On the Relevance of Extracting Macro-operators with Non-adjacent

Actions: Does It Matter?

Sandra Castellanos-Paez

, Romain Rombourg and Philippe Lalanda

Univ. Grenoble Alpes, CNRS, Grenoble INP, France

Keywords:

Automated Planning, Macro-operators, Learning, Data Mining.

Abstract:

Understanding the role that plays the extraction phase on identifying potential macro candidates to augment a

domain is critical. In this paper, we present a method to analyse the link between extracting macro-operators

from non-adjacent actions and the correctness of (1) the frequency and (2) the number of occurrences per plan.

We carried out experiments using our method on ﬁve benchmark domains and three different planners. We

found that extracting macro-operators with only adjacent actions leads to important errors in macro-operator

frequency and occurrences per plan.

1 INTRODUCTION

In AI planning, macros can model system routines. A

macro consists of a sequence of actions that occurs

frequently in solution plans. Once learned, macros

can be re-injected directly into the planning domain.

Thus, the domain beneﬁts from the knowledge ex-

tracted from previous problem solving. The planning

system applies a macro in the same way as a primi-

tive action. However, macros allow jumping into the

search space by building deep and promising states

to reach a goal state. Learning macros from previ-

ously acquired knowledge has proven to be beneﬁcial

for improving a planner’s performance (Botea et al.,

2005b; Coles and Smith, 2007; Chrpa et al., 2014).

Macros have been widely studied to speed-up

planning processes (Newton and Levine, 2010; Dulac

et al., 2013). These approaches consist of two main

phases: extraction and selection. Extraction consists

in identifying, either by using previous knowledge

or by ad-hoc construction, sequences of actions that

could be potential candidates to augment the domain.

The selection phase must ﬁnd a trade-off between the

beneﬁt expected by adding macros and the additional

cost induced by the branching factor increase.

Literature about macros presents various tech-

niques to deal with the extraction phase, ranging from

simple combination of primitive actions and the use

of chunks of plans to the use of genetic learning algo-

rithms or statistical analyses based on n-grams.

https://orcid.org/0000-0002-6241-7974

Understanding the impact of the extraction phase

on the identiﬁcation of potential macro candidates to

augment a domain is critical. Indeed, through this

phase, the inﬁnite set of possible candidates to be-

come macros passes through a ﬁrst “ﬁlter”. The aim

is to make the set ﬁnite while keeping, among that

set, the best possible macros to add to the domain

in order for them to be identiﬁed and selected in the

next phase. We are aware that the selection is the

main concern when dealing with macro-operator re-

search. However, lots of selection methods use in-

formation that can be acquired during the extraction

phase (Botea et al., 2005a; Hofmann et al., 2017;

Castellanos-Paez, 2019). For example, in (Botea

et al., 2005a), they present the CA-ED method which

ranks a given macro based on the number of plans

where it occurs (hereafter called support) and the to-

tal number of times it occurs in each plan (hereafter

called number of occurrences). It seems then crucial

to assess the correctness of such information to obtain

better results.

When extracting macro-operators from non-

adjacent actions, the number of actions possibly ig-

nored between two considered actions is called the

gap. In this paper, we investigate if a ﬁnite gap

could ensure in most cases that all macro occurrences

are found. More precisely, we analyse the link be-

tween the use of a ﬁnite gap when extracting macro-

operators and the correctness of (1) the support and

(2) the number of occurrences per plan. By correct-

ness we mean how close are the computed macro-

operator characteristics (i.e. support and number of

Castellanos-Paez, S., Rombourg, R. and Lalanda, P.

On the Relevance of Extracting Macro-operators with Non-adjacent Actions: Does It Matter?.

DOI: 10.5220/0010305110211029

In Proceedings of the 13th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2021) - Volume 2, pages 1021-1029

ISBN: 978-989-758-484-8

1021

occurrences) to the ones computed when all macro-

operator occurrences are found. Besides, we anal-

yse the effect of the gap with plans obtained through

three different planners by comparing extraction re-

sults with an inﬁnite gap and a range of ﬁnite gaps.

The paper is structured as follows. First, we in-

troduce the concepts of classical planning, macro-

operators and some concepts borrowed from sequen-

tial pattern mining. Then, we present our method to

evaluate the impact of a ﬁnite gap on the extraction of

macro-operators. After, we present our results on ﬁve

benchmark domains. Finally, we provide a discussion

of the results and a conclusion.

2 BACKGROUND THEORY

We are interested in plan synthesis, a particular form

of planning which takes a description of the world

state, all its known actions and a goal (Ghallab et al.,

2004). As a result, we get an organised set of actions

whose execution makes it possible to solve the plan-

ning task. In this work, we address sequential plan-

ning in the STRIPS framework (Fikes and Nilsson,

1971).

A state is deﬁned as a set of predicates. A planning

task is composed of a planning domain and a planning

problem and the purpose of this task is to ﬁnd a plan to

solve this problem. A planning domain describes the

world through a set of predicates and a set of planning

operators. A planning problem describes the initial

state of the world and the goal state to be attained.

A planning operator is a triple o = (name(o), pre(o),

effects(o)) where its elements are deﬁned as follows:

• name(o) is in the form name(x

, ...,x

) where

, ..., x

are the object variable symbols that ap-

pear in o.

• pre(o) is the set of predicates involving the vari-

ables of o and that must be satisﬁed when o is in-

stantiated.

• effects(o) is the set of literals (i.e. atoms and nega-

tion of atoms) involving the variables of o to be

applied to a state when o is instantiated.

A grounded operator is an instance of a planning op-

erator (i.e. a lifted operator for which variables are

instantiated). A ground operator a is applicable in a

state s if and only if all predicates in the precondi-

tions of a belong to s. A state s

is reached from s if

a grounded operator can be applied. Finally, a (solu-

tion) plan π is an ordered sequence of grounded oper-

ators to reach a goal state s

from an initial state s

Macros are based on the idea of composing a se-

quence of primitive operators and viewing the se-

quence as a single operator. For our purposes, we

distinguish two related but different terms: grounded

macros and lifted macros. A grounded macro is re-

lated to a lifted macro as a ground operator is related

to a lifted operator. We use these terms according to

literature common terminology.

In the following, we deﬁne a set of concepts

(borrowed from Sequential Pattern Mining (Fournier-

Viger et al., 2017; Han et al., 2011)) necessary for the

understanding of this work.

A sequence database C is a set of pairs < sid, s >,

where sid is a sequence identiﬁer and s is a sequence.

A sequence S

= X

, X

, . . . , X

, where X

, . . . , X

are ground operators, is a sub-sequence of another

sequence S

= Y

, . . . ,Y

, where Y

, . . . ,Y

are

ground operators, if and only if there exists integers

1 ≤ e

< e

·· · < ek ≤ m such that X

= Y

, X

, . . . , X

= Y

The absolute support of a sequence S is the num-

ber of sequences S

, in the sequence database C,

where S is a sub-sequence of S

. The relative support

of a sequence S is the absolute support of S divided

by the total number of sequences in C. A frequent se-

quence is a sequence whose relative support satisﬁes

a given relative support threshold. Thereafter, we will

call this threshold minsup.

Besides, notice that a frequent sequence can oc-

cur in several other sequences but not necessarily in a

contiguous fashion, as the deﬁnition of sub-sequence

implies. So, we deﬁne a gap as the number of op-

erators allowed between two consecutive operators of

a sequence and it can take values in [0, ∞]. A sub-

sequence S = Y

, . . . ,Y

satisﬁes a gap constraint

g if ∀i ∈ [2, k], e

− e

i−1

≤ g + 1.

3 EVALUATING THE IMPACT OF

A FINITE GAP

In planning, each plan is composed of an ordered se-

quence of grounded operators which in turn are com-

posed of parameters (objects). In Figure 1, we give

an example of plan from the depots

domain, denoted

Mining lifted macro-operators (in the following,

macro-operators) from a set of plans requires an ap-

proach that ensures to ﬁnd the frequent sequences of

operators without a loss of information about their

characteristics. The often expected characteristics for

Domain description: https://www.cs.cmu.edu/afs/cs/

project/jair/pub/volume20/long03a-html/node38.html

Domain PDDL deﬁnition: https://www.cs.colostate.edu/

meps/aips02data/depots/Strips/Depots.pddl

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Figure 1: Example plan for depots domain.

each macro-operator m are : the support, i.e. the num-

ber of plans containing at least one occurrence of m,

and the number of occurrences of m in each plan. The

former represents a general view of the frequency of

m. The latter represents a more detailed view of the

use of m through each plan.

In (Castellanos-Paez et al., 2020), we highlighted

the capital role of the gap in the extraction of macro-

operators. In this paper, we analyse the loss of in-

formation when a ﬁnite gap is used to extract macro-

operators. More precisely, we analyse the impact on

the correctness of the characteristics computed for

these macros (support, number of occurrences). Also,

we investigate if there is a different impact on correct-

ness when extracting macro-operators from optimal

or sub-optimal plans.

By correctness we mean how close are the com-

puted macro-operator characteristics (i.e. support and

number of occurrences) to the ones computed when

all macro-operator occurrences are found.

3.1 Overview of the Analysis Procedure

Our analysis focused on evaluating the impact of

a ﬁnite gap on the correctness of the support and

the number of occurrences. First, we extracted a

reference set of frequent macro-operators, M

re f

and

their characteristics. This reference set is guaran-

teed to be correct since we used the ERA algorithm

(Castellanos-Paez et al., 2020) with a minsup of 0.85,

an inﬁnite maximum length and an inﬁnite gap. By

correct, we mean that all possible occurrences were

found, thus, the computed support and number of oc-

currences are exact. Indeed, the ERA algorithm pro-

duces exhaustive results with the aforementioned gap

and length parameters.

Then, we extracted different sets of macro-

operators by varying the gap parameter. Intuitively,

using a ﬁnite gap implies that some occurrences may

be missed by the algorithm. Therefore, a minsup of

0 was used to avoid losing information on the charac-

teristics of macro-operators in the reference set.

Finally, we compared the results of the reference

set (inﬁnite gap) with the results of the sets obtained

by varying the gap parameter. The comparisons were

made by using three different metrics.

Details on the extraction algorithm and the metrics

deﬁnition are given in the following sections.

3.2 Extraction Algorithm: ERA

The ERA algorithm mines all macro-operators (re-

gardless of their length or up to a maximum length)

satisfying a frequency threshold and under a gap con-

straint from a set of solution plans. Additionally, for

each macro-operator m, this algorithm yields the fol-

lowing characteristics:

• support [integer]: the number of plans contain-

ing at least one occurrence of m.

• sequence ids [list]: the plan identiﬁers where m

appears.

• number of occurrences [list]: the number of oc-

currences of m in each plan.

The ERA algorithm checks, for each plan, every sub-

sequence satisfying the gap constraint. If an inﬁnite

gap is used, the algorithm is then guaranteed to ﬁnd all

macro-operators, satisfying the frequency threshold,

and all their occurrences. Also, macro-operator oc-

currences (even non-adjacent occurrences) registered

by ERA are valid. Indeed, to be registered, an occur-

rence must be composed of grounded operators that

can be moved contiguously in the analysed plan with-

out impacting its ﬁnal state or impeding its execution.

Finally, if the macro-operator occurrences satisfy the

frequency threshold, a macro-operator is built as a

contiguous sequence of lifted operators.

To illustrate how ERA registers occurrences,

let us consider the plan π

in Figure 1 and

a gap parameter of 1. The set of all sub-

sequences of π

satisfying the gap constraint of 1

is {(a

, a

), (a

, a

), (a

, a

), (a

, a

), (a

, a

)}. If we

focus on the sub-sequence (a

, a

), ERA will reg-

ister it as a valid occurrence of the macro-operator

lift-load

. Indeed, the grounded operators com-

posing that sub-sequence can be moved contigu-

ously in the plan π

yielding the modiﬁed plan π

, a

i, where π

is executable and leads to the

same ﬁnal state as π

. Notice that gaps are only con-

sidered during the generation of sub-sequences to be

analysed but not when building the macro-operator.

For more details about the mining and the build-

ing of macro-operators in the ERA algorithm, see

(Castellanos-Paez et al., 2020).

A hoist lifting a crate and loading it into a truck.

On the Relevance of Extracting Macro-operators with Non-adjacent Actions: Does It Matter?

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3.3 Metrics Deﬁnition

To compare the results of the reference set (inﬁnite

gap) with the results of the sets obtained by varying

the gap parameter, we deﬁned the following metrics:

The mean support error, Err

supp

, deﬁned in (1):

Err

sup

(g) =

re f

∑

m∈M

re f

(s(m;∞) − s(m; g)) (1)

represents the mean difference across each macro-

operator m in the reference set M

re f

between the terms

s(m;∞) and s(m;g). The former represents the sup-

port obtained with an inﬁnite gap for a macro m. The

latter represents the supports obtained with a ﬁnite

gap g for a macro m. The term |M

re f

| is the number

of macros in the set M

re f

The mean relative occurrence error, Err

occ

, de-

ﬁned in (2):

Err

occ

(g) =

re f

∑

m∈M

re f

o(m;∞) − o(m; g)

o(m;∞)

(2)

represents the mean relative difference across each

macro-operator m in the reference set M

re f

. In (2),

o(m;∞) represents the total number of occurrences

obtained with an inﬁnite gap, and o(m;g) represents

the total number of occurrences obtained with an ﬁ-

nite gap g.

Finally, the total relative occurrence error, Err

tot

deﬁned in (3):

Err

tot

(g) =

∑

m∈M

re f

(o(m;∞) − o(m; g))

∑

m∈M

re f

o(m;∞)

(3)

represents the fraction of occurrences missed when

using a ﬁnite gap g.

4 RESULTS

In the following, we show the impact of a ﬁnite gap on

the characteristics of the extracted macro-operators.

For this purpose, we used ﬁve benchmarks domains,

two sub-optimal planners and one optimal planner.

Keep in mind that this study does not address planner

performance or macro quality. The planner choices

were then solely based on the optimality or sub-

optimality of the plans produced.

4.1 Experimental Setup

To the best of our knowledge, no open plan database is

available. Thus, for each benchmark domain, we gen-

erated a set of 35 distinct problem instances using the

generators

from the International Planning Competi-

tion. Then, we solved the problems by using each of

the three planners (Helmert, 2006; Richter and West-

phal, 2014; Helmert and Domshlak, 2011). As a re-

sult, we obtained a total of ﬁfteen sets of solution

plans (each with 35 plans) divided as follows: ten sets

of sub-optimal plans (5 domains times 2 planners) and

ﬁve sets of optimal plans (5 domains times one plan-

ner). Indeed, we also investigated the behaviour of the

loss of information when extracting macro-operators

from optimal or sub-optimal plans.

The analysis procedure was carried out on each

one of the ﬁfteen sets of solution plans and the impact

of a ﬁnite gap was evaluated for gap values ranging

from 0 to 15. The average plan length was about 30

for the sets of sub-optimal plans and was about 25

for the sets of optimal plans. Experiments were per-

formed on dedicated machine with an Intel Core i7-

4710MQ quad-core CPU clocked at 2.5GHz and with

8GB of RAM. Table 1 presents the different parame-

ters tested.

Table 1: Parameters of the gap evaluation experiment.

Parameter Values tested

Gap 0 - 15

Planner

FastDownward (A* and FF heuristic)

LAMA

LM-Cut

Domain

Barman, Blockworld,

Depots, Rover, Satellite

4.2 Results of the Impact of a Finite

Gap

In Figures 2, 3 and 4, for each domain and for each

planner, we present respectively: Err

sup

, Err

occ

and

Err

tot

, as a function of the gap. As expected, we ob-

served that these errors decreased consistently with

the gap for all domains and planners.

We observed across almost all domains that the

errors were less important when using plans ob-

tained with the LM-Cut planner. However, between

FD(A*+FF) and LAMA, there was not a planner that

presented consistently higher errors. Compared to

the other domains, blocksworld showed signiﬁcantly

lower errors.

Additionally, we present in Table 2, for each do-

main and planner, the errors obtained with a gap value

of 0. And, in Table 3 we show the minimal gap value

required, for all domains and errors, to have an error

value below 5% for all planners.

https://bitbucket.org/planning-tools/pddl-generators

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1024

(a) Barman

(b) Blocksworld

(e) Satellite

Figure 2: Mean support error. Errors are given in percentage of the plan set.

On the Relevance of Extracting Macro-operators with Non-adjacent Actions: Does It Matter?

1025

(a) Barman

(b) Blocksworld

(e) Satellite

Figure 3: Mean relative occurrence error.

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(a) Barman

(b) Blocksworld

(d) Rover

(e) Satellite

Figure 4: Total relative occurrence error.

On the Relevance of Extracting Macro-operators with Non-adjacent Actions: Does It Matter?

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Table 2: Errors for each domain and planner when using a gap value of 0 (extraction made only from contiguous actions).

Err

sup

(%) Err

occ

(%) Err

tot

(%)

FD LAMA LM-Cut FD LAMA LM-Cut FD LAMA LM-Cut

Barman 7.9 56.2 17.8 21.4 63.2 19.3 22.9 64.1 18.2

Blocksworld 5 3.8 1.1 11.5 11 2.1 6.5 7.3 1.1

Depots 25.4 21.5 26.9 49.5 44.9 47.7 47.7 42 42.2

Rover 43.4 21.9 5.7 65.5 46.4 35 64.1 47.3 35

Satellite 27.1 38.9 27.8 35 45.6 35.4 21 27.3 21.7

Table 3: Minimal gap value required to have an error value below 5% for all planners. ’-’ indicates that the error was still

above 5% for a gap of 15.

Err

sup

Err

occ

Err

tot

FD LAMA LM-Cut FD LAMA LM-Cut FD LAMA LM-Cut

Barman 1 5 7 2 10 7 2 10 7

Blocksworld 0 0 0 2 4 0 2 2 0

Depots 4 4 3 11 8 6 10 8 5

Rover 8 1 1 13 6 3 13 5 3

Satellite 4 5 11 - - 12 9 14 9

5 DISCUSSION AND RELATED

WORKS

Our results clearly show that extracting macro-

operators by only considering adjacent groups of ac-

tions (zero gap) leads to high errors (often above

50%!) in terms of support and occurrences for every

domain and planner. Besides, errors are similar for

optimal and sub-optimal plans. Intuitively, one could

think that high gap values would not impact signiﬁ-

cantly the extraction results since two distant opera-

tors should be less related than two close operators.

Indeed, that would be true if the planning system was

focused on accomplishing sequentially disconnected

groups of sub-goals, e.g. preparing and serving a spe-

ciﬁc cocktail in barman. However, every plan (even

optimal) can be reordered in hundreds or even thou-

sands of different ways from which only a few corre-

spond to a conﬁguration where all disconnected goals

are accomplished sequentially.

An inﬁnite gap is required to guarantee correct-

ness of the support and the number of occurrences.

However, we showed that a ﬁnite and relatively small

gap value (compared to the plan length) yields accept-

able results. We also observed higher gap require-

ments for the satellite and depots domains. It

could be explained by the fact that problems from

these domains often consist in several disconnected

sub-goals e.g. take a speciﬁc picture with any satel-

lite or bring a given crate to a given truck.

One could point out that using non-adjacent ac-

tions to generate macro-operators leads to the extrac-

tion of more macros therefore possibly making the se-

lection problem more difﬁcult. However, should we

consider that with gap, more useless macros are ex-

tracted? Or, should we rather consider that without

gap, the most useful macros could be missed? We be-

lieve in the second proposition, the extraction should

never take the risk to rule out possible useful macros.

There are a number of works handling non-

adjacent operators to build macros. Botea et al.

(Botea et al., 2005b) extract macros from solution

graphs of training problems. They enumerate and se-

lect sub-graphs from the solution graphs and build one

macro for each selected sub-graph. They introduce a

k parameter as the maximal number of operators that

can be skipped between the ﬁrst and the last element

of the sub-graph (we can roughly see a sub-graph as

a sub-sequence). Their handling of the gap does not

allow an exhaustive search and can then miss many

macro-operator occurrences. Also, they set a hard

limit on the upper bound of the maximal length of the

extracted macro-operators. Chrpa et al. (Chrpa et al.,

2014) extract macros by iteratively combining opera-

tors (even non-adjacent) in plans that share some pa-

rameters while keeping the plan valid. However, their

technique is not guaranteed to ﬁnd all macro-operator

occurrences. In Castellanos-Paez et al. (Castellanos-

Paez et al., 2020), we present a pattern mining in-

spired algorithm to mine lifted macro-operators di-

rectly from a set of plans. Besides, this algorithm can

detect the occurrence of a macro-operator even if the

actions composing it are not adjacent.

To the best of our knowledge, our work is the only one

evaluating the impact of extracting macro-operators

from a set of plans by considering non-adjacent oper-

ators.

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6 CONCLUSIONS

Macro-operators for automated planning are of great

importance to improve performance by using past ex-

periences. Understanding the role that plays the ex-

traction phase on identifying potential macro candi-

dates to augment a domain is critical. In this paper, we

did not address planner performance or macro qual-

ity, however, we answered the question: ”Does con-

sidering non-adjacent actions when extracting macro-

operators matter?”. The answer is yes, and a rela-

tively small gap of 5 to 10 actions can go a long way

in reducing the error in support and number of occur-

rences below ﬁve percent. The reader can clearly take

away from this study that when dealing with macro-

operators, we should not only look at adjacent oper-

ators. For future work, we believe that the consid-

eration of these results could be advantageous when

designing new macro-learning methods.

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