David Etherington, Robert Mercer, and Raymond Reiter. On the adequacy of predicate circumscription for closed-world reasoning. Computational Intelligence, 1(1), 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....language to the set of its ground terms. For any NAT T , the models of T [H are the Herbrand models of T . McCarthy s notation is all(x) The fact that, in the early work on circumscription, McCarthy sometimes applied the circumscription operator to a subset of the theory was noted in [3]. Theorem 5.3 in that paper shows that this is, apparently, unavoidable, if the domain closure assumption is to be reduced to circumscription. 4 Two Special Cases 4.1 Minimizing a Predicate We will abbreviate blocks of the form fP : P (x) oe Ab(x) A 1 ; A n g; where P is a predicate ....

David Etherington, Robert Mercer, and Raymond Reiter. On the adequacy of predicate circumscription for closed-world reasoning. Computational Intelligence, 1(1), 1985.

....Stanford University. This paper is published in D. Nardi and P. Maes (eds. Meta level architectures and Reflection,pages 271 285, North Holland, 1988. 1 proposed circumscription that is to use second order axiomatizations for circumscribing the problem at hand to a minimal world . Reiter [EMR85] has proposed default reasoning , where partial first order theories are augmented with inference rules that allow to derive, in absence of information, some not known statements about the world. Lukaszewicz [Luk86] has proposed to express circumscription in first order logic. The ....

....to make the claim that this kind of reasoning is not non monotonic. It can be classified as inessential non monotonicity. The whole theory has been built as non monotonic because of the choice of the formalism. The implicit assumption at the basis of this approach (made, among the others, also in [Luk86, McC87, EMR85]) is that there is only one theory of the whole world which contains all the axioms and theorems. On the other hand, analogously to what happens in human reasoning, it must be possible to derive, in different situations, both A and notA, where A is a wff (in our example A is sun shining) In fact ....

D.W. Etherington, R.E. Mercer, and R. Reiter. On the adequacy of predicate circumscription for closed-world reasoning. Computational Intelligence, 1:11-- 15, 1985.

....we know that general predicate circumscription falsifies cumulativity. But it is also known that mixed predicate circumscriptions satisfy cumulativity when restricted to universal theories, and Suchenek s results concern only universal theories. Also, in predicate circumscription, it is known till [EMR85] that the equality predicate acts like a fixed predicate , and it is effectively in this way in Suchenek s results. The great restriction of Suchenek s results concerns the fact that varying predicates are forbidden. Now, our results could be extended to the predicate calculus (at the price of ....

David W. Etherington, Robert E. Mercer, and Raymond Reiter. On the adequacy of predicate circumscription for closed-world reasoning. Computational Intelligence, 1:11--15, 1985.

....formula. But this failure is no surprise, following from the incompleteness of second order logic itself. It was also noticed early on that the result of circumscribing certain predicates even in consistent theories might lead to inconsistency; a simple example, due to Etherington et al. [2], results when we consider the theory 2 , containing the sentences 9x[Nx 8y(Ny x 6= s(y) 8x(Nx Ns(x) 8xy(s(x) s(y) x = y) Any model M of 2 must assign to N an extension containing a series isomorphic to the natural numbers (with s interpreted as successor) and we can then de ....

D. Etherington, R. Mercer, and R. Reiter, \On the Adequacy of Predicate Circumscription for Closed-World Reasoning" Computational Intelligence, 1 (1985), 11-15.

....yPyx) 1) 8x(9yPyx 9yPxy) 2) 9 =1 x(9yPxy 8y:Pyx) 3) The quanti er 9 1 is an abbreviation for there exists at most one which clearly is expressible in rst order logic. P can be interpreted as a representation of the successor function in the natural numbers. As is shown in [3] and [4], there are no P minimal models of T 1 . The theory T 2 describes binary trees. To facilitate matters, we use the abbreviation x 2 Field(P) for the formula 9y(Pxy Pyx) The rst three axioms of T 2 then are: 8x(x 2 Field(P) 9 2 yPxy) 1) 8x(x 2 Field(P) 9 1 yPyx) 2) 9 =1 x(x 2 ....

....used to construct a model of T 1 . Hence the P minimal models of T are exactly the models of T 2 with nite extensions of P, particularly T j= P 9x(x 2 Field(P) 9yPxy) This sentence however, does not hold in M 0 , and is therefore not circumscriptively inferable from T. Recall from [6] and [4] that a theory T(P,Q) is called well founded with respect to P;Q i for every model M of T that is not P;Q minimal itself, there exists a P;Q minimal model N with N P;Q M. Con ning the original proposition of Perlis and Minker to well founded theories yields the following theorem. 1 Theorem ....

Etherington,D.W., Mercer,R.E. and Reiter,R., On the Adequacy of Predicate Circumscription for Closed World Reasoning, Computational Intelligence 1 (1985) 11-15.

....extent is as small as possible. Model theoretically, this means that we focus attention on those models where the extent of the circumscribed predicate is minimal. The reader who is interested in a precise definition of circumscription and the minimal models is referred to the Appendix. 10 In [Etherington et al. 1985] , Etherington, Mercer and Reiter pointed out the inability of predicate circumscription to derive new positive ground instances of any predicate. Formula circumscription, introduced in [McCarthy, 1986] does not suffer from this limitation. The main technical idea here was to vary other predicates ....

David Etherington, Robert Mercer, and Raymond Reiter. On the adequacy of predicate circumscription for closed world reasoning. Computational Intelligence, 1:11--15, 1985.

....smooth. The smoothness condition is necessary to ensure the validity of Cautious Monotonicity. It is akin to the limit assumption of Stalnaker [40] and Lewis [24] but it is defined in a more general context. Smoothness is the property called, contrary to mathematical usage, well foundedness in [8] and in [26] We shall now describe how a cumulative model defines a consequence relation. Definition 8 Suppose a cumulative model W = hS; l; OEi is given. The consequence relation defined by W will be denoted by W and is defined by: ff W fi iff for any s minimal in b ff, s j fi. Definition 9 ....

David W. Etherington, Robert E. Mercer, and Raymond Reiter. On the adequacy of predicate circumscription for closed-world reasoning. Computational Intelligence, 1:11--15, 1985.

....two terms denote distinct objects unless their equality is provable. We don t know a completely satisfactory way of doing this. Suppose that we have a language L and a theory T consisting of the consequences of a formula A. It would be most pleasant if we could just circumscribe equality, but as Etherington, Mercer and Reiter (1985) point out, this doesn t work, and nothing similar works. We could hope to circumscribe some other formula of L, but this doesn t seem to work either. Failing that, we could hope for some other second order formula taken from L that would express the unique names hypothesis, but we don t presently ....

Etherington, D., Mercer, R. and Reiter, R. (1985). On the Adequacy of Predicate Circumscription for Closed-World Reasoning, Computational Intelligence 1.

....possibility of varying the extension of predicates. Variability of predicates means that the extensions of some predicates (the varying ones) can be affected in any way (enlarged or reduced) in order to decrease the extensions of those predicates that are minimized. Etherington, Mercer and Reiter [3] noticed that with the original definition it is impossible to derive new positive facts from a knowledge base. For example, it is impossible to infer f lies(T weety) from the fact bird(Tweety) and the rule 8x:bird(x) abnormal(x) oe f lies(x) minimizing abnormal while bird and flies are fixed. ....

....cause exponential overhead. In summary, our transformation shifts the burden of handling varying predicates to predicate minimization. An application of our transformation to closed world reasoning is also possible. This technique for non monotonic reasoning is closely related to circumscription [3, 10, 5] and has been defined in several ways. In its previous versions (CWA, GCWA, EGCWA [16, 14, 19] the closed world assumption was applied to all the predicates. In its more recent definitions (CCWA, ECWA [6, 5] it is possible to vary a predicate, and the intended meaning of this variability is the ....

D. W. Etherington, R. E. Mercer, and R. Reiter. On the Adequacy of Predicate Circumscription for Closed World Reasoning. Computational Intelligence, 1:11-- 15, 1985.

....(see example 5.2 in [MR94b] thus this circumscription cannot be an X mapping. 19 that mixed predicate circumscriptions satisfy cumulativity when restricted to universal theories, and Suchenek s results concern only universal theories. Also, in predicate circumscription, it is known till [EMR85] that the equality predicate acts like a fixed predicate , and it is effectively in this way in Suchenek s results. The great restriction of Suchenek s results concerns the fact that varying objects are forbidden. Notice that theorem 2.15 2) shows that the predicate circumscriptions concerned by ....

David W. Etherington, Robert E. Mercer, and Raymond Reiter. On the adequacy of predicate circumscription for closed-world reasoning. Computational Intelligence, 1:11--15, 1985.

....Definition 1.3 (Lifschitz [5] A theory T is well founded w.r.t (P; Z) if for each model A of T, there exists a (P; Z) minimal model B such that B T P; Z A. However, there are perfectly reasonable theories, such as the simple theory of natural successor (the example given in Etherington [3]) that are not well founded. Another approach to the same problem is to weaken circumscription by generalizing the minimality criterion which will accordingly increase the class of theories that can be consistently handled. Definition 1.4 (Liau et al. 4] Given a theory T, a model A of T is ....

....all theories with a consistent set of consequences. 2 Feasible Commitment Predicate Circumscription The original definition of circumscription is based on minimal models. However, if a theory has no minimal models, then circumscribing it gives an inconsistent theory. Davis [1] and Etherington [3] provide examples of such theories for domain and predicate circumscription, respectively. Liau et al. 4] presents Etherington s example and shows how their definition of abstract circumscription solves the inconsistency problem for the given example. Our examples include theories which do not ....

D. W. Etherington, R. E. Mercer, and R. Reiter. On the adequacy of predicate circumscription for closed-world reasoning. Computational Intelligence, 1(1):11--15, 1985.

....j= LAbnormal the theory P correctly reflects our beliefs and thus is complete. As we will see below P is in fact the least (smallest) completion of P . 2 Remark 2. 1 Since for any proposition A in a complete theory P we have: P j= LA iff CIRC(P ) j= A and since in any theory we have (cf. [Lif85, EMR85, GPP89]) P j= A iff CIRC(P ) j= A in any complete theory we have: P j= LA iff P j= A: Therefore a complete theory P proves a proposition A if and only if it proves LA. In this sense, any proposition A can always be identified with its corresponding belief proposition LA. In particular, the fact ....

D. Etherington, R. Mercer, and R. Reiter. On the adequacy of predicate circumscription for closed-world reasoning. Journal of Computational Intelligence, 1:11--15, 1985.

....even though the base sentence is satisfiable; i.e. the circumscription operation itself introduces inconsistency. Circumscription and other model preference NM formalisms are well known to be susceptible to nasty unsatisfiability: when models are not well founded in the sense defined by [ Etherington et al. 1985 ] Lifschitz, 1986 ] But Lifschitz formalism permits a kind of incoherence, in the policy of varying, which results in nasty unsatisfiability even in very simple, well founded theories. Our approach does not. Fifthly, because our approach, unlike Lifschitz , is a special case of single ....

D. Etherington, R. Mercer, and R. Reiter. On the adequacy of predicate circumscription for closed-world reasoning. Computational Intelligence, 1:11--15, 1985.

....two approaches are no longer equivalent. Indeed, the minimal model semantics does not introduce any new positive information and therefore avoids the universal query problem, whereas as we pointed out in Section 4 the least Herbrand model semantics does not have this property. Theorem 5. 3 ([EMR], L2] Suppose that P is a positive logic program and F is any positive sentence. Then: MIN(P ) j= F ( P j= F: 2 Therefore, a positive formula is implied by MIN(P) iff it can be logically derived from P itself. As we mentioned already before, this seems to be a very desirable property for ....

....[L2] Parallel) circumscription CIRC(T) of a given theory T is defined as a second order formula, but it is known [L2] that its semantics in the simplest case, when all predicates are minimized is determined by the class MIN(T) of all minimal models of T. Consequently, we have Theorem 5. 5 ([EMR,L2]) Minimal model semantics of a positive logic program P coincides with the semantics of parallel circumscription of P, i.e. for any sentence F MIN(P ) j= F ( CIRC(P ) j= F: 2 Thus the choice of minimal model semantics for positive logic programs establishes a clear relationship between logic ....

Etherington, D., Mercer, R. and Reiter, R., `On the Adequacy of Predicate Circumscription for Closed-World Reasoning', Computational Intelligence 1(1985), 11-15.

....even though the base sentence is satisfiable; i.e. the circumscription operation itself introduces inconsistency. Circumscription and other model preference NM formalisms are well known to be susceptible to nasty unsatisfiability: when models are not well founded in the sense defined by [ Etherington et al. 1985 ] Lifschitz, 1986 ] But Lifschitz formalism permits a kind of incoherence in the policy which results in nasty unsatisfiability even in very simple, well founded theories. Our approach does not. Secondly, Lifschitz formalism cannot directly represent prioritization that is not layered ....

D. Etherington, R. Mercer, and R. Reiter. On the adequacy of predicate circumscription for closed-world reasoning. Computational Intelligence, 1:11--15, 1985.

....also why we can minimise equality; the minimisation occurs before the terms have assigned to individuals. We can thus affect this assignment. When minimising in the semantic domain, the minimisation occurs after terms have been assigned to individuals; thus the minimisation cannot affect equality [Etherington et al. 1985], and the unique names hypothesis Default Logic 18 is needed (as, for example, the violation set fab(a) ab(b)g can be reduced by making a = b) Theorem 4.7 can be traced to a number of sources. If we let 3 be the circumscriptive version of 3, which are the same under the unique names and ....

D. W. Etherington, R. E. Mercer, and R. Reiter. On the adequacy of predicate circumscription for closed-world reasoning. Computational Intelligence, 1(1):11--15, 1985.

....perfect model of P. According to the next result, the minimal model semantics does not introduce any new positive sentences, where by a positive sentence we mean a sentence, whose normal disjunctive form does not contain any negative literals. 5.3.2. Theorem. Etherington, Mercer and Reiter [EMR85], Lifschitz [L85] For any positive sentence F we have: MIN(P ) j= F ( P j= F: 2 The above result will help us explain why the definition of the perfect model semantics should not be limited to Herbrand models of P. 5.3.3. Example. Suppose that our program is simply p(a) Its only perfect ....

Etherington, D., Mercer, R. and Reiter, R., `On the Adequacy of Predicate Circumscription for Closed-World Reasoning', Computational Intelligence 1(1985), 11-15.

....] relating the ECWA to circumscription can be proved in a more general setting, and in a simpler way. The only thing that needs to be assumed in order to prove the the equivalence between circumscription and our reformulation of the ECWA is the very general condition known as well foundedness [ Etherington et al. 1985 ] Lifschitz, 1986 ] 2 Circumscription and well founded sentences In this note, a sentence is a sentence of a fixed second order language. We understand entailment semantically (in view of the incompleteness of secondorder logic, this is different from the deductive understanding of ....

....is well founded if it entails 9pz(A c (p; z) p P ) 2) see [ Lifschitz, 1986 ] Proposition 2.1) It is easy to show, for instance, that every sentence that has no infinite models is well founded. Other sufficient conditions for well foundedness can be found in [ Bossu and Siegel, 1985 ] Etherington et al. 1985 ] Lifschitz, 1986 ] and [ Etherington, 1988 ] We will need the following property of well founded sentences (for related results, see [ Marczewski, 1951 ] Lyndon, 1959 ] Etherington et al. 1985 ] Etherington, 1988 ] Gelfond et al. 1989 ] and [ Suchenek, 1990 ] Theorem ....

[Article contains additional citation context not shown here]

David Etherington, Robert Mercer, and Raymond Reiter. On the adequacy of predicate circumscription for closedworld reasoning. Computational Intelligence, 1(1), 1985.

....circumscription. He notes that circumscription is difficult to implement because its definition involves a second order quantifier. He introduces metamathematical results that allow, in some cases, circumscription to be replaced by an equivalent first order formula. Etherington, Mercer and Reiter [EMR85] establish results about the consistency of circumscription, showing that predicate circumscription cannot account for some kinds of default reasoning, and also provides no information about equality predicates. Perlis [Per86] shows the inadequacies of circumscription to deal with counterexamples. ....

D. Etherington, R. Mercer, and R. Reiter. On the adequacy of predicate circumscription for closed world reasoning. Computational Intelligence 1, pages 11--15, 1985.

....minimise equality; the minimisation occurs before the terms have been assigned to individuals. We can thus affect this assignment. When minimising in the semantic domain, the minimisation occurs after terms have been assigned to individuals; thus the semantic minimisation cannot affect equality [12], and the unique names hypothesis is needed. For example, the violation set fab(a) ab(b)g can be reduced by making a = b. Without the unique names assumption, from the facts fab(a) ab(b) p(a)g semantically minimizing ab (assuming :ab) will conclude p(b) The syntactic minimization does not let ....

D. W. Etherington, R. E. Mercer, and R. Reiter. On the adequacy of predicate circumscription for closed-world reasoning. Computational Intelligence, 1(1):11--15, 1985.

....way occur frequently in practice. This suggests that the problem is not an isolated baroque instance where the formalisms do not perform well but is, rather, symptomatic of fundamental difficulties. 2 Since circumscription cannot generate new equality facts without resorting to variable terms [Etherington et al. 1985], explicit inequalities are needed to rule out models where only Tweety is a bird, but she goes by various aliases. 3 A domain closure axiom (DCA) Reiter, 1980a] is a formula of the form 8x: x = t1 : x = tn , for some set of ground terms, t1 ; tn . There s Nobody Here But Us ....

.... that there is no problem, however, in believing that there are known exceptions in the scope (e.g. Bird(Opus) F lies(Opus) Scope(Opus) Is scoped circumscription consistent, however This question is important because inconsistency has plagued certain applications of circumscription [ Etherington et al. 1985 ] Etherington [ 1988 ] shows that theories without existential quantifiers have consistent circumscriptions, but counterexample axioms take us out from under this umbrella of safety. Nevertheless, scoped circumscription is consistent, regardless of the form of the original theory, provided the ....

David W. Etherington, Robert E. Mercer, and Raymond Reiter. On the adequacy of predicate circumscription for closed-world reasoning. Computational Intelligence, 1:11--15, 1985.

....follow by circumscription unless the further axiom that Tweety 6= Opus is adopted. But this amounts to assuming that Tweety is not the exceptional bird which seems to obviate the circumscription. 5 Since circumscription cannot generate new equality facts without resorting to variable terms [8], explicit inequalities are required to rule out the models where only Tweety is a bird, but she goes by various aliases. The problem surfaces in a slightly different form in default logic. Because defaults apply only to individuals in the Herbrand Universe, rather than being universally ....

....are known exceptions in the scope (e.g. Bird(Opus) F lies(Opus) 5.1.2 Consistency We now must ask whether the resulting circumscribed theory is consistent. The question of consistency is important because inconsistency has plagued certain applications of circumscription from the beginning [8]. Etherington [5] shows that the circumscription of universal theories (i.e. those without existential quantifiers) is consistent. However, counterexample axioms take us out from under this umbrella of safety. Nevertheless, we have proved that, under appropriate conditions, the circumscribed ....

David W. Etherington, Robert E. Mercer, and Raymond Reiter. On the adequacy of predicate circumscription for closed-world reasoning. Computational Intelligence, 1:11--15, 1985.

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David Etherington, Robert Mercer, and Raymond Reiter. On the adequacy of predicate circumscription for closed-world reasoning. Computational Intelligence, 1(1), 1985.

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D. Etherington, R. Mercer, and R. Reiter. On the adequacy of predicate circumscription for closed-world reasoning. Journal of Computational Intelligence, 1:11--15, 1985.

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