Journal of Arti cial Intelligence Research Submitted published
Total Order and Partial Order Planning
A Comparative Analysis
Steven Minton minton ptolemy arc nasa gov
John Bresina bresina ptolemy arc na sa gov
Mark Drummond med ptolemy arc nasa gov
Recom Technologies
NASA Ames Research Center Mail Stop
Mo ett Field CA USA
Abstract
For many years the intuitions underlying partial order planning were largely taken for
granted Only in the past few years has there b een renewed interest in the fundamental
principles underlying this paradigm In this pap er we present a rigorous comparative
analysis of partial order and total order planning by fo cusing on two sp eci c planners that
can b e directly compared We show that there are some subtle assumptions that underly
the wide spread intuitions regarding the supp osed e ciency of partial order planning For
instance the sup eriority of partial order planning can dep end critically up on the search
strategy and the structure of the search space Understanding the underlying assumptions
is crucial for constructing e cient planners
Intro duction
For many years the sup eriority of partial order planners over total order planners has b een
tacitly assumed by the planning community Originally partial order planning was intro
duced by Sacerdoti as a way to improve planning e ciency by avoiding premature
commitments to a particular order for achieving subgoals The utility of partial order
planning was demonstrated anecdotally by showing how such a planner could e ciently
solve blo cksworld examples such as the well known Sussman anomaly
Since partial order planning intuitively seems like a go o d idea little attention has b een
devoted to analyzing its utility at least until recently Minton Bresina Drummond
a Barrett Weld Kambhampati c However if one lo oks closely at
the issues involved a numb er of questions arise For example do the advantages of partial
order planning hold regardless of the search strategy used Do the advantages hold when the
planning language is so expressive that reasoning ab out partially ordered plans is intractable
e g if the language allows conditional e ects
Our work Minton et al a has shown that the situation is much more inter
esting than might b e exp ected We have found that there are some unstated assumptions
underlying the supp osed e ciency of partial order planning For instance the sup eriority of
partial order planning can dep end critically up on the search strategy and search heuristics
employed
This pap er summarizes our observations regarding partial order and total order plan
ning We b egin by considering a simple total order planner and a closely related partial
order planner and establishing a mapping b etween their search spaces We then examine
c
AI Access Foundation and Morgan Kaufmann Publishers All rights reserved
Minton Bresina Drummond
the relative sizes of their search spaces demonstrating that the partial order planner has
a fundamental advantage b ecause the size of its search space is always less than or equal
to that of the total order planner However this advantage do es not necessarily translate
into an e ciency gain this dep ends on the typ e of search strategy used For example we
describ e a domain where our partial order planner is more e cient than our total order
planner when depth rst search is used but the e ciency gain is lost when an iterative
sampling strategy is used
We also show that partial order planners can have a second indep endent advantage
when certain typ es of op erator ordering heuristics are employed This heuristic advantage
underlies Sacerdoti s anecdotal examples explaining why least commitment works However
in our blo cksworld exp eriments this second advantage is relatively unimp ortant compared
to the advantage derived from the reduction in search space size
Finally we lo ok at how our results extend to partial order planners in general We
describ e how the advantages of partial order planning can b e preserved even if highly ex
pressive languages are used We also show that the advantages do not necessarily hold for
all partial order planners but dep end critically on the construction of the planning space
Background
Planning can b e characterized as search through a space of p ossible plans A total order
planner searches through a space of totally ordered plans a partial order planner is de ned
analogously We use these terms rather than the terms linear and nonlinear b ecause
the latter are overloaded For example some authors have used the term nonlinear
when fo cusing on the issue of goal ordering That is some linear planners when solving a
conjunctive goal require that all subgoals of one conjunct b e achieved b efore subgoals of the
others hence planners that can arbitrarily interleave subgoals are often called nonlinear
This version of the linear nonlinear distinction is di erent than the partial order total
order distinction investigated here The former distinction impacts planner completeness
whereas the total order partial order distinction is orthogonal to this issue Drummond
Currie Minton et al a
The total order partial order distinction should also b e kept separate from the distinc
tion b etween world based planners and plan based planners The distinction is one
of mo deling in a world based planner each search state corresp onds to a state of the
world and in a plan based planner each search state corresp onds to a plan While total
order planners are commonly asso ciated with world based planners such as Strips several
well known total order planners have b een plan based such as Waldinger s regression plan
ner Waldinger Interplan Tate and Warplan Warren Similarly
partial order planners are commonly plan based but it is p ossible to have a world based
partial order planner Go defroid Kabanza In this pap er we fo cus solely on the
total order partial order distinction in order to avoid complicating the analysis
We claim that the only signi cant di erence b etween partial order and total order plan
ners is planning e ciency It might b e argued that partial order planning is preferable
b ecause a partially ordered plan can b e more exibly executed However execution exibil
ity can also b e achieved with a total order planner and a p ost pro cessing step that removes
unnecessary orderings from the totally ordered solution plan to yield a partial order Back
Total Order and Partial Order Planning
strom Veloso Perez Carb onell Regnier Fade The p olynomial
time complexity of this p ost pro cessing is negligible compared to the search time for plan
generation Hence we b elieve that execution exibility is at b est a weak justi cation for
the supp osed sup eriority of partial order planning
In the following sections we analyze the relative e ciency of partial order and total
order planning by considering a total order planner and a partial order planner that can
b e directly compared Elucidating the key di erences b etween these planning algorithms
reveals some imp ortant principles that are of general relevance
Terminology
A plan consists of an ordered set of steps where each step is a unique op erator instance
Plans can b e total ly ordered in which case every step is ordered with resp ect to every other
step or partial ly ordered in which case steps can b e unordered with resp ect to each other
We assume that a library of op erators is available where each op erator has preconditions
deleted conditions and added conditions All of these conditions must b e nonnegated prop o
sitions and we adopt the common convention that each deleted condition is a precondition
Later in this pap er we show how our results can b e extended to more expressive languages
but this simple language is su cient to establish the essence of our argument
A linearization of a partially ordered plan is a total order over the plan s steps that is
consistent with the existing partial order In a totally ordered plan a precondition of a plan
step is true if it is added by an earlier step and not deleted by an intervening step In a
partially ordered plan a step s precondition is possibly true if there exists a linearization in
which it is true and a step s precondition is necessarily true if it is true in al l linearizations
A step s precondition is necessarily false if it is not p ossibly true
A state consists of a set of prop ositions A planning problem is de ned by an initial
state and a set of goals where each goal is a prop osition For convenience we represent a
problem as a two step initial plan where the prop ositions that are true in the initial state
are added by the rst step and the goal prop ositions are the preconditions of the nal
step The planning pro cess starts with this initial plan and searches through a space of
p ossible plans A successful search terminates with a solution plan i e a plan in which all
steps preconditions are necessarily true The search space can b e characterized as a tree
where each no de corresp onds to a plan and each arc corresp onds to a plan transformation
Each transformation incrementally extends i e re nes a plan by adding additional steps
or orderings Thus each leaf in the search tree corresp onds either to a solution plan or
a dead end and each intermediate no de corresp onds to an un nished plan which can b e
further extended
Backstrom formalizes the problem of removing unnecessary orderings in order to pro duce a least
constrained plan He shows that the problem is p olynomial if one de nes a least constrained plan as a
plan in which no orderings can b e removed without impacting the correctness of the plan Backstrom
also shows that the problem of nding a plan with the fewest orderings over a given op erator set is a
much harder problem it is NP hard
Minton Bresina Drummond
TO P G
Termination check If G is empty rep ort success and return solution plan P
Goal selection Let c select goal G and let O b e the plan step for which c is a precondition
need
Op erator selection Let O b e an op erator in the library that adds c If there is no such O then
add add
terminate and rep ort failure Choice point al l such operators must be considered for completeness
Ordering selection Let O b e the last deleter of c Insert O somewhere b etween O and
del add del
0
O call the resulting plan P Choice point al l such positions must be considered for completeness
need
0 0
Goal up dating Let G b e the set of preconditions in P that are not true
0 0
Recursive invo cation TO P G
Figure The to planning algorithm
Plan P
S O A B O del need F
+ O add
SOO O A B F del add need
SOO A O B F del add need
SOO A B O F
del add need
Figure How to extends a plan Adding O to plan P generates three alternatives
add
A Tale of Two Planners
In this section we de ne two simple planning algorithms The rst algorithm shown
in Figure is to a total order planner motivated by Waldinger s regression planner
Waldinger Interplan Tate and Warplan Waldinger Our purp ose
here is to characterize the search space of the to planning algorithm and the pseudo co de
in Figure accomplishes this by de ning a nondeterministic pro cedure that enumerates
p ossible plans If the plans are enumerated by a breadth rst search then the algorithms
presented in this section are provably complete as shown in App endix A
Total Order and Partial Order Planning
to accepts an un nished plan P and a goal set G containing preconditions which are
currently not true If the algorithm terminates successfully then it returns a totally ordered
solution plan Note that there are two choice p oints in this pro cedure op erator selection
and ordering selection The pro cedure do es not need to consider alternative goal choices
For our purp oses the function select goal can b e any deterministic function that selects
a memb er of G
As used in Step the last deleter of a precondition c for a step O is de ned as
need
follows Step O is the last deleter of c if O deletes c O is b efore O and there is
del del del need
no other deleter of c b etween O and O In the case that no step b efore O deletes c
del need need
the rst step is considered to b e the last deleter
Figure illustrates to s plan extension pro cess This example assumes that steps A
and B do not add or delete c There are three p ossible insertion p oints for O in plan P
add
each yielding an alternative extension
The second planner is ua a partial order planner shown in Figure ua is similar to
to in that it uses the same pro cedures for goal selection and op erator selection however
the pro cedure for ordering selection is di erent Step of ua inserts orderings but only
interacting steps are ordered Sp eci call y we say that two steps interact if they are
unordered with resp ect to each other and either
one step has a precondition that is added or deleted by the other step or
one step adds a condition that is deleted by the other step
The only signi cant di erence b etween ua and to lies in Step to orders the new step
with resp ect to al l others whereas ua adds orderings only to eliminate interactions It is
in this sense that ua is less committed than to
Figure illustrates ua s plan extension pro cess As in Figure we assume that steps
A and B do not add or delete c however step A and O interact with resp ect to some
add
other condition This interaction yields two alternative plan extensions one in which O
add
is ordered b efore A and one in which O is ordered after A
add
Since ua orders all steps which interact the plans that are generated have a sp ecial
prop erty each precondition in a plan is either necessarily true or necessarily false We
call such plans unambiguous This prop erty yields a tight corresp ondence b etween the two
planners search spaces Supp ose ua is given the unambiguous plan U and to is given
the plan T where T is a linearization of U Let us consider the relationship b etween
the way that ua extends U and to extends T Note that the two planners will have the
same set of goals since by de nition each goal in U is a precondition that is necessarily
false and a precondition is necessarily false if and only if it is false in every linearization
Since the two plans have the same set of goals and since b oth planners use the same goal
selection metho d b oth algorithms pick the same goal therefore O is the same for b oth
need
Similarly b oth algorithms consider the same library op erators to achieve this goal Since T
is a linearization of U and O is the same in b oth plans b oth algorithms nd the same
need
last deleter as well When to adds a step to a plan it orders the new step with resp ect to
There is a unique last deleter in U This follows from our requirement that for any op erator in our
language the deleted conditions must b e a subset of the preconditions If two unordered steps delete
the same condition then that condition must also b e a precondition of b oth op erators Hence the two
steps interact and will b e ordered by ua
Minton Bresina Drummond
UA P G
Termination check If G is empty rep ort success and return solution plan P
Goal selection Let c select goal G and let O b e the plan step for which c is a precondition
need
Op erator selection Let O b e an op erator in the library that adds c If there is no such O then
add add
terminate and rep ort failure Choice point al l such operators must be considered for completeness
Ordering selection Let O b e the last deleter of c Order O after O and b efore O
del add del need
Rep eat until there are no interactions
Select a step O that interacts with O
int add
Order O either b efore or after O
int add
Choice point both orderings must be considered for completeness
0
Let P b e the resulting plan
0 0
Goal up dating Let G b e the set of preconditions in P that are necessarily false
0 0
Recursive invo cation UA P G
Figure The ua planning algorithm
Plan P
A S O O F del need B
+ O add
O O A add add A
S O O F S O O F del need del need
B B
Figure How ua extends a plan Adding O to plan P generates two alternatives The
add
example assumes that O interacts with step A
add
Total Order and Partial Order Planning
all existing steps When ua adds a step to a plan it orders the new step only with resp ect
to interacting steps ua considers all p ossible combinations of orderings which eliminate
interactions hence for any plan pro duced by to ua pro duces a corresp onding plan that
is less ordered or equivalent
The following sections exploit this tight corresp ondence b etween the search spaces of
ua and to In the next section we analyze the relative sizes of the two planners search
spaces and later we compare the numb er of plans actually generated under di erent search
strategies
Search Space Comparison
The search space for b oth to and ua can b e characterized as a tree of plans The ro ot
no de in the tree corresp onds to the top level invo cation of the algorithm and the remaining
no des each corresp ond to a recursive invo cation of the algorithm Note that in generating
a plan the algorithms make b oth op erator and ordering choices and each di erent set of
choices corresp onds to a single branch in the search tree
We denote the search tree for to by tr ee and similarly the search tree for ua by
TO
tr ee The numb er of plans in a search tree is equal to the numb er of times the planning
UA
pro cedure ua or to would b e invoked in an exhaustive exploration of the search space
Note that every plan in tr ee and tr ee is unique since each step in a plan is given
UA TO
a unique lab el Thus although two plans in the same tree might b oth b e instances of a
particular op erator sequence such as O O O the plans are distinct b ecause their
steps have di erent lab els We have de ned our plans this way to make our pro ofs more
concise
We can show that for any given problem tr ee is at least as large as tr ee that is
TO UA
the numb er of plans in tr ee is greater than or equal to the numb er of plans in tr ee
TO UA
This is done by proving the existence of a function L which maps plans in tr ee into sets
UA
of plans in tr ee that satis es the following two conditions
TO
Totality Prop erty For every plan U in tr ee there exists a non empty set
UA
fT T g of plans in tr ee such that L U fT T g
m m
TO
Disjointness Prop erty L maps distinct plans in tr ee to disjoint sets of plans in
UA
tr ee that is if U U tr ee and U U then L U L U fg
TO UA
Let us examine why the existence of an L with these two prop erties is su cient to prove
that the size of ua s search tree is no greater than that of to Figure provides a guide for
the following discussion Intuitively we can use L to count plans in the two search trees
For each plan counted in tr ee we use L to count a non empty set of plans in tr ee
UA TO
The totality prop erty means that every time we count a plan in tr ee we count at least
UA
P
one plan in tr ee this implies that j tr ee j j L U j Of course we must
U tr ee
TO UA
UA
further show that each plan counted in tr ee is counted only once this is guaranteed by
TO
P
the disjointness prop erty which implies that j L U j j tr ee j Thus the
U tr ee
TO
UA
conjunction of the two prop erties implies that j tr ee j j tr ee j
UA TO
We can de ne a function L that has these two prop erties as follows Let U b e a plan
in tr ee let T b e a plan in tr ee and let par ent b e a function from a plan to its parent
UA TO
Minton Bresina Drummond
ua search tree to search tree
L
h h
f g
H
H
H
L Hj
h h h
f g
A
L R
A
h h h
A
f g
R AU
L
h h h
f g
Figure How L maps from tr ee to tr ee
UA TO
plan in the tree Then T L U if and only if i T is a linearization of U and ii either
U and T are b oth ro ot no des of their resp ective search trees or par ent T L par ent U
Intuitively L maps a plan U in tr ee to all linearizations which share common derivation
UA
ancestry This is illustrated in Figure where for each plan in tr ee a dashed line is
UA
drawn to the corresp onding set of plans in tr ee
TO
We can show that L satis es the totality and disjointness prop erties by induction on the
depth of the search trees Detailed pro ofs are in the app endix To prove the rst prop erty
we show that for every plan contained in tr ee all linearizations of that plan are contained
UA
in tr ee To prove the second prop erty we note that any two plans at di erent depths in
TO
tr ee have disjoint sets of linearizations and then show by induction that any two plans
UA
at the same depth in tr ee also have this prop erty
UA
How much smaller is tr ee than tr ee The mapping describ ed ab ove provides an
UA TO
answer For each plan U in tr ee there are j L U j distinct plans in to where j L U j is
UA
the numb er of linearizations of U The exact numb er dep ends on how unordered U is A
totally unordered plan has a factorial numb er of linearizations and a totally ordered plan
has only a single linearization Thus the only time that the size of tr ee equals the size of
UA
tr ee is when every plan in tr ee is totally ordered otherwise tr ee is strictly smaller
TO UA UA
than tr ee and p ossibly exp onentially smaller
TO
Time Cost Per Plan
While the size of ua s search tree is p ossibly exp onentially smaller than that of to it do es
not follow that ua is necessarily more e cient E ciency is determined by two factors the
The reader may question why L maps U to all its linearizati ons in tr ee that share common deriva
TO
tion ancestry as opp osed to simply mapping U to all its linearizati ons in tr ee The reason is that
TO
our planners are not systematic in the sense that they may generate two or more plans with the same
op erator sequence We can distinguish such plans by their derivational history For example supp ose
two instantiations of the same op erator sequence O O O exist within a tr ee but they corre
TO
sp ond to di erent plans in tr ee L relies on their di erent derivations to determine the appropriate
UA
corresp ondence
Total Order and Partial Order Planning
Step Executions Per Plan TO Cost UA Cost
O O
O O
O O
O O e
O n O e
Table Cost p er plan comparisons
time cost p er plan in the search tree discussed in this section and the size of the subtree
explored during the search pro cess discussed in the next section
In this section we show that while ua can indeed take more time p er plan the extra
time is relatively small and grows only p olynomially with the numb er of steps in the plan
which we denote by n In comparing the relative e ciency of ua and to we rst consider
the numb er of times that each algorithm step is executed p er plan in the search tree and
we then consider the time complexity of each step
As noted in the preceding sections each no de in the search tree corresp onds to a plan
and each invo cation of the planning pro cedure for b oth ua and to corresp onds to an attempt
to extend that plan Thus for b oth ua and to it is clear that the termination check and
goal selection Steps and are each executed once p er plan Analyzing the numb er of
times that the remaining steps are executed might seem more complicated since each of
these steps is executed many times at an internal no de and not at all at a leaf However
the analysis is actually quite simple since we can amortize the numb er of executions of each
step over the numb er of plans pro duced Notice that Step is executed once for each plan
that is generated i e once for each no de other than the ro ot no de This gives us a b ound
on the numb er of times that Steps and are executed More sp eci cally for b oth
algorithms Step is executed fewer times than Step and Steps and are executed
exactly the same numb er of times that Step is executed that is once for each plan that
is generated Consequently for b oth algorithms no step is executed more than once p er
plan as summarized in Table In other words the numb er of times each step is executed
during the planning pro cess is b ounded by the size of the search tree
In examining the costs for each step we rst note that for b oth algorithms Step
the termination check can b e accomplished in O time Step goal selection can also
b e accomplished in O time for example assuming the goals are stored in a list the
select goal function can simply return the rst memb er of the list Each execution of
Step op erator selection also only requires O time if we assume the op erators are
indexed by their e ects all that is required is to p op the list of relevant op erators on each
execution
We assume that the size of the op erators the numb er of preconditions and e ects is b ounded by a
constant for a given domain
Since Steps and are nondeterministic we need to b e clear ab out our terminology We say that Step
is executed once each time a di erent op erator is chosen and Step is executed once for each di erent
combination of orderings that is selected
Minton Bresina Drummond
Steps and are less exp ensive for to than for ua Step of to is accomplished
by inserting the new op erator O somewhere b etween O and O If the p ossible
add del need
insertion p oints are considered starting at O and working towards O then each exe
need del
cution of Step can b e accomplished in constant time since each insertion constitutes one
execution of the step In contrast Step in ua involves carrying out interaction detection
and elimination in order to pro duce a new plan P This step can b e accomplished in O e
time where e is the numb er of edges in the graph required to represent the partially ordered
plan In the worst case there may b e O n edges in the plan and in the b est case O n
edges The following is the description of ua s ordering step from Figure with some
additional implementation details
Ordering selection Order O after O and b efore O Lab el all steps preceding O and
add del need add
all steps following O Let steps b e the unlab eled steps that interact with O Let O b e the
add int add del
last deleter of c Rep eat until steps is empty
int
Let O Pop steps
int int
if O is still unlab eled then either
int
order O b efore O and lab el O and the unlab eled steps b efore O or
int add int int
order O after O and lab el O and the unlab eled steps after O
int add int int
Choice point both orderings must be considered for completeness
0
Let P b e the resulting plan
The ordering pro cess b egins with a prepro cessing stage First all steps preceding or follow
ing O are lab eled as such The lab eling pro cess is implemented by a depth rst traversal
add
of the plan graph starting with O as the ro ot which rst follows the edges in one direc
add
tion and then follows edges in the other direction This requires at most O e time After
the lab eling pro cess is complete only steps that are unordered with resp ect to O are
add
unlab eled and thus the interacting steps which must b e unordered with resp ect to O
add
are identi able in O n time The last deleter is identi able in O e time
After the prepro cessing stage the pro cedure orders each interacting step with resp ect to
O up dating the lab els after each iteration Since each edge in the graph need b e traversed
add
no more than once the entire ordering pro cess takes at most O e time as describ ed in
Minton et al b To see this note that the pro cess of lab eling the steps b efore or
after O can stop as so on as a lab eled step is encountered
int
Having shown that Step of to has O complexity and Step of ua has O e complex
ity we now consider Step of b oth algorithms up dating the goal set to accomplishes this
by iterating through the steps in the plan from the head to the tail which requires O n
time ua accomplishes this in a similar manner but it requires O e time to traverse the
graph Alternatively ua can use the same pro cedure as to provided an O e top ological
sort is rst done to linearize the plan
To summarize our complexity analysis the use of a partial order means that ua incurs
greater cost for op erator ordering Step and for up dating the goal set Step Overall
ua requires O e time p er plan while to only requires O n time p er plan Since a totally
ordered plan requires a representation of size O n and a partially ordered graph requires
a representation of size O e designing pro cedures with lower costs would b e p ossible only
if the entire plan graph did not need to b e examined in the worst case
Total Order and Partial Order Planning
The Role of Search Strategies
The previous sections have compared to and ua in terms of relative search space size
and relative time cost p er no de The extra pro cessing time required by ua for each no de
would app ear to b e justi ed since its search space may contain exp onentially fewer no des
However to complete our analysis we must consider the numb er of no des actually visited
by each algorithm under a given search strategy
For breadth rst search the analysis is straightforward After completing the search to
a particular depth b oth planners will have explored their entire trees up to that depth
Both ua and to nd a solution at the same depth due to the corresp ondence b etween their
search trees Thus the degree to which ua will outp erform to under breadth rst dep ends
solely on the expansion factor under L i e on the numb er of linearizations of ua s plans
We can formalize this analysis as follows For a no de U in tr ee we denote the numb er
UA
of steps in the plan at U by n and the numb er of edges in U by e Then for each no de U
u u
that ua generates ua incurs time cost O e whereas to incurs time cost O n j L U j
u u
where j L U j is the numb er of linearizations of the plan at no de U Therefore the ratio
of the total time costs of to and ua is as follows where bf tr ee denotes the subtree
UA
considered by ua under breadth rst search
P
O n j L U j
u
cost to
u bf tr ee
bf
UA
P
cost ua O e
bf u
u bf tr ee
UA
The analysis of breadth rst search is so simple b ecause this search strategy preserves
the corresp ondence b etween the two planners search spaces In breadth rst search the two
planners are synchronized after exhaustively exploring each level so that to has explored
exactly the linearizations of the plans explored by ua For any other search strategy which
similarly preserves the corresp ondence such as iterative deep ening a similar analysis can
b e carried out
The cost comparison is not so clear cut for depth rst search since the corresp ondence is
not guaranteed to b e preserved It is easy to see that under depth rst search to do es not
necessarily explore all linearizations of the plans explored by ua This is not simply b ecause
the planners nondeterministicall y cho ose which child to expand There is a deep er reason
the corresp ondence L do es not preserve the subtree structure of the search space For a plan
U in tr ee the corresp onding linearizations in L U may b e spread throughout tr ee
UA TO
Therefore it is unlikely that corresp onding plans will b e considered in the same order by
depth rst search Nevertheless even though the two planners are not synchronized we
might exp ect that on average ua will explore fewer no des b ecause the size of tr ee is less
UA
than or equal to the size of tr ee
TO
Empirically we have observed that ua do es tend to outp erform to under depth rst
search as illustrated by the exp erimental results in Figure The rst graph compares
the mean numb er of nodes explored by ua and to on randomly generated blo cksworld
problems the second graph compares the mean planning time for ua and to on the same
problems and demonstrates that the extra time cost p er no de for ua is relatively insigni
cant The problems are partitioned into sets of problems each according to minimal
For p erspicuity we ignore the fact that the numb er of no des explored by the two planners on the last
level may di er if the planners stop when they reach the rst solution
Minton Bresina Drummond
10000
50
7500 40 TO TO
30 5000 UA 20 UA Nodes Explored 2500 Time to Solution 10
0 0 3 4 5 6 3 4 5 6
Depth of Problem Depth of Problem
Figure ua and to Performance Comparison under Depth First Search
solution length i e the numb er of steps in the plan For each problem b oth planners
were given a depth limit equal to the length of the shortest solution Since the planners
make nondeterministic choices trials were conducted for each problem The source co de
and data required to repro duce these exp eriments can b e found in Online App endix
As we p ointed out one plausible explanation for the observed dominance of ua is that
to s search tree is at least as large as ua s search tree In fact in the ab ove exp eriments
we often observed that to s search tree was typically much larger However the full story
is more interesting Search tree size alone is not su cient to explain ua s dominance in
particular the density and distribution of solutions play an imp ortant role
The solution density of a search tree is the prop ortion of no des that are solutions If the
solution density for to s search tree is greater than that for ua s search tree then to might
outp erform ua under depth rst search even though to s search tree is actually larger For
example it might b e the case that all ua solution plans are completely unordered and that
the plans at the remaining leaves of tr ee the failed plans are totally ordered In this
UA
case each ua solution plan corresp onds to an exp onential numb er of to solution plans and
each ua failed plan corresp onds to a single to failed plan The converse is also p ossible
the solution density of ua s search tree might b e greater than that of to s search tree thus
favoring ua over to under depth rst search For example there might b e a single totally
ordered solution plan in ua s search tree and a large numb er of highly unordered failed
Since the depth limit is equal to the length of the shortest solution an iterative deep ening Korf
approach would yield similar results Additional ly we note that increasing the depth limit past the
depth of the shortest solution do es not signi cantly change the outcome of these exp eriments
This de nition of solution density is ill de ned for in nite trees but we assume that a depth b ound is
always provided so only a nite subtree is explicitly enumerated
Total Order and Partial Order Planning
UA Search Tree TO Search Tree
* * * * * *
* = Solution plan
Figure Uniform solution distribution with solution density
plans Since each of these failed ua plans would corresp ond to a large numb er of to failed
plans the solution density for to would b e considerably lower
For our blo cksworld problems we found that the solution densities of the two planners
trees do es not di er greatly at least not in such a way that would explain our p erformance
results We saw no tendency for tr ee to have a higher solution density than tr ee For
UA TO
example for the problems with solutions at depth six the average solution density for
to exceeded that of ua on out of the problems This is not particularly surprising
since we see no a priori reason to supp ose that the solution densities of the two planners
should di er greatly
Since solution density is insu cient to explain ua s dominance on our blo cksworld ex
p eriments when using depth rst search we need to lo ok elsewhere for an explanation
We hyp othesize that the distribution of solutions provides an explanation We note that
if the solution plans are distributed p erfectly uniformly i e at even intervals among the
leaves of the search tree and if the solution densities are similar then b oth planners can
b e exp ected to search a similar numb er of leaves as illustrated by the schematic search
tree in Figure Consequently we can explain the observed dominance of ua over to by
hyp othesizing that solutions are not uniformly distributed that is solutions tend to cluster
To see this supp ose that tr ee is smaller than tr ee but the two trees have the same
UA TO
solution density If the solutions are clustered as in Figure then depth rst search can b e
exp ected to pro duce solutions more quickly for tr ee than for tr ee The hyp othesis
UA TO
In our exp eriments a nondeterministi c goal selection pro cedure was used with our planners which meant
that the solution density could vary from run to run We compared the average solution density over
trials for each problem to obtain our results
Even if the solutions are distributed randomly amongst the leaves of the trees with uniform probabili ty
as opp osed to b eing distributed p erfectly uniformly there will b e some clusters of no des Therefore
to will have a small disadvantage To see this let us supp ose that each leaf of b oth tr ee and tr ee
UA TO
is a solution with equal probability p That is if tr ee has N leaves of which k are solutions
UA UA UA
Minton Bresina Drummond
UA Search Tree TO Search Tree
* * * * * *
* = Solution plan
Figure Non uniform solution distribution with solution density
that solutions tend to b e clustered seems reasonable since it is easy to construct problems
where a wrong decision near the top of the search tree can lead to an entire subtree that
is devoid of solutions
One way to test our hyp othesis is to compare ua and to using a randomized search
strategy a typ e of Monte Carlo algorithm that we refer to as iterative sampling cf
Minton et al Langley Chen Crawford Baker The iterative
sampling strategy explores randomly chosen paths in the search tree until a solution is
found A path is selected by traversing the tree from the ro ot to a leaf cho osing randomly
at each branch p oint If the leaf is a solution then search terminates if not the search
pro cess returns to the ro ot and selects another path The same path may b e examined
more than once since no memory is maintained b etween iterations
In contrast to depth rst search iterative sampling is relatively insensitive to the dis
tribution of solutions Therefore the advantage of ua over to should disapp ear if our hy
p othesis is correct In our exp eriments we did nd that when ua and to b oth use iterative
sampling they expand approximately the same numb er of no des on our set of blo cksworld
problems For b oth planners p erformance with iterative sampling was worse than with
depth rst search The fact that there is no di erence b etween ua and to under iterative
sampling but that there is a di erence under depth rst search suggests that solutions are
and tr ee has N leaves of which k are solutions then p k N k N In general
TO TO TO UA UA TO TO
if k out of N no des are solutions the exp ected numb er of no des that must b e tested to nd a solution
is N k when k and approaches N k as k and N approaches This is simply the exp ected
numb er of samples for a binomial distribution Therefore since k k the exp ected numb er of
TO UA
leaves explored by to is greater than or equal to the exp ected numb er of leaves explored by ua by at
most a factor of
The iterative sampling strategy was depth limited in exactly the same way that our depth rst strategy
was We note however that the p erformance of iterative sampling is relatively insensitive to the actual
depth limit used
Total Order and Partial Order Planning
indeed non uniformly distributed Furthermore this result shows that ua is not necessarily
sup erior to to the search strategy that is employed makes a dramatic di erence
Although our blo cksworld domain may b e atypical we conjecture that our results are
of general relevance Sp eci cally for distribution sensitive search strategies like depth rst
search one can exp ect that ua will tend to outp erform to For distribution insensi tive
strategies such as iterative sampling non uniform distributions will have no e ect We note
that while iterative sampling is a rather simplistic strategy there are more sophisticated
search strategies such as iterative broadening Ginsb erg Harvey that are also
relatively distribution insensitive We further explore such strategies in Section
The Role of Heuristics
In the preceding sections we have shown that a partial order planner can b e more e cient
simply b ecause its search tree is smaller With some search strategies such as breadth
rst search this size di erential obviously translates into an e ciency gain With other
strategies such as depth rst search the size di erential translates into an e ciency gain
provided we make additional assumptions ab out the solution density and distribution
However it is often claimed that partial order planners are more e cient due to their
ability to make more informed ordering decisions a rather di erent argument For instance
Sacerdoti argues that this is the reason that noah p erforms well on problems such
as the blo cksworld s Sussman anomaly By delaying the decision of whether to stack A
on B b efore or after stacking B on C noah can eventually detect that a con ict will o ccur
if it stacks A on B rst and a critic called resolve conflicts can then order the steps
intelligently
In this section we show that this argument can b e formally describ ed in terms of our
two planners We demonstrate that ua do es in fact have a p otential advantage over to
in that it can exploit certain typ es of heuristics more readily than to This advantage is
indep endent of the fact that ua has a smaller search space Whether or not this advantage
is signi cant in practice is another question of course We also describ e some exp eriments
where we evaluate the e ect of a commonly used heuristic on our blo cksworld problems
Making More Informed Decisions
First let us identify how it is that ua can make b etter use of certain heuristics than to
In the ua planning algorithm step arbitrarily orders interacting plan steps Similarly
Step of to arbitrarily cho oses an insertion p oint for the new step It is easy to see
however that some orderings should b e tried b efore others in a heuristic search This is
illustrated by Figure which compares ua and to on a particular problem The key in
the gure describ es the relevant conditions of the library op erators where preconditions are
indicated to the left of an op erator and added conditions are indicated to the right there
are no deletes in this example For brevity the initial step and nal step of the plans
are not shown Consider the plan in tr ee with unordered steps O and O When ua
UA
intro duces O to achieve precondition p of O Step of ua will order O with resp ect to
O since these steps interact However it makes more sense to order O b efore O since O
achieves precondition q of O This illustrates a simple planning heuristic that we refer to
as the min goals heuristic prefer the orderings that yield the fewest false preconditions
Minton Bresina Drummond
UA TO
O1 O1
O1 O O 1 2 O2 O1
O2
O 1 O O O O O O3 1 2 2 3 1 O3 O2 O1 O2 O3 O1 O3
O2
KEY
p r O q q O p
O1 2 3
Figure Comparison of ua and to on an example
This heuristic is not guaranteed to pro duce the optimal search or the optimal plan but it
is commonly used It is the basis of the resolve con icts critic that Sacerdoti employed
in his blo cksworld examples
Notice however that to cannot exploit this heuristic as e ectively as ua b ecause it
prematurely orders O with resp ect to O Due to this inability to p ostp one an ordering
decision to must cho ose arbitrarily b etween the plans O O and O O b efore the
impact of this decision can b e evaluated
In the general case supp ose h is a heuristic that can b e applied to b oth partially ordered
plans and totally ordered plans Furthermore assume h is a useful heuristic i e if h
rates one plan more highly than another a planner that explores the more highly rated
plan rst will p erform b etter on average Then ua will have a p otential advantage over to
provided that h satis es the following prop erty for any ua plan U and corresp onding to
plan T h U h T that is a partially ordered plan must b e rated at least as high as any
of its linearizations Note that for unambiguous plans the min goals heuristic satis es this
prop erty since it gives identical ratings to a partially ordered plan and its linearizations
ua has an advantage over to b ecause if ua is expanding plan U and to is expanding a
corresp onding plan T then h will rate some child of U at least as high as the most highly
rated child of T This is true since every child of T is a linearization of some child of U
and therefore no child of T can b e rated higher than a child of U Furthermore there may
b e a child of U such that none of its linearizations is a child of T and therefore this child of
U can b e rated higher than every child of T Since we assumed that h is a useful heuristic
this means that ua is likely to make a b etter choice than to
Total Order and Partial Order Planning
5000
TO TO without MinGoals TO UA UA without MinGoals 4000 TO-MG TO with MinGoals UA-MG UA with MinGoals
3000 UA
2000
TO-MG Nodes Explored 1000 UA-MG
0 3 4 5 6
Depth of Problem
Figure Depth rst search with and without min goals
Illustrative Exp erimental Results
The previous section showed that ua has a p otential advantage over to b ecause it can b etter
exploit certain ordering heuristics We now examine the practical e ects of incorp orating
one such heuristic into ua and to
First we note that ordering heuristics only make sense for some search strategies In
particular for breadth rst search heuristics do not improve the e ciency of the search in a
meaningful way except p ossibly at the last level Indeed we need not consider any search
strategy in which to and ua are synchronized as de ned earlier since ordering heuristics
do not signi cantly a ect the relative p erformance of ua and to under such strategies Thus
we b egin by considering a standard search strategy that is not synchronized depth rst
search
We use the min goals heuristic as the basis for our exp erimental investigation since it is
commonly employed but presumably we could cho ose any heuristic that meets the criterion
set forth in the previous section Figure shows the impact of min goals on the b ehavior
of ua and to under depth rst search Although the heuristic biases the order in which the
two planners search spaces are explored cf Rosenblo om Lee Unruh it app ears
that its e ect is largely indep endent of the partial order total order distinction since b oth
planners are improved by a similar p ercentage For example under depth rst search on
the problems with solutions at depth six ua improved and to improved Thus
there is no obvious evidence for any extra advantage for ua as one might have exp ected
from our analysis in the previous section On the other hand this do es not contradict our
theory it simply means that the p otential heuristic advantage was not signi cant enough
to show up In other domains the advantage might manifest itself more signi cantly After
all it is certainly p ossible to design problems in which the advantage is signi cant as
Minton Bresina Drummond
100
TO-IB TO Iterative Broadening UA-IB UA Iterative Broadening TO-IB 75 TO-IS TO Iterative Sampling UA-IS UA Iterative Sampling
UA-IB 50 TO-IS
Nodes Explored 25 UA-IS
3 4 5 6
Depth of Problem
Figure Iterative sampling iterative broadening b oth with min goals
our example in Figure illustrates Our results simply illustrate that in our blo cksworld
domain making intelligent ordering decisions pro duces a negligible advantage for ua in
contrast to the signi cant e ect due to search space compression discussed previously
While the min goals heuristic did not seem to help ua more than to the results are
nevertheless interesting since the heuristic had a very signi cant e ect on the p erformance
of both planners so much so that to with min goals outp erforms ua without min goals
While the e ectiveness of min goals is domain dep endent we nd it interesting that in these
exp eriments the use of min goals makes more di erence than the use of partial orders After
all the blo cksworld originally help ed motivate the development of partial order planning
and most subsequent planning systems have employed partial orders While not deeply
surprising this result do es help reinforce what we already know more attention should b e
paid to sp eci c planning heuristics such as min goals
In our analysis of search space compression in Section we describ ed a distribution
insensitive search strategy called iterative sampling and showed that under iterative sam
pling ua and to p erform similarly although their p erformance is worse than it is under
depth rst search If we combine min goals with iterative sampling we nd that this pro
duces a much more p owerful strategy but one in which to and ua still p erform ab out
equally For simplicity our implementation of iterative sampling uses min goals as a prun
ing heuristic at each choice p oint it explores only those plan extensions with the fewest
goals This strategy is p owerful although incomplete Because of this incompleteness we
note there was one problem we removed from our sample set b ecause iterative sampling with
In Section we discuss planners that are less committed than ua For such planners the advantage
due to heuristics might b e more pronounced since they delay their decisions even longer than ua
Instead of exploring only those plan extensions with the fewest goals at each choice p oint an alternative
strategy is to assign each extension a probability that is inversely correlated with the numb er of goals
Total Order and Partial Order Planning
min goals would never terminate on this problem With this caveat in mind we turn to the
results in Figure which when compared against Figure show that the p erformance
of b oth ua and to with iterative sampling was in general signi cantly b etter than their
p erformance under depth rst search Note that the graphs in Figures and have very
di erent scales Our results clearly illustrate the utility of the planning bias intro duced by
min goals in our blo cksworld domain since on of our problems a solution exists in
the very small subspace preferred by min goals
These exp eriments do not show any advantage for ua as compared with to under the
heuristic which is consistent with our conclusions ab ove However this could equally well
b e b ecause min goals was so p owerful leading to solutions so quickly that smaller in uences
were obscured
The dramatic success of combining min goals with iterative sampling led us to consider
another search strategy iterative broadening which combines the b est asp ects of depth
rst search and iterative sampling This more sophisticated search strategy initially b ehaves
like iterative sampling but evolves into depth rst search as the breadth cuto increases
Langley Assuming that the solution is within the sp eci ed depth b ound iterative
broadening is complete In its early stages iterative broadening is distribution insen sitive
in its later stages it b ehaves like depth rst search and thus b ecomes increasingly sensitive
to solution distribution As one would exp ect from our iterative sampling exp eriments with
iterative broadening solutions were found very early on as shown in Figure Thus it is
not surprising that ua and to p erformed similarly under iterative broadening
We should p oint out that the results presented in this subsection are only illustrative
since they deal with only a single domain and with a single heuristic Nevertheless our
exp eriments do illustrate how the various prop erties we have identi ed in this pap er can
interact
Extending our Results
Having established our basic results concerning the e ciency of ua and to under various
circumstances we now consider how these results extend to other typ es of planners
More Expressive Languages
In the preceding sections we showed that the primary advantage that ua has over to is that
ua s search tree may b e exp onentially smaller than to s search tree and we also showed
that ua only pays a small p olynomial extra cost p er no de for this advantage Thus far we
have assumed a very restricted planning language in which the op erators are prop ositional
however most practical problems demand op erators with variables conditional e ects or
conditional preconditions With a more expressive planning language will the time cost
p er no de b e signi cantly greater for ua than for to One might think so since the work
required to identify interacting steps can increase with the expressiveness of the op erator
language used Dean Bo ddy Hertzb erg Horz If the cost of detecting step
and pick accordingly Given a depth b ound this strategy has the advantage of b eing asymptotical ly
complete We used the simpler strategy here for p edagogical reasons
Minton Bresina Drummond
interaction is high enough the savings that ua enjoys due to its reduced search space will
b e outweighed by the additional exp ense incurred at each no de
Consider the case for simple breadth rst search Earlier we showed that the ratio of
the total time costs of to and ua is as follows where the subtree considered by ua under
breadth rst search is denoted by bf tr ee the numb er of steps in plan a U is denoted
UA
by n and the numb er of edges in U is denoted by e
u u
P
O n j L U j
u
cost TO
U bf tr ee
bf
UA
P
cost UA O e
bf u
U bf tr ee
UA
This cost comparison is sp eci c to the simple prop ositional op erator language used so
far but the basic idea is more general ua will generally outp erform to whenever its cost
p er no de is less than the pro duct of the cost p er no de for to and the numb er of to no des
that corresp ond under L Thus ua could incur an exp onential cost p er no de and still
outp erform to in some cases This can happ en for example if the exp onential numb er of
linearizations of a ua partial order is greater than the exp onential cost p er no de for ua In
general however we would like to avoid the case where ua pays an exp onential cost p er
no de and instead consider an approach that can guarantee that the cost p er no de for ua
remains p olynomial as long as the cost p er no de for to also remains p olynomial
The cost p er no de for ua is dominated by the cost of up dating the goal set Step and
the cost of selecting the orderings Step Up dating the goal set remains p olynomial as
long as a plan is unambiguous Since each precondition in an unambiguous plan is either
necessarily true or necessarily false we can determine the truth value of a given precondition
by examining its truth value in an arbitrary linearization of the plan Thus we can simply
linearize the plan and then use the same pro cedure to uses for calculating the goal set
As a result it is only the cost of maintaining the unambiguous prop erty i e Step that
is impacted by more expressive languages One approach for e ciently maintaining this
prop erty relies on a conservative ordering strategy in which op erators are ordered if they
even possibly interact
As an illustration of this approach consider a simple prop ositional language with con
ditional e ects such as if p and q then add r Hence an op erator can add or delete
prop ositions dep ending on the state in which it is applied We refer to conditions such as
p in our example as dependency conditions Note that like preconditions dep endency
conditions are simple prop ositions Chapman showed that with this typ e of lan
guage it is NP hard to decide whether a precondition is true in a partially ordered plan
However as we p ointed out ab ove for the sp ecial case of unambiguous plans this decision
can b e accomplished in p olynomial time
Formally the language is sp eci ed as follows An op erator O as b efore has a list of pre
conditions pre O a list of unconditional adds adds O a list of unconditional deletes
dels O In addition it has a list of conditional adds cadds O and a list of conditional
deletes cdels O b oth containing pairs hD ei where D is a conjunctive set of dep en
e e
dency conditions and e is the conditional e ect either an added or a deleted condition
Analogous with the constraint that every delete must b e a precondition every conditional
delete must b e a memb er of its dep endency conditions that is for every hD ei cdels O
e
e D
e
Total Order and Partial Order Planning
Figure shows a version of the ua algorithm called ua c which is appropriate for this
language The primary di erence b etween the ua and ua c algorithms is that in b oth Steps
and b an op erator may b e specialized with resp ect to a set of dep endency conditions
The function sp ecialize O D accepts a plan step O and a set of dep endency conditions
D it returns a new step O that is just like O but with certain conditional e ects made
unconditional The e ects that are selected for this transformation are exactly those whose
dep endency conditions are a subset of D Thus the act of sp ecializing a plan step is the
act of committing to expanding its causal role in a plan Once a step is sp ecialized ua c
has made a commitment to use it for a given set of e ects Of course a step can b e further
sp ecialized in a later search no de but sp ecializations are never retracted
More precisely the de nition of O sp ecialize O D where O is a step D is a con
junctive set of dep endency conditions in O and n is the set di erence op erator is as follows
pre O pre O D
adds O adds O fe j hD ei cadds O D D g
e e
dels O dels O fe j hD ei cdels O D D g
e e
0 0
D nD g ei j hD ei cadds O D D D cadds O fhD
e e e
e e
0 0
D nD g ei j hD ei cdels O D D D cdels O fhD
e e e
e e
The de nition of step interaction is generalized for ua c as follows We say that two
steps in a plan interact if they are unordered with resp ect to each other and the following
disjunction holds
one step has a precondition or dependency condition that is added or deleted by the
other step or
one step adds a condition that is deleted by the other step
The di erence b etween this de nition of step interaction and the one given earlier is indi
cated by an italic font This mo di ed de nition allows us to detect interacting op erators
with a simple inexp ensive test as did our original de nition For example two steps that
are unordered interact if one step conditionally adds r and the other has precondition r
Note that the rst step need not actually add r in the plan so ordering the two op erators
might b e unnecessary In general our de nition of interaction is a su cient criterion for
guaranteeing that the resulting plans are unambiguous but it is not a necessary criterion
Figure shows a schematic example illustrating how ua c extends a plan The pre
conditions of each op erator are shown on the left of each op erator and the unconditional
adds on the right We only show the preconditions and e ects necessary to illustrate the
sp ecialization pro cess no deletes are used in the example Conditional adds are shown
For simplicity the mo di cations used to create ua c are not very sophisticated As a result ua c s space
may b e larger than it needs to b e in some circumstances since it aggressively commits to sp ecializati ons
A more sophisticated set of mo di cations is p ossible however the subtlies involved in e ciently planning
with dep endency conditions Pednault Collins Pryor Penb erthy Weld are largely
irrelevant to our discussion
Minton Bresina Drummond
UA C P G
Termination check If G is empty rep ort success and return solution plan P
Goal selection Let c select goal G and let O b e the plan step for which c is a precondition
need
Op erator selection Let O b e an op erator schema in the library that possibly adds c that is
add
either c adds O or there exists an hD ci cadds O In the former case insert step O and in
c add
the latter case insert step sp ecialize O D If there is no such O then terminate and rep ort
add c add
failure Choice point al l ways in which c can be added must be considered for completeness
a Ordering selection Let O b e the unconditional last deleter of c Order O after O and
del add del
b efore O
need
Rep eat until there are no interactions
Select a step O that interacts with O
int add
Order O either b efore or after O
int add
Choice point both orderings must be considered for completeness
0
Let P b e the resulting plan
b Op erator role selection While there exists a step O with unmarked conditional add hD ci
cadd c
and a step O with precondition c such that O is after O and there is no unconditional
use use cadd
deleter of c in b etween O and O
use cadd
Either mark hD ci or replace O with sp ecialize O D
c cadd cadd c
Choice point Both options must be considered for completeness
0 0
Goal up dating Let G b e the set of preconditions in P that are necessarily false
0 0
Recursive invo cation UA C P G
Figure The ua C planning algorithm
underneath each op erator For instance the rst op erator in the plan at the top of the
page has precondition p This op erator adds q and conditionally adds u if t is true The
gure illustrates two of the plans pro duced as a result of adding a new conditional op erator
to the plan In one plan the conditional e ects u s and t u are selected in the
sp ecialization pro cess and in the other plan they are not
The new step Step b requires only p olynomial time p er plan generated and the time
cost of the other steps are the same as for ua Hence as with our original ua algorithm
the cost p er no de for the ua c algorithm is p olynomial
to can also handle this language given the corresp onding mo di cations changing Step
and adding Step b and the time cost p er plan also remains p olynomial Moreover
the same relationship holds b etween the two planners search spaces tr ee is never larger
UA
than tr ee and can b e exp onentially smaller This example illustrates that the theoretical
TO
advantages that ua has over to can b e preserved for a more expressive language As we
p ointed out our de nition of interaction is a su cient criterion for guaranteeing that the
resulting plans are unambiguous but it is not a necessary criterion Nevertheless this
conservative approach allows interactions to b e detected via a simple inexp ensive syntactic
test Essentially we have kept the cost p er no de for ua c low by restricting the search space
it considers as shown in Figure ua c only considers unambiguous plans that can b e
generated via its conservative ordering strategy ua c is still a partial order planner and
In fact Step b b e implemented so that the time cost is O e using the graph traversal techniques
describ ed in Section As a result the ua c implementation and the corresp onding to c implementation
have the same time cost p er no de for this new language as they did for the original language O e and
O n resp ectively
Total Order and Partial Order Planning
q r O s p q O []t u O
r Add Operator: O [u s]
q r u r q O r O O r O p q [u s] s p q s s O O []t u t u
O O
Figure An example illustrating the ua c algorithm
it is complete but it do es not consider all partially ordered plans or even all unambiguous
partially ordered plans
The same trick can b e used for other languages as well provided that we can devise
a simple test to detect interacting op erators For example in previous work Minton et al
b we showed how this can b e done for a language where op erators can have variables in
their preconditions and e ects In the general case for a given ua plan and a corresp onding
to plan Steps and of the ua algorithm cost the same as the corresp onding steps of
the to algorithm As long as the plans considered by ua are unambiguous Step of the
ua algorithm can b e accomplished with an arbitrary linearization of the plan in which case
it costs at most O e more than Step of the to algorithm Thus the only p ossibility for
additional cost is in Step In general if we can devise a lo cal criterion for interaction
such that the resulting plan is guaranteed to b e unambiguous then the ordering selection
step can b e accomplished in p olynomial time By lo cal we mean a criterion that only
considers op erator pairs to determine interactions i e it must not examine the rest of the
plan
Although the theoretical advantages that ua has over to can b e preserved for more
expressive languages there is a cost The unambiguous plans that are considered may have
more orderings than necessary and the addition of unnecessary orderings can increase the
size of ua s search tree The magnitude of this increase dep ends on the sp eci c language
domain and problem b eing considered Nevertheless we can guarantee that ua s search
tree is never larger than to s
The general lesson here is that the cost of plan extension is not solely dep endent on
the expressiveness of the op erator language it also dep ends on the nature of the plans that
Minton Bresina Drummond
partially ordered plans
unambiguous partially ordered plans
unambiguous partially ordered plans produced by conservative ordering strategy
totally ordered
plans
Figure Hierarchy of Plan Spaces
the planner considers So although the extension of partially ordered plans is NP hard for
languages with conditional e ects if the space of plans is restricted e g only unambiguous
plans are considered then this worst case situation is avoided
Less Committed Planners
We have shown that ua a partial order planner can have certain computational advantages
over a total order planner to since its ability to delay commitments allows for a more
compact search space and p otentially more intelligent ordering choices However there
are many planners that are even less committed than ua In fact there is a continuum
of commitment strategies that we might consider as illustrated in Figure Total order
planning lies at one end of the sp ectrum At the other extreme is the strategy of maintaining
a total ly unordered set of steps during search until there exists a linearization of the steps
that is a solution plan
Compared to many well known planners ua is conservative since it requires each plan
to b e unambiguous This is not required by noah Sacerdoti NonLin Tate
Totally Completely Ordered Unordered
TO UA
Figure A continuum of commitment strategies
Total Order and Partial Order Planning
MT P G
Termination check If G is empty rep ort success and stop
Goal selection Let c select goal G and let O b e the plan step for which c is a precondition
need
Op erator selection Let O b e either a plan step p ossibly b efore O that adds c or an op erator
add need
in the library that adds c If there is no such O then terminate and rep ort failure
add
Choice point al l such operators must be considered for completeness
Ordering selection Order O b efore O Rep eat until there are no steps p ossibly b etween
add need
O and O which delete c
add need
Let O b e such a step cho ose one of the following ways to make c true for O
del need
Order O after O
del need
Cho ose a step O p ossibly O that adds c that is p ossibly b etween O and O
k nig ht add del need
order it after O and b efore O
del need
Choice point both alternatives must be considered for completeness
0
Let P b e the resulting plan
0 0
Goal up dating Let G b e the set of preconditions in P that are not necessarily true
0 0
Recursive invo cation MT P G
Figure A Prop ositional Planner based on the Mo dal Truth Criterion
nor Tweak Chapman for example How do these less committed planners compare
to ua and to One might exp ect a less committed planner to have the same advantages
over ua that ua has over to However this is not necessarily true As an example in
this section we intro duce a Tweak like planner called mt and show that its search space
is larger than even to s in some circumstances
Figure presents the mt pro cedure mt is a prop ositional planner based on Chapman s
Mo dal Truth Criterion Chapman the formal statement that characterizes Tweak s
search space It is straightforward to see that mt is less committed than ua The algorithms
are quite similar however in Step whereas ua orders all interacting steps mt do es not
Since mt do es not immediately order all interacting op erators it may have to add additional
orderings b etween previously intro duced op erators later in the planning pro cess to pro duce
correct plans
The pro of that ua s search tree is no larger than to s search tree rested on the two
prop erties of L elab orated in Section By investigating the relationship b etween mt and
to we found that the second prop erty the disjointness prop erty do es not hold for mt
and its failure illustrates how mt can explore more plans than to and consequently than
ua on certain problems The disjointness prop erty guarantees that ua do es not generate
overlapping plans The example in Figure shows that mt fails to satisfy this prop erty
b ecause it can generate plans that share common linearizations leading to considerable
redundancy in the search tree The gure shows three steps O O and O where each O
i
has precondition p and added conditions g p p and p The nal step has preconditions
i i
g g and g but the initial and nal steps are not shown in the gure At the top of the
gure in the plan constructed by mt goals g g and g have b een achieved but p p
and p remain to b e achieved Subsequently in solving precondition p mt generates plans
which share the linearization O O O among others In comparison b oth to and
We use Tweak for this comparison b ecause like ua and to it is a formal construct rather than a realistic
planner and therefore more easily analyzed
Minton Bresina Drummond
O1
O2
O3
O2 O1 O3 O1
O3 O2
O3 O2 O1 O3 O2 O1
KEY g g g 1 p p2 p3 1 p 1 1 p O 2 O2 p O 1 1 p p 3 3 p p2 p2 p2
3 3 3
Figure Overlapping plans
ua only generate the plan O O O once In fact it is simple to show that under
breadth rst search mt explores many more plans than to on this example and also more
than ua by transitivity due to the redundancy in its search space
This result may seem counterintuitive However note that the search space size for a
partial order planner is p otentially much greater than that of a total order planner since
there are many more partial orders over a set of steps than there are total orders Thus
when designing a partial order planner one may preclude overlapping linearizations in order
to avoid redundancy as discussed by McAllester Rosenblitt and Kambhampati
c
Of course one can also construct examples where mt do es have a smaller search space
than b oth ua and to Our example simply illustrates that although one planner may b e
less committed than another its search space is not necessarily smal ler The commitment
strategy used by a planner is simply one factor that in uences overall p erformance In
particular the e ect of redundancy in a partial order planner can overwhelm other con
siderations In comparing two planners one must carefully consider the mapping b etween
their search spaces b efore concluding that less committed smaller search space
Related Work
For many years the intuitions underlying partial order planning were largely taken for
granted Only in the past few years has there b een renewed interest in the fundamental
principles underlying these issues
Total Order and Partial Order Planning
Barrett et al and Barrett and Weld describ e an interesting and novel
analysis of partial order planning that complements our own work They compare a partial
order planner with two total order planners derived from it one that searches in the space
of plans and the other that searches in the space of world states Their study fo cuses
on how the goal structure of the problem a ects the e ciency of partial order planning
Sp eci cally they examine how partial order and total order planning compare for problems
with indep endent serializable and non serializable goals when using a resource b ounded
depth rst search They re ne Korf s work on serializable goals Korf intro ducing a
distinction b etween trivially serializable subgoals where the subgoals can b e solved in any
order without violating a previously solved subgoal and lab oriously serializable subgoals
where the subgoals are serializable but at least n of the orderings can cause a previously
solved subgoal to b e violated Their study describ es conditions under which a partial order
planner may have an advantage For instance they show that in a domain where the goals
are trivially serializable for their partial order planner and lab oriously serializable for their
total order planners their partial order planner p erforms signi cantly b etter
Our study provides an interesting contrast to Barret and Weld s work since we investi
gate the relative e ciencies of partial order and total order planning algorithms indep endent
of any particular domain structure Instead we fo cus on the underlying prop erties of the
search space and how the search strategy a ects the e ciency of our planners Nevertheless
we b elieve there are interesting relationships b etween the forms of serializabili ty that they
investigate and the ideas of solution density and clustering that we have discussed here
To illustrate this consider an arti cial domain that Barret and Weld refer to as D S
where in each problem the goals are a subset of fG G G g the initial conditions
are fI I I g and each op erator O has precondition I adds G and deletes
i i
i f g
I It follows that if a solution in D S contains op erators O and O where i j then O
i i j i
must precede O In this domain the goals are trivially serializable for their partial order
j
planner and lab oriously serializable for their total order planners thus the partial order
planner p erforms b est But note also that in this arti cial domain there is exactly one
solution p er problem and it is totally ordered Therefore it is immediately clear that if
we give ua and to problems from this domain then ua s search tree will generally b e
much smaller than to s search tree Since there is only single solution for b oth planners
the solution density for ua will clearly b e greater than that for to Thus the prop erties
we discussed in this pap er should provide a basis for analyzing how di erences in subgoal
serializibil i ty manifest their e ect on the search This sub ject however is not as simple as
it might seem and deserves further study
In other related work Kambhampati has written several pap ers Kambhampati a
b c that analyze the design space of partial order planners including the ua
planner presented here Kambhampati compares ua Tweak snlp McAllester Rosen
blitt ucpop Penb erthy Weld and several other planners along a variety of
dimensions He presents a generalized schema for partial order planning algorithms Kamb
hampati c and shows that the commitment strategy used in ua can b e viewed as a
way to increase the tractability of the plan extension or re nement pro cess Kambhampati
also carries out an empirical comparison of the various planning algorithms on a particu
lar problem Kambhampati a showing how the di erences in commitment strategies
a ects the e ciency of the planning pro cess He distinguishes two separate comp onents
Minton Bresina Drummond
of the branching factor b and b the former resulting from the commitment strategy for
t e
op erator ordering or in his terms the tractability re nements and the latter resulting
from the choice of op erator establishment re nements Kambhampati s exp eriments
demonstrate that while eager commitment strategies tend to increase b sometimes they
t
also decrease b b ecause the numb er of p ossible establishers is reduced when plans are more
e
ordered This is of course closely related to the issues investigated in this pap er
In addition Kambhampati and Chen have compared the relative utility of reusing
partially ordered and totally ordered plans in learning planners They showed that the
reuse of partially ordered plans rather than totally ordered plans result in storage com
paction b ecause they can represent a large numb er of di erent orderings Moreover partial
order planners have an advantage b ecause they can exploit such plans more e ectively than
total order planners In many resp ects these advantages are fundamentally similar to the
advantages that ua derives from its p otentially smaller search space
Conclusions
By fo cusing our analysis on a single issue namely op erator ordering commitment we have
b een able to carry out a rigorous comparative analysis of two planners We have shown
that the search space of a partial order planner ua is never larger than the search space of
a total order planner to Indeed for certain problems ua s search space is exp onentially
smaller than to s Since ua pays only a small p olynomial time increment p er no de over
to it is generally more e cient
We then showed that ua s search space advantage may not necessarily translate into
an e ciency gain dep ending in subtle ways on the search strategy and heuristics that are
employed by the planner For example our exp eriments suggest that distribution sensitive
search strategies such as depth rst search can b ene t more from partial orders than can
search strategies that are distribution insensi tive
We also examined a variety of extensions to our planners in order to demonstrate
the generality of these results We argued that the p otential b ene ts of partial order
planning may b e retained even with highly expressive planning languages However we
showed that partial order planners do not necessarily have smaller search spaces since
some less committed strategies may create redundancies in the search space In particu
lar we demonstrated that a Tweak like planner mt can have a larger search space than
our total order planner on some problems
How general are these results Although our analysis has considered only two sp eci c
planners we have examined some imp ortant tradeo s that are of general relevance The
analysis clearly illustrates how the planning language the search strategy and the heuristics
that are used can a ect the relative advantages of the two planning styles
The results in this pap er should b e considered as an investigation of the p ossible b ene ts
of partial order planning ua and to have b een constructed in order for us to analyze the
total order partial order distinction in isolation In reality the comparative b ehavior of two
planners is rarely as clear as witnessed by our discussion of mt While the general p oints
we make are applicable to other planners if we chose two arbitrary planners we would not
exp ect one planner to so clearly dominate the other
Total Order and Partial Order Planning
Our observations regarding the interplay b etween plan representation and search strat
egy raise new concerns for comparative analyses of planners Historically it has b een
assumed that representing plans as partial orders is categorically b etter than represent
ing plans as total orders The results presented in this pap er b egin to tell a more accurate
story one that is b oth more interesting and more complex than we initially exp ected
App endix A Pro ofs
A De nitions
This section de nes the terminology and notation used in our pro ofs The notion of plan
equivalence is intro duced here b ecause each plan step is by de nition a uniquely lab eled
op erator instance as noted in Section and Section Thus no two plans have the same
set of steps Although this formalism simpli es our analysis it requires us to de ne plan
equivalence explicitly
A plan is a pair h i where is a set of steps and is the b efore relation on
i e is a strict partial order on Notationally O O if and only if O O
For a given problem we de ne the search tree tr ee as the complete tree of plans
TO
that is generated by the to algorithm on that problem tr ee is the corresp onding
UA
search tree generated by ua on the same problem
Two plans P h i and P h i are said to b e equivalent denoted
P P if there exists a bijective function f from to such that
for all O O and f O are instances of the same op erator and
for all O O O O if and only if f O f O
A plan P is a step to extension or step ua extension of a plan P if P is
equivalent to some plan pro duced from P in one invo cation of to or ua
A plan P is a to extension or ua extension if either
P is the initial plan or
P is a step to extension or step ua extension of a to extension or ua
extension
It immediately follows from this de nition that if P is a memb er of tr ee or tr ee
TO UA
then P is a to extension or ua extension In addition if P is a to extension or
ua extension then some plan that is equivalent to P is a memb er of tr ee or
TO
tr ee
UA
P is a linearization of P h i if there exists a strict total order such that
and P h i
Given a search tree let par ent b e a function from a plan to its parent plan in the tree
Note that P is the parent of P denoted P par ent P only if P is a step
extension of P
Minton Bresina Drummond
Given U tr ee and T tr ee T L U if and only if plan T is a linearization
UA TO
of plan U and either b oth U and T are ro ot no des of their resp ective search trees or
par ent T L par ent U
The length of the plan is the numb er of steps in the plan excluding the rst and last
steps Thus the initial plan has length A plan P with n steps has length n
P is a subplan of P h i if P h i where
and
is restricted to i e
P is a strict subplan of P if P is a subplan of P and the length of P is less than
the length of P
A solution plan P is a compact solution if no strict subplan of P is a solution
A Extension Lemmas
TO Extension Lemma Consider totally ordered plans T h i and T h i
such that fO g and Let G b e the set of false preconditions in T
add
Then T is a step to extension of T if
c select goal G where c is the precondition of some step O in T and
need
O adds c and
add
O O and
add need
O O where O is the last deleter of c in T
del add del
Pro of Sketch This lemma follows from the de nition of to Given plan T with false
precondition c once to selects c as the goal to will consider all op erators that achieve c
and for each op erator to considers all p ositions b efore c and after the last deleter of c
UA Extension Lemma Consider a plan U h i pro duced by ua and plan
U h i such that fO g and Let G b e the set of false
add
preconditions of the steps in U Then U is a step ua extension of U if
c select goal G where c is the precondition of some step O in U and
need
O adds c and
add
is a minimal set of consistent orderings such that
and
O O and
add need
O O where O is the last deleter of c in U and
del add del
no step in U interacts with O
add
Total Order and Partial Order Planning
Pro of Sketch This lemma follows from the de nition of ua Given plan U with false
precondition c ua considers all op erators that achieve c and for each such op erator ua then
inserts it in the plan such that it is b efore c and after the last deleter ua then considers
all consistent combinations of orderings b etween the new op erator and the op erators with
which it interacts No other orderings are added to the plan
A Pro of of Search Space Corresp ondence L
Mapping Lemma Let U h i b e an unambiguous plan and let U h i
u u
b e a step ua extension of U If T h i is a linearization of U then there exists
t
a plan T such that T is a linearization of U and T is a step to extension of T
Pro of Since U is a step ua extension of U there is a step O such that
add
fO g Let T b e the subplan pro duced by removing O from T that is T h i
add add t
where Since it follows that T
t t u u t t
is a linearization of U
Using the TO Extension lemma we can show that T is a step to extension of T
First T is a linearization of U so the two plans have the same set of goals Therefore if
ua selects some goal c in expanding U to selects c in extending T Second it must b e
the case that O adds c since O is the step ua inserted into U to make c true Third
add add
O is b efore O in T since O is b efore O in U by de nition of ua and since
add need add need
T is a linearization of U Fourth O is after the last deleter of c O in T since O
add del add
is after O in U by de nition of ua and since T is a linearization of U Therefore the
del
conditions of the TO Extension lemma hold and thus T is a step to extension of T
Q E D
Totality Prop erty For every plan U in tr ee there exists a non empty set fT T g
m
UA
of plans in tr ee such that L U fT T g
m
TO
Pro of It su ces to show that if plan U is a ua extension and plan T is a linearization
of U then T is a to extension The pro of is by induction on plan length
Base case The statement trivially holds for plans of length
Induction step Under the hyp othesis that the statement holds for plans of length n we
now prove that the statement holds for plans of length n Supp ose that U is a ua
extension of length n and T is a linearization of U Let U b e a plan such that U is a
step ua extension of U By the Mapping lemma there exists a plan T such that T is a
linearization of U and T is a step to extension of T By the induction hyp othesis T
is a to extension Therefore by de nition T is also a to extension Q E D
Disjointness Prop erty L maps distinct plans in tr ee to disjoint sets of plans in tr ee
UA TO
that is if U U tr ee and U U then L U L U fg
UA
Pro of By the de nition of L if T T L U then T and T are at the same tree depth
d in tr ee furthermore U is also at depth d in tr ee Hence it su ces to prove that if
TO UA
plans U and U are at depth d in tr ee and U U then L U L U fg
UA
Base case The statement vacuously holds for depth
Induction step Under the hyp othesis that the statement holds for plans at depth n we
prove by contradiction that the statement holds for plans at depth n Supp ose that
Minton Bresina Drummond
there exist two distinct plans U h i and U h i at depth n in
tr ee such that T L U L U Then by de nition of L par ent T L par ent U
UA
and par ent T L par ent U Since par ent U par ent U contradicts the induction
hyp othesis supp ose that U and U have the same parent U Then by the de nition
of ua either i or ii and In the rst case since the two
plans do not contain the same set of plan steps they have disjoint linearizations and
hence L U L U fg which contradicts the supp osition In the second case
hence b oth plans resulted from adding plan step O to the parent plan Since
add
there exists a plan step O that interacts with O such that in one plan O
int add int
is ordered b efore O and in the other plan O is ordered b efore O Thus in either
add add int
case the linearizations of the two plans are disjoint and hence L U L U fg
which contradicts the supp osition Therefore the statement holds for plans at depth n
Q E D
A Completeness Pro of for TO
We now prove that to is complete under a breadth rst search control strategy To do so it
su ces to prove that if there exists a solution to a problem then there exists a to extension
that is a compact solution Before doing so we prove the following lemma
Subplan Lemma Let totally ordered plan T b e a strict subplan of a compact solution T
s
Then there exists a plan T such that T is a subplan of T and T is a step to extension
s
of T
Pro of Since T is a strict subplan of T and T is a compact solution the set of false
s s
preconditions in T G must not b e empty Let c select goal G let O b e the
need
step in T with precondition c and let O b e the step in T that achieves c Consider the
add s
totally ordered plan T h fO g i where Clearly T is a subplan of
add s
T Furthermore by the TO Extension Lemma T is a step extension of T by to To
s
see this note that O is ordered b efore O in T since it is ordered b efore O in T
add need need s
Similarly O is ordered after the last deleter of c in T since any deleter of c in T is a
add
deleter of c in T and O is ordered after the deleters of c in T Thus the conditions of
s add s
the TO Extension Lemma hold Q E D
TO Completeness Theorem If plan T is a totally ordered compact solution then T
s s
is a to extension
Pro of Let n b e the length of T We show that for all k n there exists a subplan of T
s s
with length k that is a to extension This is su cient to prove our result since any subplan
of exactly length n is equivalent to T The pro of is by induction on k
s
Base case If k the statement holds since the initial plan which has length is a
subplan of any solution plan
Induction step We assume that the statement holds for k and show that if k n the
statement holds for k By the induction hyp othesis there exists a plan T of length k
that is a strict subplan of T By the Subplan Lemma there exists a plan T that is b oth a
s
subplan of T and a step to extension of T Thus there exists a subplan of T of length
s s
k Q E D
Total Order and Partial Order Planning
A Completeness Pro of for UA
We now prove that ua is complete under a breadth rst search strategy The result follows
from the search space corresp ondence de ned by L and the fact that to is complete In
particular we show b elow that for any to extension T there exists a ua extension U such
that T is a linearization of U Since ua pro duces only unambiguous plans it must b e the
case that if T is a solution U is also a solution From this it follows immediately that ua
is complete
Inverse Mapping Lemma Let T h i b e a totally ordered plan Let T
t
h i b e a step to extension of T Let U h i b e a plan pro duced by ua such
t u
that T is a linearization of U Then there exists a plan U such that T is a linearization
of U and U is a step ua extension of U
Pro of By the de nition of to fO g where O added some c that is a
add add
false precondition of some plan step O in U Consider U h i where is a
need u u
minimal subset of such that
t
and
u u
O O and
add need u
O O where O is the last deleter of c in U and
del add u del
no step in U interacts with O
add
Since T is a linearization of U In addition U is an extension of U since
u t
it meets the three conditions of the UA Extension Lemma as follows First since c must
have b een the goal selected by to in extending T c must likewise b e selected by ua in
extending U Second O adds c since O achieves c in T Finally by construction
add add
satis es the third condition of the UA Extension Lemma Q E D
u
UA Completeness Theorem Let T b e a totally ordered compact solution Then there
s
exists a ua extension U such that T is a linearization of U
s s s
Pro of Since to is complete it su ces to show that if T is a to extension then there
exists a ua extension U such that T is a linearization of U The pro of is by induction on
plan length
Base case The statement trivially holds for plans of length
Induction step Under the hyp othesis that the statement holds for plans of length n we
now prove that the statement holds for plans of length n Assume T is a to extension
of length n and let T b e a plan such that T is a step to extension of T By
the induction hyp othesis there exists a ua extension U of length n such that T is a
linearization of U By the Inverse Mapping Lemma there exists a plan U that is b oth a
linearization of T and a step ua extension of U Since U is a step ua extension of
U it has length n Q E D
Minton Bresina Drummond
Acknowledgements
Most of the work present in this pap er was originally describ ed in two conference pap ers
Minton et al a We thank Andy Philips for his many contributions to this
pro ject He wrote the co de for the planners and help ed conduct the exp eriments We also
thank the three anonymous reviewers for their excellent comments
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