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Session 15 Robot Problem Solving PLANNING IN A HIERARCHY OF ABSTRACTION SPACES*
by Earl D. Sacerdott Stanford Research I n s t i t u t e A r t i f i c i a l I n t e l l i g e n c e Center Menlo Park, C a l i f o r n i a 94025
Abstract
A problem domain can be represented as a hierarchy of a b s t r a c t i o n spaces in which successively f i n e r levels of d e t a i l are introduced. The problem solver ABSTRIPS, a m o d i f i c a t i o n of STRIPS, can define an a b s t r a c t i o n space hierarchy from the STRIPS representation of a problem domain, and it can u t i l i z e the hierarchy in s o l v i n g problems. Examples of the system's performance are presented t h a t demonstrate the s i g n i f i c a n t i n - creases in problem-solving power that t h i s approach p r o v i d e s. Then some f u r t h e r I m p l i c a t i o n s of the h i e r - a r c h i c a l planning approach are explored.
Key Words: Problem s o l v i n g , h e u r i s t i c search, r e p r e s e n t a t i o n , a b s t r a c t i o n space, robot p l a n n i n g , h i e r a r c h i c a l p l a n n i n g.
I n t r o d u c t i o n
General purpose problem s o l v e r s , such as STRIPS s t
or GPS,^3 must do t h e i r work using general purpose search h e u r i s t i c s. U n f o r t u n a t e l y , by using such h e u r i s t i c s , it is not possible to solve any reasonably complex set of problems in a reasonably complex domain. Regardless of how good such h e u r i s t i c s are at d i r e c t i n g search, attempts to traverse a complex problem space can be caught in a combinatorial quagmire.
This paper presents an approach to augmenting the power of the h e u r i s t i c search process. The essence of t h i s approach is to u t i l i z e a means f o r d i s c r i m i n a t i n g between important i n f o r m a t i o n and d e t a i l s in the prob- lem space. By planning in a hierarchy of a b s t r a c t i o n spaces in which successive l e v e l s of d e t a i l are i n t r o - duced, s i g n i f i c a n t increases in problem-solving power have been achieved.
Section II sketches the h i e r a r c h i c a l planning ap- proach and gives m o t i v a t i o n f o r i t s use. Sections I I I and IV describe the d e f i n i t i o n and use of a b s t r a c t i o n spaces by ABSTRIPS (Abstraction-Based STRIPS), a modi- f i c a t i o n of the STRIPS problem-solving system that i n - corporates t h i s approach. Section V describes the per- formance of the system. Section VI discusses the i m p l i - c a t i o n s of t h i s approach f o r problem s o l v i n g and f o r r o b o t i c s.
The work reported herein was sponsored by the Advanced Research P r o j e c t s Agency of the Department of Defense under Contract DAHC04-72-C-0008 with the U.S. Army Research O f f i c e.
References are l i s t e d at the end of t h i s paper.
II The M o t i v a t i o n f o r Using Abstract i o n Spaces in Problem Solving
It was not q u i t e f a i r to assert in the previous sec- t i o n that a complex problem domain is beyond the combi- n a t o r i a l c a p a b i l i t y of general purpose problem solverB. A problem solver deals not w i t h the problem domain it- s e l f , but w i t h some r e p r e s e n t a t i o n of that domain. So it would be more c o r r e c t to state t h a t a complex r e p r e - sentation exceeds the scope of general purpose problem s o l v e r s.
U n f o r t u n a t e l y , a s t r a i g h t f o r w a r d t r a n s c r i p t i o n of a complex problem domain w i l l y i e l d a complex represen- t a t i o n. However, a well-chosen t r a n s c r i p t i o n can lead to a simpler r e p r e s e n t a t i o n. By choosing such a s i m p l i - f y i n g r e p r e s e n t a t i o n , one can have the problem solver do i t s work in a context t h a t is simple enough f o r some u s e f u l problem s o l v i n g to take p l a c e.
In other words, the h e u r i s t i c search through the s i m p l i f y i n g representation w i l l be of s u f f i c i e n t l y short d u r a t i o n t h a t a goal s t a t e in the problem space can be reached. Such a representation d i s p l a y s what McCarthy and Hayes term " h e u r i s t i c adequacy,"
Attempts to achieve s i m p l i f y i n g r e p r e s e n t a t i o n s , such as the "macro o p e r a t o r , " or MACROP, of the STRIPS problem s o l v e r , 2 have heretofore t r i e d to preserve, in McCarthy and Hayes' terminology, "epistemological ade- quacy"; t h a t i s , the s i m p l i f y i n g representations had to preserve a l l the d e t a i l t h a t was needed to solve the problem at hand. A MACROP s i m p l i f i e s the r e p r e s e n t a t i o n of a problem domain by p r o v i d i n g a means of s e l e c t i n g at one time an e n t i r e sequence of p r i m i t i v e o p e r a t o r s , l i n k e d in a semantically sensible manner. But it p r e - serves every d e t a i l of the preconditions and e f f e c t s of i t s c o n s t i t u e n t o p e r a t o r s.
Such s i m p l i f y i n g representations can provide only l i m i t e d enhancement to the power of a problem-solving system because of a somewhat dismaying f a c t ; For a suf- f i c i e n t l y complex problem domain, no e p i s t e m o l o g i c a l l y adequate r e p r e s e n t a t i o n can be h e u r i s t i c a l l y adequate.
Epistemological adequacy implies t h a t every r e l e - vant d e t a i l i s properly d e a l t w i t h. But a t t e n t i o n t o d e t a i l is p r e c i s e l y what defeats h e u r i s t i c adequacy. A good h e u r i s t i c e v a l u a t i o n f u n c t i o n w i l l enable a prob- lem solver to r e j e c t most of the possible paths in a s i t u a t i o n space. But i f a l l the d e t a i l s are attended t o , the e v a l u a t i o n f u n c t i o n must be applied at a l l the nodes at which the d e t a i l s are a f f e c t e d. The combina- t o r i c s of the expanding search space w i l l enable the problem solver to solve only r a t h e r simple problems.
A superior approach to problem s o l v i n g would be to search f i r s t through an a b s t r a c t i o n space, a s i m p l i f y i n g
representation of the problem space in which unimportant (2) A Set of Operator Descriptions—Each a c t i o n d e t a i l s are ignored. When a s o l u t i o n to the problem in in the problem domain is represented by an the a b s t r a c t i o n space is discovered, a l l that remains "operator" f o r changing one model i n t o an- is to account f o r the d e t a i l s of the linkup between the other. An operator is defined by a precon- steps of the s o l u t i o n. This can be regarded as a se- d i t i o n w f f , an add l i s t , and a delete l i s t. quence of subproblems in the o r i g i n a l problem space. For an operator to be applicable in a given If they can be solved, a s o l u t i o n to the o v e r a l l prob- m o d e l , i t s precondition wff must be s a t i s f i e d , lem w i l l have been achieved. If they cannot be solved, The add and delete l i s t s describe which wffs more planning in the a b s t r a c t i o n space is required to (^) a r e changed when an a p p l i c a t i o n of the opera- discover an a l t e r n a t i v e s o l u t i o n. (^) t o r transforms the world model.
PolyaB^ c i t e s the importance of t h i s approach f o r A problem is stated to STRIPS as a goal w f f. human problem s o l v i n g. It has been used by computer STRIPS must develop a sequence of operator a p p l i c a t i o n s programs to f i n d proofs in symbolic l o g i c 6 (ignoring that w i l l lead to a world model in which the goal wff the nature of the connectives and the ordering of sym- is t r u e. A GPS-like means-ends analysis s t r a t e g y 3 is bols as d e t a i l s ) and to detect edges in scenes^7 (using employed to generate the operator sequence. a shrunken p i c t u r e w i t h less d e t a i l ). A " d i f f e r e n c e " between the i n i t i a l model and the The concept can r e a d i l y be extended to a hierarchy goal model is e x t r a c t e d. STRIPS determines which i n - of spaces, each dealing with fewer d e t a i l s than the stances of which operators would reduce the d i f f e r e n c e ; ground space below it and w i t h more d e t a i l s than the the instance that most reduces the d i f f e r e n c e is se- i b s t r a c t i o n space above i t. B y considering d e t a i l s l e c t e d. I f i t i s applicable i n the i n i t i a l state ( i. e. , only when a successful plan in a higher l e v e l space i t s precondition wff is true in the i n i t i a l world gives strong evidence of t h e i r importance, a h e u r i s t i c model), the operator is a p p l i e d , and a new world model search process w i l l i n v e s t i g a t e a g r e a t l y reduced por- created. If the goal wff is true in the new model, t i o n of the search space. STRIPS is done. If not, the d i f f e r e n c e between the new state and the goal state is e x t r a c t e d , and the process The process of a b s t r a c t i o n defined in Section I I I continues, i s general i n t h a t i t i s not domain-dependent. But i t is h i g h l y s t r u c t u r e d and very dependent on the syntax if the operator instance that most reduced the o f the problem domain. I t i s a f i r s t step, providing d i f f e r e n c e i s not applicable i n the i n i t i a l s t a t e ( i. e. , no c a p a b i l i t y f o r a " r e p r e s e n t a t i o n a l s h i f t " that would i t s Drecondition wff is not provable in the world r e s t a t e a d i f f i c u l t problem in terms that render i t s model), the precondition is set up as a subgoal w f f. s o l u t i o n markedly e a s i e r. Rather, it employs a series STRIPS w i l l then t r y to develop a sequence of operator of r e p r e s e n t a t i o n a l nudges that increase the power of a p p l i c a t i o n s that w i l l lead to a world model in which the h e u r i s t i c search process over a problem space. the subgoal wff is t r u e. If the subgoal is achieved, the operator instance can be applied as before. If I I I Automated D e f i n i t i o n not, another operator instance is selected, and the of A b s t r a c t i o n Spaces process continues as b e f o r e.
The f o l l o w i n g sections describe the ABSTRIPS sys- A b s t r a c t i o n Spaces in the STRIPS Context tem, a m o d i f i c a t i o n of the STRIPS problem s o l v e r. 1 ' 2 A b r i e f d e s c r i p t i o n of the aspects of STRIPS that are For a p r a c t i c a l problem-solving system, one would relevant to the discussion to f o l l o w is presented be- l i k e to have an a b s t r a c t i o n space d i f f e r from i t s low.* The reader is encouraged to see Section II of ground space enough to achieve a s i g n i f i c a n t improve- Ref. 2 f o r a b r i e f but thorough summary of the opera- ment in problem-solving e f f i c i e n c y , but yet not so much t i o n of STRIPS or Ref. 1 f o r a f u l l d e s c r i p t i o n. as to make the mapping from a b s t r a c t i o n space to ground space complex and time-consuming. B r i e f l y , the r e p r e s e n t a t i o n of a problem domain w i t h which STRIPS deals consists o f : For the STRIPS system, t h i s c r i t e r i o n is met by having the a b s t r a c t i o n spaces d i f f e r from t h e i r ground (1) A World Model—The world model is a set of spaces only in the l e v e l of d e t a i l used to specify the wffs in the predicate c a l c u l u s , describing preconditions of operators. Although the change in f a c t s ( e. g. , CONNECTS(D00R1.R00M1,R00M2)) or representation provided by t h i s choice may seem i n t u i - laws ( e. g. , VRx,Ry,Dx CONNECTS(Dx,Rx,Ry) < = > t i v e l y i n s u f f i c i e n t , i t s a t i s f i e s the c r i t e r i o n w e l l. CONNECTS(Dx.Ry.Rx)) of the problem domain. The world model can remain unchanged; there Is no need to delete unimportant d e t a i l s from it because they can simply be ignored. No operators need be deleted in t h e i r e n t i r e t y ; i f a l l they d o i s achieve d e t a i l s , they w i l l never be selected as r e l e v a n t. Any change to the
- l n the i n t e r e s t s of b r e v i t y and c l a r i t y , no f u r t h e r add or delete l i s t s of the operators would cause the mention w i l l be made of the MACROPs in the STRIPS sys- operators' e f f e c t s to be very d i f f e r e n t in d i f f e r e n t tem. A MACROP is the r e s u l t of g e n e r a l i z i n g a p r e v i - spaces. Since the a p p l i c a b i l i t y of a p a r t i c u l a r oper- ously completed p l a n. Most of i t s v a l i d subsequences ator at some intermediate s t a t e might depend on any of operators can be extracted f o r use in f u r t h e r p l a n - e f f e c t s of any previously applied o p e r a t o r s , the map- n i n g. Each such subsequence could be treated by Ping of plans among spaces would be rendered too com-
p l a n. When a new problem is posed to ABSTRIPS, the e x t e r n a l I n t e r f a c e program sets the preconditions of a dummy operator to the goal wtt. The domain's maximum c r i t i c a l i t y , which was determined when c r i t i c a l i t i e B were assigned, is r e t r i e v e d. The executive is c a l l e d w i t h the c r i t i c a l i t y set to the maximum and the s k e l e - ton c o n s i s t i n g of the dummy operator.
W i t h i n the highest a b s t r a c t i o n space, the execu- t i v e plans to achieve the preconditions of the dummy step in the skeleton p l a n , i. e. , the main g o a l. When a plan is found, the executive computes the c r i t i c a l i t y of the next lowest space in which planning Is needed, and it b u i l d s a skeleton of nodes along the path of the successful p l a n. The executive then invokes i t s e l f r e - c u r s i v e l y. The new i n v o c a t i o n solves in t u r n the sub- problems of b r i d g i n g the gaps between steps in the skeleton plan and of ensuring t h a t the steps in the skeleton plan are s t i l l a p p l i c a b l e a t the appropriate p o i n t s In the new p l a n. The f i n a l step in the skeleton is always the dummy o p e r a t o r , and so the f i n a l a p p l i c a - b i l i t y check ensures that the o r i g i n a l goal has been reached. Mien a l l subproblems have been solved, the executive invokes i t s e l f f o r planning i n a s t i l l lower space. This r e c u r s i o n continues u n t i l a complete plan is b u i l t up in the problem space i t s e l f.
This search strategy might be termed a " l e n g t h - f i r s t " search. It pushes the planning process in each a b s t r a c t i o n space a l l the way to the o r i g i n a l goal s t a t e before beginning to plan in a lower space. This enables the system to recognize as e a r l y as possible the steps that would lead to dead ends or very i n e f f i c i e n t plans.
If any subproblem in a p a r t i c u l a r space cannot be solved, c o n t r o l i s returned t o the process i n i t s ab- s t r a c t i o n space. The search t r e e is restored to i t s s t a t e p r i o r to the s e l e c t i o n of the node t h a t led to f a i l u r e in the ground space. That node is eliminated from c o n s i d e r a t i o n , and the search f o r a successful plan at the higher l e v e l continues.
This f a i l u r e mechanism is analogous to the auto- matic backtracking feature of the PLANNER language. It has the major defect t h a t when a f a i l u r e of a lower l e v e l process is r e p o r t e d , the process and the context in which the f a i l u r e occurred are no longer around f o r a n a l y s i s. So ABSTRIPS r e l i e s h e a v i l y on being able to produce good plans at the highest l e v e l.
This requirement has led to two m o d i f i c a t i o n s to the search a l g o r i t h m o r i g i n a l l y employed by STRIPS. The f i r s t is an a l t e r a t i o n of the e v a l u a t i o n f u n c t i o n used to select which node in the search t r e e to expand next. STRIPS emphasizes the estimated cost of achiev- ing the goal from the given node and deemphasizes the cost of a r r i v i n g at the node from the i n i t i a l s t a t e. Thus, it has a tendency to f i n d a s l i g h t l y longer plan q u i c k l y , r a t h e r than the cheapest plan more s l o w l y. But each e x t r a step in a high a b s t r a c t i o n space is l i k e l y to lead to many e x t r a steps in the corresponding plan in the problem space. Thus, f o r ABSTRIPS, the e v a l u a t i o n f u n c t i o n has i t s e l f been made a f u n c t i o n of the l e v e l of a b s t r a c t i o n. At the highest l e v e l , AB- STRIPS gives equal weight to the cost of reaching a given node and to the estimated cost of reaching the goal from t h a t node. This e v a l u a t i o n f u n c t i o n changes
incrementally as the l e v e l of a b s t r a c t i o n decreases, u n t i l it reaches the old STRIPS f u n c t i o n at the l e v e l of the problem space.
The second m o d i f i c a t i o n involves postponing the s e l e c t i o n of one among several equivalent instances of a r e l e v a n t operator. During the process of s e l e c t i n g relevant operators to reduce a p a r t i c u l a r d i f f e r e n c e , a p a r t i a l i n s t a n t i a t i o n of the o p e r a t o r s ' parameters may occur. For example, if the d i f f e r e n c e were t h a t the robot was not in Room 3, then the operator "Go through a door i n t o a room" might be selected and i n - s t a n t i a t e d to "Go through a door i n t o Room 3. " The p r e - c o n d i t i o n s of t h i s operator would then be analyzed by the theorem prover to determine which door to choose. If several choices seem equally good to STRIPS ( 3 , e , , the states in which the various choices can be applied are e q u a l l y d i f f i c u l t t o reach), then i t would a r b i t r a r - i l y p i c k a door.
For ABSTRIPS, a l t e r n a t i v e i n s t a n t i a t i o n s in an ab- s t r a c t i o n space might appear e q u i v a l e n t , and yet one choice might be s u b s t a n t i a l l y superior when f u r t h e r de- t a i l s are considered. So ABSTRIPS defers i t s decision when more than one equivalent "best choice" of a r e l e - vant operator is found. The p a r t i a l l y i n s t a n t i a t e d r e l e v a n t operator (e.g., "Go through a door i n t o Room 3") is used in p l a n n i n g. When subsequent a n a l y s i s in a lower a b s t r a c t i o n space reveals a p r e f e r r e d i n s t a n t i a - t i o n , that i n s t a n t i a t i o n i s then chosen. I f t h i s selec- t i o n should eventually lead t o f a i l u r e , the other i n - s t a n t i a t i o n s can s t i l l be chosen through the backtrack- ing mechanism.
In summary, h i e r a r c h i c a l planning using a b s t r a c t i o n spaces in a " l e n g t h - f i r s t " search technique postpones extending the search tree through the l e v e l s concerned w i t h the d e t a i l e d preconditions o f a n operator u n t i l i t knows t h a t doing so w i l l be h i g h l y e f f e c t u a l in reaching the goal (because the operator l i e s along an almost c e r - t a i n l y successful p a t h ). By avoiding work on f r u i t l e s s branches of the search t r e e , the technique achieves s i g - n i f i c a n t e f f i c i e n c i e s in the f o r m u l a t i o n of complex p l a n s.
V Examples of ABSTRIPS' Performance
To c l a r i f y the issues raised and the way in which the ABSTRIPS system works, the system's performance is traced through some examples below. The ABSTRIPS sys- tem c o n s i s t s of some 370 BBN-LISP f u n c t i o n s , which run as compiled code on a PDP-10 computer. A l l the examples presented were drawn from the environment of the Stan- f o r d Research I n s t i t u t e mobile r o b o t. The domain con- s i s t s of seven rooms interconnected by doorways. Oper- a t o r s have been defined that model the r o b o t ' s a b i l i t y to navigate to any o b j e c t or l o c a t i o n w i t h i n a room, to push boxes w i t h i n a room or through a doorway, to n a v i - gate through a doorway, to block a doorway using a box, and to unblock a doorway. In a d d i t i o n , f i c t i t i o u s oper- ators have been defined to model the opening and c l o s i n g of doors; these actions are beyond the r o b o t ' s c a p a b i l i - t i e s. In a l l , 167 predicate c a l c u l u s w f f s have been de- f i n e d as axioms to model the robot domain.
The d e f i n i t i o n of the domain is e s s e n t i a l l y i d e n - t i c a l to the one used f o r the examples in the l a t e s t r e p o r t on the STRIPS system.^2
i n s t a n c e s were c o m p u t e d. The f i r s t o f t h e s e , PUSHB (B0X2.B0X1), was e x a m i n e d. I t s p r e c o n d i t i o n w f f i n t h i s a b s t r a c t i o n space was t r u e i n t h e i n i t i a l s t a t e ; B O t h e o p e r a t o r was a p p l i e d. T h i s r e s u l t e d i n a new s t a t e i n w h i c h t h e r o b o t , BOX1, and BOX2 were n e x t t o each o t h e r. T h e d i f f e r e n c e between t h i s s t a t e and t h e g o a l s t a t e was computed and f o u n d t o b e INR00M(R0B0T, R U N I ). Two r e l e v a n t o p e r a t o r i n s t a n c e s were f o u n d , and t h e f i r s t , G0THRUDR(Parl2,RUNI), was e x a m i n e d. ( P a r l 2 i s a n u n i n s t a n t i a t e d p a r a m e t e r. ) I t s p r e c o n d i t i o n w f f i n t h i s a b s t r a c t i o n s p a c e , TYPE(RUNI,ROOM) A TYPE (Parl2,DO0R> A ( E r y ) C 0 N N E C T S ( P a r l 2 , r y , R U N I ) , was s a t i s - f i e d when P a r l 2 was i n s t a n t i a t e d t o DUNIMYS. S o GUTHRUDR(DUNIMYS,RUNI) was a p p l i e d , and t h i s g e n e r a t e d a s t a t e i r w h i c h t h e g o a l w f f was t r u e. F i g u r e 4 ( b ) d e p i c t s t h e s e a r c h t r e e i n t h e h i g h e s t a b s t r a c t i o n s p a c e. The p o s i t i o n i n g o f t h e nodes s u g g e s t s t h e c o r - r e s p o n d e n c e t o t h e nodes i n t h e STRIPS s e a r c h t r e e.
A s k e l e t o n p l a n was b u i l t c o n s i s t i n g o f t h e nodes a t w h i c h t h e two o p e r a t o r s were a p p l i e d. The p l a n was:
PUSHB(BOX2,BOX1); GOTHRUDR(DUNIMYS,RUNI)
P l a n n i n g t h e n began i n t h e space o f c r i t i c a l i t y 5.
The f i r s t s u b g o a l was t h e p r e c o n d i t i o n w f f i n t h i s a b s t r a c t i o n space o f t h e f i r s t o p e r a t o r , PUSHB(B0X1, B 0 X 2 ). The d i f f e r e n c e b e t w e e n the i n i t i a l s t a t e and t h e one i n w h i c h t h e w f f was t r u e was INR0OM(ROBOT, RPDP). O p e r a t o r i n s t a n c e s r e l e v a n t t o r e d u c i n g t h i s d i f f e r e n c e were GOTHRUDR(Parl7,RPDP) and PUSHTHRUDR (ROBOT,Par20,RPDP). The p r e c o n d i t i o n w f f o f t h e f i r s t was t e s t e d , b u t i t was n o t c o m p l e t e l y s a t i s f i e d. T h e r e were s t i l l d i f f e r e n c e s INROOM(ROBOT,HMYS) o r INROOM (ROBOT,RCUC) b e f o r e GOTHRUDR(Parl7,RPDP) c o u l d be a p - p l i e d ( i. e. , t h e r o b o t was n o t y e t i n a room a d j o i n i n g RPDP). The PUSHTHRUDR o p e r a t o r was c o m p l e t e l y i n a p p l i - c a b l e because t h e r o b o t i s n o t a p u s h a b l e o b j e c t.
Then ABSTRIPS t r i e d t o r e d u c e t h e d i f f e r e n c e s t h a t w o u l d r e n d e r G0THRUDR(Parl7,RPDP) a p p l i c a b l e. F o u r r e l e v a n t o p e r a t o r s were f o u n d. The f i r s t was GOTKRUDR (Par22,RMYS), and i t s p r e c o n d i t i o n w f f was n o t s a t i s f i e d
e i t h e r ( t h e r o b o t was n o t i n a room a d j o i n i n g RMYS). The second r e l e v a n t o p e r a t o r was G0THRUDR(Par22,RCLK), and i t s p r e c o n d i t i o n w f f was s a t i s f i e d when P a r 2 2 was i n s t a n t i a t e d to DCLKRIL. So GOTHRUDR(DCLKRIL,RCLK) was a p p l i e d , p r o d u c i n g a s t a t e i n w h i c h GOTKRUDR(DPDPCLK, RPDP) was a p p l i c a b l e. T h a t o p e r a t o r was a p p l i e d , p r o - d u c i n g a s t a t e i n w h i c h t h e i n i t i a l s u b g o a l , t h e p r e - c o n d i t i o n w f f of PUSHB(B0X2,B0X1), was t r u e. The PUSHB o p e r a t o r was t h e n a p p l i e d.
Then a new s u b g o a l was s e t u p , i n w h i c h t h e p r e c o n - d i t i o n s o f GOTHRUDR(DUNIMYS,RUNI) i n t h i s space were t r u e. The d i f f e r e n c e b e t w e e n t h e c u r r e n t s t a t e and t h e s u b g o a l s t a t e was INR00M(ROBOT,RMYS). G0THRUDR(Par27, RMYS) was s e l e c t e d a s a r e l e v a n t o p e r a t o r , and i t s p r e - c o n d i t i o n s were s a t i s f i e d when Par27 was bound t o DMVSPDP. So GOTHRUDR(DMYSPDP,RMYS) was a p p l i e d , p r o - d u c i n g a s t a t e i n w h i c h t h e s u b g o a l was s a t i s f i e d. The o p e r a t o r a s s o c i a t e d w i t h t h i s s u b g o a l , GOTHRUDR(DUNIMYS, R U N I ) , was a p p l i e d , and t h e g o a l s t a t e was a g a i n r e a c h e d. F i g u r e 4 ( c ) shows t h e s e a r c h t r e e s i n t h i s s p a c e.
The f o l l o w i n g new s k e l e t o n p l a n was b u i l t u p : GOTHRUDR(DCXKRIL,RCLK); GOTHRUDR(DPDPCLK,RPDP); PUSHB (B0X2.BOX1); G0THRUDR(DMYSPDP,RMYS);GOTHRUDR(DUNIMYS, R U N I ). The p l a n n i n g p r o c e s s was t h e n r e i n v o k e d i n a n a b s t r a c t i o n space o f c r i t i c a l i t y 2.
The f i r s t s u b g o a l , t h e p r e c o n d i t i o n w f f o f t h e f i r s t s t e p i n t h e s k e l e t o n p l a n , GOTHRUDR(DCLKRIL,RCLK), was n o t s a t i s f i e d i n t h e i n i t i a l m o d e l. The d i f f e r e n c e was STATUS(DCLKRIL,OPEN). A n a n a l y s i s showed t h a t i t c o u l d be e l i m i n a t e d by a p p l y i n g GOTOD(DCLKRIL) and t h e n OPEN ( D C L K R I L ). T h i s r e s u l t e d i n a s t a t e t h a t s a t i s f i e d t h e f i r s t s u b g o a l. S o GOTHRUDR(DCLKRIL,RCLK) was a p p l i e d.
Each o f t h e r e m a i n i n g s u b g o a l s o f t h e p r o c e s s I n t h i s a b s t r a c t i o n space were i m m e d i a t e l y s a t i s f l a b l e , and s o each s t e p o f t h e s k e l e t o n p l a n was a p p l i e d i n t u r n , r e s u l t i n g i n a s t a t e i n w h i c h t h e o r i g i n a l g o a l was s a t i s f i e d. The s k e l e t o n p l a n p r o d u c e d was GOTOD (DCUCRIL); OPEN (DCUCRIL), f o l l o w e d b y a l l t h e s t e p s o f t h e p r e v i o u s s k e l e t o n p l a n. F i g u r e 4 ( d ) shows t h e s e a r c h t r e e s i n t h i s s p a c e.
F i n a l l y , planning took place in the ground space, the space i n c l u d i n g l i t e r a l s of c r i t i c a l i t y 1. The f i r s t three steps of the skeleton plan were applied in t u r n. But the preconditions of GOTHRUDR(DPDPCLK.RPDP) were not s a t i s f i a b l e in a state in which the robot had j u s t come through DCIKRIL. The d i f f e r e n c e was NEXTTO (ROBOT,DPDPCUC). and analysis i n d i c a t e d t h a t it could be eliminated by applying GOTOD(DPDPCLK), enabling GOTHRUDH(DPDPCIiC,RPDP) to be a p p l i e d.
The next subgoal, the preconditions of PUSHB(BOX2, BOXl), was not s a t i s f i e d at t h i s p o i n t. The d i f f e r e n c e was NEXTTO(ROBOT,BOX2), which could be eliminated by an a p p l i c a t i o n of the first r e l e v a n t operator selected, GOTOB(BOX2). A f t e r P0SHB(B0X2,B0X1) was a p p l i e d , the next two subgoals f a i l e d because the robot was not next to the appropriate door. An analysis s i m i l a r to the one t h a t occurred w i t h DPDPC1X was performed, enabling ABSTRIPS to f i n i s h the plan w i t h an operator to go to and an operator to go through DMYSPDP and DUNIMYS.
Note that the planning in t h i s space is Just as if STRIPS were given seven small problems to solve consec- u t i v e l y , without the b e n e f i t of MACROPS. The search trees f o r the ground space are shown in Figure 4 ( e ). The e n t i r e planning process f o r ABSTRIPS produced 60 nodes, 54 of which were on the successful path in one space or another. This process required 5:28 of com- puter time. This is less than o n e - f i f t h of the time required by the nonhierarchical STRIPS.
Other ExampleB
The set of tasks from the most recent r e p o r t on STRIPS^3 was run on ABSTRIPS. The running times and the search trees are compared w i t h those from the STRIPS system in Table 1. Figure 5 p l o t s the planning time as a f u n c t i o n of plan length f o r STRIPS and ABSTRIPS on an extended set of problems from the robot domain.
i s t h e d e s c r i p t i o n o f t h e p r o b l e m d o m a i n ) , can i n c o r - p o r a t e t h a t knowledge i n t o i t s s e a r c h h e u r i s t i c s?
The p r o c e s s o f automated d e f i n i t i o n o f a b s t r a c t i o n space o f f e r s a p o s s i b l e a p p r o a c h. By a p p l y i n g a g e n e r a l p u r p o s e p r o b l e m s o l v e r t o a p a r t i c u l a r domain i n t h e most g e n e r a l manner d e s c r i b e d i n S e c t i o n 1 1 1 , a t a s k - s p e c i f i c d e t a i l h i e r a r c h y can b e b u i l t u p. The a b i l i t y o f a system t o d i s c r i m i n a t e i m p o r t a n t c o n s i d e r a t i o n s f r o m mere d e t a i l s i s a n i m p o r t a n t a s p e c t o f t a s k - s p e c i f i c k n o w l e d g e.
A f u r t h e r a s p e c t o f t a s k - s p e c i f i c knowledge i s t h e f a c i l i t y f o r n e g o t i a t i n g those a r e a s o f t h e s e a r c h space t h a t a r e e a s i l y t r a v e r s i b l e. i n t h e h i e r a r c h i c a l r e p r e s e n t a t i o n f r a m e w o r k , e a s i l y t r a v e r s i b l e a r e a s c o r - r e s p o n d t o s u b p r o b l e m s o f a c h i e v i n g d e t a i l s , once the more c r i t i c a l a s p e c t s o f a p r o b l e m have been s o l v e d.
The AliSTRIPS system d e t e r m i n e s t h a t a R i v e n l i t e r a l i s a d e t a i l when i t has b u i l t a s m a l l n l a n t o a c h i e v e a s t a t e i n w h i c h i t i s t r u e. T h a t s m a l l p l a n can b e saved as a MACROP, to be used as t h e f i r s t - c h o i c e r e l e v e n t o p e r a t o r whenever t h e d e t a i l needs t o b e a c h i e v e d. The r e l a t i v e l y s m a l l number o f MACROPs lormed i n t h i s way, when added t o t h e s e t o f b a s i c o p e r a t o r s , c o n s t i t u t e a b a s i c body o f knowledge about how t o s o l v e problems i n a p a r t i c u l a r t a s k d o m a i n.
P l a n n i n g w i t h M u l t i p l e Outcome O p e r a t o r s
The use of a h i e r a r c h i c a l r e p r e s e n t a t i o n can g r e a t l y s i m p l i f y t h e p r o c e s s o f c r e a t i n g c o n d i t i o n a l p l a n s , p l a n s w i t h i n f o r m a t i o n g a t h e r i n g o p e r a t o r s , and p l a n s w i t h l o o p s. T h i s i s because the outcomes o f these o p e r a t o r s a r e u n c e r t a i n o n l y t o a p a r t i c u l a r l e v e l o f d e t a i l. T h u s , i n a h i g h e r a b s t r a c t i o n space a s i m p l e s p e c i f i c a t i o n can a d e q u a t e l y model the p r e c o n d i t i o n s and e f f e c t s o f t h e o p e r a t o r s , a l t h o u g h some o f t h e e f - f e c t s may have t o b e d e s c r i b e d i n terms o f u n i n s t a n t i - a t e d p a r a m e t e r s. A drawback t o t h i s approach i s t h a t , a s n o t e d i n S e c t i o n I I I , the mapping o f p l a n s among spaces becomes d i f f i c u l t when t h e e f f e c t s o f o p e r a t o r s are a b s t r a c t e d , N e v e r t h e l e s s , t h e s i m p l i c i t y o f r e p r e - s e n t a t i o n o f t h e s e r a t h e r complex o p e r a t o r s r e n d e r s t h i s scheme a t t r a c t i v e.
A s a n e x a m p l e , i n p l a n n i n g t o d r i v e t o t h e a i r p o r t to c a t c h a p l a n e , one would use a " P a r k the c a r " o p e r a - t o r. Such a n o p e r a t o r m i g h t have t h e e f f e c t o f " I f L o t A i s n o t f u l l , p a r k i n s i d e L o t A. E l s e i f L o t B i s n o t f u l l , p a r k i n s i d e L o t B. E l s e d r i v e a r o u n d , and t h e n p a r k t h e c a r. " I f one p l a n s a t a h i g h l e v e l o f a b s t r a c - t i o n t o d r i v e t o t h e a i r p o r t , h e does n o t c o n s i d e r t h e " P a r k t h e c a r " o p e r a t o r i n i t s f u l l c o m p l e x i t y. R a t h e r , h e c o n s i d e r s a n image o f t h e o p e r a t o r i n a n a b s t r a c t i o n space i n w h i c h n o u n c e r t a i n t i e s e x i s t. I t m i g h t have t h e s i m p l e p r e c o n d i t i o n A t ( C a r , A i r p o r t ) and m i g h t cause t h e c l a u s e P a r k e d - i n - l o t ( C a r , P a r a m e t e r 3 7 ) t o b e added t o t h e m o d e l. F u r t h e r p l a n n i n g c o u l d c o n t i n u e w i t h o u t c o n s i d e r i n g a s s e p a r a t e cases s t a t e s i n w h i c h P a r k e d - i n - l o t ( C a r , L o t A ) o r P a r k e d - i n - l o t ( C a r , L o t B ) were t r u e.
A n I n t e g r a t e d Robot System
A p r i m a r y m o t i v a t i o n f o r b u i l d i n g t h e STRIPS s y s - t e m , and I t s o f f s p r i n g ABSTRIPS, was t o b u i l d p l a n s f o r
a m o b i l e r o b o t. I n the S t a n f o r d Research I n s t i t u t e r o b o t s y s t e m , t h e o p e r a t o r d e s c r i p t i o n s a r e models f o r a c t i o n s t h a t t h e r o b o t can a c t u a l l y t a k e. The a c t i o n s modeled are termed " i n t e r m e d i a t e l e v e l a c t i o n s " ( I L A s ). When t h e y are e x e c u t e d , t h e y i n v o k e " l o w l e v e l a c t i o n s " ( L L A s ) , w h i c h are c o n c e r n e d w i t h i n i t i a t i n g and m o n i t o r - i n g m o t i o n o f t h e r o b o t. Those r o u t i n e s i n t u r n pass commands t o , and r e c e i v e i n f o r m a t i o n f r o m , a program i n a PDP-15 c o m p u t e r , w h i c h communicates w i t h t h e r o b o t i t s e l f v i a a r a d i o l i n k.
The ground space as viewed by ABSTRIPS is in f a c t j u s t a n o t h e r a b s t r a c t i o n space f r o m t h e p o i n t o f v i e w o f p l a n s b u i l t u p f r o m b a s i c o p e r a t i o n s a t l o w e r l e v e l s. The p r o b l e m s o l v e r can be e x t e n d e d to h a n d l e s u c c e s s - i v e l y f i n e r l e v e l s o f d e t a i l u n t i l a ground space i s reached i n w h i c h t h e o n l y r e m a i n i n g d e t a i l s are t o r o l l the r o b o t a r o u n d. T h i s o f f e r s the e n t i c i n g p o s s i b i l i t y o f a f u l l y i n t e g r a t e d p l a n n i n g and e x e c u t i o n s y s t e m. But t h e i n t e r a c t i o n o f p l a n n i n g and e x e c u t i o n would r e - q u i r e t h a t t h e p l a n s t h a t such a system b u i l t b e d i f - f e r e d f r o m the t r a d i t i o n a l f o r m o f p l a n b u i l t b y p r o b - lem s o l v e r s.
For a system t h a t d e a l s w i t h complex p r o b l e m s i n u r e a l w o r l d , a s opposed t o a s i m u l a t e d o n e , i t i s u n - d e s i r a b l e t o s o l v e a n e n t i r e problem w i t h a n e p i s t e m o - l o g i c a l l y adequate p l a n. There are t o o many r e a s o n a b l y l i k e l y outcomes f o r each r e a l - w o r l d o p e r a t i o n. The num- b e r o f h y p o t h e t i e a l l y p o s s i b l e s t a t e s o f the w o r l d a t - t a i n a b l e b y a p a r t i c u l a r p l a n w i l l grow e x p o n e n t i a l l y w i t h t h e l e n g t h o f the p l a n. Most o f the e f f o r t o f such a system would be Spent r e a s o n i n g a b o u t w o r l d s t a t e s t h a t would never b e a c h i e v e d , and v e r y l i t t l e o f i t would b e spent moving the r o b o t t o w a r d i t s R o a l s.
I t i s d e s i r e d t h a t the s y s t e m ' s p l a n n i n g e f l o r t s f o c u s o n r e a s o n i n g a b o u t s t a t e s o f t h e w o r l d t h a t a r e l i k e l y t o b e t r a v e r s e d i n t h e c o u r s e o f r o b o t e x e c u t i o n. T h u s , t h e o v e r a l l p l a n n i n g s h o u l d b e roughed o u t i n a n a b s t r a c t i o n space t h a t i g n o r e s enough l e v e l s o f d e t a i l s o t h a t the rough p l a n i s f a i r l y c e r t a i n t o s u c c e e d.
A few s t e p s of the p l a n can be used as a s k e l e t o n , t o w h i c h more d e t a i l e d s t e p s are added i n a manner s i m - i l a r t o ABSTRIPS. These new s t e p s a r e f a i r l y c e r t a i n t o succeed a t t h e l e v e l o f d e t a i l t o w h i c h t h e y a r e s p e c i f i e d. Even more d e t a i l e d s t e p s can b e f i l l e d i n f o r the b e g i n n i n g p o r t i o n o f t h i s s u b p l a n , and t h e p r o c e s s can c o n t i n u e u n t i l a s h o r t s u b p l u n o f l o w - l e v e l r o b o t commands i s b u i l t. These can b e e x e c u t e d i n s e - q u e n c e. Any d e v i a t i o n s between t h e a c t u a l s t a t e o f t h e w o r l d and t h e h y p o t h e s i z e d r e s u l t s o f the s u b p l a n w i l l h o p e f u l l y b e mere d e t a i l s t o the space t h a t i s a n a b - s t r a c t i o n o f t h e r o b o t commands. T h u s , the r e m a i n i n g s t e p s o f t h e p l a n i n t h i s space, a s w e l l a s a l l h i g h e r s p a c e s , a r e s t i l l o n t h e s o l u t i o n p a t h.
F u r t h e r b u i l d i n g and e x t e n d i n g o f the v a r i o u s s u b - p l a n s can t h e n t a k e p l a c e , i n c l u d i n g a new b o t t o m - l e v e l s u b p l a n t o move t h e r o b o t. T h i s s u b p l a n w i l l a c c u r a t e l y r e f l e c t t h e p r e c i s e r e s u l t s o f p r e v i o u s e x e c u t i o n , and s o i t w i l l b e f u l l y a p p r o p r i a t e f o r a c h i e v i n g t h e u l t i - mate g o a l. The p r o c e s s o f a l t e r n a t i v e l y a d d i n g d e t a i l e d s t e p s t o t h e p l a n and t h e n a c t u a l l y e x e c u t i n g some s t e p s can c o n t i n u e u n t i l t h e g o a l i s a c h i e v e d.
If a grievous f a i l u r e occurB at some point in exe- cution and nondetalls in higher models no longer r e f l e c t the actual state of the world, subplans at affected levels of d e t a i l can propagate the f a i l u r e up to an abstraction space in which the deviation from the pre- dicted world model was a d e t a i l. Replannlng can be i n i t i a t e d from t h i s level of abstraction, thus reusing the results of as much as possible of the previous planning.
Therefore, by using a hierarchy of abstraction spaces to mask uncertainties in the real world effects of planned operations, an e f f e c t i v e l y integrated robot planning and executing system can be created. By deal- ing with a hierarchy of short, simple plans, such a system w i l l be able to cope e f f e c t i v e l y with t r u l y com- plex problems.
Acknowledgements
The author is indebted to Richard Fikes, Peter Hart, and Nils Nilsson for t h e i r enthusiastic encour- agement and i n t e l l e c t u a l support. The research r e - ported in t h i s paper was supported by the Advanced Research Projects Agency of the Department of Defense under Contract DAHC04-72-C-0008 with the U.S. Army Re- search Office.
References
- R. E. Fikes and N. J. Nilsson, "STRIPS: A New Approach to the Application of Theorem Proving to Problem Solving," A r t i f i c i a l I n t e l l i g e n c e , Vol. 2, Nos. 3/4, pp. 189-208 (1971).
- R. E. Fikes, P. E. Hart, and N. J. Nilsson, "Learn- ing and Executing Generalized Robot Plans," A r t i f i - c i a l I n t e l l i g e n c e , Vol. 3, pp. 251-288 (1972). 3. G. Ernst and A. Newell, GPS; A Case Study in Gen- e r a l i t y and Problem Solving, ACM Monograph Series (Academic Press, New York, New York, 1969).
- J. McCarthy and P. Hayes, "Some Philosophical Prob- lems from the Standpoint of A r t i f i c i a l I n t e l l i - gence," in Machine Intelligence 4, B. Meltzer and D. Michie, eds., pp. 463-502 (American Elsevier Publishing Company, New York, New York, 1969.
- G. Polya, How to Solve I t , p. 8 (Princeton Univer- s i t y Press, Princeton, New Jersey, 1945).
- A. Newell, J. C. Shaw, and H. A. Simon, "Report on a General Problem Solving Program," Proceedings of the International Conference on Information Process- ing, UNESCO, Paris, pp. 256-264 (1960),
- M. D. Kelly, "Edge Detection in Pictures by Com- puter Using Planning," in Machine Intelligence 6, B. Meltzer and D. Michie, eds., pp. 397-409 (Amer- ican Elsevier Publishing Company, New York, New York, 1971).
- C, Hewitt, "Description and Theoretical Analysis (Using Schemata) of PLANNER: A Language f o r Proving Theorems and Manipulating Models in a Robot," Ph.D.
Thesis, Department of Mathematics, Massachusetts I n s t i t u t e of Technology, Cambridge, Massachusetts (1972).
9, B. Buchanan, G. Sutherland, and E. Feigenbaum, "HEURISTIC DENDRAL: A Program for Generating Explanatory Hypotheses in Organic Chemistry," in Machine Intelligence 4, B. Meltzer and D. Michie, eds., pp. 209-254 (American Elsevier Publishing Company, New York, New York, 1969.
