Computed answer based on fuzzy knowledgebase a model for handling uncertain information

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1 Computed answer based on fuzzy knowledgebase a model for handling uncertain information Ágnes Achs Departement of Computer Science University of Pécs, Faculty of Engineering Pécs, Boszorkány u. 2, Hungary achs@witch.pmmf.hu Abstract The basic question of our study is how we can give a possible model for handling uncertain information. This model is worked out in the framework of DATALOG. The concept of fuzzy knowledge-base will be defined as a quadruple of any background knowledge; a deduction mechanism; a connecting algorithm, and a decoding set of the program, which help us to determine the uncertainty level of the results. A possible evaluation strategy is given also. Keywords: fuzzy knowledgebase, fuzzy DATALOG. 1. Introduction The large part of human knowledge can t be modelled by pure inference systems, because this knowledge is often ambiguous, incomplete and vague. When knowledge is represented as a set of facts and rules, this uncertainty can be handled by means of fuzzy logic. A few years ago in [1] and [3] there was given a possible combination of Datalog-like languages and fuzzy logic. In these works there was introduced the concept of fuzzy Datalog by completed the Datalogrules and facts with an uncertainty level and an implication operator. In [2] there was given an extension of fuzzy Datalog to fuzzy relational databases. Parallel with these works, there were researches on possible combination of Prolog language and fuzzy logic also. Several solutions were arisen for this problem. These solutions propose different methods for handling uncertainty. Most of them use the concept of similarity, but in various ways. More essays deal with fuzzy unification and fuzzy resolution, for example [5], [9], [10], [12]. In this paper, continuing our former concept, we give other possible model for handling uncertain information, based on the extension of fuzzy Datalog. According to our former papers, in the next chapter we describe the concept of fuzzy Datalog, in the further chapters we discuss our actually new idea about fuzzy knowledgebase. We build the model of fuzzy knowledge-base as a quadruple of any kind of background knowledge; a deduction mechanism; a connecting algorithm and a decoding set. 2. The fuzzy Datalog A Datalog program consists of facts and rules. In fuzzy Datalog, we can complete the facts by an uncertainty level, and the rules by an uncertainty level and an implication operator. Applying this operator, the uncertainty level of rule s head can be determined from the uncertainties of rule s body and the rule. For example if in the program beautiful( Mary ); 0.7 likes( John,x) beautiful(x);0.8;i. the implication operator is the Gödel-operator (I(x,y) = 1 if x y; otherwise I(x,y) = y), then the level of the rule-head can be calculate as the minimum of the level of the rule-body and the level of the rule. For John and Mary it does: likes( John, Mary ), 0.7, that is John likes Mary at least 0.7 level. 94

2 In [1], [2] there is given an extension of Datalog-like languages to fuzzy relational databases. In these papers lower bounds of degrees of uncertainty in facts and rules are used. This language is called fuzzy Datalog (fdatalog). In this language the rules are completed by an implication operator and a level. We can deduce the level of a rule-head from the level of the body and the level of the rule by the implication operator of the rule. Now we are going to summarise the concept of fdatalog based on [1], [2]. Similarly to DATALOG programs, a fdatalog program consists of rules and facts. The notion of fuzzy rule is given in definition below: Definition 1. A fdatalog rule is a triplet r;β;i, where r is a formula of the form Q Q 1,...,Q n (n 0). Q is an atom (the head of the rule), Q Q 1,...,Q n are literals (the body of the rule); I is an implication operator and β (0,1] (the level of the rule). For getting finite result, all the rules in the program must be safe, that is all variables occurring in the head also occur in the body, and all variables occurring in a negative literal also occur in a positive literal. A fdatalog program is a finite set of safe fdatalog rules. There is a special type of rule, called fact. A fact has the form A ;β;i. Further on we refer to facts as (A,β). The semantics of fdatalog is defined as the fixpoints of consequence transformations. Depending on these transformations we can define two semantics for fdatalog based on a deterministic and a nondeterministic transformation. According to the deterministic transformation, the rules of a program are evaluated parallel, while in nondeterministic case the rules are considered independently one after another. In [2] it is proved, that starting from the set of facts, both deterministic and nondeterministic transformation have fixpoint, which are the least fixpoints (lfp) in the case of positive program P. It was also proved, that this fixpoints are models of P, so we could define this fixpoints as the deterministic and the nondeterministic semantics of fdatalog programs. For function- and negation-free fdatalog, the two semantics are the same, but they are different if the program has any negation. In this case the least fixpoint of deterministic transformation is not always a minimal model, but the nondeterministic semantics is minimal under certain conditions. This condition is the stratification. Stratification gives an evaluating sequence in which the negative literals are evaluated first. (Detailed in [2].) Further on we deal only with the nondeterministic semantics. This transformation is the following: Definition 2. Let B P the Herbrand base of the program P, and let F(B P ) denote the set of all fuzzy sets over B P. The nondeterministic consequence transformation NT P : F(B P ) F(B P ) is defined as NT P (X)={(A, α A )} U X, where (A A 1,,A n ; β; I) ground(p), ( A i,α Ai ) X, 1 i n, α A =max(0,min{γ I(α body,γ) β}) A denotes p(c) if either A=p(c) or A= p(c) where p is a predicate symbol with arity k and c is a list of k ground terms. Example 1. Given the next fdatalog program: 1. (r(a), 0.8). 2. p(x) r(x), q(x); 0.6; I. 3. q(x) r(x); 0.5; I. 4. p(x) q(x); 0.8; I. The stratification is: P 1 = {r,q}, P 2 = {p}, so the evaluation order is: 1., 3., 2., 4. Then in the case of Gödelian implication operator, the nondeterministic semantics of the program is lfp(nt P ) = {(r(a),0.8); (p(a),0.5); (q(a),0.5)}. 3. Background knowledge The facts and rules of a fdatalog program can be regarded as any kind of knowledge, but sometime as in the case of our model we need other information in order to get answer for a query. In this paragraph we give a model of background knowledge. We define similarity between predicates and between constants, and these structures of similarity will serve for the background knowledge. 95

3 Definition 3. A similarity on a domain D is a fuzzy subset S D : D D [0,1] such that the following properties hold: S D (x,x) = 1 for any x D (reflexivity) S D (x,y) = S D (y,x) for any x,y D (symmetry). In our model the background knowledge is a set of similarity sets: Definition 4. Let d D any element of domain D. The similarity set of d is a fuzzy subset over D: S d = {(d 1, λ 1), (d 2, λ 2),, (d n, λ n)}, where d i D and S D (d, d i ) = λ i for i = 1, n. Based on similarities we can construct the background knowledge, which is information about the similarity of terms and predicate symbols. Definition 5. Let T be any set of ground terms and P any set of predicate symbols. Let ST and SP any similarity over T and P respectively. The background knowledge is: Bk = {ST t t T} U {SP p p P} 4. Fuzzy knowledge-base We made two steps on the way leading to the concept of fuzzy knowledgebase: we defined the concept of fuzzy Datalog program and the concept of background knowledge. Now the question is: how we can connect this program with the background knowledge? In [4] the author gave a possible connecting algorithm, replacing all predicates and all ground terms of the program by theirs similarity sets. Now we show a new and other possibility: instead of modifying the program, we ll modify the evaluation transformation of the program. To solve this problem, we ll define the concept of modified fdatalog program, which is the extension of the original one according to similarity; and the concept of decoding functions, which compute the final value of the uncertainty. Evaluating an extended fdatalog program, the facts and the result atoms are completed by an uncertainty level. The final uncertainty level can be computed from this level and from the similarity values of actual predicate and its arguments. It is expectable, that in the case of identity the level must be unchanged, but in other cases it is to be less or equal then the original level or then the similarity values. Furthermore we require the decoding function to be monotone increasing. Definition 6. A decoding function is an (n+2)-ary fuzzy function : ϕ(α,λ,λ 1,,λ n ) : (0,1] (0,1] (0,1] [0,1], so that ϕ(α, λ, λ 1,, λ n ) min (α, λ, λ 1,, λ n ), ϕ (α, 1, 1,, 1) = α and ϕ(α, λ, λ 1,, λ n ) is monotone increasing in all arguments. Example 2. ϕ 1 (α, λ, λ 1,, λ n ) = min (α, λ, λ 1,, λ n ); ϕ 2 (α, λ, λ 1,, λ n ) = min (α, λ, (λ 1 λ n )); ϕ 3 (α, λ, λ 1,, λ n ) = α λ λ 1 λ n are decoding functions. We have to order decoding functions to all predicates of the program. The set of decoding functions will be the decoding set of the program: Definition 7. Let P be a fuzzy DATALOG program. The decoding set of P is: Φ P = {ϕ q (α q, λ q, λ t1,, λ tn ) q(t 1, t 2,, t n ) B P } Let P be a fuzzy Datalog program, Bk be any background knowledge and Φ P be the decoding set of P. Now we want to connect the program with background knowledge. For this purpose we decide the modified fdatalog program, mp. This is the original program with modified consecution transformation. The original consequence transformation is defined over the set of all fuzzy sets of P s Herbrand base, that is over F(B P ). To define the modified transformation s domain, let us extend P s Herbrand universe with all possible ground terms occurring in background knowledge so we get the modified Herbrand universe, mh P. Let the modified Herbrand base, mb P be the set of all possible ground atoms whose predicate symbols occur in P U Bk and whose arguments are elements of mh P. So: Definition 8. The modified consequence transformation mnt P : F(mB P ) F(mB P ) is defined as mnt P (X)= {(q(s 1,...,s n ), φ p (α, λ q, λ s1,, λ sn ) (q, λ q ) SP p ; (s i, λ si ) ST ti, 1 i n} U X, 96

4 where (p(t 1,...,t n ) A 1,...,A k ; I ; β) ground(p), ( A i, α Ai ) X, 1 i k, α = max(0,min{γ I(α body, γ) β}). ( A denotes r(c) if either A=r(c) or A= r(c) where r is a predicate symbol with arity k and c is a list of k ground terms.) Then starting from the facts of the program and creating the powers of the transformation mnt P, finally we reach the fixpoint. The next proposition can be easily prove: Proposition 1. In the case of positive or stratified program P, mnt P has least fixpoint. It can be shown, that this fixpoint is a model of P, but lfp(nt P ) lfp(mnt P ), so it is not a minimal model. Now there are all together to define the concept of a fuzzy knowledge-base: Definition 9. A fuzzy knowledge-base (fkb) is a quadruple {Bk, P, Φ P, ma}, where Bk is a background knowledge, P is a fuzzy DATALOG program, Φ P is a decoding set of P and ma is any modifying (or connecting) algorithm. Definition 10. Let {Bk, P, Φ P, ma} be a fuzzy knowledge-base. Then lfp(mnt P ) is the consequence of the knowledge-base, denoted by C(Bk, P, Φ P, ma). Example 3. Let us consider the next dialogue: - Do you know: whether Mary likes Bach? - I think, because her favourite composer is Vivaldi. Now we try to give a computed answer for this question. Let us suppose that the musicians generally love the good composers, Mary fairly is a music founder, and Vivaldi is nearly the favourite composer. Let us denote love by lo, good composer by gc, musician by mu, favourite by fv, music founder by mf. Then let the fdatalog program, the background knowledge and the decoding set be as follows (according to the modifying algorithm, it is enough to consider only the decoding functions of headpredicates). The implication operator of the program be the Gödelian. (I(x,y) = 1 if x y; otherwise I(x,y) = y) lo(x,y) gc(y), mu(x); 0.7; I. (fv(v), 0.9). (mf(m),0.8). B V M B V M 1 lo li gc fv mu mf lo li gc fv mu mf φ lo := φ = φ(α,x,y,z) := min(α,x,y,z) φ fv := θ = θ(α,x,y) := α x y φ mf := ω = ω(α,x,y) := min(α, x y) Let us apply the modification algorithm, and start from the facts, {(fv(v),0.9), (mf(m),0.8)}, we get: according to similarity: {(fv(v),0.9), (gc(v), θ(0.9,0.75,1) = = 0.675), (fv(b), θ(0.9,1,0.9) = 0.81), (gc(b), θ(0.9,0.75,0.9) = ), (mf(m),0.8), (mu(m), ω(0.8,0.6,1) = min(0.8, 0.6 1) = 0.6)} applying the rules on the above set of facts : lo(m,v) :- gc(v), mu(m); 0.7; I. lo(m,b) :- gc(b), mu(m); 0.7; I. we get: (lo(m,v), min(0.675, 0.6, 0.7) = 0.6) (lo(m,b), min(0.6075, 0.6, 0.7) = 0.6) and according to similarity: (li(m,v), φ(0.6,0.8,1,1) = min(0.6,0.8,1,1) = 0.6), (li(m,b), φ(0.6,0.8,0.9,1) = min(0.6,1,0.9,1) = 0.6) So the consequence of knowledge-base is: C(Bk, P, Φ P, ma) = {(fv(v),0.9); (gc(v),0.675); (fv(b),0.81); (gc(b),0.6075); (mf(m),0.8); (mu(m),0.6); (lo(m,v),0.6); (lo(m,b),0.6); (li(m,v),0.6); li(m,b),0.6)}. That is Mary likes Bach at least 0.6 levels. 97

5 5. Evaluation strategies According to the previous example (especially in the case of enlarging the program with other facts and rules), it can be seen, that the fixpoint-query that is the bottom-up evaluation maybe involves many superfluous calculations, because sometimes we want to give an answer for a concrete question, and we are not interested in the whole consequence. If a goal (query) is specified together with the fuzzy knowledgebase, it is enough to consider only the rules and facts being necessary to reach the goal. In this chapter we deal with the top-down evaluation of knowledgebase, which starts from the goal, and applies the suitable rules and similarities to find the required starting facts and rules to get answer for this query. A goal is a pair (q(t 1, t 2, t n ), α), where q(t 1, t 2, t n ) is an atom, α is the level of the atom. It is possible, that among the arguments of q there are variables or constants, and α can be either a constant or a variable. In some cases by the top down evaluation the goal is evaluated through sub-queries. This means, that there are selected all possible rules, whose head can be unify with the given goal, and the atoms of the body are considered as new sub-goals. This procedure continues until obtaining the facts. [3] deals with the evaluation strategies of fuzzy Datalog. The top down evaluation of fuzzy Datalog doesn t terminate obtaining the facts, because we need to determine the uncertainty level of the goal. The algorithm, given in [3], calculates this level in a bottom up manner: starting from the leaves of evaluating graph, going backward to the root, and calculating the actual uncertainty-level along the suitable path of this graph, finally we get the uncertainty level of the root. In recent model, we rather rely on the bottom up evaluation, but the selection of required starting facts takes place in a top-down way. As only the required starting facts are searched, therefore in the top-down part of the evaluation there are no need for the uncertainty levels, so we search only among the ordinary facts and rules. To do this, we need the concept of substitution and unification which are detailed for example in [3], [6], [11], etc. But now sometime we need also other kind of substitutions: to substitute some predicate p or term t for their similarity sets S p and S t, and to substitute some similarity sets for their members. In the next, for the sake of simpler terminology, we mean by goal, rules and facts these concepts without uncertainty levels. An AND/OR tree arise during the evaluation, this is the searching tree. Its root is the goal; its leaves are one of YES or NO. The parentnodes of YES are the required starting facts. This tree is build up by alternation of similarity-based and rule-based unification. The rule-based unification unifies the sub-goals with the head of suitable rules, and continues the evaluating by the bodies of these rules. This unification is special, because during this unification the similarity sets of terms are considered as ordinary constants, and a constant can be unify with its similarity set. The similarity-based unification unifies the predicates of sub-goals by the members of its similarity set, excepting the first and last unification. The first similarity-based unification unifies the ground terms of the goal with their similarity sets, and the last one unifies the similarity sets among the parameters of resulting facts with theirs members. The searching graph according to its depth is build up in the following way: If the goal is on depth 0, then every successor of any node on depth 3k+2 (k=0,1, ) are in AND connection, the others are in OR connection. Detailing: The successors of goal g(t 1,t 2, t n ) be all possible g (t 1,t 2, t n ), where g S g ; t i = t i if t i is some variable and t i = S t i if t i is a ground term. If the atom p(t 1,t 2, t n ) is in depth 3k (k=1,2, ), then the successor nodes be all possible p (t 1,t 2, t n ), where p S p. If the atom L is in depth 3k+1 (k=1,2, ), then the successor nodes be the bodies of suitable unified rules, or the unified facts, if L is unifiable with any fact of the program, or NO, if there is not any unifiable rule or fact. That is, if the head of rule M M 1,...,M n (n>0) is unifiable with L, then the successor of L be M 1 θ,...,m n θ, where θ is the most general unification of L and M. If n=0, that is in the program, there is any fact with the predicate of L, then the successors be the unified facts. If L = p(t 1,t 2, t n ) and in the program there is any fact with predicate p, then the successor nodes be all possible p(t 1,t 2, t n ), where t i S ti if t i = S ti or t i = t i θ, if t i is a variable, and θ is a suitable unification. 98

6 According to the previous paragraph, there are three kind of nodes in depth 3k+2 (k=1,2, ): an unified body of a rule; an unified fact with ordinary ground term arguments; or the symbol NO. In the first case the successors are the members of the body. They are in AND connection, which is not important in our context, but maybe important for a possible future development. If the body has only one literal, then the length of evaluating path would be reduce one, but it would be damage for the view of homogeneous treatment. In the second case the successors are the symbol YES or NO, depending on whether the unified fact is among the ground atoms of the program. These nodes have not successors. From the construction of searching graph, it is obvious: Proposition 2. Let X 0 be the set of ground facts being in parentnodes of symbols YES. Starting from X 0, the fixpoint of mnt P contains the answer for the query. From the viewpoint of the query, this fixpoint may contains more superfluous ground atom, but generally it is smaller then the consequence of knowledge-base. More reduction of the number of superfluous resulting facts is the work of a possible further development. Example 4. Let us consider the knowledge-base of example 3! (Now it is enough to consider only the program and the background knowledge.) Let the goal be li(m,x), where x is a variable. Then the searching graph is: The algorithm of searching the starting facts based on the above construction is being in press. Summary In this paper there is given a possible model of handling uncertain information by defining fuzzy knowledge-base as a quadruple of a background knowledge, a deduction mechanism, a decoding set and some modifying algorithm, which connects the background knowledge to the deduction mechanism. There was also given a possible evaluation strategy. Possibly this evaluation strategy can be improved, or maybe there is a better modifying algorithm. To find these better solutions will be the work of a possible further development. References [1] Á. Achs, A. Kiss: Fixpoint query in fuzzy Datalog Annales Univ. Sci. Budapest, Sect. Comp. 15 (1995) pp [2] Á. Achs, A. Kiss: Fuzzy extension of Datalog Acta Cybernetica,Szeged 12 (1995), pp [3] Á Achs: Evaluation Strategies of Fuzzy Datalog Acta Cybernetica, Szeged 13 (1997), pp [4] Á. Achs: Fuzzy knowledge-base with fuzzy Datalog ISDA 2004 IEEE 4th International Conference on Intelligent System Design and Application, August 26-28, 2004, Budapest pp [5] F.Arcelli, F.Formato, G.Gerla, Fuzzy Unification as Foundations of Fuzzy Logic programming in Logic Programming and Soft Computing, Ed. RSP-Wiley, England, [6] S.Ceri, G.Gottlob, L.Tanca: Logic Programming and Databases Springer-Verlag Berlin, 1990 [7] D. Dubois, H. Prade: Fuzzy sets in approximate reasoning, Part 1 Fuzzy Sets and Systems 40 (1991) pp [8] J.W.Lloyd: Foundations of Logic Programming Springer-Verlag, Berlin, [9] M. Schroeder, R. Schweimeier: Arguments and Misunderstandings: Fuzzy Unification for Negotiating Agents Proceedings of the ICLP workshop CLIMA02, Copenhagen, Aug [10] M. I. Sessa: Approximate reasoning by similaritybased SLD resolution Theoretical Computer Science 275(2002) pp [11] J.D.Ullman: Principles of database and knowledgebase systems Computer Science Press, Rockville [12] H. E. Virtanen: Fuzzy unification 5th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems, July 4-8, Paris 99