V.S.Subrahmanian. Mechanical proof procedures for many valued lattice-based logic programming. Journal of Non-Classical Logic 7, pp. 7-41, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....multiple valued semantics used here) There are several reasons for using this setting. The most important ones for our purposes are the following: In the context of four valued semantics it is possible to de ne consequence relations that are not degenerated w.r.t. any theory (see, e.g. [2, 3, 25, 26, 29]) the fact that every theory has a nonempty set of four valued models implies that four valued reasoning may be useful for properly handling inconsistent theories. As shown e.g. in [2, 3] this indeed is the case. Analysis of four valued models can be instructive to pinpoint the causes of the ....

.... classical models of can be represented by Circ f ; g : a valuation in L L is a most classical model of i ( 2DL and (Circ f ; g ) sup(L) Particular cases in which the representations above may be used are the bilattice based logics introduced in [2, 3, 4] and the annotated logic [29] RI, introduced in [16, 17] provided that the underlying many valued structure is of the form L L. 5 Conclusion In this paper we have introduced a method of representing paraconsistent reasoning by classical second order formulae. This method touches upon several di erent aspects: 1. It ....

V.S.Subrahmanian. Mechanical proof procedures for many valued lattice-based logic programming. Journal of Non-Classical Logic 7, pp. 7-41, 1990.

....may remain inconsistent, but the set of conclusions implied by it is not explosive, i.e. not every fact follows from an inconsistent database. Paraconsistent procedures for integrating data (e.g. 14, 41] are often based on a paraconsistent reasoning process, such as LFI [13] annotated logics [30, 40], or other non classical proof systems [5, 37] Coherent (consistency base) methods, in which the amalgamated data is revised in order to restore consistency (see, e.g. 6, 8, 11, 25, 31] In many cases the underlying formalism of these approaches are closely related to the theory of belief ....

....many truth values, which allow to decode within the language itself some metainformation such as con dence factors, amount of belief for or against a speci c assertion, etc. These approaches combine corresponding formalisms of knowledge representation (such as annotated logic programs [40, 41] or bilattice based logics [5, 21, 33] together with non classical refutation procedures [20, 30, 40] that allow to detect inconsistent parts of a database and maintain them. A closely related topic is the problem of giving consistent query answers in inconsistent database [3, 10, 25] The idea ....

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V.S.Subrahmanian. Mechanical proof procedures for many valued lattice-based logic programming. J. Non-Classical Logic 7, pp.7-41, 1990.

.... ones (e.g. de ning priorities on di erent sources of information or preferring certain data over another [2, 4, 5] Other approaches are based on rewriting rules for representing the information in a speci c form [14] or use multiple valued semantics (e.g. annotated logic programs [28, 29] and bilattice based formalisms [12, 22] together with non classical refutation procedures [11, 19, 28] that allow to decode within the language itself some meta information such as con dence factors, amount of belief for against a speci c assertion, etc. Each one of the techniques mentioned ....

.... data over another [2, 4, 5] Other approaches are based on rewriting rules for representing the information in a speci c form [14] or use multiple valued semantics (e.g. annotated logic programs [28, 29] and bilattice based formalisms [12, 22] together with non classical refutation procedures [11, 19, 28] that allow to decode within the language itself some meta information such as con dence factors, amount of belief for against a speci c assertion, etc. Each one of the techniques mentioned above has its own limitations and or drawbacks. For instance, in order to properly translate the ....

V.S.Subrahmanian. Mechanical proof procedures for many valued lattice-based logic programming. Journal of Non-Classical Logic 7, 7-41, 1990.

....one can infer a disjunction, p : r 1 : p : rn . Therefore, the class of Horn clauses is not closed under the inference rules, and no ecient proof strategy, such as SLD resolution, is likely to exist. In contrast, an analogue of SLD resolution is known to exist for Horn clauses in APC [31] Thus, our restriction on BSL to be a semilattice was motivated by the desire to keep the proof theory simple. 8 An Alternative Semantics It was noted in Section 5 that Herbrand s theorem does not hold for APC in its full generality, and the restrictions in Theorems 2 and 3 are necessary. In ....

V.S. Subrahmanian. Mechanical proof procedures for many-valued lattice-based logic programming. Contemporary Mathematics, 1989.

.... taking as the only maximal element (and all the other truth values are incomparable) The preferential order for j= 4 I2 obtains by taking and as greater than t and f (see [2] for a justification of these choices) ffl The logic RI of Kifer and Lozinskii [9] This is an annotated logic [22]. The preferred models of RI minimize the assignments w.r.t. a certain set Delta aeL. Thus, the preferential order in this case obtains by considering every element in Delta as strictly greater than every element in Ln Delta. 4 Useful properties of j= L;D 4.1 Non monotonicity and ....

V.S.Subrahmanian. Mechanical proof procedures for many valued lattice-based logic programming. Journal of Non-Classical Logic 7, pages 7--41, 1990.

....that do not contradict any previously drawn conclusions are allowed. For achieving the goals above we consider an algorithmic approach that is based on a four valued semantics [3, 4] Using a multiple valued semantics is a common way to overcome the shortcomings of classical calculus (see, e.g. [3, 6, 7, 12 14]) and as we shall see in what follows, four valued semantics is particularly suitable for our purpose. A similar algorithmic approach for recovering stratified knowledge base, which is also based on a four valued semantics, was introduced in [1, 2] Here we generalize and improve that approach ....

V.S.Subrahmanian. Mechanical proof procedures for many valued lattice-based logic programming. Journal of Non-Classical Logic 7, pages 7--41, 1990.

....(multi valued) case there are many others. Kifer and Lozinskii ( KL92] also propose a similar relation (denoted there j Delta , where Delta stands for the values that are considered as representing inconsistent knowledge) This relation is considered in the framework of annotated logics ([Su90a, Su90b]) See [AA96] for a discussion on the similarities and the differences between j= B;F I and j Delta . Some of the basic properties of j= B;F I are considered in [AA96, AA97b] We just note here that this relation is not only nonmonotonic, paraconsistent, and a plausibility logic [Le92] but ....

V.S.Subrahmanian. Mechanical proof procedures for many valued lattice-based logic programming. Journal of Non-Classical Logic, Vol.7, pages 7--41, 1990.

....general (multi valued) case there are many others. Kifer and Lozinskii [KL92] also propose a similar relation (denoted there j Delta , where Delta stands for the values that are considered as representing inconsistent knowledge) This relation is considered in the framework of annotated logics [Su90a, Su90b]. See [AA96, AA98] for a discussion on the similarities and the differences between j= B;F I and j Delta . We now show that again everything that one can infer by using j= B;F I may be inferred in hFOURi together with either I 1 or I 2 as the inconsistency set: Theorem 6.27 For every logical ....

V.S.Subrahmanian. Mechanical proof procedures for many valued lattice-based logic programming. Journal of Non-Classical Logic, Vol.7, pages 7--41, 1990.

....f; g it is Sobocinski s implication ( Sobo] which is the implication of RM 3 the strongest logic in the family of relevance logics (see also [AnBe] Dunn] and [Hind] Notes: 1. Using we can sometimes translate annotated atomic formulae from Subrahmanian s annotated logic (see [CHLS] [Sub1], Sub2] KiLo] KiSu] Thus, the translation to BL(4) of : b when b 2 FOUR, and the partial order is t , is simply b . 2. In FOUR, OE) 2D k (FOUR) iff ( t (OE) Moreover, oe OE is equivalent to OE( OE) The next example demonstrates the potential use of j= BL as well as of ....

V.S.Subrahmanian. Mechanical Proof Procedures for Many Valued Lattice-Based Logic Programming. Journal of Non-Classical Logic, Vol.7 (pp. 7-41); 1990.

....A preliminary version of this paper appears in [AA95] One possible solution to this problem is to use some kind of a paraconsistent logic ( dC74] i.e. a logic in which trivial reasoning from a contradiction is not allowed. Several candidates has been suggested in the literature (see, e.g. [BCDLP, BDP95, CSHL, KL92, Lo94, Pr89, Pr91, Su90a, Su90b, Su94]) The use of such logic has, however, its own drawbacks. The major one is the fact that it forces us to give up classical reasoning altogether. This is certainly not justified in case the given knowledge base is consistent. Moreover: the classical calculus is a very convenient framework to work ....

....not contradict any data entailed by the original knowledge base. 20 The last example also shows that two valued semantics is not sufficient even in cases where there are no spoiled literals. 7. 2 Annotated logics; Kifer and Lozinskii s treatment Annotated logics were introduced by Subrahmanian [Su90a, Su90b], and further developed by him and others (see, e.g. CSHL, KL92, KS92, Su94] They also use multi valued algebraic structures in order to provide a semantics for rule based systems with uncertainty. As we have already noted, KL92] use annotated logic for similar purposes as ours. However, the ....

V.S.Subrahmanian. Mechanical proof procedures for many valued lattice-based logic programming. Journal of Non-Classical Logic, Vol.7, pages 7--41, 1990.

....of the logic 1 . This relation can be viewed as a kind of a closed word assumption , since it considers only the most consistent models 1 See [AA94c] for a detailed discussion on the advantages of the present logic with respect to the logic of [KL92] in particular, and to annotated logics [Su90a, Su90b, KS92] in general. mcms) of a given set of assertions. As was shown in [AA94a, AA94b] this relation enjoys some appealing features, such as being non monotonic, paraconsistent ( dC74] and a plausibility logic ( Le92] By using j= con we are able to: a) discover easily the core of the ....

V.S.Subrahmanian. Mechanical proof procedures for many valued lattice-based logic programming. Journal of Non-Classical Logic, Vol.7 (pp. 7-41); 1990.

....all scientific investigations into reasoning with uncertainty and chance, probability theory is perhaps the best understood paradigm. Nevertheless, all studies conducted thus far into the semantics of quantitative logic programming (cf. van Emden [51] Fitting [18, 19, 20] Blair and Subrahmanian [5, 6, 49, 50], Kifer et al. [29, 30, 31] have restricted themselves to non probabilistic semantical characterizations. In this paper, we take a few steps towards rectifying this situation. We define a logic programming language that is syntactically similar to the annotated logics of [5, 6] but in which the ....

....In a related context, Keisler [26] has shown that even finite logics with oe additive probability distributions (cf. Halmos [25] over the domain of an interpretation are not compact. We are currently working on this problem. 7 Conclusions Thus far, quantitative logic programming languages [51, 18, 19, 5, 48, 49] have been unable to deal with probabilistic information. As probabilistic and statistical information is widely used in everyday decision making, it is essential that logic programs have the ability to represent probabilistic information. We have proposed, in this paper, a probabilistic framework ....

V. S. Subrahmanian. (1988) Mechanical Proof Procedures for Many Valued Lattice Based Logic Programming, to appear in: Journal of Non-classical Logic.

.... X W j j=F and W j 22 B L p j 1 C A max LC(S P (h 1 ) 0 B X W j j=F and W j 22 B L p j 1 C A = d 1 . Hence it is true that [c 1 ; d 1 ] c 2 ; d 2 ] and thus T P (h 1 ) F ) T P (h 2 ) F ) Since this subset relation is true for all F 2 bf(BL ) T P (h 1 ) T P (h 2 ) 2 In [24, 26, 32, 42] additional conditions are imposed to prove the monotonicity of their fixpoint operators associated with a program. In fact, those conditions are only needed to prove that the operators are continuous. Since our T P operator is not continuous (cf. Example 11) such conditions are dropped in the ....

V. S. Subrahmanian. (1988) Mechanical Proof Procedures for Many Valued Lattice Based Logic Programming, to appear in a technical journal.

....k) A) t(T P ) A) For Claim (2) R P (R P ) R P follows from the monotonicity of R P and since R P R P i, for all i. The other inclusion is proved similarly to the earlier proof that R P (R P k) A) t(T P ) A) in Claim (1) 2 4 Constrained Queries and GAPs Following [44], we are now going to develop an SLD style proof theory for GAPs based on the general semantics. Definition 12 Suppose P is a GAP and C 1 ; C n are renamed versions of clauses in P such that no pair C i and C j of clauses shares common variables. Further, let each C r , 1 r n, be of the ....

....later, there is also a close connection between annotated programs and certain fragments of temporal logics. Thus, there is hope that in the future a single unifying framework for multivalued, temporal, and constraint logic programming will emerge. In related works, Morishita [36] and Subrahmanian [44] have also studied multivalued logic programming where annotations were associated with clauses, rather than with individual atoms. Morishita s framework is as follows: Associated with each atom is a lattice, and associated with each clause is a function that maps the product of the lattices ....

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V. S. Subrahmanian. (1990) Mechanical Proof Procedures for Many Valued Lattice Based Logic Programming, to appear in: Journal of Non-Classical Logic, 1990.

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