Automated Synthesis of Constrained Generators

Knowledge compilation is an emerging research area that focuses on "compi l ing" a problem solver's inefficient, explicit knowledge representat ion into more efficient, impl ic i t forms. This paper presents a technique that transforms a declarative problem description (specifying the problem but not how to solve i t ) into a reasonably efficient, generate-and-test problem solver. Our technique performs constraint incorporation, modifying the parameter generators so they only generate values that satisfy the problem constraints. Successful constraint incorporation depends upon choosing the right solut ion representation (i.e., the set of parameters). Having expressed a constraint in terms of a particular set of parameters, incorporation fails if the constraint is not factorable into constraints on the individual parameter generators. R ICK, a Refinement-based constraint Incorporator for Compil ing Knowledge, is a prototype program that compiles a problem specification into a problem solver using least commitment, topdown refinement to achieve constraint incorporat ion. R ICK refines an abstract solution representation to avoid premature commitment to representations that hinder constraint incorporat ion. R I C K is able to incorporate local constraints that constrain relatively small portions of the entire solution. We have tested these ideas by having RICK automatical ly construct a house floor planning problem solver. *The research reported here was supported in part by the Defense Advanced Research Projects Agency (DARPA) under Contract Number N00014-85-K-0116, in part by the National Science Foundation (NSF) under Grant Number DMC-8610507, and in part by the Center for Computer Aids to Industrial Productivity (CAIP), Rutgers University, with funds provided by the New Jersey Commission on Science and Technology and by CAIP's industrial members. The first author has also received support from IBM. The opinions expressed in this paper are those of the authors and do not reflect any policies, either expressed or implied, of any granting agency. * currently on leave at IBM Watson Research Center. 1 I n t r o d u c t i o n Many AI systems perform run time evaluation of explicitly represented domain knowledge to find a solution to a problem. Such systems make a particular tradeoff between the "explicitness of the representation and the efficiency of the computat ion" [Dietterich (ed.), 1986]. Expl ic i t ly represented knowledge is easier to gather, but more costly to evaluate. This observation has spawned a research area called knowledge compilation, which includes methods for "compil ing" an inefficient, explicit knowledge representation into more efficient, impl ic i t forms. The work described in this paper is being conducted wi th in the context of the KBSDE project [Tong, 1986]. The purpose of the project is to develop compilat ion techniques whose input is a declarative problem description that does not specify how to solve the problem. The problem description includes a set of problem constraints that must be satisfied by a solution. The compilation techniques we are developing produce a generateand-test problem solver. Such a problem solver must generate a solution that passes a set of tests, one for each problem constraint. We have attempted to develop a compilation method that places as few requirements as possible on the domain, so that the method is broadly applicable. Thus, we have made the very weak requirement that the problem be solvable using a generate-andtest algori thm. In addit ion to insisting that our compiler be general, we have also required that it produce reasonably efficient algorithms. Like computer language compilers, knowledge compilers can use optimization techniques to improve the performance of the resulting problem-solving system. One optimization technique for a generate-and-test architecture is test incorporation [Dietterich and Bennett, 1986]. Test incorporation involves test movement, constraint incorporation, or both. Test movement regresses tests back into the generator process to achieve early pruning wi thout affecting the correctness of the problem solver. Constraint incorporation [Tappel, 1980] modifies the generator so that it enumerates only those values which satisfy a particular problem constraint; we wi l l call such a generator a constrained generator. The test corresponding to the incorporated constraint can be removed from the generate-and-test problem solver. Constraint incorporation reduces the size of the problem solver's search space and results in a more efficient problem solver. ConBraudaway and Tong 583 s t ra in t i n c o r p o r a t i o n makes use o f e x p l i c i t l y represented d o m a i n knowledge (e.g. , spec i f ica t ions for sys tem c o m ponen ts ) , t hus d i s t i n g u i s h i n g i t f r o m more conven t i ona l code o p t i m i z a t i o n approaches w h i c h re ly solely on syn tac t i c know ledge (e .g . , d a t a dependencies a m o n g sys tem componen t s ) . T h i s paper focuses on c o n s t r u c t i n g cons t ra ined genera tors of hierarchical so lu t ions , i.e., so lu t ions t h a t have par ts (e.g. , a house w i t h r o o m s ) . H ie ra rch i ca l so lu t ions can be genera ted us ing co r respond ing l y h ie ra rch ica l genera tors (e.g. , F i g u r e 3) . T h a t is , so l u t i on pa r t s are genera ted b y sub-genera to rs , a n d p r i m i t i v e so lu t i on p a r a m eters are assigned values by pa rame te r genera tors . A p a r t i c u l a r solution representation is def ined by a set of p r i m i t i v e pa rame te rs . To i n c o r p o r a t e a cons t ra in t i n t o a h ie ra rch ica l genera tor requi res t h a t i t be loca l izab le to a n d i n c o r p o r a t e d i n one o r more o f the ( p r i m i t i v e ) parame te r genera tors . I n c o r p o r a t i n g cons t ra in t s i s d i f f i cu l t w h e n the so lu t i on rep resen ta t i on is " i n a p p r o p r i a t e " . A cons t ra in t expressed in te rms of a p a r t i c u l a r so lu t i on represen ta t ion can have a s t r u c t u r e t h a t does no t " m a t c h " t h a t o f the genera to r ; t h a t is , the cons t ra i n t m a y no t be fac to rab le i n t o cons t ra in t s on the i n d i v i d u a l pa rameter generators . We w i l l ca l l th i s d i f f i cu l t y the structure mismatch problem. T h i s paper demons t ra tes a sys tem wh i ch i nco rpo ra tes the cons t ra in t s wh i l e a v o i d i n g th i s s t r u c t u r e m i s m a t c h p r o b l e m . R I C K , a Ref inement -based cons t ra in t I n c o r p o r a t o r for C o m p i l i n g K n o w l e d g e , is a p r o t o t y p e p r o g r a m t h a t compi les a p r o b l e m solver us ing least c o m m i t m e n t , top-down refinement to achieve cons t r a i n t i n c o r p o r a t i o n . T h e least c o m m i t m e n t app roach helps to avo id a p r e m a t u r e c o m m i t m e n t to a representat i o n t h a t m a y lead t o the s t r u c t u r e m i s m a t c h p r o b l e m . Sect ion 2 of th is paper defines the class of doma ins for w h i c h our m e t h o d appl ies and i l l us t ra tes an example f r o m th i s class: the house f loor p l a n n i n g d o m a i n . Sect i o n 3 i l l us t ra tes t he s t r u c t u r e m i s m a t c h p r o b l e m in the house f l oo r p l a n n i n g d o m a i n . T h e f o u r t h sect ion discusses the concepts b e h i n d our app roach and i l l us t ra tes the i r i m p l e m e n t a t i o n i n R I C K . W e compare our wo rk w i t h re la ted research in Sect ion 5. F i na l l y , Sect ion 6 summar izes t he paper and some o f the l i m i t a t i o n s o f our app roach . 2 T h e p r o b l e m doma in P a r a m e t e r i n s t a n t i a t i o n d e s i g n p r o b l e m s . T h e K B S D E p ro jec t has focused on design doma ins where the p r o b l e m solver cons t ruc ts an a r t i f a c t t h a t satisfies a set o f p r o b l e m cons t ra in t s . M a n y design p rob lems can be v iewed as parameter instantiation p rob lems . T h e hierarchical structure of the a r t i f a c t is usua l l y p re -de te rm ined for a class of pa rame te r i n s t a n t i a t i o n p rob lems ; for i n s tance, a l l house f loor p lans consist o f rec tangu la r r ooms , w h i c h , i n t u r n , consist o f sides and corners . T h e design task r e m a i n i n g for the p r o b l e m solver i s to " f i l l i n " and " i n t e r c o n n e c t " the s t r u c t u r e by ass igning values to the unspeci f ied a r t i f a c t pa ramete rs in a w a y t h a t is cons istent w i t h the p r o b l e m cons t ra i n t s . R I C K cons t ruc ts a p r o b l e m solver for pa rame te r i n s t a n t i a t i o n p rob lems whose so lu t ions are compos i te ob jec ts (e.g. , house f loor p lans ) . T h e p r o b l e m cons t ra in t s are p resumed to be h a r d cons t ra in t s t h a t def ine feasible so lu t ions ( ra ther t h a n soft cons t ra in t s t h a t def ine the re la t ive o p t i m a l i t y o f feasible so lu t i ons ) . We also presume t h a t the design p r o b l e m is no t ove r -cons t ra ined : a so lu t i on t h a t satisfies a l l the cons t ra in t s can be f o u n d . A h o u s e f l o o r p l a n n i n g d o m a i n . T o i l l u s t r a te our ideas, we w i l l use the pa rame te r i n s t a n t i a t i o n p r o b l e m of c o n s t r u c t i n g "house floor p l a n s " . A house floor p lan is a t w o d i m e n s i o n a l , rec tangu la r house p laced a t the o r ig in o f an x y g r i d and h a v i n g rec tangu la r rooms as par ts ; such a f loor p lan is a b s t r a c t l y dep ic ted in F igu re 1. A l l leng ths a n d coord ina tes are m u l t i p l e s o f 1 f oo t . T h e solut i o n is a f loor p l an t h a t con ta ins a p rob lem-spec i f i c n u m ber o f r o o m s , and satisf ies a l l o f the p r o b l e m cons t ra in ts shown in F igu re 2 . C o n s t r a i n t s S C I , SC2 and SC3 are " l o c a l " cons t ra in t s since t hey cons t ra in each i n d i v i d u a l r o o m . C o n s t r a i n t s SC4 a n d SC5 are more " g l o b a l , " cons t r a i n i n g pairs o f rooms, and all r ooms , respect ive ly . O u r examples w i l l focus on the i n c o r p o r a t i o n o f the local con-