^{\prime})$;

  * •

the constraints in $\mathcal{C}$ are of the form

$\hbox{\bf executable}(sa,\varphi)$ (70)

where $sa$ is a set of actions in $\bigcup_{r\in R}D_{r}$ and $\varphi$ is a
set of literals. This can be used to represent parallel actions, non-
concurrent actions, etc.

For a multi-agent planning problem $\mathcal{M}=(\mathcal{P}_{r\in
R},\mathcal{C})$ where $\mathcal{P}_{r}=\langle
D_{r},\Delta_{r},\Gamma_{r}\rangle$, the program $\Pi(\mathcal{M},n)$ that
computes solution for $\mathcal{M}$ consists of

  1. 1.

the set of agent declarations, $agent(r)$ for $r\in R$;

  2. 2.

the collection151515 We keep only one instance of the domain-independent
rules.  of the rules from $\Pi(\mathcal{P}_{r},n)$ with the following
modifications:

     * •

the action specification of the form $action(a)$ is replaced by $action(r,a)$;

     * •

the action generation rule is replaced by

$1\\{occ(A,T):action(R,A)\\}1\leftarrow time(T),agent(R)$

where, without the loss of generality, we assume that every $D_{r}$ contains
the action _noop_ that is always executable and has no effect;

  3. 3.

for each constraint of form (70), we create a new “collective” action named
$sa_{id}$, add $action(sa_{id})$ to the set of actions in $\mathcal{M}$ (and
make the (70) its executability condition, translated into ASP as for any
other action) and, for each $a\in sa$, the following rule is added to
$\Pi(\mathcal{P}_{r},n)$:

$\displaystyle occ(a,T)$ $\displaystyle\leftarrow$ $\displaystyle
occ(sa_{id},T)$ (71)

###  8.2 Distributed Planning

A main drawback of centralized planning is that it cannot exploit the
structural organization of agents (e.g., hierarchical organization of agents)
in the planning process. Distributed planning has been proposed as an
alternative to centralized planning that aims at exploiting the independence
between agents and/or groups of agents. We discuss distributed planning in two
settings: fully collaborative agents and partially-cooperative or non-
cooperative agents.

####  8.2.1 Fully Collaborative Agents

When agents are fully collaborative, a possible way to exploit structural
relationships between agents is to allow each group of agents to plan for
itself (e.g., using the planning system described in Section 3 and then employ
a centralized post-planning process (a.k.a. the _controller/scheduler_) to
create the joint plan for all agents. The controller takes the output of these
planners—individual plans—and merges them into an overall plan. One of the
main tasks of the controller is to resolve conflicts between individual plans.
This issue arises because individual groups plan without knowledge of other
groups (e.g., robot $r_{1}$ does not know the location of robot $r_{2}$). When
the controller is unable to resolve all possible conflicts, the controller
will identify plans that need to be changed and request different individual
plans from specific individual groups.

Any implementation of distributed planning requires some communication
capabilities between the controller and the individual planning systems. For
this reason, a client-server architecture is often employed in the
implementation of distributed planning. A client plans for an individual group
of agents and the server is responsible for merging the individual plans from
all groups. Although specialized parallel ASP solvers exist (e.g., the systems
discussed in the papers by Le and Pontelli (2005) and Schneidenbach et al.
(2009)), there has been no attempt to use parallel ASP solvers in distributed
planning. Rather, distributed planning using ASP has been implemented using a
combination of Prolog and ASP, where communication between server and clients
is achieved through Prolog-based message passing, and planning is done using
ASP (e.g., the system described in the paper by Son et al. (2009)).

Observe that the task of resolving conflicts is not straightforward and can
require multiple iterations with individual planner(s) before the controller
can create a joint plan. Consider again the two robots in Figure 5. If they
are to generate their own plans, then the first set of individual solutions
can be

${\begin{array}[]{l}occ(move(r_{1},p_{2},p_{4}),0),occ(move(r_{1},p_{4},p_{5}),1)\end{array}}$
(72)

and

${\begin{array}[]{l}occ(move(r_{2},p_{4},p_{2}),0),occ(move(r_{2},p_{2},l_{3}),1)\end{array}}$
(73)

A parallel execution of these two plans will result in a violation of the
constraint stating that two robots cannot be at the same location at the same
time. One can see that the controller needs to insert a few actions into both
plans (e.g., $r_{1}$ must move to either $l_{1}$ or $l_{3}$ before moving to
$l_{4}$).

Let $\mathcal{M}$ be a multi-agent planning problem and $P_{\\{r\in R\\}}$ be
the plans received by the controller. The feasibility of merging these plans
into a single plan for all agents can be checked using ASP. Let $\pi_{n}$ be
the program obtained from $\Pi(\mathcal{M},n)$ (described in Subsection 8.1)
by adding to $\Pi(\mathcal{M},n)$

  * •

the set of action occurrences in $P_{\\{r\in R\\}}$, i.e.,

$\bigcup_{\\{r\in R\\}}\\{occurs(a,t)\mid occ(a,t)\in P_{r}\\}$

  * •

for $r\in R$, rules mapping time steps from 0 to $n$ to time steps used in
$P_{r}$ ($r\in R$),

$\begin{array}[]{l}1\\{map(r,T,J):time(J)\\}1\leftarrow
time(T),T<n,T\leq\max_{r}\\\ \leftarrow
map(r,T,J),map(p,T^{\prime},J^{\prime}),T<T^{\prime},J>J^{\prime}\end{array}$

where $\max_{r}$ is the maximal index in $P_{r}$. Intuitively, $map(r,i,j)$
indicates that the $i^{th}$ action in $P_{r}$ should occur in the $j^{th}$
position in the joint plan. This mapping must conform to the order of action
occurrences in $P_{r}$.

  * •

a rule ensuring that an atom $occurs(a,j)\in P_{r}$ must occur at the
specified position:

$\leftarrow occ(a,t),map(r,t,j),{not\>}occurs(a,j).$

It can be checked that $\pi_{4}$ would generate an answer set consisting of

$\begin{array}[]{l}occ(move(r_{1},p_{2},p_{1}),0),occ(move(r_{2},p_{4},p_{2}),0),\\\
occ(move(r_{2},p_{2},l_{3}),1),occ(move(r_{1},p_{1},p_{2}),1),\\\
occ(move(r_{1},p_{2},p_{1}),2),\\\ occ(move(r_{2},p_{4},p_{2}),3)\end{array}$

which corresponds to the mapping $map(1,0,2),map(1,1,3),map(2,0,0),map(2,1,1)$
and is a successful merge of the two plans in (72)–(73).

Observe that the program $\pi_{n}$ might have no answer sets, which indicates
that the merging of the plans $P_{\\{r\in R\\}}$ is unsuccessful. For
instance, $\pi_{3}$ has no answer set, i.e., the two plans in (72)–(73) cannot
be merged with less than four steps.

####  8.2.2 Non/Partially-Collaborative Agents

Centralized planning or distributed planning with an overall controller is
most suitable in applications with collaborative (or non-competitive) agents
such as the robots in the MAPF problems. In many applications, this assumption
does not hold, e.g., agents may need to withhold certain private information
and thus do not want to share their information freely; or agents may be
competitive and have conflicting goals. In these situations, distributed
planning as described in the previous sub-section is not applicable and
planning will have to rely on a message passing architecture, e.g., via peer-
to-peer communications. Furthermore, an online planning approach might be more
appropriate. Next, we describe an ASP approach that is implemented centrally
by Son et al. (2009) but could also be implemented distributedly.

In this approach, the planning process is interleaved with a negotiation
process among agents. As an example, consider the robots in Figure 5 and
assume that the robots can communicate with each other, but they cannot reveal
their location. The following negotiation between $r_{2}$ and $r_{1}$ could
take place:

  * •

$r_{2}$ (to $r_{1}$): “can you ($r_{1}$) move out of $l_{2}$, $l_{3}$, and
$l_{4}$?” (because $r_{2}$ needs to make sure that it can move to location
$l_{2}$ and $l_{3}$). This can be translated to the formula $\varphi_{1}=\neg
at(r_{1},l_{2})\wedge\neg at(r_{1},l_{3})\wedge\neg at(r_{1},l_{4})$ sent from
$r_{2}$ to $r_{1}$.

  * •

$r_{1}$ (to $r_{2}$): “I can do so after two steps but I would also like for
you ($r_{2}$) to move out of $l_{2}$, $l_{4}$, and $l_{5}$ after I move out of
those places.” This means that $r_{1}$ agrees to satisfy the formula sent by
$r_{2}$ but also has some conditions of its own. This can be represented by
the formula $\varphi_{1}\supset\varphi_{2}=\neg at(r_{2},l_{2})\wedge\neg
at(r_{2},l_{4})\wedge\neg at(r_{2},l_{5})$.

  * •

$r_{2}$ (to $r_{1}$): “that is good; however, do not move through $l_{4}$ to
get out of the area.”

  * •

etc.

The negotiation will continue until either the agent accepts (or refutes) the
latest proposal from the other agent. A formal ASP based negotiation framework
(e.g., the system described by Son et al. (2014)) could be used for this
purpose.

Observe that during a negotiation, none of the robots changes its location or
executes any action. After a successful negotiation, each robot has some
additional information to take into consideration in its planning. In this
example, if the two robots agree after the second proposal by $r_{2}$, robot
$r_{1}$ agrees to move out of $l_{2}$, $l_{3}$, and $l_{4}$ but should do so
without passing by $l_{4}$; robot $r_{2}$ knows that he can have $l_{2}$,
$l_{3}$, and $l_{4}$ for itself after sometime and also knows that it can
stand at $l_{4}$ until $r_{1}$ is out of the requested area; etc. Note,
however, that this is not yet sufficient for the two robots to achieve their
goals. To do so, they also need to agree on the timing of their moves. For
example, $r_{1}$ can tell $r_{2}$ that $l_{2}$, $l_{3}$, and $l_{4}$ will be
free after two steps; $r_{2}$ responds that, if it is the case, then $l_{2}$,
$l_{4}$, and $l_{5}$ will be free after 2 steps; etc. This information will
help the robots come up with plans for their own goals.

To the best of our knowledge, only a prototype implementation of the approach
to interleaving negotiation and planning has been presented Son et al. (2009).
It is also not implemented distributedly.

###### Remark 7

  1. 1.

There are two different ways to enforce the collision-free constraint in the
MAPF encoding. One can, for example, replace (3) with the rule

$\begin{array}[]{rcl}&\leftarrow&agent(R),agent(R^{\prime}),R\neq
R^{\prime},\\\ &&holds(at(R,V),T),holds(at(R^{\prime},V^{\prime}),T),\\\
&&occ(move(R,V,V^{\prime}),T),occ(move(R^{\prime},V^{\prime},V),T+1)\end{array}$

  2. 2.

ASP-based solutions for various extensions of the MAPF problems have been
discussed by Nguyen et al. (2017) and Gómez et al. (2020). The encoding
proposed by Gómez et al. (2020) is special in that its grounded program has a
linear size to the number of agents. An ASP-based solution for this problem
has been applied in a real-world application Gebser et al. (2018). An
environment for experimenting with MAPF has been developed by Gebser et al.
(2018). A preliminary implementation of a MAPF solver on distributed platform
can be found in the paper by Pianpak et al. (2019).

  3. 3.

We observe that little attention has been paid to answer set programming based
distributed planning. This also holds for the answer set programming based
distributed computing platforms. Perhaps the need to attack problems in multi-
agent systems will eventually lead to a truly distributed platform that could
push the investigation of using answer set programming in this research
direction to the next level. We note that the need for such platform exists
and ad-hoc combinations with other programming language have been developed by
Le et al. (2015).

###  8.3 Context: Planning in Multi-Agent Environments

Planning in multi-agent environments has been extensively investigated by the
multi-agent research community. There exists a broad literature in this
direction which addresses several issues, such as coordination, sharing of
resources, use of shared resources, execution of joint actions, centralized or
distributed computation of plans, sharing of tasks, etc. Earlier works in
multi-agent planning (e.g., see the papers Allen and Zilberstein (2009);
Brafman and Domshlak (2008); Bernstein et al. (2002); Brenner (2003); Crosby
et al. (2014); Durfee (1999); de Weerdt et al. (2003); de Weerdt and Clement
(2009); Goldman and Zilberstein (2004); Guestrin et al. (2001); Nair et al.
(2003); Nissim and Brafman (2012); Peshkin and Savova (2002); Shoham and
Leyton-Brown (2009); Torreño et al. (2012); Vlassis (2007)) focus on
generating plans for multiple agents, coordinating the execution of plans, and
does not take into consideration knowledge, beliefs, or privacy of agents.
Work in planning for multiple self-interested agents can be found in the
papers Gmytrasiewicz and Doshi (2005); Rathnasabapathy et al. (2006); Poupart
and Boutilier (2003); Sonu and Doshi (2015).

The planning problem in multi-agent environments discussed in this section
focuses on the setting in which each agent has its own goal, similar to the
setting discussed in earlier work on multi-agent planning. It should be noted
that MAPF has attracted a lot of attention in recent years due to its
widespread applicability such as in warehouse or airtraffic control, leading
to the organization of the yearly MAPF workshop at IJCAI and/or ICAPS
conferences and several tutorials on the topic. A good description of this
problem can be found in the papers Barták et al. (2019); Stern et al. (2019).
Challenges and opportunities in MAPF and its extensions have been described by
Salzman and Stern (2020). As with planning, search-based approaches to solving
MAPF are frequently used. Early MAPF solvers, such as the ones described in
the papers Goldenberg et al. (2014); Wagner and Choset (2015); Sharon et al.
(2015); Boyarski et al. (2015); Cohen et al. (2016); Wang and Botea (2011);
Luna and Bekris (2011); de Wilde et al. (2014), can compute optimal,
boundedly-suboptimal, or suboptimal solutions of MAPF. Erdem et al. (2013), Yu
and LaValle (2016), and Surynek et al. (2016) applied answer set programming,
mixed-integer programming, and satisfiability testing, respectively, to solve
the original MAPF problem. Suboptimal solutions of MAPF using SAT is discussed
by Surynek et al. (2018). Surynek (2019b) presents an SMT-based MAPF solver.

Several extensions of the MAPF problem have been introduced. Ma and Koenig
(2016) generalize MAPF to _combined Target Assignment and Path Finding
(TAPF),_ where agents are partitioned into teams and each team is given a set
of targets that they need to reach. MAPF with deadlines is introduced by Ma et
al. (2018). Extensions of the MAPF problem with delay probabilities have been
described by Ma et al. (2017). The answer set planning implementation by
Nguyen et al. (2017) shows that TAPF can be efficiently solved by answer set
planning in multi-agent environments.

Andreychuk et al. (2019) investigate MAPF with continuous time, which removes
the assumption that transitions between nodes are uniform. Barták and Svancara
(2019) present a SAT-based approach to deal with this extension, while Surynek
(2019a) describe an SMT-based MAPF solver for MAPF with continuous time and
geometric agents.

Atzmon et al. (2020) focus on the issue of unexpected delays of agents and
introduce the notion of $k$-robust MAPF plan, which can still be successfully
executed when at most $k$ delays happen. This paper also studies a
probabilistic extension of $k$-robust MAPF plan, called $pk$-robots MAPF plan.

A more realistic version of MAPF, which allows agents to exchange packages and
transfer payload, is considered by Ma et al. (2016). Discussion of the
problems where robots have kinematic constraints can be found in the paper
Hönig et al. (2016).

It is worth noting that all of the aforementioned approaches to solving MAPF
are centralized. Pianpak et al. (2019) propose a distributed ASP-based MAPF
solver. In a recent paper, Gómez et al. (2021) present a compact ASP encoding
for solving optimal sum-of-cost MAPF that is competitive with other
approaches.

##  9 Planning with Scheduling and Extensions of ASP

Research on applications of ASP planning has also given impulse to, and is
intertwined with, work on extensions of ASP. Let us consider the scenario from
the following example.

######  Example 10 (From the paper by Balduccini (2011))

In a typical scenario from the domain of industrial printing, orders for the
printing of books or magazines are more or less continuously received by the
print shop. Each order involves the execution of multiple jobs. First, the
pages are _printed_ on (possibly different) press sheets. The press sheets are
often large enough to accommodate several (10 to 100) pages, and thus a
suitable layout of the pages on the sheets must be found. Next, the press
sheets are _cut_ in smaller parts called _signatures_. The signatures are then
_folded_ into booklets whose page size equals the intended page size of the
order. Finally the booklets are _bound_ together to form the book or magazine
to be produced. The decision process is made more complex by the fact that
multiple models of devices may be capable of performing a job. Furthermore,
many decisions have ramifications and inter-dependencies. For example,
selecting a large press sheet would prevent the use of a small press. The
underlying decision-making process is often called _production planning_.
Another set of decisions deals with _scheduling_. Here one needs to determine
_when_ the various jobs will be executed using the devices available in the
print shop. Multiple devices of the same model may be available, thus even
competing jobs may be run in parallel. Conversely, some of the devices can be
offline—or go suddenly offline while production is in progress – and the
scheduler must work around that. Typically, one wants to find a schedule that
minimizes the tardiness of the orders while giving priority to the more
important orders. Since orders are received on a continuous basis, one needs
to be able to update the schedule in an incremental fashion, in a way that
causes minimal disruption to the production, and can satisfy _rush orders_ ,
which need to be executed quickly and take precedence over the others.
Similarly, the scheduler needs to react to sudden changes in the print shop,
such as a device going offline during production. $\Diamond$

This problem involves a combination of planning, configuration (of the devices
involved) and scheduling. While ASP can certainly be used to _represent_ the
problem, computation presents challenges. In particular, the presence of
variables with large domains has a tendency to cause a substantial increase in
the size of the grounding of ASP programs. Under these conditions, both the
grounding process itself and the following solving algorithms may take an
unacceptable amount of time and/or memory. Similar challenges have been
encountered in the ASP encoding of planning problems with large number of
actions and steps (see, e.g., the discussion by Son and Pontelli (2007)).

Various extensions of ASP have been proposed over time to overcome this
challenge. Some approaches, e.g., in response to large planning problems, rely
on the avoidance of grounding, as illustrated in systems with lazy grounding
(see, e.g., the papers by Palù et al. (2009), Cat et al. (2015), and Taupe et
al. (2019)) or on the use of top-down execution models (see, e.g., the papers
by Bonatti et al. (2008) and Marple and Gupta (2013)). At the core of the
attempts focused on combination of planning and scheduling is the integration
of ASP with techniques from constraint solving; this approach enables the
effective ability to handle variables with large domains (especially
numerical) efficiently.

Elkabani et al. (2004) provided an initial exploration of the combination of
answer set programming with constraint logic programming, mostly focused on
supporting the introduction of aggregates in answer set programming. Baselice
et al. (2005) were among the first to propose a methodology for achieving such
an integration as a way to extend the capabilities of the answer set
programming framework. In their approach, the syntax and semantics of ASP is
extended to enable the encoding of numerical constraints within the ASP
syntax. Mellarkod et al. (2008) and Gebser et al. (2009) proposed solvers that
support variants of ASP defined along the lines of the approach by Baselice et
al. (2005), and specific ASP and constraint solvers are modified and
integrated. In particular, the approach discussed in the _clingcon_ systems
Ostrowski and Schaub (2012); Banbara et al. (2017) explores the integration of
constraint solving techniques with techniques like clause learning and back-
jumping.

Experimental results show that these approaches lead to increased scalability,
enabling the efficient resolution of planning domains of larger size. Later
research extends and generalizes the language Bartholomew and Lee (2013),
increasing the efficiency of the resolution algorithms; these extensions have
been eventually integrated in the mainstream clingo solver Gebser et al.
(2016). Notably, its extension with difference constraints, viz. clingo[dl],
is operationally used by Swiss Railway for routing and scheduling train
networks Abels et al. (2019).

All of these approaches rely on a _clear-box_ architecture Balduccini and
Lierler (2013), i.e., an architecture where the ASP components of the
algorithm and its constraint solving components are tightly integrated and
modified specifically to interact with each other. A different approach is
proposed by Balduccini (2009) and later extensions. In that line of research,
the goal is to enable the reuse, without modifications, of existing ASP and
constraint solving algorithms. The intuition is that such an arrangement
allows one to employ the best solvers available and makes it easy to analyze
the performance of different combinations of solvers on a given task. This
enabled researchers to propose a _black-box_ architecture (and, later, a more
advanced _gray-box_ architecture), where the ASP and the constraint solving
components are unaware of each other and are connected only by a thin “upper
layer” of the architecture, which is responsible for exchanging data between
the components and triggering their execution. The architecture proposed by
Balduccini and Lierler (2017) is illustrated in Figure 6. Intuitively, answer
sets are computed first by an off-the-shelf ASP solver. Special atoms are
gathered by the “upper layer” and translated into a constraint satisfaction
problem, which is solved by an off-the-shelf constraint solver. Solutions to
the overall problem correspond to pairs formed by an answer set and a solution
to the constraint satisfaction problem extracted from that answer set.

Figure 6: Architecture of the EZCSP solver Balduccini (2011)

This approach features an embedding of constraint solving constructs directly
within the ASP language—that is, without the need to extend the syntax and
semantics of ASP—by means of pre-interpreted relations known to the “upper
layer” of the architecture (and some syntactic sugars for increased ease of
formalization). For instance, constraint variables for the start time of jobs
from Example 10 can be declared in the language described in the paper
Balduccini (2011) by means of the rule

$cspvar(st(D,J),0,MT)\leftarrow job(J),\ job\\_device(J,D),max\\_time(MT)$
(74)

where $cspvar$ is a special relation that the “upper layer” knows how to
translate into a variable declaration for a numerical constraint solver.
Similarly, the effect on start times of precedences between jobs can be
encoded by the ASP rule

$\begin{array}[]{l}required(st(D2,J2)\geq st(D1,J1)+Len1)\leftarrow\\\ \hskip
72.26999ptjob(J1),\ job(J2),\ job\\_device(J1,D1),\ job\\_device(J2,D2),\\\
\hskip 72.26999ptprecedes(J1,J2),\ job\\_len(J1,Len1)\end{array}$ (75)

where “$\geq$” is a syntactic sugar for the predicate $at\\_least$, written in
the infix notation, and it is replaced by a pre-interpreted function symbol
during pre-processing. As before, the “upper layer” is aware of the relation
$required$ and translates the corresponding atoms to numerical constraints. A
problem involving a combination of planning and scheduling such as that
described in Example 10 can be elegantly and efficiently solved by extending a
planning problem $\langle D,\Gamma,\Delta\rangle$ with statements such as (74)
and (75).

Later research on this topic Balduccini and Lierler (2013, 2017) uncovered an
interesting result: contrary to what one might expect, there is no clear
winner in the performance comparison between black-box and white-box
architectures (and gray-box as well); different classes of problems are more
efficiently solved by a different architecture. Furthermore, Balduccini et al.
(2017) showed that the EZCSP architecture can be used in planning with PDDL+
domains.

##  10 Conclusions and Future Directions

This paper surveys the progress made over the last 20+ years in the area of
_answer set planning._ It focuses on the encoding in answer set programming of
different classes of planning problems: when the initial state is complete,
incomplete, and with or without sensing actions. In addition, the paper shows
that answer set planning can reach the level of scalability and efficiency of
state-of-the-art specialized planners, if useful information which can be
exploited to guide the search process in planning, such as heuristics, is
provided to the answer set solver. The paper also reviews some of the main
research topics related to planning, such as planning with preferences,
diagnosis, planning in multi-agent environments, and planning integrated with
scheduling. We note that research related to answer set planning has been
successfully applied in different application domains, often in combination
with other types of reasoning, such as planning for the shuttle spacecraft by
Nogueira et al. (2001), planning and scheduling Balduccini (2011), robotics
Aker et al. (2011), scheduling Abels et al. (2019); Dodaro et al. (2019);
Gebser et al. (2018), and multi-agent path findings Gómez et al. (2021);
Nguyen et al. (2017). Section 9 also discusses the potential impacts that the
use of answer set planning in real-world applications can have on the
development of answer set programming.

In spite of this extensive body of research, there are still several
challenges for answer set planning.

##### Performance of ASP Planning:

We note that extensive experimental comparisons with state-of-the-art
compatible planning systems have been conducted. For example, Gebser et al.
(2013) experimented with classical planning, Eiter et al. (2003b) and Tu et
al. (2011) worked with incomplete information and non-deterministic actions,
and Tu et al. (2007) with planning with sensing actions. The detailed
comparisons can be found in the aforementioned papers and several other
references that have been discussed throughout the paper. These comparisons
demonstrate that ASP-planning is competitive with other approaches to
planning, such as heuristic based planning or SAT-based planning.

The flexibility and expressiveness of answer set programming provide a simple
way for answer set planning to exploit various forms of domain knowledge. To
the best of our knowledge, no heuristic planning system can take advantages of
all well-known types of domain knowledge, such as hierarchical structure,
temporal knowledge, and procedural knowledge, whist they can be easily
integrated into a single answer set planning system, as demonstrated by Son et
al. (2006).

It is important to note that the performance of answer set planning systems
depends heavily on the performance of the answer set solvers used in computing
the solutions. As such, it is expected that these answer set planing systems
can benefit from the advancements made by the ASP community. On the other
hand, this hand-off approach also gives rise to limitations of ASP-based
planning systems, such as scalability, heuristics, and ability to work with
numeric values. It is worth noting that heuristics can be specified for
guiding the answer set solver (e.g., as done by Gebser et al. (2013) in
planning) and there are considerable efforts in integrating answer set solvers
and constraint solvers (see Section 9). However, there exists no answer set
planning system that works with PDDL+, similarly to the system SMTPlan by
Cashmore et al. (2020) which employs SMT planning for hybrid systems described
in PDDL+. An ASP-based planning system for PDDL+ is proposed by Balduccini et
al. (2017). Yet, this system cannot deal with all features of PDDL+ as
SMTPlan. The issue of numeric constraints is also related to the next
challenge.

##### Probabilistic planning:

This research topic is the objective of intensive research within the
automated planning community. Similarly to other planning paradigms,
competitions among different probabilistic planning systems are organized
within ICAPS (International Conference on Automated Planning and Scheduling,
e.g., https://ipc2018-probabilistic.bitbucket.io/) and attract several
research groups from academia and industry.

Probabilistic planning is concerned with identifying an optimal policy for an
agent in a system specified by a _Markov decision problem (MDP)_ or a _Partial
observable MDP (POMDP)._ While algorithms for computing an optimal policy are
readily available (e.g., value iteration algorithm by Bellman (1957),
topological value iteration by Dai et al. (2011), ILAO* by Hansen and
Zilberstein (2001), LRTPD by Bonet and Geffner (2003), UCB by Kocsis and
Szepesvári (2006), as well as the work of Kaelbling et al. (1998); the
interested readers is also referred to the survey by Shani et al. (2013)),
scalability and efficiency remain significant issues in this research area.

Computing MDP and POMPD in logic programming will be a significant challenge
for ASP, due to the fact that answer set solvers are not developed to easily
operate with real numbers. In addition, the exponential number of states of a
MDP or POMPD require a representation language suitable for use with ASP. This
challenge can be addressed using probabilistic action languages, such as the
ones proposed by Baral et al. (2002) or by Wang and Lee (2019). On the other
hand, working with real numbers means that the _grounding-then-solving_ method
to compute answer sets is no longer adequate. First of all, the presence of
real numbers implies that the grounding process might not terminate. While
discretization can be used to alleviate the grounding problem, it could
increase the size of the grounded program significantly, creating problems
(e.g., lack of memory) for the solving process. A potential approach to
address these challenges is to integrate constraint solvers into answer set
solvers, creating _hybrid systems_ that can effectively deal with numeric
constraints. Research in this direction has been summarized in Section 9. The
recent implementation by Abels et al. (2019) demonstrates that answer set
programming based system can work effectively with a huge number of numeric
constraints.

##### Epistemic planning:

In recent years, epistemic multi-agent planning (EMP) has gained significant
interest within the planning community. Löwe et al. (2011) propose a general
epistemic planning framework. Complexity of EMP has been studied in the papers
Aucher and Bolander (2013); Bolander et al. (2015); Charrier et al. (2016).
Studies of EMP can be found in several papers Bolander and Andersen (2011);
Engesser et al. (2017); van der Hoek and Wooldridge (2002); Huang et al.
(2017); Löwe et al. (2011); Eijck (2004) and many planners have been developed
Burigana et al. (2020); Fabiano et al. (2020); Le et al. (2018); Muise et al.
(2015); Kominis and Geffner (2015, 2017); Wan et al. (2015). With the
exception of the planner developed by Burigana et al. (2020), which employs
answer set programming, the majority of the proposed systems are heuristic
search based planners. Some EMP planners, such as those proposed by Muise et
al. (2015), Kominis and Geffner (2015) and 2017, translate an EMP problem into
a classical planning problem and use classical planners to find solutions.

A multi-agent planning problem of this type is different from the planning
problems discussed in Section 8, in that it considers the knowledge and
beliefs of agents, and explores the use of actions that manipulate such
knowledge and beliefs. This is necessary for planning in non-collaborative and
competitive environments. The difficulty in this task lies in that the result
of the execution of an action (by an agent or a group of agents) will change
the state of the world and the state of knowledge and belief of other agents.
Inevitably, some agents may have false beliefs about the world. Semantically,
the transitions from the state of affair, that includes the state of the world
and the state of knowledge and beliefs of the agents, to another state of
affair could be modeled by transitions between Kripke structures (see, e.g.,
the books Fagin et al. (1995); Van Ditmarsch et al. (2007)). A Kripke
structure consists of a set of worlds and a set of binary accessibility
relations over the worlds. A practical challenge is related to the size of the
Kripke structures, in terms of the number of worlds—as this can double after
the execution of each action. Intuitively, this requires the ability to
generate new terms in the answer set solvers during resolution. Multi-shot
solvers (see, e.g., the paper Gebser et al. (2019)) could provide a good
platform for epistemic planning. Preliminary encouraging results on the use of
answer set programming in this context have been recently presented by
Burigana et al. (2020).

##### Explainable Planning (XAIP):

This is yet another problem that has only recently been investigated but
attracted considerable attention from the planning community, leading to the
organization of a a yearly workshop on XAIP associated with ICAPS (e.g.,
https://icaps20subpages.icaps-conference.org/workshops/xaip/). Several
questions for XAIP are discussed by Fox et al. (2017). In XAIP, a human
questions the planner (a robot’s planning system) about its proposed solution.
The focus is on explaining _why_ a planner makes a certain decision. For
example, why an action is included (or not included) in the plan? Why is the
proposed plan optimal? Why can a goal not be achieved? Why should (or should
not) the human consider replanning?

Chakraborti et al. (2017), for example, describe model reconciliation problem
(MRP) and propose methods for solving it. In a MRP, the human and the robot
have their own planning problems. The goal in both problems are the same, but
the action specifications and the initial states might be different. The robot
generates an optimal plan and informs the human of its plan. The human
declares that it is not an optimal plan according to their planning problem
specification. The robot, which is aware of the human’s problem specification,
needs to present to the human the reasons why its plan is optimal and what is
wrong in the human’s problem specification. Often, the answer is in the form
of a collection of actions, actions’ preconditions and effects, and literals
from the initial states that should be added to, or removed from, the human’s
problem specification so that the robot’s plan will be an optimal plan of the
updated specification. While explanations have been extensively investigated
by the logic programming community, explainable planning using answer set
programming has been investigated only recently by Nguyen et al. (2020).

### Acknowledgement

Tran Son and Enrico Pontelli have been partially supported by NSF grants
1914635, 1833630, 1757207, and 1812628. Tran Son’s and Marcello Balduccini’s
contribution was made possible in part through the help and support of NIST
via cooperative agreement 70NANB21H167. Torsten Schaub was supported by DFG
grant SCHA 550/15, Germany.

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