Scott Domains Generalized Ultrametric Spaces and Generalized Acyclic Logic Programs

Every Scott domain can be viewed as a generalized ultrametric space with proper ties which allow to apply a generalization of the Banach contraction mapping theorem We will give this construction in detail and apply it to a class of programs which is stricly larger than the class of all acyclic programs The paper is taken from Hit Chapter Domains as Generalized Ultrametric Spaces We rst introduce Scott domains and generalized ultrametric spaces The following is taken from SLG De nition Scott Ershov domain A partially ordered set A is called consistent if it has an upper bound and is called directed if every nite subset of A has an upper bound in A A partial ordered set D is called a complete partial order cpo if there exists D such that for all a D we have a is called the bottom element of D and if A D is a directed set then supA exists in D An element c of a cpo is called compact or nite if for every directed set A D with c supA there is some a A with c a We denote the set of all compact elements of D by Dc A cpo D is called a Scott Ershov domain if for every a D the set approx a fc Dc j c ag is directed a sup approx a D is algebraic and every consistent set in D has a supremum D is consistently complete Intuitively x y in a domain can be interpreted as x approximates y Compact elements can be considered as practically implementable objects in a computer system so that every object of interest can be arbitrarily closely approximated by those The following is taken from PR De nition generalized ultrametric space Let X be a set and let be a partial order with least element We call X d a generalized ultrametric space if d X X is a function such that for all x y z X d x y if and only if x y d x y d y x and if d x y d y z then d x z For and x X the set B x fy X j d x y g is called a ball in X A generalized ultrametric space is called spherically complete if for any chain C of balls in X T C A function f X X is called contractive if d f x f y d x y for all x y X strictly contracting on orbits if d f x f x d f x x for every x X with x f x and strictly contracting if d f x f y d x y for all x y X with x y We will need the following observations which are well known for ultrametric spaces Lemma Let X d be a generalized ultrametric space For and x y X the following statements hold If and B x B y then B x B y If B x B y then B x B y Bd x y x Bd x y y Proof Let a B x and b B x B y Then d a x and d b x hence d a b Since d b y we have d a y hence a B y which proves the rst statement The second follows by symmetry and the third by replacing by d x y The following theorem was given in PR in a more general form Theorem Priess Crampe and Ribenboim Let X d be a spherically com plete generalized ultrametric space and let f X X be contractive and strictly contracting on orbits Then f has a xed point Moreover if f is strictly contracting on X then f has a unique xed point Proof Assume that f has no xed point Then d x f x for all x X We de ne the set B fBd x f x x j x Xg Now let C be a maximal chain in B Since X is spherically complete there exists z T C We show that Bd z f z T C Let Bd x f x x C Since z Bd x f x x we get d z x d x f x and d z f x d x f x By non expansiveness of f we get d f z f x d z x d x f x It follows that d z f z d x f x and therefore Bd z f z z Bd x f x x by Lemma Since x was chosen arbitrarily Bd z f z z T C Now since f is strictly contracting on orbits d f z f z d z f z and there fore z Bd f z f z f z Bd z f z f z By Lemma this is equivalent to Bd f z f z f z Bd z f z z which is a contradiction to the maximality of C So f has a xed point Now let f be strictly contracting on X and assume that x y are two distinct xed points of f Then we get d x y d f x f y d x y which is impossible So the xed point of f is unique in this case Note that the above given proof is not constructive so it does not indicate a means by which one can actually nd a xed point In order to apply this result we show rst how every domain can be viewed as a spher ically complete generalized ultrametric space For some countable ordinal let be the set f j g of symbols with ordering if and only if De nition see SH Let D be a domain and r Dc a function called a rank function and denote by De ne dr D D by dr x y inff j c x if and only if c y for every c Dc with r c g Then D dr is called the generalized ultrametric space induced by r It is straightforward to see that D dr is indeed a generalized ultrametric space We proceed to show that D dr is spherically complete For every generalized ultra metric which is induced by some rank function we will denote the ball B x in the following by B x Lemma see SH Let B x B y so Then the following statements hold fc approx x j r c g fc approx y j r c g B supfc approx x j r c g and B supfc approx y j r c g both exist B B Proof Since dr x y the rst statement follows immediately from the de nition of dr The second statement follows from the fact that every domain is consistently com plete The third statement follows from the observation that B supfc approx y j r c g supfc approx x j r c g supfc approx x j r c g B Theorem see SH D dr is spherically complete Proof By the previous lemma every chain B x of balls in D gives rise to a chain B in D in reverse order Let B supB Now let B x be an arbitrary ball in the chain It su ces to show that B B x Since B B x we have dr B x and since dr is an ultrametric it remains to show that dr B B For every c B we have c B by construction of B Now let c B with c Dc and r c We have to show that c B Since D is a domain hence an algebraic cpo there exists B in the chain with c B Now suppose B B otherwise c B immediately Then by the above lemma and the fact that the collection B x is a chain we have B x B x and therefore c fc approx x j r c g fc approx x j r c g Since B is the supremum of the right hand set c B It should be noted that we needed both algebraicity and consistent completeness of domains to prove the previous theorem Application to Generalized Acyclic Logic Pro grams We apply this result to logic programming We next introduce level mappings on IP which will be used for de ning rank functions For the following we denote the set of all nite subsets of IP which is the set of all compact elements in IP by Ic De nition level mapping Let P be a normal logic program and let A mapping l BP is called a level mapping We call l an level mapping if We set L fA BP j l A g for and L We de ne the rank function induced by the level mapping l by r I maxfl A j A Ig for every I Ic A generalized ultrametric obtained by such a rank function will further be denoted by dl The following proposition makes calculation of distances easier Proposition Let P be a normal logic program let l be a level mapping for P and let I J IP Then dl I J inff j I L J L g Proof Immediate by the observation that for every I IP I supffAg j A Ig The results obtained so far will be applied to the semantics of a class of programs which is introduced next De nition see SH Let P be a normal logic program We call P gener alized acyclic if there exists a level mapping l such that for every clause H B Bn C Cn in ground P l Bi l H and l Cj l H hold for every i n and j n Acyclic programs i e programs with the property given in the previous de nition where the level mapping is an level mapping were studied in AP in the context of termination problems If we weaken the condition in De nition to for positive body literals we obtain locally strati ed programs as introduced in Prz It was shown in Fit that acyclic programs have a unique supported model We will see that this in fact carries over to generalized acyclic programs Note that locally strati ed programs in general do have more than one supported model since every de nite program is locally strati ed Theorem see SH Let P be a generalized acyclic program with respect to a level mapping l Then TP is strictly contracting on IP dl Proof Let I I IP with d I I Let so I and I di er on some element of BP with level Let A TP I with l A Since P is generalized acyclic Amust be the head of a clause in ground P and so A TP I By the same argument if A TP I with l A then A TP I So TP I L TP I L and it follows that d TP I TP I d I I as required Let so I and I di er on some element of BP with level but agree on all ground atoms of lower level Let A TP I with l A Then there is a clause A A Ak B Bl ground P where k l with Ak I and Bl I for all k k l l Since P is generalized acyclic and I L I L it follows that Ak I and Bl I for k k l l Therefore A TP I By the same argument if A TP I with l A then A TP I So TP I L TP I L and it follows that d TP I TP I d I I as required Theorem see SH Let P be a generalized acyclic logic program Then TP has a unique xed point and hence P has a unique supported model Proof Immediate by Theorem and the previous theorems Program Consider the following program P q p X p s X p p s X p X De ne l BP by l p s n and l q s as a level mapping By Theorem P has a unique supported model which is the set fp s n j n Ng Program Let P be the following program p p s Y p Y X p Y s X p Y s X p Y X De ne a level mapping on BP by l p s s k j Then P is strictly level decreasing and hence has a unique supported model which turns out to be fp s n j n Ng fp s s k j k n Ng Note that Theorem only yields the existence of a unique model for generalized acyclic programs Its proof does not provide a method for actually nding it In SH such a method is given and is expanded on in Hit Again in Hit computational adequacy of generalized acyclic programs is studied