From: AAAI Technical Report SS-93-03. Compilation copyright © 1993, AAAI (www.aaai.org). All rights reserved.
ON THE UTILITY
REDUNDANCY
OF SYSTEMATICITY:
AND COMMITMENT
IN
UNDERSTANDING
TRADEOFFS
BETWEEN
PARTIAL-ORDERING
PLANNING
Subbarao Kambhampati*
Departmentof ComputerScience and Engineering, Arizona State University
Tempe, AZ 85287-5406, Emaih rao@asuvax.asu.edu
1.
INTRODUCTION
Althoughthe idea of generatingplans throughnonlinear or partially
ordered partially instantiated (POPI) planningI has been around
for almost twenty years, it is only recently that the search space
characteristics of POPIplanners have received particular attention. A big thrust in this workhas been on reducing the redundancy
in the search space of POPIplanners. This was largely motivated
by the belief that redundancyreduction will lead to improvements
in planning efficiency[10, 8]. Oneapproach towards redundancy
elimination, that turned out to be particularly influential (as evidencedbyseveral closely related extensions[8, 18, 16, 2, 17]), was
that of McAllester’s[8]. McAllestershowedthat it is possible to
design a POPIplanner that is systematic in the strong sense that
it never visits two equivalent plans or plans having overlapping
linearizations. Suchsystematic planners were claimed to be more
efficient than planners that admit redundancyin their search space.
Whilesearch space redundancyis an important factor affecting
the efficiency of a P O PI planner, another (perhaps) equally important one is the level of commitment
in the planner. After all, avoiding premature commitmentwas one of the primary motivations
for POPIplanning. There is a tradeoff between the redundancy
elimination and least commitment,in that often the redundancyis
eliminated at the expense of increased commitmentin the planner. For example,McAllester’s planner achieves systematicity by
keepingtrack of the causal structures of the plans generatedduring
search, and ensuring that each branch of the search space commits
to and protects mutuallyexclusive causal structures for the partial
plans. Wewill see that such protection amountsto a strong formof
premature commitment,which increases the amountof backtracking as well as the solution depth, and can have an adverse effect on
the performanceof the planner.
In this paper we shall argue that the performance of a POPI
planner dependsmoreclosely on the wayit deals with the tradeoff
between redundancy and commitment,than with the systematicity of its search. Wewill start with a rational reconstruction of
the motivations behind systematicity in POPIplanning and show
that systematicity is just one extremesolution for the tradeoff between redundancyand commitment.Wewill show that there are a
spectrumof solutions to this tradeoff, and identify the dimensions
along whichthey vary. Wewill explore the relative utility of the
different solutions through a comparativestudy of seven planners
that fall at different points on the spectrum.Ourstudies showthat
*This research is supportedin part by NationalScienceFoundation
undergrant ]RI-9210997.
WethankSteveMinion,AustinTare, MadcPeot,
Will Harvey,DavidMcAllesterand DanWeldfor helpful comments.
1Throughout
this paper, we shall use the term POPIplanningrather
than nonlinearplanning,as the formeravoids confusionwith iinearity
assumption.
67
both SNLP- which concentrates on redundancy elimination, and
TWEAK
[1 ] - which concentrates on least commitment,could be
very inefficient. They also showthat the planners that strike a
more judicious balance between redundancy and commitmentcan
outperform both less committed,and more systematic planners.
Organization: The paper is organized as follows: The rest of this
section presents some terminology for POPI planning. Section
2. reviews the motivations behind systematic nonlinear planning,
and brings out the tradeoff between redundancyand commitment.
Section 3. describes seven different planners, whichbetweenthem
take a spectrum of approaches for dealing with the redundancycommitment
tradeoff, and analyzes their effectiveness. Section 4.
supports the analysis through an empirical comparisonof the seven
planners. Section 5. summarizesthe contributions of this paper
and discusses relations to past work.
Terminology: Many POPI planners use causal links (also
knownas protection intervals, we shall use these namesinterchangeably) to systematize the planning process (see below).
say that ‘5 ~ w is a generalized (multi-contributor) causal link
P if the step w and the set of steps ‘5 (called the contributor set)
belong to ~’, such that all the steps in S are ordered to precede
w and each of them is capable of giving the precondition p to w.
Given a step v, and causal link S ~ w of the plan P such that
Vs E ,5 <>(s -.4 v -4 w), v is called a negative threat for the causal
link if v deletes p. v is called a positive threat for the causal link
if v adds p. A plan 7’ is provablycorrect if there is a causal link
supporting each of the prerequisites of the plan, and none of the
causal links haveany negative threats.2 Theset of causal links of a
(partial) plan is also referred to as its causal or establishmentstruc¯
.
ture. Acausallink,5 v wof a plan ~ Is¯ said
to be exhaustive(c.f.
[4]), if for every groundlinearization ~’ of 7~, some step s’ E,5
will be the last step precedingw that gives p in that linearization. It
is easy to see that a correct plan with exhaustive causal links, none
of the causal links will havepositive or negative threats. If every
causal link of a plan has a singletoncontributor set (i.e., 1,5[ = 1),
then the plan is said to havesingle contributorcausal sla’ucture.
2.
A RATIONAL RECONSTRUCTION OF
MOTIVATIONS FOR SYSTEMATICITY
POPIplanning was motivated by the observation that total ordering planners waste search effort by committingprematurely to the
orderings and bindings of the steps. POPIplanners avoid this
premature commitmentby keeping plans as partially ordered partiaily instantiated sequencesof actions, and searching in the space
of these partial plans. Chapman’sTWEAK
truth criterion specifies necessary and sufficient conditions for checkingthe truth of a
2Notethat the converseof this propositionis not tree - there maybe
correctplansthat donot satisfy these conditions
propositionin a P O P I plan, whichin turn can be used to checkthe
correctness of a plan. A plan is said to be complete(or correct)
everypreconditionof every action in the plan is necessarily true by
MTC.MTC
can be used to enumerate all legal ways of refining a
plan to makea proposition(that is not necessarilytrue) true. It thus
implicitly defines the state space of POPIplans. TWEAK,
which
searches in this space, is completein the sense that it will eventually generate every legal POPIplan as a result of somesequence
of refinementsof the null plan.
A major problem with TWEAK
is that its search space contains
two significant forms of redundancy:
1. Multiple paths to a (complete/incomplete) plan: TWEAK’s
search space is a graphrather than a tree - that is, a plan (complete
or incomplete) in the search space can be reached through more
than one set of refinements of the null plan?
2. (Complete/incomplete) plans with overlapping linearizations in the search space: TWEAK’s
search space contains complete/incomplete plans that have overlappinglinearizations.
Thefirst type of redundancyis typical of any graph search problems, and is normallyhandled by keeping a closed list and checking
for duplicates. Unfortunately, however,this approachtoms out to
be infeasible for POPIplanning, both because of the excessive
storage requirements, and because of the high cost of duplicate
checking (checking for equivalence of two plans involves computing and comparingthe transitive closure of the ordering and
binding relations of both the plans, and reduces to graph isomorphism problem).
Theinfeasibility of maintaininga closed list forces POPIplanners to do a tree search in a graphical search space, leading to a
significant amountof repetition/redundancyin the search [6]. One
way of avoiding this redundancyis to equip the planner with a
generator that automaticallytraverses the search graph via a spann/ng tree. To see howthis can be done, we first note that part of
the duplication comesfrom the in-built redundancyin the planning
decisions faced by TWEAK.
For example, it can be shownthat any
planner that backtracks on operator choices, ordering choices and
binding choices can safely ignore goal ordering choices without
losing completeness[15,8].
Oncesuch redundant choice points are "cut" from the planner’s
search space, the planner can generate all partial plans in its search
space in a systematic fashion with the help of protection briervals (or causal links). Protection intervals record the producerconsumerdependencies that have been established by the planner.
They allow the planner to keep track of its ownprogress, and to
avoid unnecessary undoing of previous establishments or redoing
of previously failed establishment structures. The idea of protection intervals has been around from Sussman’s HACKER
[19] and
Waldinger’s regression planner [21], but was first used in POPI
planning by NONLIN
[20]. All these planners organize their search
in such a waythat in every search branch, the causal links are protected from negative threats. Whensuch protection cannot be
enforced by adding ordering/binding constraints to the plan in a
given branch, the planner abandonsthat branch (i.e., backtracks).
Comingto the second type of redundancymentionedabove, althoughtraversing the search space via a spanningtree ensures that
no partial plan is visited twice, it will still not precludevisiting
plans having overlapping linearizations. Since the primary motivation for POPIplanning is to speedup TOplanning, rather than
to find the least constrained plans, allowingplans with overlapping
linearizations into the search space of POPIplans will be tan3Noticethat"cycling"between
planstates will not occursince the search
operatorsare all refinementoperatorsandgoingbackto a less constrained
plan froma moreconstrainedplan requiresretraction.
68
tamount to looking at some TOplans more than once during the
search process. To avoid this type of redundancy, the planner
needsto restrict its attention to a subset of the plans in its search
space, such that every complete TOplan is a linearization of one
and only one POPIplan in the search space.
In [8] McAllestershowedthat it is possible to write a planner
that is systematic in the strong sense that it avoids both types
of redundancy discussed above.4 His planner (implemented as
SNLPby Barret et. al. [18]) achieves systematicity by using the
samecausal link/protection interval based organization of search
discussed earlier, but with one extension: Unlike planners such
as Nonlinwhichprotect the establishment structures only against
possible deleters, SNIPalso protects the establishment structures
against possible adders. In the terminology of Section 1., SNLP
maintainsexhaustive causal structures and thus avoids both positive
and negative threats for every causal link. By doing this, SNLP
ensures that at every decision point, the possible completions of
a partial plan are split into mutuallyexclusive sets with different
causal stnlctures, thereby achievingsystematicity.
2.1. The Tradeoffbetween Redundancy and Commitment
In the foregoing we reviewed the motivations and approaches for
cutting downredundancy in the search space of POPI planners.
Whileredundancyin the search space is an important factor affecting the performanceof a POPI planner, another (perhaps equally)
important factor is the level of commilment
in the planner. Indeed,
as mentionedearlier, the original impetus for POPIplanning was
the realization that total ordering planners such as STRIPS[13]
committoo soon to specific action orderings. It turns out that
all the methods for reducing redundancy that we have discussed
above involve a corresponding increase in commitment.For example, although traversing search space via a spanningtree eliminates
redundancy, it may not always necessarily improve performance.
For one thing, the particular spanningtree that is selected maynot
be the best wayof traversing the search space to find the solution?
Secondly,using +re threats increases the depth of search- a plan
that is complete and correct
according to McNonlinand TWEAK
4Thereseemsto be someconfusionregardingthe exact definitionof systematicityusedin McAllester’s
paper[8]. McAllesteroriginallymotivates
systematicnonlinearplanningby sayingthat it providesa wayof speeding
up total orderingplanningby searchingin the spaceof equivalence
classes.
Underthis motivation,systematicitywouldmeanthat the plannerwill not
generateanypartial or completeplanswithoverlappinglinearizations(as
describedby the secondtype of redundancy
above).Indeed,as discussed
in [11], McAllester’s
plannerdoeshavethis propertyas longas threat detection is foldedinto childrengeneration.However,
McAllesteralso seems
to implya different, weaker,notionof systematicityin his sketchyproof
of systematicity.According
to this notion, a systematicplannerneedonly
guaranteethat no completeplans withoverlappinglinearizations is produeed.Notsurprisingly,this definition will holdirrespectiveof whether
the threat resolutionis foldedinto childrengeneration,or deferredarbitrarily (see Harveyet. al. for a proof of this). However,
underthis notion,
the search space of the POPIplanner is no longer boundedfrom above
by the searchspaceof corresponding
total orderingplanner. Ouroriginal
experiments
dealt withan implementation
of SNLP
that resolvesthreats as
andwhentheyare detected(as is describedin McAllester’s
originalpaper),
but wealso repeatedthe experiments
with a versionof SNLP
that postpones
thethreats.
SForexample,supposeweare trying to find a plan for achievingthe
goals{gl, g2, ¯ "", ft,}. Suppose
further that making
g,~ tree wouldalso
makegl "" "g,-I true. Now,a TWEAK
based planner doing breadthfirst
searchwouldfind the solutionat level one,ff it looksat all goalorders;and
mayhaveto goall the wayto level n of the searchspaceff it doesn’tlookat
all goal orders andhappensto select andcommit
to the wronggoal order.
Pcpi
SNLP
NONLIN
MP
MP-I
UA
NOTWEAK TWEAK
Figure 1: A speemunof solutions to the tradeoff between redundancy and commitment in POPI planning
maystill needto be refined further to makeit completeand correct
6for SNLP.This inturn increases the cost of planning
Moreimportantly,the use of protection intervals and causal links,
while systematizing the search, also results in increased commitmentto particular establishment structures. Increased commitment
leads to higher backtrackingas well as increased depth of the solution, whichin turn adversely affect the performance.In particular,
SNLP’spractice of committing to and protecting establishment
slructures could cause cosily backtracking whenthe initial commitment turns out to be wrong. Similarly, SNLP’smaintenanceof
exhaustive causal links increases the effective solution depth since
it has to consider positive threats in addition to negativethreats.
The tradeoff between redundancyand commitmentsuggests that
neither SNLPwhich eliminates redundancywithout worrying about
commitment, nor TWEAK
which guarantees least commitment
without worryingabout redundancy,is guaranteed to be efficient
all the time. In the next two sections, we will expore this tradeoff
further by comparinga specmunof solutions that fall in the middle
of SNLP and TWEAK.
3.
A SPECTRUM OF SOLUTIONS FOR THE TRADEOFF
BETWEEN REDUNDANCY AND COMMITMENT
The SNLPand TWEAK
planners, discussed in the previous seetion, are in fact just twoextremesin the spectrumof solutions to the
tradeoff between redundancy and commitment. Figure 1 shows
other soundand completeP O PI planners that fall in the middleof
these two extremes.In this section, we will characterize the dimensions along whichthese planners vary, with particular emphasison
the waythey deal with redundancy-commitment
tradeoff.
SNLP, McNonlin, MPand MP-I, the four planners on the
bottomleft in Figure 1 use causal links or protection intervals to
organize their search. They introduce causal links to support each
open condition of the plan, and ensure that none of the causal links
are threatened. In each ease, planning is considered completewhen
each prerequisite in the plan is supportedby a causal link, and all
the causal links are safe (i.e, unthreatened). Theplanners differ
only in their definition andtreatment of threatened causal links.
Twoof them, SNLPand McNonlin,use the traditional single
contributor causal links to guide planning. SNLPconsiders a
causal link {s} ~ w to be threatened if there exists a step v of the
plan whichis either a positive or a negative threat (see Section 1)
6In fact, it is trivial to comeup withdomains
whereSNLP’s
obsession
withpositivethreats slowsit down
drastically.Considera variantof blocks
worldwhereeveryaction has additional effects used(?x)for each block
?x that is usedby the action. Suppose
weuse the usual blocksworldtest
problems,exceptthat weaugmentthe goal state with a bunchof used(?x)
assertions. This trivial syntactic changewill leave NONI..IN
andTWEAK
unaffectedwhiledrastically slowingdownSNLP.
This is becauseused(?x)
will be madetrue bymanypossibleactionsin the plan andthus causallinks
protecting used(?x)will have manypositive threats, whichslow down
SNIP,but have no effect on McNonlinand TWEAK
69
7 considers a causal link
of the causal link. In contrast McNonlin
to be threatened only whenthere exists a step v that is a negative
threat. In either case, the link is madesafe by ordering v to come
either before s or after w.
The other two planners, MPand MP-I, use multi-contributor
causal links [4] to guide planning. MPconsiders a causal link
S ~ w to be threatened if there exists a step v in the plan whichis
either a positive threat or a negative threat to the causal link. The
rink is madesafe by ordering v to comeeither after w or before
somestep s E S. In addition, if v is adding the condition p, we
can also make the link safe by ordering v to come before w and
merging v with S. MP-Iconsiders a causal link to be threatened
onlywhenthere exists a step v that is a negative threat of the causal
link. The causal link is madesafe by either promotingv to come
after w or demoting it to comebefore somes E S. Additionally
if there exists a step s’ such that it followsv and adds p, then the
canal link can also be madesafe by ordering v~ to comebefore w
and mergings’ with S.
Clearly, the four planners balance the tradeoff betweenredundancy and commitmentin different ways. In particular, SNLPand
McNonlinreduce redundancyin the search space by increasing the
commitmentto specific contributors, while MPand MP-I reduce
commitment
to individual contributors at the expense of increased
redundancy. MPand SNLPmaintain exhaustive causal structures
whichallow themto split the possible partial plans into sets of
plans with mutually exclusive establishment structures, there by
controlling the redundancyin the search space. However,the same
exhaustivenessalso increases effective solution depth, since unlike
McNordinand MP-I, MPand SNLPhave to deal with positive as
well as negativethreats to the causal links.
UA, TWEAK
and NGTWEAK,
the three planners on the bottom
right in Figure 1, do not use causal links (or protection intervals)
their search. Thus, they completely eliminate the commitment
to
contributors. Of these three, TWEAK
closely follows the idea of
inverting Chapman’s
MTC
(with the minordifference that it doesn’t
use external white-knightsfor declobbering). It is the only planner
among the seven that backtracks on goal orderings. NGTWEAK
reduces some of the redundancy in the TWEAK
search space by
avoiding backtracking on goal ordering decisions (similar to the
implementation described in [22]). UA[10, 11], reduces the redundancyfurther by maintainingpartial plans that are unambiguous
in the sense that each prerequisite in the plan is either necessarily
true or necessarily false. UAachieves this by ordering every newly
introduced step with respect to all possibly interacting steps. This
policy allows UAto eliminate plans with overlappinglinearizations
from its search space, but does not ensure systematicity [10].
4.
EMPIRICAL ANALYSIS OF THE TRADEOFF
BETWEEN REDUNDANCY AND COMMITMENT
To understand howthe various solutions to the tradcoff between
redundancy and commitmentaffect the planning efficiency, we
performedempirical study on the performanceof the seven planners
discussed in the previous section. This section describes the study
and analyzesthe results.
4.1. Experimental Setup
Our test domains included classical toy-worlds such as blocks
world, as well as the synthetic domainsused in Weldet. al.’s
work[18]. In this paper, we will concentrate on the results from
Weldet. al.’s synthetic domainsand our variants of them, as they
provide for a more controlled testing of the tradeoffs. Weldet.
7 McNonlin
is a considerablesimplificationof Tate’soriginalimplemen
tation of NONLIN[20],
whichwasa hierarchicalnonlinearplanner.
al.’s original domains include ART-MD
and ART-1D
(also called
DInS1, D1S1 respectively), which are designed to contain easily serializable and laboriously serializable sub-goals respectively.
The domainsare defined as:
ART-MD
Ai prec : Ii add : Gi del : {I, Ij < i}
ART-1DAi precond : h add : Gi del : 1i-1
To these, we also added our own variants: ART-MD-RD
and
ART-1D-RD,
which introduce preconditions achieved and deleted
by multiple operators. The variants are produced by makingevery
even numberedaction require an additional precondition he, delete
that precondition and add an additional postcondition hfi s The
odd numberedactions similarly require and delete hf and add he.
The specification of ART-MD-RD
is shown below. ART-1D-RD
is
produced similarly from ART-1D
(Note: although we experimented with ART-MD-NS
domainand its variants, space limitations
preclude us fromdiscussing the results).
even/ Aiprec:
h,headd:
Gi,hf dei: {Ijlj<i}U{he
}
oddi Ai prec : ii,hf add : Gi,he del : {l~lj < i} u {hf}
In each of the domains, we comparedall seven planners, over
solvable problemswith 1 to 8 goals from the set { GI... Gs }. Since
the effect of commitment
to contributors dependslargely upon the
order in whichthe various goals and subgoals are addressedbythe
planner, wetested with two types of goal order strategies: strategy
L whichis a LIFOstrategy wherea goal andall its reeursive subgoals
are addressed before the next higher level goal is addressed; and
strategy GbyG,whichis a FIFOstrategy whereall the top level goals
are addressed before their subgoals are considered by the planner
(strategy L correspondsto a depth-first traversal of the goal-subgoal
tree, while strategy GbyGcorrespondsto a breadth-first tmversal).
4.2. Experimental Results and Discussion
Wewill describe the experiments in two stages. First, we will
concentrate on the comparisonbetweenthe four causal link based
planners. This will be followed by a comparison of all seven
planners.
4.2.1. Comparisonbetween the Causal Link Based Planners
Westarted by comparing the cpu time (in m.sec, on a SUN
SPARe-f1),taken by the four causal link based planners for each of
the goal ordering strategies in ART-MD
and ART-1D
respectively.
Wefound that there is no appreciable difference in time taken by
the planners for solving problemsin these domains(plots omitted
due to space limitations).
Next, we studied the performanceof these planners in ART-MDRD, and ART-1D-RD
domains. Westarted by comparing the sizes
of the overall search spaces of all the planners. Theleftmost plot
in Figure 2 comparesthe total search space sizes of these four
planners for one of the goal orderings. 9 They showthat the four
planners have varying amountsof redundancyin the search space,
with SNLPhaving the smallest search space, and MP-I having
the largest. If the search space size were to be sole indicator
of performance, we would expect that SNLPwould perform best,
followed by McNonlin, MPand MP-I.
Theplots in Figure 2 comparethe performanceof the planners in
ART-MD-RD
and ART-1D-RDdomains which contain the easily
achieved and deleted conditions hfand he. (Fhe missing data points
on a plot correspondto the problemsthat couldn’t be solved before
the lime bound,whichfor us wasthe time it took for the lisp to run
She and hf are supposed to be mnemonicsfor handempty and
handfull conditions in the traditional blockswodddomain[13], which
are achievedor deletedby manyof the actions in the domain.
9 Thesize of the overallsearchspaceis foundbysetting the rennin
a tion
conditionfor eachplannerto be uniformlyfalse, thus forcingthe planners
to visit everynodemthe searchspacebeforegivingup.
70
out of memoryand fail). A comparison of the numberof partial
plans expandedby each of the planners yields very similar patterns.
Similar performance was also observed with two other weighted
heuristics.
Analysts: The behavior of the four causal link based planners in
our experimentscan be explained in terms of the tradeoff between
redundancy and commitmentthat we discussed earlier. In particular, the near identical performanceof the four planners in the
original ART-MD
and ART-1Ddomains can be explained by the
fact that in these domainseach precondition is ultimately provided
by a single action in the plan, and there is no penalty for committing
prematurely to that action as the contributor. Premature commitment does hurt performance In ART-x-RD
domains, and we find
that:
MPand MP-I perform better than SNLPand McNonlin in
the case of the LIFOgoal ordering strategy L (see plots in
Figure 2) even though the latter two search in exponentially
smaller search spaces (see plots in Figure 2), with SNLP
particular having the smallest search space, as guaranteed by
its systematicity property.
MP-I and McNonlin perform better than SNLPand MPin
the case of GbyGstrategy, even though the former two don’t
split their search spaces into brancheswith mutuallyexclusive
establishment structures and thus search in more redundant
Onceagain, this behavior can be explained in terms of the way
each planner handles the tradeoff between redundancy and commitment. Since both MPand MP-I can change their commitment
to establishment structures within the same branch, they mayvisit a plan more than once. They thus have larger search spaces
compared to SNLP. Moreover, MPand SNLP, which maintain exhaustive causal structures, have smaller search spaces than MP-I
and McNonlinrespectively.
However,the performance is not directly correlated with the
search space size. In the LWOordering strategy L, SNLPand
McNonlin
are forced to committo a specific contributor for the hf
and he subgoals (preconditions) of a top level goal G, before the
other top level goals are expanded.Since both hfand he are easily
added and deleted by manyactions in the domains,such premature
commitmenthas a high probability of being wrong. Since both
SNLPand McNonlinprotect causal links to eliminate redundancy,
the only waythey can get rid of a wrongcausal link is to backtrack
over it (i.e., go over to another branch of the search space). Such
backtracking turns out to be very costly in terms of performance.
MPand MP-Iavoid problematic backtracking as they can deal with
their initial wrongcommitment
by mergingadditional contributors
m
into the contributor list as and whenthey becomeavailable,
Premature commitmentturns out to be less of a problem when
the FIFOordering strategy GbyGis used, since in this case Gi and
h are addressed before hf and he, and each action Ai is capable
of giving only one of the goals Gi. Since only the initial state is
capable of giving all h, the orderings imposedto deal with the
deletions of li’s by individual actions of the plans constrain the
t°In view of the note in Section 2. regarding twodifferent notions
of systematicity,wehavealso repeatedthese experimentswitha version
of SNLPwhichuniformlypostponesresolution of conflicts to the end.
Althoughthis version performedslightly better than the original SNLP,
therewasnoqualitativechangein the relative performance
of the planners.
For example,mART-MD-RD
with the Lifo goal orderingstrategy, a 5 goal
problemis solved by MP(I)in 490 ms, by SNLP
in 7180ms, and by the
alternative version of SNIPin 3660ms.Both versions of SNLPfail to
solvea size 8 problem,whileMP(I)hasno difficulty.
1.0e+06 .
i
,
. ,
.
,
r
.
.
. ,
.
.
1,0e+06
1,0e+06
o_.oMp(~)
v..vup.t
((;~s~)
1.0~o4 a--aS~(~)
o--oNon~(e~)
-
1.00+08
],
+%/
1.0e+04
/’)"
o
i
1.0e+O1 0
1.0e4~2
i , t ,
1’0e+00t.0, 2.0
3.0
t
4.0
8.0
6.0
7.0
8.0
Num.
~mlsfromART-MD-RD
,.o.+o,
1+0
1.0~2
3.0
,/.o ’ ’/.o
Num.
Goals
fromART-MD-RD
,.o,,+o.,.o ’
+’.o ’ ,to ’ ;.o
Num.
Goals
fromART-1
D-RD
Figure 2: Performance of MP, MP-I, SNLPand McNonlin in ART-MD-RD
and ART-1D-RD
and Domains. The first plot compares the
fun search space sizes in ART-MD-RD,
and the other two compareperformance on solvable problems in both domains
plan enoughso that by the time hf and he are addressed, the only
available contributor choice also happensto be the correct choice.
Although premature commitment is not a problem, SNLPand
MPconsider positive as well as negative threats for causallinks and
thus have higher solution depth. In particular, note that plans that
are complete for MP-Iand McNonlinmay still need to be refined
further to ensure exhaustivenessof their validation structure and
thereby make them complete with respect to MPand SNLP.This
explains the higher planning cost of SNLPand MPcompared to
McNonlinand MP-I respectively.
tematicity of its search. Weshowedthat there are a specmmlof
solutions to this tradeoff, and identified the type of causal links
used (single vs. multi-conlxibutor vs. none), and the level of commitmentto the causal links (exhaustive vs. non-exhaustive)as the
important dimensions of variation. Weconducted focused empirical studies to understand howthese different dimensionsaffect
the performanceof a planner. Themain objective of the empirical
study has been to establish the presence of tradeoffs. Thestudies
characterize SNLPand TWEAK
as two extreme solutions to the
redundancy-commitmenttradeoff, and demonstzate that planners
whichstrike a morejudicious balance in this tradcoff can outperfor4.2.2. Comparison between Causal Link based and Nonm both less committed, and more systematic planners. Although
causal Link based Planners
our empirical studies were mainlydonein artificial domains,we beGiven that the reduced commitment
to contributors provided by
lieve that the hypotheses regarding tradeoffs betweenredundancy
multi-contributor causal links allows themto strike a better balance
and commitmentwill also apply to other domains.
between redundancyand commitmentthan is the case with single
Our studies showthat although the size of the overall search
conaibutor causal links, a natural question arises as to whether
space is significantly smaller for systematic planners as expected
planners that completely avoid causal links strike an even better
(see Figure 2), their performance on solvable problems is not
balance. To answer this question, we comparedthe performance
directly correlated with the search space size (see Figures 2 and
of all seven planners shown in Figure 1. The plots in Figure 3
3). This should not be surprising. Thesearch space size will have
show the results of these comparisons in ART-MD-RD.
a significant bearing on the cost of planning only whenthe solution
As expected, both SNLP,which completely eliminates redundandensity is so low that the planner is forced to search a significant
cy at the expense of increased commitment, and TWEAK
which
part of its search space. Howeverin situations involving solvable
avoids all forms of commiUnent,fare badiy in this domain. The
problems, wherethe solution density is not sufficiently low, the
former gets penalized for the overcommitment(especially in the
average case performance is influenced more by factors such as
LIFOgoal ordering strategy), and the latter gets penalized for the
effective depth of the solution and the amountof backtracking
excessive redundancy(high branching factor). For the LIFOgoal
done by the planner. The latter factors are clearly correlated with
ordering strategy, the planners using multi-conlributorcausal links,
the planner’s level of commitment,and the extent to which the
and those using no causallinks (except TWEAK),
do better than the
domain penalizes overcommitment.
planners using single contributor causal links, including SNLPand
Ourconclusions aboveare similar in spirit to Langlcy’s[7] work
TWEAK.
This can be explained by the fact that LIFOgoal ordering
on the comparativeutility of systematic and nonsystematic search
heavily penalizes planners that committoo early to specific constrategies. They also have someparallels with the use of macrotributors and protect the commitments.The best performancefor
operators to improve performanceof a base-level problem solver
both goal orderings is shown by UAand MP-I with NGTWEAK
(c.f. [9, 12]). Macropsincrease the effective branching factor
comingthird.
adding redundancyto the search space of the base level problem
solver to reducethe depthof the solution. It is well recognizedthat
5. CONCLUSIONSANDRELATED
WORK
in most cases the performanceof base level problemsolver declines
In this paper we provided a rational reconstruction of the motiwith both complete elimination and indiscriminate accumulation
vations for systematicity, and argued that the performance of a
of macrops. Best performanceof the base-level problemsolver is
POPIplanner is correlated more closely with the wayit balances
attained by carefully controlling the accumulationof macrops. In
the tradeoff betweenredundancyand commitment,than on the systhe similar vein, MPand MP(I)look at every plan that is visited
71
~hLIFO{L)Goal OMedng
1.0e+06
1,0e~05
a.~l:Jm.p(L)
O--<>t~(L)
O---OUP
(L)
~F---VMP-I
(L)
NGTweak
(i.)
A-mA
Twed¢
~.)
~4~
//.A
I
1.0e+04
E
o.
U
’3
1.0o+03
lE
1.01~2
1.0e~O1
3,0
8.0
7.0
Num. Goals from ART-MO-RD
~io ’
~Io ’
7’.o
Num. Goals from ART-MD-RD
Figure3: Comparison
of all seven plannersin ART-MD-RD
Domain
SNLP,
plussomemore.As wehaveseen, this increasedredundancy [12] RJ. Mooney.The effect of role use on the utility of explanation-based
learning. In Proceedingsof the I lth IJCAI, pages 725-730,1989.
doesprovidethemmoreefficiency undercertainconditions.
Ourstudyalso clarifies someof the conjecturesmadein the
[13] NJ. Nilsson. Principles of Artificial Intelligence. Tioga Publishers,
Palo Alto, CA,1980.
literatureon the relativeperformance
of the causallink-based
and
non-causal
link basedplanners.Theresults in Figure3 showthat
[14] J. Pearl. Heuristics: Intelligent Search Strategies for Computer
Problem Solving. Addison-Wesley,Reading, Massachusettes (1984).
contrary
to the conjecture
in [8], nonsystematic
planners,includ[15] E.ED.Pednanlt. Synthesizing Plans that contain actions with Contexting thosethat don’tuse causallinks, canoutperform
systematic
Dependent Effects Computational Intelligence, Vol. 4, 356-372
planners
like SNLP.
Atthe sametime,contrary
to the conjecture
in
(1988).
[10, 11], maintenance
of causallinks doesnotnecessarilyincrease
[16]
J.S.
Penberthy and D. Weld. UCPOP:
A Sound, Complete, Partial Orthe cost of planning.In fact, as ourstudyshowed,
in the ART-MDder Planner for ADL.In Proceedings of KnowledgeRepresentation,
RDdomain
the best performance
is exhibitedby MP-Ia causallink
KR-92, November1992.
basedplannerthat uses multi-contributor
non.exhaustive
causal
[17] M. Peot and D. Smith. Conditional Nonlinear Planning. In Prolinks, andUAa non-causal
link basedplannerthatrestricts search
ceedings of First Intl. Conference on ,41 Planning Systems, June
1992.
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plans.
[18] A. Barrett and D. Weld. Partial Order Planning: Evaluating Possible
Efficiency Gains Technical Report 92-05-01, Departmentof Computer Science and Engineering, University of Washington, Seattle, WA,
June 1992.
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Elsevier, NewYork, 1975
[20] A. Tate. Generating Project Networks. In Proceedings oflJCAI-77,
pages 888-893, Boston, MA,1977.
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