Abductive reasoning through Filtering - Semantic Scholar

Abductive reasoning through Filtering Chitta Baral Department of Computer Sc. Univ. of Texas at El Paso El Paso, TX, 79968, USA [email protected] 915-747-6952/5030 (ph/fax)

Abstract Abduction is an inference mechanism where given a knowledge base and some observations, the reasoner tries to nd hypotheses which together with the knowledge base explain the observations. A reasoning based on such an inference mechanism is referred to as abductive reasoning. Given a theory and some observations, by ltering the theory with the observations, we mean selecting only those models of the theory that entail the observations. Entailment with respect to these selected models is referred to as lter entailment. In this paper we give necessary and sucient conditions when abductive reasoning with respect to a theory and some observations is equivalent to the corresponding lter entailment. We then give suciency conditions for particular knowledge representation formalisms that guarantee that abductive reasoning can indeed be done through ltering and present examples from the knowledge representation literature where abductive reasoning is done through ltering. We extend the notions of abductive reasoning and lter entailment to allow preferences among explanations and models respectively and give conditions when they are equivalent. Finally, we give a weaker notion of abduction and show it to be equivalent to lter entailment under less restrictive conditions.

1 Introduction and Motivation Abduction is an inference mechanism where given a knowledge base and some observations, the reasoner tries to nd hypotheses which together with the knowledge base explain the observations. These hypotheses are then referred to as explanations of the observations with respect to the knowledge base. Abduction was introduced by Peirce [Pei58,Pei92] in the beginning of the century and has been used in various AI applications [PMG98], including: temporal explanations [Sha89,DMB92,Sha93]; diagnosis [Reg83,Rei87]; planning [Esh88,AKPT91,MD95]; natural language understanding [HSAM90]; Preprint submitted to Elsevier Science

4 April 1998

default reasoning [PGA87,Poo88,EK89,KKT93], belief revision and updates [BB95,Bou96]; and formulation of negation as failure [EK89,KKT93]. Although, abduction is often used as a `backward reasoning' method where observations are explained, we can also do `forward reasoning' with abduction. This happens when, certain new conclusions are entailed by each of the explanations (or each of the preferred explanations) of an observation together with the knowledge base. By new conclusions we mean conclusions that are not entailed by the knowledge base without using abduction. Moreover, these conclusions may not be entailed by the theory obtained by simply adding the observations to the knowledge base. The following example makes this point clear.

Example 1 Consider the knowledge base T represented by the following logic

program:

fly(X ) bird(X ) fly(X ) aeroplane(X ) haswings(X ) aeroplane(X ) haswings(X ) bird(X ) Now, suppose we observe ffly(tweety)g. If we would like to explain this observation in terms of the predicates bird and aeroplane, then this observation has three explanations:

E1 = fbird(tweety)g, E2 = faeroplane(tweety)g, E3 = fbird(tweety); aeroplane(tweety)g. It is easy to see that for i = 1; 2; 3, T [ Ei entails the new conclusion haswings(tweety), which is not entailed by T . However, the theory T [ ffly(tweety)g obtained by simply adding the observation to the theory, does not entail haswings(tweety). 2 We refer to the entailment of haswings(tweety) from the theory T and the observation ffly(tweety)g as abductive entailment. Such entailments were used in [DDS93,Dun93] while formalizing reasoning about actions using abductive logic programming and one such entailment was formally de ned { using generalized stable models [KM90,EK88] { with respect to abductive logic programming in [BG94]. Shanahan in Chapter 17 of his book [Sha97] refers to such entailments as `knowledge assimilation'. In fact, throughout the book he views explanation as an aspect of knowledge assimilation. We believe abductive entailment and the corresponding knowledge assimilation to be important in knowledge representation. As discussed in the above example we can not (always) just literally add the observations to the theory.

2

Console et al. [CDT91] and Konolige [Kon92] explore this issue and described methods to transform the original theory so that abduction with respect to the original theory can be achieved by simply adding the observation to the transformed theory. (In [CDT91] the transformed theory is in classical logic.) Shanahan [Sha97,Sha89] refers to this as doing abduction through deduction. Assimilation of observations into a theory is also done through ltering [San89]. Intuitively, the models in Filter(T; Q) where T is a (possibly non-monotonic) theory and Q is a set of observations is the set of `models' of T that are also `models' of Q. Here, we use the word `models' in a generic sense. (For example, if T is a normal logic program by `models' of T we will mean stable models of T .) If T is a monotonic theory then the models in Filter(T; Q) are the same as the models of T [ Q. (That means for monotonic theories the notion of ltering is same as the notion of `abduction through deduction' in [CDT91,Kon92].) On the other hand when T is non-monotonic the set of models of T [ Q may not be equivalent to and Filter(T; Q). For example: if T is the logic program fa not bg and Q = fbg, then T [ Q has the model fbg while Filter(T; Q) = ;. Hence, Filter(T; Q) is a more general notion than the notion of `abduction through deduction' described in the previous paragraph and explored in [CDT91,Kon92]. In [San89], Sandewall introduces the notion of lter-preferential entailment and discusses why it is intuitive to take a theory of action expressed as preferential models and ` lter' it with the observations. In that paper Sandewall frequently refers to `observations' and `explanations' and it may give the impression that `abduction' { in a formal sense { can be always done using ltering. It seems to us that his references to `observations' and `explanations' are in an intuitive sense, and are not based on any formal theory of abduction. Our goal in this paper is to give conditions on theories where abductive reasoning can indeed be done using ltering. The motivation behind looking for conditions when abductive reasoning can be done through ltering can be described as follows. Abductive reasoning is widely considered as a fundamental mode of reasoning. It was motivated by philosophical logic and can be de ned in intuitive terms. Research in abductive reasoning is focussed on formalizing this intuition and making it precise. On the other hand, ltering is a precise mathematical method of constructing an entailment relation, which is perhaps more easier to implement. (Reiter in [Rei96] discusses possible problems and ineciency in computing explanations during abduction.) The question here is opposite; when does the entailment relation de ned using ltering makes intuitive sense? In this paper we try to provide some answer to this question and give conditions on theories when ltering corresponds to abductive reasoning. Finally, since [San89], ltering in some form or other has been used in many 3

action formalizations. Baral and Gelfond [BG97] and Turner [Tur95,Tur97] use logic program rules with empty head (or false in the head) to represent observations as lters to a theory of action expressed using logic programs. Turner also uses default rules with empty consequents (or false as the consequent) to represent observations as lters to a theory of action expressed using default logic. Also, the formalization of action theories using NATs [KL94] use multiple application of ltering. The results in this paper will make it easier to identify instances of abductive reasoning (and the corresponding knowledge assimilation) achieved through ltering in these formulations. Moreover, it will guide us in future to develop representations so that we can achieve abductive reasoning and the corresponding knowledge assimilation through ltering.

2 Basic De nitions: Simple abduction and Filtering In the rest of the paper we will use a simpler form of abduction where we are given a set of possible hypothesis from which we can select explanations. Moreover, we will also restrict this set to be a set of ground atoms and their negations and denote this set by Abd. (The set of ground atoms in Abd will be referred to as Abda.) We will refer to the elements of Abd as abducible literals or abducibles. Similarly, we will have a set of ground atoms denoted by Obsa, which we will call observable atoms. Any formula constructed using atoms in Obsa, and logical connectives will be referred to as an observable formula. By Obs, we will denote the set of all observable formulas. We will often refer to an element of Obs as an observation, and denote the set of ground literals in Obs as Obsl. In this simpler form, abduction can now be thought of as a method of reasoning which given a knowledge base T { whose language contains the atoms in Abda and Obsa { and an observation Q, nds possible explanations of Q in terms of a complete subset 1 of the abducibles. We would now like to formally de ne explanations and abductive entailment. In this paper T denotes a possibly non-monotonic theory. In particular we will be considering (i) extended and disjunctive logic programs, (ii) default theories, and (iii) theories in propositional and rst-order logic possibly augmented with circumscription. For each of this theories we will now list, what we mean by a model, what observations are allowed and how entailment is de ned. { In case of a disjunctive logic program (or an extended logic program), by 1 We

say a set S is a complete set w.r.t. Abd, i for any atom a in Abda { the set of atoms in Abd, either a or :a is in S . Often, we will just say S is complete.

4

`model' we will mean an answer set [GL91]. When T is a disjunctive logic program, observations (Q) are allowed to be a (possibly in nite) collection of ground formulas, constructed using atoms in Obsa , and classical logic connectives. Q may be represented by a set of formulas Qf with variables, where the variables serve as schema variables and are substituted with ground terms in the language to obtain Q. We say an answer set entails a formula F if A entails F in the sense of classical logic. We say a disjunctive logic program T entails Q { denoted by T j= Q { if all answer sets of T entail all formulas in Q. Given a set of literals L and a logic program T , by T [ L we mean the logic program T [ fl : l 2 Lg. { In case of a default theory, by `model' we will mean an extension. The allowed observations (Q) is as in the previous case. We say an extension E entails a formula F if E entails F in the sense of classical logic. We say a default theory T entails Q { denoted by T j= Q { if all extensions of T entail all formulas in Q. { In case of propositional theory (or a rst order theory) { possibly augmented with circumscription, by `model' we will mean a classical model. The allowed observations (Q) are formulas in propositional logic (or rst order logic) and entailment between T and Q is the classical entailment relation. Given a set of formulas Q and a classical theory T = fT1; : : :; Tng, where each Ti is a classical theory (possibly augmented with circumscription) by T [ Q we will mean the theory fQ; T1; : : :; Tng.

De nition 1 (Explanation) Let T be a (possibly nonmonotonic) theory with an entailment relation j=, and Q be an observation. A complete set of abducibles E is said to be an explanation of Q w.r.t. a theory T if T [ E j= Q and T [ E is consistent. 2 We would now like to de ne abductive entailment (j=abd) with respect to a pair hT; Qi, which we refer to as an abductive theory. De nition 2 (Abductive Entailment) (i) M is a model of hT; Qi if there exists an explanation E of Q w.r.t. T such that M is a model of T [ E . (ii) For any formula f , hT; Qi j=abd f if f is true in all models of hT; Qi. 2 Proposition 1 Abductive theories are monotonic with respect to addition of

2

observations.

Proof: Suppose we have Q1  Q2. Then any explanation of Q2 with respect to T is an explanation of Q1 with respect to T . Thus models of hT; Q2i are models of hT; Q1i and hence, j=abd is monotonic with respect to Q. 2 De nition 3 Let T be a (possibly nonmonotonic) theory and Q be an ob-

servation. By Filter(T; Q), we refer to the set of models of T which entail Q. 5

2

Proposition 2 Entailment with respect to Filter(T; Q) is monotonic with

respect to Q.

2

Proof: Follows directly from the de nition of Filter(T; Q).

2

3 Abductive Reasoning through ltering { Main results The main goal of this paper is to identify conditions on theories, abducibles and observables such that abductive reasoning can be done through ltering. We now formally de ne such triplets.

De nition 4 (Filter-abducible) A theory T , a set Abd, and a set Obs are said to be lter-abducible if for all possible observations Q 2 Obs; Filter(T; Q) is the set of models of hT; Qi. 2 Before we de ne conditions for lter-abducibility, we rst show an example where Filter(T; Q) is dierent than the set of models of hT; Qi, and then show an example when they are same.

Example 2 Consider the extended logic program [GL91] T1: p a p b Let Abd = fa; b; :a; :bg, and Obs = fpg. Let Q = fpg. it is easy to see that the models of hT1; Qi are ffp; a; :bg; fp; b; :ag; fp; a; bgg, while Filter(T1; Q) = ;. Now consider T2 to be the following extended logic program:

p p a

a b

not :a :a not a b not :b :b not b It is easy to see that the set of models of hT2; Qi is ffp; a; :bg, fp; b; :ag, fp; a; bgg which is same as Filter(T2; Q). Note that the set of models of T2 [ Q is ffp; a; :bg, fp; b; :ag, fp; a; bg, fp; :b; :agg and is dierent from Filter(T2; Q). 2 6

We now describe some conditions which guarantee lter-abducibility.

Condition A

If M is a model of theory T then M is the model of the theory T [ (M \ Abd). Intuitively, Condition A means that the models of a theory T can be characterized by just the abducible literals in that model.

Condition B

For any complete subset E of Abd if T [ E is consistent then there exists a model M of T such that M \ Abd = E Intuitively, Condition B means that the theory T has models corresponding to each valid input. I.e., the models of the theory T enumerate the valid inputs.

Lemma 1 Let T be a theory satisfying Conditions A and B. Let E be any complete subset of Abd. If M is a model of T [ E , then M \ Abd = E . 2 Proof:

Let E be any complete subset of Abd. From Condition B we have that there exists a model M 0 of T such that M 0 \ Abd = E . But From Condition A, we have that M 0 is the model of T [ E . Thus if M is a model of T [ E , then M = M 0 and thus M \ Abd = E . 2

Theorem 1 If a theory T , observables Obs, and abducibles Abd satisfy conditions A and B then they are lter-abducible.

2

Proof:

(a) We rst show that if T , Obs and Abd satisfy conditions A and B then all elements of Filter(T; Q) are models of hT; Qi. Let M 2 Filter(T; Q) ) M is a model of T and M entails Q. ) M is the model of T [ (M \ Abd) and M entails Q. (From Condition A) Let E = M \ Abd. We now have M is the model of T [ E and M entails Q. ) T [ E j= Q, T [ E is consistent, and M is the model of T [ E . ) There exists an explanation E of Q w.r.t. T such that M is a model of T [ E. ) M is a model of hT; Qi. (b) We will now show that if T , Obs and Abd satisfy conditions A and B then all models of hT; Qi are in Filter(T; Q). 7

Let M be a model of hT; Qi. ) There exists an explanation E of Q w.r.t. T such that M is a model of T [ E. ) There exists a complete set of abducibles E such that T [ E j= Q and T [ E is consistent and M is a model of T [ E . From Condition B we have that there exists a model M 0 of T such that M 0 \ Abd = E . But from Condition A, we have that M 0 is the model of T [ E . Thus M = M 0 and M is the model of T [ E , and M is a model of T . Since T [ E j= Q we have M entails Q. Thus M 2 Filter(T; Q). 2 Condition A requires that if M is a models of a theory T then M should be the only model of the theory T [ (M \ Abd). This is a strong condition, as in many cases T [ (M \ Abd) may have multiple models. To take into account such cases, we can weaken condition A, by the following condition. But we will need to use Obs as part of our condition.

Condition A0

If M is a model of theory T then: (i) M is a model of T [ (M \ Abd); (ii) all models of T [ (M \ Abd) are also models of T ; and (iii) all models of T [ (M \ Abd) agree on Obs; where two models M1 and M2 of a theory are said to agree on Obs if for all Q 2 Obs, we have M1 j= Obs i M2 j= Obs.

Lemma 2 If T , Obs and Abd satisfy condition A then they also satis es Condition A0. 2 Proof: Straightforward. Theorem 2 If T , Obs and Abd satisfy conditions A0 and B then they are

2

lter-abducible.

Proof:

(a) We rst show that if T , Obs and Abd satisfy conditions A0 and B then all elements of Filter(T; Q) are models of hT; Qi. Let M be an element of Filter(T; Q) ) M is a model of T and M entails Q. From A0 we have T [ (M \ Abd) is consistent and all models of T [ (M \ Abd) are also models of T and agree on the observables. Let E = M \ Abd. We then have T [ E j= Q and T [ E is consistent. ) There exists an explanation E of Q w.r.t. T such that M is a model of T [ E. ) M is a model of hT; Qi. 8

(b) We will now show that if T , Obs and Abd satisfy conditions A0 and B then all models of hT; Qi are in Filter(T; Q). Let M be a model of hT; Qi. ) There exists an explanation E of Q w.r.t. T such that M is a model of T [ E. ) There exists a complete set of abducibles E such that T [ E j= Q and T [ E is consistent and M is a model of T [ E . From Condition B we have that there exists a model M 0 of T such that M 0 \ Abd = E . But from Condition A0 all models of T [ E are also models of T and agree on Q. Thus M is a model of T . Since M is also a model of T [ E and T [ E j= Q, we have M entails Q. Thus M 2 Filter(T; Q). 2 We now give some examples where we can verify lter-abducibility by verifying the above mentioned conditions.

Example 3 The propositional theory T1 = p , a _ b with Abd = fa; b; :a; :bg and observables = fp; :pg satis es Conditions A and B. This is because the models of T1 are ff:a; :b; :pg, f:a; b; pg, fa; :b; pg, fa; b; pgg and it is easy to see that for each model M in this set, T1 [ M 2

has the only model M .

Example 4 The following logic program T2 q_r a

p a p b a _ :a b _ :b with abducibles fa; b; :a; :bg and observables fpg satis es Conditions A0 and B. This can be veri ed as follows. The program T2 has six models, which are: fa; b; p; qg; fa; b; p; rg; fa; :b; p; qg; fa; :b;p; rg; f:a; b; p; qg, and f:a; :bg. Let us refer to them as M1, M2, M3, M4, M4 and M6 respectively. Consider the model M1. Let us verify that it satis es Condition A0. It is easy to see that M1 is a model of T2 [ (M1 \ Abd) = T2 [ fa; bg, and all models of T2 [ fa; bg are models of T2 and they agree on Obs. We can similarly verify that the other models of T2 satisfy the conditions of A0. In this example there are four complete subsets of Abd. These are: fa; bg; fa; :bg; f:a; bg; f:a; :bg. Consider E = fa; bg. Since, T2 [ E is consistent we need to verify that there exists a model M of T2 such that M \Abd = E . M1 is such a model of T2. We can similarly verify that the other complete sub9

2

sets of Abd satisfy condition B .

4 Filter-abducibility in monotonic and non-monotonic theories Depending on whether a theory is a monotonic or a non-monotonic theory ltering can be achieved in dierent ways. We now list how ltering can be achieved in several dierent knowledge representation formalisms. { If T is in a monotonic language to lter T by Q we just add Q to T . Otherwise, i.e. if T is in a non-monotonic language we can not just add Q to T . { If T is a logic program to lter T by Q we can view Q as an integrity constraint. (Often integrity constraints are expressed through rules with empty head or with false in its head.) Such a view is used in [Tur97,BG97] to assimilate observations to action theories expressed in a logic program. In [Llo87] an algorithm is given to translate general formulas to the restricted syntax of integrity constraints in logic programs. { If T is a default theory to lter T by Q we can express Q as a default rule with an empty consequent or false as its consequent. Such a view is used in [Tur97] to assimilate observations to action theories expressed in a logic program. { If T is a nested abnormality theory (NAT) then we can lter T w.r.t. Q by the NAT consisting of two blocks; one of which is T and the other is Q. As a non-monotonic knowledge representation formalism NAT has the advantage that it makes it easy to have multiple levels of ltering. Our goal in this section is to develop and describe speci c suciency conditions that can be used to show lter-abducibility in particular knowledge representation formalisms. The ones we consider are: circumscriptive theories; rst order theories; disjunctive logic programming; and Reiter's default logic. But rst we de ne the notion of a theory encoding a function, which will play an important role when we de ne the speci c suciency conditions, for each of the knowledge representation formalism that we will be considering.

De nition 5 A theory T is said to encode a function (or is functional) from a set of literals, called input, to a set of literals called output if for any subset E of input such that T [ E is consistent, all models of T [ E agree on the 2

literals from output.

10

4.1 Filter abducibility of rst-order and circumscriptive theories

The following proposition states conditions for the lter-abducibility of propositional and rst-order theories.

Proposition 3 A propositional or rst-order theory (possibly augmented

with circumscription) T is lter-abducible with respect to abducibles Abd and observables Obs if T is functional from Abd to Obs. 2

Proof:

The proof is based on showing that the if part of the proposition implies the conditions A0 (i, ii, and iii) and B. Earlier in Theorem 2 we proved that A0 and B guarantee lter-abducibility. It is easy to see that, because of monotonicity of the theory T , conditions A0(i) and A0(ii) are true. Condition A0(iii) is true due to the condition that T is functional from Abd to Obs. We now only need to show that Condition B holds. Let E be a complete subset of Abd such that T [ E is consistent. Let M be a model of T [ E . Obviously, E  M . Since E is a complete subset of Abd we have that M \ Abd = E . Thus Condition B holds. 2 We now discuss two examples whose lter-abducibility can be deduced using Proposition 3.

Example 5 Consider the theory Tc: r(p) r(p)

q(a) q(b)

and let Obs = fr(p)g and Abd = fq(a); q(b); :q(a); :q(b)g. The theory Tc with Obs and Abd is not lter-abducible. Suppose our observation is r(p). Then the models of Tc are fq(a); q(b); r(p)g; f:q(a); q(b); r(p)g; fq(a); :q(b); r(p)g; f:q(a); :q(b); r(p)g; and f:q(a); :q(b); :r(p)g. It is easy to see that the elements of Filter(Tc; fr(p)g) are fq(a); q(b); r(p)g; f:q(a); q(b); r(p)g; fq(a); :q(b); r(p)g; and f:q(a); :q(b); r(p)g, while the models of hTc; fr(p)gi are fq(a); q(b); r(p)g; f:q(a); q(b); r(p)g; and fq(a); :q(b); r(p)g. Thus, Tc, Obs and Abd are not lter-abducible. Since Tc is a rst order theory, using Proposition 3 we can conclude that Tc does not encode a function from Abd to Obs. Indeed we can verify this by considering the input E = f:q(a); :q(b)g. The theory Tc [ E has two models fr(p); :q(a); :q(b)g and f:r(p); :q(a); :q(b)g with dierent values of 11

the observable r(p), and hence, is not functional from Abd to Obs. Let us now consider T1 = CIRC (Tc; r; ;), then Filter(T1; fr(p)g), and the set of models of hT1; fr(p)gi are the same. What happens is that by minimizing r in T1, we have T1 to be equivalent to T2 = (r(p) , q(a) _ q(b)), which is functional from Abd to Obs. 2 In the above example, T2 is obtained by doing completion [Cla78] of T1 with respect to the predicate r. In [CDT91], Console et al. describe more general results where they transform a hierarchical propositional and rst-order theories so that abduction with respect to the original theory is achieved by simply adding the observation { which in this case is same as ltering { to the transformed theory. We discuss this in the next subsection. 4.1.1 Knowledge assimilation through completion

A hierarchical propositional theory is a collection of clauses of the form

L1 ^ : : : Ln ! p where each Li is a literal and p is an atom, and where each atom in the theory is assigned a level ( a natural number) such that in a clause of the above form, the level of p is greater than the level of Li, for any 1  i  n. The set Abda is the set of atoms that do not appear in the right hand side of any of the clauses. Consider such a theory T with pi ; : : :; pn as all the non-abducible atoms in it. The transformation Tc is the set of equivalences fpi $ Qi1 _ : : : _ Qim : i = 1; : : : ; ng, where fQij ! pi; j = 1; : : :; mg are the set of clauses in T having pi as their head. The reason observations can now be directly added to Tc can now be explained in terms of Proposition 3 and the fact that the restriction on T and the construction of Tc guarantees that Tc is functional from abducibles to observables. This gives us additional insight into the method used by Console et al. and why after completion of a given theory { in the above mentioned form { abduction can be done through deduction. We will now give another example from the domain of reasoning about actions, where we have lter-abducibility. 12

4.1.2 Filter-abducibility of action theories

Let us consider a subset of the action description language A [GL93] where a domain descriptions may have two kinds of propositions: initial value propositions and eect propositions. We will refer to this language as A0 . (The alphabet of A0 consists of three disjoint nonempty sets of symbols called uents, actions, and situations. We will also assume that S0 is one of the situations in the language of A0 and by a uent literal we will mean a uent possibly preceded by :.) An initial value proposition is a proposition of the form

initially F where F is a uent literal. An eect proposition is a proposition of the form

A causes F if P1 ; : : :Pn where F; P1; : : : ; Pn are uent literals and A is an action. We refer to fP1; : : :; Pn g as the precondition of the above eect proposition. We will be restricting domain descriptions to collections of eect propositions and initial value propositions such that for any two eect propositions that describe the eect of the same action a on complementary f 's, and have P1; : : : ; Pn and Q1; : : : ; Qm as their preconditions we have that fP1; : : :; Pn g \ fQ1; : : :; Qmg 6= ;. (This will guarantee that an action does not cause a uent and its complement in the same situation.) Following is a translation of domain descriptions in this language to a manysorted nested abnormality theory [Lif95] in situation calculus notation. (The sorts in the theory are: actions, uents, are situations. The variables for the three sorts will be denoted by possibly indexed letters a, f and s, respectively, unless otherwise stated. The language includes the actions and uents in A as action constants, and uent constants, and S0 as one of the situation constants. In addition, the theory has the predicate constants: Causes { with arguments of sort actions, uents and situations, Holds { with arguments of sort uents and situations, and Initially { with arguments of sort uents, and the function constant Res from actions and situations to situations. The theory also has the standard Unique name axioms (UNA) 2 and Domain closure axioms (DCA) 3 (say as in [Lif95,KL94]). In the theory, we use the notation H as a short hand, 2 If A1; : : :; An are the only actions in the language, then UNA[actions] stands for A1 6= A2; A2 6= A3, etc. 3 If A1 ; : : :; An are the only actions in the language, then DCA[actions] stands for (8a):a = A1 _ : : : _ a = An 13

where for a uent F , H (F; S ) denotes Holds(P; S ) and H (:F; S ) denotes :Holds(F; S ).)

SC 8 c(Dl ) = >> UNA[actions]; UNA[fluents];UNA[situations] >> >> DCA[actions]; DCA[fluents]; DCA[situations] >> >> (1):Causes+(a; f; s) ^ :Causes?(a; f; s)  [Holds(f; s)  Holds(f; Res(a; s))] >> >> (2)Causes+(a; f; s)  Holds(f; Res(a; s)) >> (3)Causes?(a; f; s)  :Holds(f; Res(a; s)) >> >> (4)Initially(f )  Holds(f; S0) >< (for each initially F 2 D) >> (5)Initially(F ) >> (6):Initially(F ) (for each initially :F 2 D) >> >> f min Causes+(?) : >> >> (7)H (P1 ; s) ^ : : : ^ H (Pn ; s)  Causes+(?)(A; F; s) >> (for each A causes (:)F if P1; : : :; Pn 2 D) >> >> g >> >: In the above theory let us have Abda as the set of atoms using Initially and Obsa as the set of atoms using Holds. By showing that the above theory is functional from abducibles to observables, we can use Proposition 3 4 to conclude that assimilation of observations with respect to the above theory can be done by simply adding the observation to the theory. Let us now brie y argue why the above theory is functional from abducibles to observables. From axiom (4) it is clear that given a complete set of `Initially' literals, we have a unique (and complete) set of Holds literals at the situation s0, in all models of the theory. We will now argue that given a complete set of Holds literals at the situation s, for any action A, their is a unique (and complete) set of Holds literals at the situation Res(A; s), in all models of the theory. (A detailed proof of this is given in [BGPml].) The NAT block de ning Causes+(?) guarantees that given a complete set of `Holds' literals at a situation s, we have a unique (and complete) set of Causes+(?) literals at the situation s, w.r.t. any action A, and any uent F . Because of our restrictions on domain descriptions, we will never have 4 Actually we need to extend Proposition 3 so as to be applicable

maliy theories. This extension is straightforward.

14

to nested abnor-

Causes+(a; f; s) and Causes?(a; f; s; ) to be true in the same model. This fact together with the axioms (1) - (3) guarantee that given a complete set of Holds literals at any situation s, and for any action A, their is a unique (and complete) set of Holds literals at the situation Res(A; s) in all models of the theory. Thus SCc is functional from its abducibles to its observables and hence it is lter abducible. 4.2 Filter abducibility of Disjunctive Logic Programs

In this subsection we will discuss the lter-abducibility of disjunctive logic programs. In particular, we will give some suciency conditions that guarantee that conditions A0 and B holds. But rst, for the sake of completeness, we will give the de nitions and results related to the notion of splitting [LT94] which will be used in the suciency conditions. A disjunctive logic program (DLP) is a collection of rules of the form

l1 or : : : or lk

lk+1; : : : lm; not lm+1; : : : ; not ln

(1)

where k  0, and each li is a literal, i.e., an atom possibly preceded by :, and not is the negation as failure operator. Expression on the left hand (right hand) side of is called the head (the body) of the rule. Both, the head and the body of (1) can be empty. Unless otherwise stated, we assume that rules with variables are used as shorthand for the set consisting of all their ground instantiations. Intuitively the rule can be read as: if lk+1; : : : ; lm are believed and it is not true that lm+1; : : : ; ln are believed then at least one of fl1; : : :; lk g is believed. For a rule r of the form (1) the sets fl1; : : : ; lkg, flk+1; : : :; lmg and flm+1; : : :; lng are referred to as head(r), pos(r) and neg(r) respectively. lit(r) S stands for head(r) [ pos(r) [ neg(r). For any DLP , head() = r2 head(r). For a set if predicates S , Lit(S ) denotes the set of literals with predicates from S . For a DLP , Lit() denotes the set of literals with predicates from the language of . When it is clear from the context we write Lit instead of Lit().

De nition 6 (Splitting set) [LT94] A splitting set for a program  is any set U of literals such that, for every rule r 2 , if head(r) \ U 6= ; then lit(r)  U . If U is a splitting set for , we also say that U splits . The set of rules r 2  such that lit(r)  U is

called the bottom of  relative to the splitting set U and denoted by bU (). The subprogram  n bU () is called the top of  relative to U . 2

De nition 7 (Partial evaluation) [LT94] 15

The partial evaluation of a program  with splitting set U w.r.t. a set of literals X is the program eU (; X ) de ned as follows. For each rule r 2  such that: (pos(r) \ U )  X and (neg(r) \ U ) \ X = ; put in eU (; X ) all the rules r that satisfy the following property: 0

head(r0) = head(r); pos(r0 ) = pos(r) n U; neg(r0) = neg(r) n U

2

De nition 8 (Solution) [LT94]

Let U be a splitting set for a program . A solution to  w.r.t. U is a pair hX; Y i of literals such that: { X is an answer set for bU (); { Y is an answer set for eU ( n bU (); X ); { X [ Y is consistent.

2

Lemma 3 (Splitting Lemma) [LT94] Let U be a splitting set for a program . A set A of literals is a consistent answer set for  if and only if A = X [ Y for some solution hX; Y i to  w.r.t. 2

U.

We are now ready to give suciency conditions that guarantee lterabducibility for disjunctive logic programs.

Proposition 4 An extended logic program T is lter-abducible with respect to abducibles Abd and observables Obs if (i) T is functional from Abd to Obs, (ii) for all l; :l 2 Abd, l _ :l is in T , and (iii) Abd is a splitting set for T .

2

Proof:

We prove this by showing that the conditions (i) { (ii) above imply the conditions A0 and B which in turn guarantee lter abducibility of a theory. (a) Showing A0(i) We now show that the conditions (i) { (ii) implies A0(i). When T is inconsistent this result trivially holds. Let us consider the case when T is consistent. Let M be a consistent answer set of T ) M is a minimal answer set of T M . ) M is a minimal answer set of T M [ (M \ Abd). 16

) M is a minimal answer set of (T [ (M \ Abd))M . ) M is an answer set of T [ (M \ Abd). ) Condition A0(i) holds. (b) Showing A0(ii) Let M be an answer set of T . It is clear that M \ Abd is a complete set of abducible literals and is an answer set of Tbot. Thus by Lemma 3, all answer set of T [ (M \ Abd) are answer sets of T . Thus condition A0(ii) holds. (c) Showing A0(iii) Since T is functional from Abd to Obs and M \ Abd is a complete set of abducible literals, it is clear that condition A0(iii) holds.

(d) Showing B Let E be any arbitrary complete subset of Abd. From condition (ii) of the proposition, E is an answer set of Tbot. Hence by Lemma 3, there exists an answer set M of T , such that M \ Abd = E . Thus condition B is satis ed. 2 Let us again consider the domain description in the previous section and show how it can be formalized into a lter-abducible disjunctive logic program. Here we use the logic programming notation, with variables in capital letters, and constants in small letters. 1. Inertia Axioms:

9> (1a) holds(F; Res(A; S )) holds(F; S ); not ab(A; F; S ); = ; (1a) ? (1b) (1b) :holds(F; Res(A; S )) :holds(F; S ); not ab(A; F; S ); >

2. Translating e{propositions: The translation of an e{proposition \a causes f if p1; : : : ; pn" consists of

9

= (2a) h(f; res(a; S )) h(p1; S ); : : :; h(pn ; S ) > ; (2b) ab(a; f; S ) h(p1 ; S ); : : :; h(pn ; S ) > where for a uent g, h(g; s) denotes the literal holds(g; s) and h(:g; s) denotes the literal :holds(g; s). The eect axiom allows us to prove that f will hold after a, if the preconditions are satis ed. 3. Initial value proposition 17

An initial proposition \ initially f " is translated as (3) h(f; s0) 4. Full awareness about the initial situation rule (4) holds(F; s0) or :holds(F; s0) For a given domain description D, the lter-abducibility of a program generated by the steps 1-4 can be easily veri ed by showing that the program satis es the conditions of Proposition 4 when abducibles are literals about holds in the situation s0 and observables are literals about holds in any situation. (The detailed analysis of an extension of this program is given in [BG97].) To assimilate observations of the form \f after a1; : : :; am" we need to lter the above logic program. In logic programming ltering can be done using integrity constraints (or through clauses with empty heads). Thus in this case assimilation is done by adding integrity constraints of the form below to the logic program. (5)

not h(f; [a1; : : : ; am])

where [a1; : : : ; am] stands for Res(am; Res(am?1; : : :; Res(a1; s0) : : :)).

the

ground

term

4.3 Filter abducibility of Default theories

Suciency conditions for the lter-abducibility of Reiter's default theory is very similar to the suciency conditions for disjunctive logic programs that we gave in the last subsection.

Proposition 5 A default theory T is lter abducible with respect to ab-

ducibles Abd and observables Obs if (i) T is functional from Abd to Obs, (ii) f :ll ; :::ll : l 2 Abdg is a subset of T , and (iii) Abd is a splitting set [Tur96] for T .

Proof:

Similar to the proof of Proposition 4. 18

2 2

The logic program in the previous section can be written as a default theory satisfying the conditions in Proposition 5. The main trick is to write the `full awareness about the initial situation rule' as defaults of the following form: : :holds(F; s0) :holds(F; s0)

: holds(F; s0) holds(F; s0) Turner in [Tur97] uses such defaults.

Observations of the form \f after a1; : : :; am" can then be assimilated by adding

:holds(f; [a1; : : :am])

false to the corresponding default theory. Such defaults with false in their conclusion satisfy the role of ltering in default theories.

5 Are the conditions A0 and B necessary? In Section 3 we show that the conditions A0 and B are sucient for lterabducibility. In this section we show that they are also necessary.

Theorem 3 Let T be a theory, and Obs and Abd be observables and abducibles such that, for all Q 2 Obs, Filter(T; Q) is equivalent to the set of models of hT; Qi. Then T , Obs and Abd satisfy the conditions B , A0(i); A0(ii) and A0(iii). 2

Proof: (i) Suppose T , Obs and Abd do not satisfy condition B . That means there exists an E  Abd, such that T [ E is consistent, but there does not exist a model M of T such that M \ Abd = E . Since T [ E is consistent it has at least one model. Let M  be a model of T [ E . Let Q be the conjunction of the literals in M  \ Obsl. Obviously M  is a model of hT; Qi. Since M  is a model of T [ E , M  \ Abd = E . But then from our initial assumption, M  can not be a model of T . Hence, M  is not in Filter(T; Q). This contradicts the assumption in the lemma that Filter(T; Q) is equivalent to the set of models of hT; Qi. Hence T must satisfy condition B. (ii) Suppose T , Obs and Abd do not satisfy condition A0(i). That means there is a model M of T which is not a model of T [ (M \ Abd). Let Q = M \ Abd. 19

Obviously, M is in Filter(T; Q). We will now show that M is not a model of hT; Qi. Suppose M is a model of hT; Qi. That means there is an E  Abd, such that M is a model of T [ E and M j= Q. But then M \ Abd = E , and this contradicts our initial assumption that M is not a model of T [ (M \ Abd). Hence, M is not a model of hT; Qi. But this contradicts the assumption in the lemma that Filter(T; Q) is equivalent to the set of models of hT; Qi. Hence T must satisfy condition A0(i). (iii) Suppose T , Obs and Abd do not satisfy condition A0(ii). That means there is a model M of T such that all models of T [ (M \ Abd) are not models of T . Let Q = M \ Abd. Let M 0 be a model of T [ (M \ Abd) which is not a model of T . Obviously, M 0 is a model of hT; Qi. But it is not an element of Filter(T; Q). This contradicts the assumption in the lemma that Filter(T; Q) is equivalent to the set of models of hT; Qi. Hence T must satisfy condition A0(ii). (iv) Suppose T , Obs and Abd do not satisfy condition A0(iii). That means there is a model M of T such that all models of T [ (M \ Abd) do not agree on the observables. This means Obsl n Abd 6= ;. Let Q = (M \ Obsl ) [ (M \ Abd). Obviously, M is in Filter(T; Q). We will now show that M is not a model of hT; Qi. Suppose M is a model of hT; Qi. That means there is a complete subset E of Abd, such that M is a model of T [ E and T [ E j= Q. Since E is a complete subset of Abd, E = M \ Abd. Since all models of T [ (M \ Abd) do not agree on observables T [ E 6j= Q. This contradicts our assumption and hence M is not a model of hT; Qi. But then, we have a contradiction to the assumption in the lemma that Filter(T; Q) is equivalent to the set of models of hT; Qi. Hence T must satisfy condition A0(iii). 2 We would like to mention that the known ways to satisfy condition B in default theories and logic programs are to have rules of the form l _ :l (or have two rules of the form l not :l; :l not l) for all abducible atoms l in logic programs and defaults of of the form :ll and :::ll for all abducible atoms l in default theories. (The former was rst used in [Ino91] to relate semantics of abductive logic programs { based on the generalized stable models [KM90], and extended logic programs. The later was used in [Tur97,MT93].) Hence, the necessity of condition B for lter-abducibility makes it necessary (to the best of our knowledge) to have such rules and defaults in lter-abducible logic programs and default theories, respectively.

6 Preferential abduction and Filtering So far we have considered simple abduction and ltering. But often (as in [San89]) both abduction and ltering is accompanied by some preference cri20

teria. In this section we extend the de nition of abductive entailment to preferential abductive entailment and also extend the de nition of ltering to preferential ltering. We then give conditions (suciency) when they are equivalent. We now de ne preferential abductive entailment and preferential ltering.

De nition 9 (Preferential abductive entailment) Suppose we have a partial ordering e that encodes preferences between explanations. We say: M is a model of hT; Q; ei if there exists an e -minimal explanation E of Q w.r.t. T such that M is a model of T [ E .

hT; Q; ei j=abd f if f is true in all models of hT; Q; ei. 2 De nition 10 Suppose we have a partial ordering m that encodes preferences between models.

Filter(T; Q; m) = The m models among the models of T that entail Q. 2 For the equivalence of preferential abductive entailment and preferential ltering we need two additional conditions that relate the partial ordering e between explanations and m between models.

Condition C If E e E 0 then for any model M of T [ E and any model M 0 of T [ E 0 we have M m M 0. Condition D If M m M 0 then M \ Abd e M 0 \ Abd. De nition 11 (Preferential-Filter-abducible) A theory T , observables Obs and abducibles Abd are said to be preferential lter abducible w.r.t. orderings e and m if for all possible observations Q; Filter(T; Q; m) is equivalent to the set of models of hT; Q; ei. 2 Proposition 6 If T , Obs, Abd, e and m satisfy conditions A, B, C, and D then they are preferential lter-abducible.

2

Proof: (a) We show that all elements of Filter(T; Q; m) are models of hT; Q; ei. Let M be an element of Filter(T; Q; m). That means M is in Filter(T; Q) and there does not exist an M 0 in Filter(T; Q) such that M 0 m M . (1) Since M is in Filter(T; Q) it is clear that M is a model of hT; Qi. ) There exists E such that T [ E j= Q, T [ E is consistent and M is a model of T [ E . 21

We now need to show that there does not exist E 0 dierent from E such that T [ E 0 j= Q, T [ E 0 is consistent, and E 0 e E . Suppose such an E 0 exist. Let M 0 be a model of T [ E 0. ) M 0 is a model of hT; Qi. ) M 0 2 Filter(T; Q). From Condition C and the fact that M and M 0 are models of T [ E and T [ E 0 resp. and E 0 e E , we have M 0 m M . This contradicts (1). Hence, we have shown that there does not exist E 0 dierent from E such that T [ E 0 j= Q, T [ E 0 is consistent, and E 0 e E . Hence, M is a model of hT; Q; ei. (b) We will now show that all models of hT; Q; ei are also elements of Filter(T; Q; m). Let M be a model of hT; Q; ei. That means there exists an E such that T [ E j= Q, T [ E is consistent, M is a model of T [ E and there does not exist an E 0 such that T [ E 0 j= Q, T [ E 0 is consistent and E 0 e E . It is clear that M is a model of hT; Qi, and hence is in Filter(T; Q). We only need to show that there does not exist an M 0 such that M 0 is in Filter(T; Q) and M 0 m M . Let us assume to the contrary. I.e. there exists an M 0 such that M 0 is in Filter(T; Q) and M 0 m M . By Condition D, we have M 0 \ Abd e M \ Abd. Since, M is a model of T [ E , from Lemma 1 we have M \ Abd = E . Now consider the fact that we have assumed M 0 to be in Filter(T; Q). This means M 0 is a model of hT; Qi. This means there exists an E  such that M 0 is a model of T [ E . But then using Lemma 1 we have M 0 \ Abd = E . Now we have E  e E . This contradicts our initial assumption that there does not exist an E 0 such that T [ E 0 j= Q, T [ E 0 is consistent and E 0 e E . Hence, we can conclude that there does not exist an M 0 such that M 0 is in Filter(T; Q) and M 0 m M . Therefore, M is in Filter(T; Q; m). 2

Example 6 Let us reconsider Example 3, where T1 = p , a _ b with Abd = fa; b; :a; :bg and Obs = fp; :pg. We showed that satis es T1, Abd and 22

Obs satisfy conditions A and B. Let us now de ne e as E1  E2 if atoms(E1)  atoms(E2). The intuition behind the de nition of e is that we prefer explanations where less (based on the subset ordering) abducible atoms are true. The notion of minimal explanations commonly uses this ordering. Now one simple ordering m between models that will satisfy conditions C and D is as follows: M1  M2 if atoms(M1) \ Abd  atoms(M2) \ Abd. (Here, we are saying that we prefer models with less abducibles.) It is easy to see that T1, Obs, Abd, e , and m satisfy conditions A; B; C and D. Thus they are preferential lter-abducible. Let us now verify this w.r.t. a particular observation Q = fpg. The models of hT1; Qi are ffp; a; :bg; fp; b; :ag; fp; a; bgg which is same as Filter(T1; Q). Now using m and e we obtain that the models of hT1; Q; mi are ffp; a; :bg; fp; b; :agg which is same as Filter(T1; Q; e). 2 As mentioned earlier Condition A requires that if M is a model of a theory T then M should be the only model of the theory T [ (M \ Abd). This is a strong condition, as in many cases T [ (M \ Abd) may have multiple models. To take into account such cases, we weakened condition A to A0. But with preferential ltering and abduction we also need to weaken condition D.

Condition D0 If M m M 0 then for any E , and E 0 such that M is a model of T [ E and M 0 is a model of T [ E 0, E e E 0. Lemma 4 If T , Obs, Abd, e and m satisfy conditions A, B, and D then

2

they also satisfy Condition D0 .

Proposition 7 If T , Obs, Abd, e and m satisfy conditions A0, B, C, and D0 then they are preferential lter-abducible.

2

Proof: (a) We show that all elements of Filter(T; Q; m) are models of hT; Q; ei. The proof is exactly the same as the proof of the Proposition 6. (b) We will now show that all models of hT; Q; ei are in Filter(T; Q; m). Let M be a model of hT; Q; ei. That means there exists an E such that T [ E j= Q, T [ E is consistent, M is a model of T [ E and there does not exist an E 0 such that T [ E 0 j= Q, T [ E 0 is consistent and E 0 e E . 23

It is clear that M is a model of hT; Qi, and hence is in Filter(T; Q). We only need to show that there does not exist an M 0 such that M 0 is in Filter(T; Q) and M 0 m M . Let us assume to the contrary. I.e. there exists an M 0 such that M 0 is in Filter(T; Q) and M 0 m M . Now consider the fact that we have assumed M 0 to be in Filter(T; Q). This means M 0 is a model of hT; Qi. This means there exists an E  such that M 0 is a model of T [ E . Since M is a model of T [ E , M 0 is a model of T [ E 0 and M 0 m M , using Condition D0 we have E  e E . This contradicts our initial assumption that there does not exist an E 0 such that T [ E 0 j= Q, T [ E 0 is consistent and E 0 e E . Hence, we can conclude that there does not exist an M 0 such that M 0 is in Filter(T; Q) and M 0 m M . Therefore, M is in Filter(T; Q; m). 2

7 Weak abduction vs ltering Several instances of ltering used in the literature that de ne an intuitively meaningful entailment relation do not satisfy the conditions described earlier in this paper. In particular, when actions have non-deterministic eects (as in [Tur97]) ltering may still make intuitive sense, but our current de nition of abduction is too strong to match the entailment de ned through ltering. The following example illustrates our point. Consider the disjunctive logic program: T

a_b p p _ :p where Abd = fp; :pg, and Obsa = fa; bg. Suppose we observe a. Using ltering we would be able to conclude { in this case, intuitively explain our observation by { p. (I.e., p will be true all models of Filter(T; fag).) But the current de nition of abduction is too strong to explain this observation by p. (Note that the above theory will violate our condition A0(iii).) This rigidity of abduction has been noticed earlier and several suggestions for weaker versions of abduction have been made; for example in [Gel90,Rei87,Sha97]. In this section we de ne a weaker notion of abduction and show that it is equivalent to ltering under less restrictive conditions than given in the earlier sections; in particular, we no longer need condition A0(iii). As a result we can also weaken 24

the suciency conditions in Propositions 3{ 5. We now formally de ne weak abductive entailment and state theorems and propositions similar to the ones in the previous sections. (The proofs of these theorems and propositions are very similar to the earlier proofs and for brevity we omit them.)

De nition 12 (Weak abductive Entailment) Let T be a (possibly nonmonotonic) theory with an entailment relation j=, and Q be an observation. (i) M is a w-model (or weak-model) of hT; Qi if there exists a complete subset E of abducibles such that M is a model of T [ E and M j= Q. (ii) For any formula f , hT; Qi j=wabd f if f is true in all w-models of hT; Qi.

2

De nition 13 (Weak- lter-abducible) A theory T , a set Abd, and a set Obs are said to be weak- lter-abducible if for all possible observations Q 2 Obs; Filter(T; Q) is the set of w-models of hT; Qi. 2 Theorem 4 (Suciency) Let T be a theory, and Obs and Abd be observables. If T , Obs and Abd satisfy conditions A0(i), A0(ii) and B then they are weak- lter-abducible. 2 Theorem 5 (Necessity) Let T be a theory, and Obs and Abd be observables and abducibles such that, for all Q 2 Obs, Filter(T; Q) is equivalent to the set of weak-models of hT; Qi. Then T , Obs and Abd satisfy the conditions B , 2

A0(i); andA0(ii).

Proposition 8 A propositional or rst-order theory (possibly augmented

with circumscription) T is weak- lter-abducible with respect to any abducibles Abd and observables Obs. 2

Proposition 9 An extended logic program T is weak- lter-abducible with respect to abducibles Abd and observables Obs if (i) for all l; :l 2 Abd, l _ :l is in T , and

2

(ii) Abd is a splitting set for T .

Proposition 10 A default theory T is weak- lter-abducible with respect to abducibles Abd and observables Obs if (i) f :ll ; :::ll : l 2 Abdg is a subset of T , and

(ii) Abd is a splitting set [Tur96] for T .

25

2

8 Conclusion and Future Work In this paper we gave conditions on theories that guarantee the equivalence of abductive reasoning and ltering for assimilation of observations into theories. We also showed why these conditions are necessary. We gave some suf ciency conditions on theories expressed in several knowledge representation formalisms so as to guarantee the equivalence of ltering and abductive reasoning in those formalisms. Finally, we illustrated several examples from the knowledge representation literature where the theories that use ltering indeed satisfy the suciency conditions and thus can be shown to be encoding abductive reasoning. One direction of future work is to explore more syntactic and easily veri able suciency conditions for particular knowledge representation formalisms. It will be also interesting to examine additional formalizations in the literature where ltering is used and examine if indeed some form of abductive reasoning is encoded there.

References [AKPT91] J. Allen, H. Kautz, R. Pelavin, and J. Tenenberg. Reasoning about plans. Morgan Kaufmann, 1991. [BB95] C. Boutilier and V. Becher. Abduction as belief revision. Arti cial Intelligence, 77:43{94, 1995. [BG94] C. Baral and M. Gelfond. Logic programming and knowledge representation. Journal of Logic Programming, 19,20:73{148, 1994. [BG97] C. Baral and M. Gelfond. Reasoning about eects of concurrent actions. Journal of Logic Programming, 31(1-3):85{117, May 1997. [BGPml] C. Baral, A. Gabaldon, and A. Provetti. Formalizing Narratives using nested circumscription. In AAAI 96, pages 652{657, 1996 (Extended version with proofs available at http://cs.utep.edu/chitta/chitta.html). [Bou96] C. Boutilier. Abduction to plausible causes: an event based model of belief update. Arti cial Intelligence, 83:143{166, 1996. [CDT91] L. Console, D. Dupre, and P. Torasso. On the relationship between abduction and deduction. Journal of Logic and Computation, 2(5), 1991. [Cla78] K. Clark. Negation as failure. In Herve Gallaire and J. Minker, editors, Logic and Data Bases, pages 293{322. Plenum Press, New York, 1978. [DDS93] M. Denecker and D. De Schreye. Representing incomplete knowledge in abductive logic programming. In Proc. of ILPS 93, Vancouver, pages 147{164, 1993.

26

[DMB92] M. Denecker, L. Missiaen, and M. Bruynooghe. Temporal reasoning with abductive event calculus. In Proc. of ECAI92, 1992. [Dun93] P. Dung. Representing actions in logic programming and its application in database updates. In D. S. Warren, editor, Proc. of ICLP-93, pages 222{238, 1993. [EK88] K. Eshghi and R. Kowalski. Abduction through deduction, 1988. unpublished paper. [EK89] K. Eshghi and R. Kowalski. Abduction compared with negation as failure. In Giorgio Levi and Maurizio Martelli, editors, Logic Programming: Proc. of the Sixth Int'l Conf., pages 234{255, 1989. [Esh88] K. Eshghi. Abductive planning with event calculus. In Proc. of the Fith ICLP., pages 562{579, 1988. [Gel90] M. Gelfond. Belief sets, deductive databases and explanation based reasoning. Technical report, University of Texas at El Paso, 1990. [GL91] M. Gelfond and V. Lifschitz. Classical negation in logic programs and disjunctive databases. New Generation Computing, pages 365{387, 1991. [GL93] M. Gelfond and V. Lifschitz. Representing actions and change by logic programs. Journal of Logic Programming, 17(2,3,4):301{323, 1993. [HSAM90] J. Hobbs, M. Stickel, D. Appelt, and P. Martin. Interpretation as abductions. Technical report, SRI, Menlo Park, California, 1990. [Ino91] K. Inoue. Extended logic programs with default assumptions. In Proc. of ICLP91, 1991. [KKT93] A. Kakas, R. Kowalski, and F. Toni. Abductive logic programming. Journal of Logic and Computation, 2(6):719{771, 1993. [KL94] G. Kartha and V. Lifschitz. Actions with indirect eects: Preliminary report. In KR 94, pages 341{350, 1994. [KM90] A. Kakas and P. Mancarella. Generalized stable models: a semantics for abduction. In Proc. of ECAI-90, pages 385{391, 1990. [Kon92] K. Konolige. Abduction versus closure in causal theories. Arti cial Intelligence, 53:255{272, 1992. [Lif95] V. Lifschitz. Nested abnormality theories. Arti cial Intelligence, 74:351{ 365, 1995. [Llo87] J. Lloyd. Foundations of logic programming. 1987. Second, extended edition. [LT94] V. Lifschitz and H. Turner. Splitting a logic program. In Pascal Van Hentenryck, editor, Proc. of the Eleventh Int'l Conf. on Logic Programming, pages 23{38, 1994.

27

[MD95] M. Missiaen, L.and Bruynooghe and M. Denecker. CHICA: A planning system based on event calculus. Lournal of Logic and Computation, 5(5):579{602, 1995. [MT93] W. Marek and M. Truszczynski. Nonmonotonic Logic: Context dependent reasoning. Springer Verlag, 1993. [Pei58] C. Peirce. Collected papers of Charles Sanders Peirce (Volume 1-8, Edited by Hartshorn et al.). Harvard University Press, 1931-1958. [Pei92] C. Peirce. Reasoning and the logic of things. Harvard University Press, 1992. [PGA87] D. Poole, R. Goebel, and R. Aleliunas. Theorist: a logical reasonig system for defaults and diagnosis. In N. Cercone and G. McCalla, editors, The Knowledge Frontier: Essays in the Representation of Knowledge, pages 331{352. Springer-Verlag, New York, 1987. [PMG98] D. Poole, A. Mackworth, and R. Goebel. Computational Intelligence. Oxford University Press, 1998. [Poo88] D. Poole. A logical framework for derfault reasoning. Arti cial Intelligence, 36(1):27{48, 1988. [Reg83] R. Reggia. Diagnostic expert system based on a set covering model. International journal of man machine studies, 19(5):437{460, 1983. [Rei87] R. Reiter. A theory of diagnosis from rst principles. Arti cial Intelligence, 32(1):57{95, 1987. [Rei96] R. Reiter. Natural actions, concurrency and continuous time in the situation calculus. In L. Aiello, J. Doyle, and S. Shapiro, editors, KR 96, pages 2{13, 1996. [San89] E. Sandewall. Combining logic and dierential equations for describing real-world systems. In R. Brachman, H. Levesque, and R. Reiter, editors, Proc. of the First Int'l Conf. on Principles of Knowledge Representation and Reasoning, pages 412{420, 1989. [Sha89] M. Shanahan. Prediction is deduction but explanation is abduction. In Proc. of IJCAI-89, pages 1055{1060, 1989. [Sha93] M. Shanahan. Explanation in the situation calculus. In Proc. of IJCAI93, pages 160{165, 1993. [Sha97] M. Shanahan. Solving the frame problem: A mathematical investigation of the commonsense law of inertia. MIT press, 1997. [Tur95] H. Turner. Representing actions in default logic: A situation calculus approach. In Proceedings of the Symposium in honor of Michael Gelfond's 50th birthday (also in Common Sense 96), 1995. [Tur96] H. Turner. Splitting a default theory. In Proceedings of AAAI, 1996.

28

[Tur97] H. Turner. Representing actions in logic programs and default theories. Journal of Logic Programming, 31(1-3):245{298, May 1997.

29