Appears in: 14th Int. Conf. on Foundations of Software Technology & Theoretical Computer Science, Dec. 1994.
An Epistemic Foundation for Logic Programming with Uncertainty Laks V.S. Lakshmanan Department of Computer Science Concordia University, Montreal, Quebec email:
[email protected]
1 Introduction
The need for uncertainty management in database and knowledge-base systems has motivated much of the work on the logical foundations of reasoning with uncertain knowledge. Numerous formalisms have been proposed for modeling uncertainty (see Section 6) and our concern in this paper is primarily with probabilistic extensions to logic programming and deductive databases. Substantial work has been done in this area (see Section 6 for a brief survey). One criticism (e.g. see [17]) leveled against probabilistic approaches for uncertainty management is how the probabilities representing degree of likelihood can be derived. We note, following Pearl [17], that when beliefs (and doubts) are formed by an agent, the agent must have had underlying scenarios or situations in the context of which the facts or rules are believed (or doubted). In transferring one's beliefs and doubts these contextual scenarios are often omitted, or suppressed, for simplicity or economy. An interesting question is whether we can develop an abstract framework in which these scenarios underlying beliefs and doubts are given rst class status. The potential bene ts are twofold. Firstly, such an approach would serve to highlight the structure of reasoning that probabilistic reasoning is supposed to be about. Secondly, a link between such a structural approach and the actual probabilities can be established by associating probability distributions over the space of scenarios. What is more, since we maintain the probabilities and the underlying scenarios at two di erent levels, any necessary updates to the actual probabilities can be made while keeping the underlying reasoning structure intact. In this paper, we develop such a framework by associating propositional formulas representing belief and doubt, with the facts and rules in a knowledge-base. The following example illustrates the framework.
Example 1.1 Consider a medical application where uncertain knowledge about certain diseases and symptoms is speci ed as follows.
r0 : (disease(X; D) r1 : (disease(X; D) This
has(X; S) ^ symptom(D; S)) : (f(1; ?; 1)g; f(1; ?)g). familyhistory(X; D) ^ hereditary(D)) : (f(1; 1; ?g; f(1; ?1)g).
research was supported in part by grants from the Natural Sciences and Engineering Research Council of Canada and the Fonds Pour Formation De Chercheurs Et L'Aide A La Recherche of Quebec.
f0 : has(john; unclear vision) : (f(?; 1; ?)g; f(?;1)g). f1 : symptom(achromatopsia; unclear vision) : (f(1; ?; ?)g; f(?; ?)g). f2 : familyhistory(john; achromatopsia) : (f(?; 1; 1)g; fg). f3 : hereditary(achromatopsia) : (f(1; 1; ?)g; f(?;1)g). Rule r0 says if X has any symptom S for disease D, then X is likely to have D. Rule r1 says if D is hereditary and X's family has a history of disease D, then X is likely to have D as well. The facts f0 ; : : :; f3 relate to an individual john and an eye disease achromatopsia, and are self-explanatory. Each fact/rule is annotated with a pair of sets of tuples over f1; ?1; ?g. In an annotation of the form (B; D), B represents the scenarios under which the fact/rule is believed and D the scenarios in which it is doubted (by the agent). In this example, we are assuming a xed set of distinct proposition symbols p1; p2; p3; q1; q2 representing basic scenarios. There is a natural correspondence between the sets B; D and propositional formulas over fp1; p2; p3g and fq1; q2g respectively. For instance, the belief formula associated with r1 is p1 ^ p2 while the doubt formula is q1 ^ :q2. Sets of tuples correspond to disjunctions of formulas associated with the individual tuples. We let form(B) (form(D)) denote the propositional formula associated with B (D). Finally, two special cases are noteworthy: the tuple consisting of only ? entries represents universal truth; the empty set of tuples fg represents universal falsehood. For example, the fact symptom(achromatopsia; unclear vision) in f1 is always doubted, while familyhistory(john; achromatopsia) in f2 is never doubted. These special cases lead to interesting modalities2 within our framework, as illustrated below: (f(?; ?; ?)g; fg) corresponds to the modality of necessity; (fg; f(?; ?)g) corresponds to impossibility; each annotation of the form (B; D) where B 6= fg captures (di erent degrees of) possibility; nally, each annotation of the form (B; D) where D 6= fg corresponds to (di erent degrees of) non-necessity. In addition, this framework captures possible inconsistency in the acquired knowledge. For instance, in f1 : symptom(achromatopsia; unclear vision) : (f(1; ?; ?)g; f(?; ?)g), the annotation indicates that the fact is always doubted, while the belief is not totally absent. In an application such as knowledge acquisition, where it is important to assimilate information acquired from various sources, there is a potential for inconsistency in the information obtained. The framework developed in this paper a ords the capability of localizing the e ects of inconsistency thereby making it \inconsistency-tolerant". Section 2 develops the necessary machinery. Section 3 deals with the model theory and establishes an epistemic foundation for logic programming with belief and doubt. In Section 4, we develop the xpoint semantics of logic programs with belief and doubt and establish the equivalence of alternate semantics of programs developed in this paper. Section 5 addresses the issue of calculating numerical measures representing probabilities of belief and doubt. Section 6 compares this work with similar works and brings out the novelty and originality of this approach. Section 7 discusses future research. For lack of space, proofs of results are omitted. The full paper [11] contains the proofs and complete details about the framework, a sound and complete proof procedure, and the complexity results on some fundamental query optimization problems in this framework.
2 A Model for Uncertainty
Let T = f1; ?1; ?g be any set. For intuitive purposes, we can regard 1 as representing truth, ?1 as falsity, and ? as \don't care". (As we shall see, for our purpose, ? is more naturally thought of as \don't care" than as \unde ned".) Consider the upper semilattice 2
Relative to the beliefs/doubts of the agent.
L = (T; ), where the partial order is de ned by 1 ?, ?1 ?. This is depicted in Figure 1. Notice that ? is the greatest element (and the least upper bound (lub) of 1 and ?1). We denote the meet and join on this semilattice as and . In this paper, we shall extend the basic (semi)lattice structure in various ways. In the context of the extensions, we will use the same notation (e.g. ; ; ) for the ordering and operations as in the basic setting, for simplicity and clarity. ⊥
-1
1
Figure 1: The Basic Semilattice L of Truth Values. Direct products preserve the partial ordering above and naturally generate upper semilattices in the obvious manner: Lk = (Tk ; ) is an upper semilattice where is the obvious pointwise extension of the partial order on L to k-tuples: (a1; : : :; ak ) (b1 ; : : :; bk) i ai bi; 8i. The greatest element is the k-tuple (?; : : :; ?). The meet and join operations on Lk can be obtained by similar pointwise extensions of the corresponding operations on the lattice L: (a1 ; : : :; ak) (b1; : : :; bk ) = (a1 b1 ; : : :; ak bk ), where is or . Note that for a pair of k-tuples u; v 2 Tk , uv is always de ned, while u v could be unde ned, a property inherited from the original upper semilattice L. (Clearly, when ai bi is unde ned for any i, (a1 ; : : :; ak ) (b1 ; : : :; bk ) is unde ned.) k Next, consider the powerset 2T under the cover ordering de ned as follows. For k S; T 2 2T , S T provided 8u 2 S; 9v 2 Tk such that u v in the sense de ned above. It is possible to have distinct sets S; T 2 2T such that S T and T S, and thus is k not a partial order on 2T . (E.g. , consider f(1; ?)g and f(1; ?); (1; ?1)g.) We de ne the k operations ; on 2T as follows. k De nition 2.1 Let S; T 2 2T . Then 1. S T =def fu v j u 2 S; v 2 T; u v is de nedg. 2. S T =def fu 2 S [ T j6 9v 2 S [ T such that v 6= u and u vg. k De ne a binary relation on 2T by S T i S T and T S. Clearly, is an equivalence relation. Let [S] denote the equivalence class generated by S. We have k Proposition 2.1 The relation is a congruence w.r.t. the algebra (2T ; ; ). The k k ordering is a partial order on the quotient 2T = and Lk = (2T = ; ; ; ) is a complete distributive lattice with [fg] as the least element and [f(?; : : :; ?)g] as the greatest element.
In the following, for simplicity, we drop the square brackets representing equivalence classes (as in [S]) and simply write sets of tuples (S) in their place. This can be formalized by choosing representatives from the equivalence classes and working with the representatives. We refer to Lk de ned above as a situation lattice with parameter k. In fact, Lk is actually a Boolean algebra, but in this paper, we will not need this property of Lk . Now, in order
to create the structure appropriate for reasoning with beliefs and doubts, we can consider the direct product of a pair of situation lattices Lm and Ln, with parameters m and n, as de ned above. This gives rise to a distributive bilattice in the sense of Fitting [6]. The following de nition formalizes this construction.
De nition 2.2 Let Lm and Ln be any situation lattices as de ned above. Then mthe state lattice determined by the parameters m; n is the algebraic structure L = (2T n T 2 ; t; t ; mt; k ; nk ; k ), where t ; t; t; k ; k ; k are de ned as follows. Let (S1 ; T1 ); (S2 ; T2) 2 2T 2T be any pair of elements. Throughout below, refers to the cover ordering, and and to the meet and join operations, on sets of tuples, de ned earlier. 1. (S1 ; T1)t (S2 ; T2 ) i S1 S2 &T2 T1 . 2. (S1 ; T1 )k (S2 ; T2) i S1 S2 &T1 T2 . 3. (S1 ; T1) t (S2 ; T2 ) = (S1 S2 ; T1T2 ). 4. (S1 ; T1 ) k (S2 ; T2) = (S1 S2 ; T1 T2 ). 5. (S1 ; T1)t (S2 ; T2 ) = (S1 S2 ; T1 T2 ). 6. (S1 ; T1 )k (S2 ; T2) = (S1 S2 ; T1T2 ). m n We refer to members of 2T 2T as states. W.r.t. the order t , the greatest element of L is the state >t = (f(?; : : :; ?)g; fg) and the least element is the state ?t = (fg; f(?; : ::; ?)g). W.r.t. the order k , the greatest element is >k = (f(?; : : :; ?)g; f(?; : : :; ?)g) and the least element is ?k = (fg; fg). We have the following result.
Proposition 2.2 Let L be any state lattice as de ned above. Then the meet (join) w.r.t. any order distributes over the join (meet) of any (other) order. Furthermore, the meet and join operations w.r.t. any order are monotone w.r.t. any (other) order.
3 Logic Programs
Syntax Let L be a rst order language with in nitely many function symbols f; g; : : : of various
arities, in nitely many predicate symbols p; q; : : : of various arities, and with in nitely many individual variables X; Y; : : :. Terms, atoms, and literals are de ned as usual. In this paper, we con ne attention to de nite clause programs. Clauses and de nite clauses are de ned as usual. In the sequel, we shall consider an arbitrary, but xed, state lattice L. A program rule (rule for short) is of the form (A B1 ^ ^ Bk ) : (S; T), where A; Bi are atoms, k 0, and (S; T) 2 L. We refer to (A B1 ^ ^ Bk ) as the underlying de nite clause of the above rule. When k = 0, we refer to the rule as a fact, A being its underlying unit clause.
Model Theoretic Semantics
Let P be any program. The notions of Herbrand universe (HP ) and Herbrand base (BP ) of programs are de ned as usual. The Herbrand instantiation P of a program P is obtained by substituting ground terms for all the variables occurring in P. Note that each (ground) rule in P has a state associated with it. The appropriate model theoretic framework for programs in our framework is obtained by relating them to a version of the modal logic S4 [8]. Let P be an in nite set of propositions. A formula of the logic S4 is either a proposition or is of one of the following forms, where ; are S4 formulas: _ , ^ , : , ! , 2 , 3 . A partial S4 Kripke structure [8] is a 3-tuple M = hW; ; 7!i where (i) W is a non-empty set of worlds; (ii) : W ![P ; ftrue; falseg], is a function that maps each world to a partial function on the set of propositions: for w 2 W and A 2 P , whenever (w)(A) is de ned, it says whether A is true or false at w, and otherwise its truth value is unde ned at w; and (iii) 7! is a re exive transitive relation on W. The following de nition formalizes the notion of satisfaction of formulas in partial S4 structures.
De nition 3.1 Let M = hW; ; 7!i be a structure. Then satisfaction of modal formulas by M is de ned as follows. Let w 2 W be any world. 1. For a proposition A, M; w j= A i (w)(A) = true. 2. M; w j= _ i M; w j= or M; w j= . 3. M; w j= ^ i M; w j= and M; w j= . 4. M; w j= ! i either M; w j= : or M; w j= . 5. M; w j= 2 i 8w00 2 W , whenever w7!w0 0, we have M; w0 j= . 0 6. M; w j= 3 i 9w 2 W such that w7!w , and M; w j= . 7. Satisfaction for negation is de ned recursively as follows. For a proposition A, M; w j= :A i (w)(A) = false; M; w j= :( _ ) i M; w j= : and M; w j= : ; M; w j= :( ^ ) i M; w j= : or M; w j= : ; M; w j= :( ! ) i M; w j= and M; w j= : ; M; w j= :2 i 9w0 2 W such that w7!w0 and M; w0 j= : ; M; w j= :3 i 8w0 2 W such that w7!w0, M; w0 j= : . The negation used here is strong negation: note the distinction between M; w 6j= and M; w j= : . This distinction is important for correctly capturing beliefs and doubts. Also notice that it is possible to have, e.g. , M; w 6j= and M; w 6j= : . We now de ne
structures for programs and satisfaction of programs by structures. It turns out we cannot relate S4 structures directly to programs. Programs in our framework are sets of facts/rules with associated (belief, doubt) pairs and in order to de ne satisfaction of programs by structures, we need to know the scenarios (w.r.t. belief and doubt) underlying the worlds in a structure. We formalize this below by considering special (partial) S4 structures. Let F and G represent the sets of equivalence classes of satis able propositional formulas over fp1; : : :; pm g and fq1; : : :; qng respectively. For simplicity, below, we shall treat members of F and G as formulas (rather than as equivalence classes of formulas). This can be made precise by using representative formulas from the equivalence classes. The intention will be clear from the context. De nition 3.2 Let P be a logic program. A structure for P is a 3-tuple M = hW; ; 7!i
satisfying the following conditions. 1. W = F G . 2. For wi = ( i ; i) and wj = ( j ; j ), wi 7!wj i either wi and wj are identical (i.e. their associated formulas are equivalent, componentwise), or j ! i and j ! i are both propositional tautologies. 3. : W ![BP ; ftrue; falseg] is a function that maps each world to a partial function on the set of ground atoms, which indicates the truth status of each atom at the world.
Remarks: (1) Each world w in a structure M is identi ed with a pair of propositional formulas ( ; ) over fp1; : : :; pm g and fq1; : : :; qng, respectively. The formula ^ essentially brings out what0 scenarios are known to hold in the world w. (2) In a structure M, for a pair of worlds w; w such that w7!w0 , we say, following custom, that w0 is reachable from w. (3) Consider w; w0 2 W such that w7!w0. Intuitively, this means that no more things 0 are known0 to hold at w than at w , and when w0 6= w, strictly more things are known to hold at w . Thus, reachability can be thought of as progression in available knowledge at given worlds. Excluding worlds associated with unsatis able formulas corresponds to the idea that the scenarios \known to hold" cannot be unsatis able! (4) It is straightforward to see that 7! is a re exive transitive relation on W. Thus structures as de ned in De nition 3.2 are indeed (partial) S4 Kripke structures. (5) Formulas obtained by closing BP ,
the set of ground atoms associated with P, under the usual boolean connectives and the modal operators 2; 3 can be interpreted against structures de ned above, along the lines of De nition 3.1. The following de nition relates the worlds of a structure to the beliefs and doubts associated with facts and rules. (Recall that form(B) and form(D) are the formulas associated with B and D as described in Section 1.) De nition 3.3 Let M be a structure as de ned in De nition 3.2. Consider a state (B; D) and a world w = ( ; ) of M . We say that w veri es the state (B; D) provided !form(B) and !:form(D) are both propositional tautologies. The world w contradicts the state (B; D) provided !:form(B) and !form(D) are both propositional tautologies. The following proposition shows that once a state is veri ed or contradicted in a world, then it remains that way in all worlds accessible from that world in the sense that the additional knowledge available in the \future" worlds does not go against the (\convictions" of the) original world. Proposition 3.1 Let M be a structure for a program P and let w be a world of M that veri es a state (B; D). Then every world reachable from w in M also veri es (B; D). Similarly, whenever w contradicts the state (B; D), then every world reachable from w also contradicts (B; D). The next de nition formalizes satisfaction of programs by structures. De nition 3.4 Let M = hW; ; 7!i be a structure as de ned in De nition 3.2. For a ground atom A and a state (B; D), M j= A : (B; D) i (i) for every world w 2 W which veri es (B; D), M; w j= 2A, and (ii) for every world w 2 W that contradicts (B; D), M; w j= 2:A. For a ground rule A B1 ^ ^ Bk and a state (B; D), M j= (A B1 ^ ^ Bk ) : (B; D) i (i) for every world w 2 W that veri es (B; D), M; w j= 2(A B1 ^ ^ Bk ), and (ii) for every world w 2 W that contradicts (B; D), M; w j= 2:(A B1 ^ ^ Bk ). For a program P , M j= P i for every ground rule r in the Herbrand instantiation of P , M j= r.
Least Model Semantics
A structure M is a model of a program P provided M j= P. Note that all structures for a program coincide on the worlds W and the reachability relation 7!. They only di er in the function . To obtain the model theoretic meaning of programs, we need the analogue of least Herbrand model for classical logic programming. In the classical case, this is simpli ed by the fact that both the existence of models and the model intersection property are straightforward. In our framework, these require work. First, we de ne intersection of structures as follows. Let M = hW; M ; 7!i and N = hW; N ; 7!i be any structures for P. Then the intersection of M and N is the structure M \ N = hW; M \N ; 7!i, where W and 7! are as before, and M \N is de ned as follows. For each world w 2 W, and for each ground atom A 2 BP , M \N (w)(A) is de ned exactly when M (w)(A) and N (w)(A) are both de ned and are equal: in this case, M \N (w)(A) = M (w)(A) (= N (w)(A)). In particular, note that when (M (w)(A) and N (w)(A) are both de ned and) M (w)(A) 6= N (w)(A), M \N (w)(A) is unde ned. The following lemma extends the model intersection property of classical logic programs to our framework. Lemma 3.1 Let P be a program and M; N any models of P . Then M \ N is also a model of P .
The next question is the existence of models for programs. The following construction shows how to obtain a model for any (de nite clause) program. Let P be any program and P its Herbrand instantiation. Construct a structure Mg = hW; ; 7!i such that W and 7! are de ned as for any structure. is de ned as follows. (1) For every rule of the form (A B1 ^ ^ Bk ) : (B; D) in P , and for every world w contradicting (B; D), de ne (w)(Bi ) = true, 1 i k, and (w)(A) = false. (2) For every world w and every ground atom A 2 BP such that (w)(A) is not de ned by step (1), set (w)(A) = true. This completes the construction. The following lemma shows that this construction always yields a model of P. Lemma 3.2 Let P be any program. The structure Mg constructed above is a model of P . Analogously to the classical case, it now follows that every program P has a least model. De nition 3.5 Let P be a program. Then the least model of P , denoted MP , is the intersection of all models of P . We can take the model theoretic semantics of programs as given by their least model. Intuitively, the least model of a program ensures that each rule (or fact) (A B1 ^ ^ Bk ) : (B; D) in the Herbrand instantiation of a program is true in exactly those worlds that verify (B; D) and is false in exactly those worlds that contradict (B; D). Note that each world that veri es (B; D) con rms the beliefs B and precludes the doubts D. Similarly, each world that contradicts (B; D) con rms the doubts D and precludes the beliefs B. Thus, the least model of a program provides a simple and intuitive semantics while giving an epistemic foundation for logic programming with the notions of belief and doubt. Let P be a program and A : (B; D) be any (annotated) ground atom. We say that P logically implies A : (B; D), denoted P j= A : (B; D), provided A : (B; D) is true in every model of P. We can prove the following analogue of the van Emden-Kowalski Theorem for classical logic programming (see Lloyd [14]). Theorem 3.1 Let P be a program and A : (B; D) be any annotated ground atom. Then P j= A : (B; D) i MP j= A : (B; D). We illustrate the concepts introduced in this section with the following example. Example 3.1 Revisit Example 1.1 of Section 1. The model Mg for the program P of Example 1.1, obtained using the above construction, is as follows. W and 7! are constructed as usual. The following table summarizes for which pairs of worlds (speci ed in terms of the states they contradict { see column 1) and atoms the value of is false. Here, c; d are arbitrary ground terms (which in this case are any constants appearing in the program). world contradicting atom (world)(atom) (f(1; 1; ?)g; f(?;1)g) disease(c, d) false (f(1; 1; ?g; f(1; ?1)g) disease(c, d) false (f(?; 1; ?g; f(?; 1)g) has(john, unclear vision) false (f(1; ?; ?)g; f(?; ?)g) symptom(achromatopsia, unclear vision) false (f(?; 1; 1)g; fg) familyhistory(john, achromatopsia) false (f(1; 1; ?)g; f(?;1)g) hereditary(achromatopsia) false For every world w and every ground atom A, whenever (w)(A) is not de ned by the above table, the construction sets (w)(A) = true. It can be veri ed that the structure so constructed is indeed a model of P . The least model MP of the program P is characterized as follows. For each of the facts f0; : : :; f3 in P , the corresponding ground atom is true at all worlds verifying the associated
states, false at all worlds contradicting the associated states, and unde ned at all other worlds. The atom disease(john; achromatopsia) is true in all worlds verifying the state (f(1; 1; 1)g; f(1; ?1); (?; 1)g), false at all worlds contradicting this state, and unde ned elsewhere. The truth value of all other ground atoms is unde ned at every world.
4 Fixpoint Semantics
In this section, we develop an alternative semantics of programs based on the notion of valuations. We shall later establish the equivalence between the various semantics of programs. De nition 4.1 Let P be a program and BP its Herbrand Base. Let L be the state lattice as in De nition 2.2. Then a valuation for P is a function v : BP !L. The basic structure of the state lattice L of Section 2 can be lifted to the set of valuations, giving rise to a corresponding distributive bilattice. De nition 4.2 Let P be any program. Let V be the set of all valuations for P . De ne the algebraic structure Lv = hV ; t ; t; t; k ; k ; k i as follows. Let u; v 2 V be any valuations.
u v i u(A) v(A); 8A 2 BP , where is t or k . (u v)(A) = u(A) v(A); 8A 2 BP , where is t ; t ; k , or k . The greatest and least elements of Lv w.r.t. t and k are the valuations given below, where the subscript identi es the rank of the valuation and its order. v>t (A) = >t; 8A 2 BP ; v?t (A) = ?t; 8A 2 BP ; v>k (A) = >k; 8A 2 BP ; v?k (A) = ?k; 8A 2 BP . De nition 4.3 Let P and BP be as before and let v be a valuation for P . 1. v satis es a ground annotated atom A : (B; D) i (B; D)t v(A). 2. v satis es ^ i v satis es and v satis es . 3. v satis es a rule (A B1 ^ ^Bk ) : (S; T) i v(B1 ) t t v(Bm ) t (S; T)t v(A). 4. v satis es a program P i v satis es each ground rule in P .
Intuitively, a fact or a rule is satis ed by a valuation provided the truth value assigned to the rule head is no less than \implied" by the rule. The distributive nature of the bilattice plays an important role in the following result. Proposition 4.1 The valuation v>t always satis es every program. Furthermore, whenever u; v are valuations satisfying a program P , the valuation u k v also satis es P . De nition 4.4 Let P be a program. De ne an operator TP associated with P , as follows. TP : V!V , where V is the set of all valuations for P . For any valuation v 2 V and any ground atom A 2 BP , TP (v)(A) = t fv(B1 ) t t v(Bk ) t (B; D) j (A B1 ^ ^ Bk ) : (B; D) 2 P g. Intuitively, given a valuation v and a ground atom A, TP \evaluates" all rules with head A and collects together the individual (belief, doubt) pairs so obtained. In evaluating rules, it makes use of the lattice meet operation t and for collecting together the results of the individual derivations it uses the join operation t . We have the following results. Proposition 4.2 TP is monotone and continuous w.r.t. both the orders t and k of the valuation lattice Lv .
We de ne an increasing sequence of approximations to the intended semantics of programs. Let v?k be the valuation that assigns the state ?k to every ground atom in BP . TP " 0 = v?k : TP " n + 1 L = TP (TP " n): TP " ! = t (TP " n): n
p1 ^:p2 ^ p3 ^ q1 ^:q2. Suppose } : fp1; : : :; pm ; q1; : : :; qng![0; 1] is a probability distribution function that associates with each basic scenario the probability of its occurrence (in the \real world"). In [11], we give an algorithm for generating the belief and doubt probabilities associated with ground atoms, by capturing the constraints on the probabilities in the form of a linear program. Here, for lack of space, we only illustrate the method.
Example 5.1 We revisit Example 1.1 again. In Section 4, we determined the least xpoint of the operator TP associated with the program P there. Suppose } : p17!0:6, p27!0:7, p37!0:65, q17!0:45, q2 7!0:55 is a given probability distribution on the basic scenarios. Then the linear program P(}) determined by } is the following. We adopt a convenient representation for the variables representing probabilities of the (scenarios corresponding to the) various truth assignments. Arrange the propositions in the order (p1; p2; p3; q1; q2). Denote each truth assignment as a binary vector of length 5, where a 1 in a position corresponding to pi (qj ) indicates pi (qj ) is assigned the value true, while a zero there indicates it is assigned the value false. Converting the binary vector into decimal representation, we get a compact encoding of the truth assignments as subscripts of the appropriate probability variables. The linear program P(}) follows.
!16 + + !31 !8 + + !15 + !24 + + !31 !4 + + !7 + !12 + + !15 + !28 + + !31 !2 + !3 + !6 + !7 + !10 + !11 + !14 + !15+ !19 + !20 + !22 + !23 + !26 + !27 + !30 + !31 !1 + !3 + + !31 0 !i 1; !0 + + !31
= 0:6: = 0:7: = 0:65:
= 0:45: = 0:55: i = 0; : : :; 31: = 1: Suppose we want to determine the probabilities of belief and doubt for the atom disease(john; achro-motopisa) against the given program. From Section 4, we know the state associated with this atom by the least xpoint lfpk (TP ) is (f(1; 1; 1)g; f(1; ?1); (?;1)g). The objective function associated with the belief is (!28 + + !31), while that associated with the doubt is (!2 + !6 + !10 + !14 + !18 + !22 + !26 + !30) +(!1 + !3 + + !31). We can calcu-
late the bounds (maximum and minimum) for the objective functions, giving us ranges of probabilities associated with belief and doubt.
The methodology developed in this section is rather general in that no a priori assumptions about the interrelationships among p's and q's are necessary. Also, incremental knowledge about the relationships among the basic scenarios can be easily incorporated into the framework without any disturbance to the underlying reasoning structure. For example, suppose at some point it becomes known that whenever p1 occurs q2 cannot occur. The inferences made with the existing program are still robust in the sense that only the probabilities of belief and doubt need to be recomputed. The new information can be incorporated in calculating probabilities by adding the following constraint to the linear program P(}). w17 + w19 + + w31 = 0: This equation is obtained by considering all truth assignments where p1 and q2 are both assigned the value true and insisting that the probabilities of such truth assignments (equivalently, their corresponding scenarios) be zero. The new linear program can now be used to calculate the belief and doubt probabilities to re ect the e ect of the new piece of knowledge. Thus, the methodology is very exible. We remark that techniques developed (see, e.g., Bell et. al. [2]) for minimizing redundancies in linear programs could be profitably employed in our context. A nal note is that information about the probability distribution of the basic scenarios might be obtained from statistical technqiues.
6 Comparison with Related Work
Logic (programming) with uncertainty management is an extensively researched area. For brevity, we cite only representative works and those directly relevant to this paper. Most of the proposals in the literature employ one of the following formalisms: (1) a form of fuzzy logic (programming) (e.g. van Emden's quantitative logic programming [19], and Fitting's logic programming with bilattices [6, 5]), (2) annotated logic programming (e.g. , see Blair and Subrahmanian [3], Kifer and Li [9], and Kifer and Subrahmanian [10]), (3) evidence theoretic logic programming (e.g. see Baldwin [1] which uses Dempster's theory as the basis), and (4) probabilistic logic programming (see below). Ng and Subrahmanian [15, 16] have recently proposed an interesting scheme for probabilistic logic programming. They developed xpoint and model-theoretic semantics, and provided a sound and (weakly) complete proof procedure. Guntzer et. al. [7] have proposed a sound (propositional) probabilistic calculus based on conditional probabilities, for reasoning in the presence of incomplete information. Lakshmanan and Sadri [12] have recently proposed an approach to probabilistic deductive databases, based on the the trilattice of probabilistic truth values, an extension to bilattices. This framework allows for reasoning with facts and rules with associated ranges of probabilities indicating belief and doubt. Besides these works, a wealth of literature is available on probabilistic logic alone (e.g. , see Fagin et. al. [4]). A conceptually di erent approach to uncertainty management was proposed by Sadri [18] in the form of the so called information source tracking method. This approach allows for the representation of the contributing sources of each piece of information in a database and allows this source information to be manipulated in query processing. Mechanisms are developed whereby reliabilities associated with the contributing information sources can be used to calculate the reliability of answers to queries. This framework was extended to deductive databases by Lakshmanan and Sadri [13]. As mentioned in the introduction, the practical diculty with directly associating speci c numbers (indicating degrees of likelihood) with facts/rules has given rise to a serious criticism of probability based approaches to uncertainty [17]. The work presented in this paper is intended to provide a link between conceptual notions of likelihood in an expert's mind and quantitative measures of likelihood. This is achieved by explicitly allowing facts/rules together with belief and doubt, where a propositional language is used to capture the scenarios in which a fact/rule is believed (doubted). Calculation of probabilities is accomplished by compiling the belief and doubt information into a linear program and then using it to calculate bounds on belief and doubt probabilities. In this respect, the approach of this paper is similar in spirit to [15, 18, 13]. However, there are major di erences with those works. Firstly, these earlier works do not capture the notions of belief and doubt. Secondly, while [18, 13] make speci c assumptions about the independence between information contributed by di erent sources, our approach is quite general, and does not need any a priori assumptions on the interrelationships among the basic scenarios. Unlike [15], our framework can take advantage of any available knowledge about the nature of interaction between events, in that it does not force us to work in total ignorance about the scenarios. Our work is also related to Fitting [6, 5] and Lakshmanan and Sadri [12] in the sense that these papers have dealt with logic programming with belief and doubt, in di erent contexts. However, the novelty of this paper is the treatment of belief and doubt in abstract conceptual terms together with a mechanism to migrate from this representation to quantitative measures whenever there is a need. Besides, we also develop an intuitive epistemic basis for programming with belief and doubt, by relating the framework to extended modal logic S4.
7 Future Research
(1) As discussed in Section 4 the framework developed in this paper allows for di erent types of operators to be associated with logic programs with belief and doubt. It would be interesting to investigate the relationships between the di erent operators (under di erent xpoint semantics) and various forms of negation such as negation as failure and classical negation. (2) An important advantage of basing uncertainty management on a sound formalism like (extended) deductive databases is the possibility of encouraging high level logic programmability for the user while eciency issues are relegated to the system in the form of query optimization. Indeed, ecient query processing and query optimization in logic systems with uncertainty is a very important topic for future study. Some results in the context of the framework of this paper appear in [11]. We are currently working on a generic programming environment for logic programming with uncertainty.
References
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