Labelled Logics of Defeasible Goals (Extended

Labelled Logics of Defeasible Goals (Extended Paper)

Leendert W.N. van der Torre

Max Planck Institute for Computer Science, Im Stadtwald, D-66123 Saarbrucken, Germany [email protected]

Abstract

In this paper we study con icts between goals. In line with negative results obtained in the logic of preference, we argue that most proof rules of the logic of goals { such as strengthening of the antecedent (monotony), transitivity and the conjunction rule { only hold in a restricted sense. We study restricted applicability in Gabbay's labelled deductive systems, and we show how to resolve con icts in labelled logics of conditional goals. We also prove two phasing theorems.

1 Introduction

In the usual approaches to planning in AI, a planning agent is provided with a description of some state of aairs, a goal state, and charged with the task of discovering (or performing) some sequence of actions to achieve that goal. Recently several logics for goals (and the closely related desires) have been proposed [Doyle and Wellman, 1991; Doyle et al., 1991; Pearl, 1993; Boutilier, 1994; Tan and Pearl, 1994b; 1994a; Lang, 1996; Huang and Bell, 1997], and it has been observed that con icts between goals deserve special attention. Consider the following two examples given by Bacchus and Grove [1996]. Surgery. A person may prefer not having surgery over having surgery, but this preference might be reversed in the circumstances where surgery improves one's long term health. We say that the more speci c preference overrides the more general one, which can be formalized with speci city-based principles developed in logics of defeasible reasoning. We can defeasibly infer that the person prefers no surgery only as long as it is not known that surgery improves his or her long term health. Marriage. A girl named Sue prefers to be married to John over not, she prefers to be married to Fred over Short version appeared in Proceedings of the Fourth Dutch-German Workshop on Nonmonotonic Reasoning Techniques and Their Applications (DGNMR'99). 

not, and at the same time she reasonably prefers to be married to neither over being married to both. If all we know is that she prefers to be married to John and that she prefers to be married to Fred, then it is reasonable to defeasibly infer that she likes to be married to both. However, if we additionally know that she prefers to be married to neither of them, then this inference is defeated. The examples illustrate that intuitively most proof rules of the logic of defeasible goals { i.e. goals which can be overridden by other goals { only hold in a restricted sense. This is in line with negative results obtained in the logic of preference, where counterexamples have been given to nearly all proposed rules [Mullen, 1979]. In semantic terms the problem of the logic of preference is that there is no straight forward way to lift a preference relation between worlds, for example represented by a utility function, to a preference relation between sets of worlds or propositions. Pearl [1993] also observed counterexamples to most proof rules when utilities and probabilities are combined in a so-called decision-theoretic logic of preference. This restricted applicability of rules explains why the proposed logics of defeasible goals are either too strong { the logic supports the proof rules { or very weak { it does not support them. In this paper we study restricted applicability in Gabbay's labelled deductive systems [Gabbay, 1996] and we show how to incorporate a con ict resolution method in the labelled logic of conditional goals [van der Torre, 1998a; 1998b]. The resulting logic has the following desirable properties. The identity (i.e.  is a goal if ) does not hold; Strengthening of the antecedent (i.e. monotony) is restricted to incorporate, amongst others, the speci city principle; Con ict averse strategies do not hold;1 In the surgery example above it seems reasonable to infer that the agent prefers not to be in a situation where surgery improves his long term health, but when rewards are higher than penalties it is bene cial to be in a con ict. 1

The conjunction and transitivity rules are de ned

in terms of strengthening of the antecedent and a new generalized cut rule, and therefore only hold in a restricted sense; Weakening of the consequent (i.e. consequential closure) holds to support reasoning about sub-goals. Restricted applicability is of course well-known from logics of defeasible reasoning, where monotony and transitivity only hold in a restricted sense and where restricted alternatives of monotony have been developed, e.g. cautious and rational monotony. However, it is important to note that the logic of defaults is quite dierent from the logic of defeasible goals. For example, according to the generally accepted Kraus-Lehmann-Magidor paradigm [Kraus et al., 1990] the identity (called supraclassicallity) and the conjunction rule both hold in the logic of defaults. This dierence supports Mullen [1979]'s claim that there is no generally applicable preference logic, and that (as he concluded) a logic of preference can only be tested in the process of testing the more general theory in which it is embedded. This paper is organized as follows. In Section 2 we discuss several reasons to restrict proof rules of the logic of goals and we introduce a labelled logic of defeasible goals. In Section 3 we discuss and illustrate the desirable properties and in Section 4 we discuss how goal independence can be used to make the logic more adventurous. In Section 5 we introduce phasing in the logic of defeasible goals and we prove two phasing theorems.

2 Labelled goals

Before we introduce the labelled logic of defeasible goals, we discuss restricted applicability. 2.1 Restricted proof rules We are interested in a logical system that tells us which conditional goals (the output) can be derived from a set of conditional goals (the input) called the goal base. Even in this simple setting { in which we do not consider logical connectives between the conditional goals { reasoning about goals is non-trivial, for the following three reasons. Goals can be overridden by other goals. The surgery and marriage examples illustrate that the addition of a goal to the goal base may result in new goals coming into force and, more importantly, old goals losing their force. Goals can con ict. Goals only impose partial preferences, i.e. preferences given some objective and given some context. Objectives can con ict and, as a consequence, goals with overlapping contexts can con ict. Goals are context dependent. When the robot's goal to get me some coee cannot be achieved, for example because the coee machine is broken, then its alternative goal may be to get me some tea. In preference-based approaches, the goal to get me

some tea if the coee machine is broken is a preference for tea if I do not get coee, i.e. a preference of tea without coee over neither tea nor coee. The three problems lead to restricted applicability of proof rules, which we study in this paper with versions of Gabbay's labelled deductive system [Gabbay, 1996]. In particular, we use and extend the labelled logics introduced by Van der Torre [1998a; 1998b] and Makinson [Makinson, 1998]. The distinction between the approaches of Van der Torre and Makinson is that the rst is more cautious with respect to the treatment of con icting contexts. We follow this more cautious approach, because its more constructive character has attractive computational properties. This so-called phasing is discussed in detail in Section 5. The language of the labelled logics of conditional goals (llg) consists of formulas of the type G( j )L , to be read as ` is preferred if in the proof context L.' Roughly speaking,2 the label L of a goal G( j )L consists of a record of the ful llments of the premises, where the ful llment of a goal G(j ) is  ^ , together with strengthenings of the antecedent, where the strengthening 2 derives G( j 1 ^ 2 ) from G(j 1 ), that are used in the derivation of G( j ). The consistency check on the label L blocks derivations from con icting goals (consistency check on consequents), as well as derivations from goals with con icting contexts (consistency check on antecedents). In this paper we introduce labelled logics of defeasible conditional goals (lldg), in which the consistency check not only covers the premises a goal is derived from, but also all other goals of the goal base. To formalize this dependence on the goal base we de ne an inference relation relative to the goal base, as is explained in detail later. With this extension we do not only cover con icts and context dependence, but also overriding as it occurs in the surgery and marriage examples.

Overriding. Consider the goals `Sue prefers to marry John over not' G(j ) and `Sue prefers to marry Fred over not' G(f ), where G() is short for G(j>) and > stands for any tautology like p _ :p. The llg derivation below shows how the goal `Sue prefers to marry John and Fred' G(j ^ f ) can be derived from the two goals via the goal `Sue prefers to marry Fred if she marries John' G(f jj ). The logic llg does not have the conjunction rule as a proof rule, but this rule can be derived from strengthening of the antecedent and a new generalized cut rule (cta),

2 There is a complication with reasoning by cases, which forces us to use sets of sets of formulas instead of sets of formulas, as is discussed in detail in the following sections.

which is discussed in the following section. G(j j>)fjg

? ? ? ? ? sa G(j jf )fj;f g G(f j>)ff g G(j ^ f j>)fj;f g

cta

Now consider the additional goal `Sue prefers not to be married to both' G(:(j ^ f )). In llg the derivation above is still valid, because additional premises do not in uence valid derivations. However, this llg derivation is not a lldg derivation, because in the labelled logic of defeasible goals introduced in this paper the goal G(j j f ) cannot be derived. The strengthening of the antecedent rule is blocked, because the label fj; f g is inconsistent with the ful llment of the additional premise, i.e. :(j ^ f ). In the derivation above, as well as in following gures, the rst blocked derivation step is represented by a dashed line. Con icts. Both llg and lldg can reason about con icting goals, because we have G(p); G(:p) 6`llg G(p ^ :p) and G(p); G(:p) 6`llg G(q) if q is not logically implied by p or :p. In particular, the consistency check blocks the second derivation step in the following counterintuitive derivation. G(pj>)fpg wc G(p _ qj>)fpg

? ? ? ? ? ? ? ? sa G(p _ qj:p)fp;:pg G(:pj>)f:pg G(q ^ :pj>)fp;:pg wc G(qj>)fp;:pg

cta

The label of the goal potentially derived by sa contains besides the ful llment p also the contradictory strengthening :p. Consequently sa cannot be applied, because it would result in a goal with contradictory formulas in its label. In general, it is obvious that the two premises G(p) and G(:p) cannot be combined in any derivation, because the derived goal would contain the contradictory p and :p. Context. Consider the two goals `I prefer coee' G(c) and `I prefer tea if there is no coee' G(tj:c). The logics llg and lldg combine strengthening of the antecedent (sa) with weakening of the consequent (wc) without validating the following counterintuitive derivation of `I prefer coee or no tea if there is no coee' G(c _ :tj:c) from the rst premise. G(cj>)fcg wc G(c _ :tj>)fcg

????????

G(c _ :tj:c)fc;:cg

sa

The sa rule cannot be applied, because the label of the potentially derived goal contains besides the ful llment of the premise c also the strengthening :c. Before we discuss more complex examples we introduce the details of the labelled logic of defeasible goals lldg.

2.2 Labelled logic of defeasible goals (lldg) The rst labelled logic of the type considered in this paper was proposed in [van der Torre and Tan, 1995] for obligations, see also [van der Torre and Tan, 1997; van der Torre, 1997]. In this rudimentary logic the label of a premise only contains the consequent, and the consistency check considers the label together with the antecedent. The logic does not have the disjunction rule, because there is a complicating factor with this consistency check for reasoning by cases. When reasoning by cases we intuitively want to combine cases from con icting contexts, and sometimes we even want to combine cases of con icting goals. For example, we want to derive G(a j>) from the con icting goals G(a ^ b j c) and G(a ^ :bj:c) (this example is discussed in detail in Section 3.4). However, a consistency check on the consequents would obviously block the derivation. Two solutions have been proposed to cover the parallel tracks created through reasoning by cases. First, in [van der Torre, 1998a] the label of premises G( j ) contains the material conditional ! . Second, in [Makinson, 1998] the label contains a separate set of formulas for each case. Hence, in the latter more exible solution, which we adopt here too, the label contains sets of sets of boolean formulas. As far as reasoning by cases is concerned in the latter solution the labels of the premises G(j ) can either contain material implications ! , consequents  only, or ful llments  ^ . De nition 1 (Language) Let L be a propositional base logic. The language of lldg consists of the labelled dyadic goals G(j )L , with  and sentences of L, and L a set of sets of sentences of L. G(j )L is read as ` is preferred if in the proof context L.' We write j= for entailment in L. In this paper we choose the most conservative option for the labels of the premises. Each formula G( j )L occurring as a premise has a label that consists of its own (propositionally consistent) ful llment. De nition 2 (Premise) A formula G( j )ff^ gg , where  ^ is consistent in L, is called a premise of lldg. The distinctive property of labelled deductive systems with respect to other logical systems is that we have to de ne how proof rules operate on the labels, and how the proof rules use the information in the labels. The label of a derived goal is the union (or) or the product (sa, cta) of strengthenings and the labels of the premises used in this proof rule, where the product is de ned by fS1; : : : ; Sn g  fT1 ; : : : ; Tm g = fS1 [ T1; : : : ; S1 [ Tm ; : : : ; Sn [ Tm g. For non-defeasible goals [van der Torre, 1998b] a consistency check ensures that G( j )L may only be derived if each F 2 L is consistent: it must always be possible to ful ll a derived goal together with each of the goals it is derived from, though { to support reasoning by cases { not necessarily all of them at the same time.

In the logic of defeasible goals developed here, a goal can be overridden by other goals. The inference relation is relative to a set of goals (the goal base, intuitively the set of premises) and we may only apply a rule if the label of the derived goal does not make the ful llment of a goal of the goal base impossible. If the goal base is xed, then the inference relation (which we represent by `Blldg for the goal base B ) is monotonic. However, the intended interpretation of this goal base is that it re ects the set of premises, and it thus grows when the set of premises grows. Under this intended interpretation, the inference relation (which we represent by  j lldg ) is non-monotonic. It is well-known from logics of defeasible reasoning that there are many dierent ways to de ne non-monotonic closures, ranging from safe to adventurous. When applying a safe or cautious rule, such as System Z [Pearl, 1990], the set of derived conclusions is relatively small. The proposed construction here is cautious in the sense that it blocks the inheritance of goals to sub-ideal subclasses. This is discussed in detail in Section 4, where we also show how to de ne more adventurous extensions. De nition 3 (lldg) Let B be a set of lldg-formulas called the goal base. The labelled logic of defeasible goals lldg consists of the inference rules below, extended with the following condition R. R: G( j )L may only be derived if for each F 2 L (of the form F1 [ f g, F1 [ F2 or F1 ) and for each goal G(0 j 0 ) 2 B , F [ f0 ^ 0 g is only inconsistent if F1 [ f0 ^ 0 g (and F2 [ f0 ^ 0 g) is already inconsistent. The inference rules of lldg are replacements by logical equivalents and the following four rules. G( j 1 )L ; R saR : G( j 1 ^ 2 )Lf 2 g G( j ^ )L1 ; G( j )L2 ; R ctaR : G( ^ j )L1 L2 G(1 j )L ; R wcR : G(1 _ 2 j )L G( j 1 )L1 ; G( j 2 )L2 ; R orR : G( j 1 _ 2 )L1 [L2 For a goal base B , we say fG(i j i ) j 1  i  ng `Blldg G(j ) if there is a labelled goal G( j )L that can be derived from the set of goals fG(i j i )ff ^ gg j 1  i  ng. Moreover, we say B j lldg G(j ) i B `Blldg G(j ). Before we consider the relation between lldg and llg in Proposition 2 we rst prove the following useful proposition. It says that each element of the label implies the ful llment of the goal, i.e. the conjunction of its antecedent and consequent. Proposition 1 For each goal G(j )L derived in lldg we have for each F 2 L that F j=  ^ . i

i

Proof By induction on the structure of the proof tree.

The property trivially holds for the premises, and it is easily seen that the proof rules retain the property. The following proposition shows that lldg derives a subset of llg introduced in [van der Torre, 1998b]. Proposition 2 Let llg be the lldg in which the consistency check R is replaced by the following check Rllg on the consistency of all elements of L. Note that the inference relation of llg is not relative to the goal base. Rllg: G( j )L may only be derived if each F 2 L is consistent. If we have B j lldg , then we have B `llg  (but obviously not necessarily vice versa).

Proof We prove that if R holds, then Rllg also holds. So assume that R holds for a derived goal G(j )L . For each F 2 L there exists a goal G(0 j 0 ) 2 B used in the proof of G( j )L , and we have F j= 0 ^ 0 and F j=  ^ (Proposition 1). We have F [ f0 ^ 0 g and F [ f ^ g are consistent according to R, and therefore F is consistent. Consequently Rllg holds.

The following proposition shows how lldg can be de ned as llg with additional restrictions. Proposition 3 Let the ful llment conditions of a goal G(i j i ) in goal base B be F C (G(i j i ) j B ) = ffi ^ i ^ j ^ j g j G(j j j ) 2 B and i ^ i ^ j ^ j is consistentg We have B j lldgG(j ) if and only if there is a labelled goal G(j )L that can be derived from the set of goals fG(i j i )ff ^ gg[FC (G( j )jB) j 1  i  ng in llg, i.e. in lldg in which the consistency check R is replaced by the check Rllg of Proposition 2. i

i

i

i

Proof We prove that R implies Rllg and vice versa. ) Assume a derivation for which Rllg does not hold. There is an F 2 L which is inconsistent, and given Proposition 2 the inconsistency is a result of the ful llment conditions. Consequently R does not hold. ( If Rllg

holds then the ful llment conditions imply R. The only unusual rule of llg and lldg is cta, a combination of cumulative transitivity (cut or ct) and the conjunction rule (and). In the following sections we will see that this unusual rule leads to several desired properties of the logic.

3 The desirable properties of lldg

In this section we discuss and illustrate the desirable properties of lldg we already mentioned in the introduction: the identity and con ict averse strategies do not hold, but restricted strengthening of the antecedent (to support the speci city principle), restricted conjunction, restricted transitivity, and the disjunction rule (to support reasoning by cases) do hold.

3.1 The identity From the de nitions follows immediately that G( j ) does not hold, and that we cannot derive G( ^ j ) from G( j ). This contrasts lldg with logics of goals de ned as follows in an underlying preference logic, in which 1  2 is read as `1 is preferred to 2 .' G(j ) =def ( ^ )  (: ^ ) Note that the fact that the identity is an undesirable property of the logic of goals also makes it dicult to de ne a semantics for such logics, because we cannot use results obtained in e.g. conditional, counterfactual and default logics. 3.2 Restricted strengthening of the antecedent and con ict averse strategies Strengthening of the antecedent is restricted to formalize, amongst others, overriding of general preferences by more speci c ones. This is illustrated by Bacchus and Grove's surgery example. Example 1 (Surgery) Consider the two goals `A person prefers not having surgery over having surgery' G(:s), and `this preference is reversed in the circumstances where surgery improves one's long term health' G(sji). We have the following. G(:s) j lldgG(:sji) G(:s); G(sji) 6  j lldg G(:sji)

Hence, by addition of a premise we loose conclusions. The circumstances where surgery improves one's long term health (i) represent a con ict situation, because the rst goal prefers :s whereas the second goal prefers the contradictory s. We say that the latter more speci c goal overrides the former more general one, because G(:sji) can be derived from the rst goal, but not from both goals. The surgery example can also be used to illustrate another aspect in which the logic of goals is dierent from, for example, the logic of defaults and the logic of prima facie obligations. When in con ict, i.e. when i is true, it is sure that some rule will be achieved and another one will be violated, and the relevant question is what we can say about this con ict situation. In the logic of defaults it seems unlikely to be in a situation in which a default is violated, and therefore it is unlikely to be in a con ict situation. In the logic of prima facie obligations, it is forbidden to violate obligations and it is therefore forbidden to be in a con ict situation (this derivation also depends on the capabilities of the agent to evade the con ict). However, in the logic of goals it may be desirable to be in a con ict situation, namely when the rewards of the achieved goal are higher than the penalties of the violated goal. We have, as desired, the following non-derivation in lldg. G(:sj>); G(sji) 6  j lldgG(:ij>)

Moreover, in several logics of goals the combination of con icting contexts leads to the counterintuitive result to prefer one of these contexts [van der Torre and Tan, 1998]. Obviously in lldg these counterintuitive derivations are blocked, because con icting contexts can only be combined when reasoning by cases.

3.3 Restricted conjunction and transitivity The following proposition shows that the conjunction and transitivity rules are de ned in terms of strengthening of the antecedent, and they are therefore also restricted. Proposition 4 The following conjunction rule, transitivity rule and extended disjunction rule are implied by the logic lldg. G(1 j )L1 ; G(2 j )L2 ; R andR : G(1 ^ 2 j )L1 L2 G( j )L1 ; G( j )L2 ; R transR : G( j )L1 L2 G(1 j 1 )L1 ; G(2 j 2 )L2 ; R or2R : G(1 _ 2 j 1 _ 2 )L1 [L2

Proof For the conjunction rule, we can rst strengthen

G(1 j ) to G(1 j ^ 2 ), and then apply cta as follows to derive G(1 ^ 2 j ). The derived goal is equivalent to the goal derived by the proof rule above, because L2 implies 2 (Proposition 1). G(1 j )L1 sa G(1 j ^ 2 )L1 f2 g G(2 j )L2 cta G(1 ^ 2 j )L1 f2 gL2

For the transitivity rule, we have the following derivation. Again, L2 implies (Proposition 1). G(j )L1 sa G(j ^ )L1 f g G( j )L2 cta G( ^ j )L1 f gL2 wc G(j )L1 f gL2 For the extended disjunction rule, we have the following derivation. G(1 j 1 )L1 G(2 j 2 )L2 wc wc G(1 _ 2 j 1 )L1 G(1 _ 2 j 2 )L2 or G(1 _ 2j 1 _ 2 )L1 [L2 The following example illustrates the derived restricted conjunction rule by Bacchus and Grove's marriage example. Example 2 (Marriage) We have G(j ); G(f ) j lldgG(j ^ f ) and G(j ); G(f ); G(:(j ^ f )) 6  j lldgG(j ^ f ),

because in the following derivation the goal G(j jf ) cannot be derived from G(j j >) due to the existence of G(:(j ^ f )). G(j j>)ffjgg

? ? ? ? ? ? sa G(j jf )ffj;f gg G(f j>)fff gg G(j ^ f j>)ffj;f gg

cta

Hence, the goal to marry both is only derived nonmonotonically from the rst two goals, and the derivation is blocked by addition of the third goal. Notice that the derivation is not blocked by a speci c goal but by a most general goal (with antecedent >), and that consequently the speci city principle cannot be used to block the derivation. The logic lldg is very cautious, because the counterintuitive goal cannot be derived if we replace the rst goal of the goal base by G(j jf ). We have the following. G(j jf ); G(f ) j lldgG(j ^ f ) G(j jf ); G(f ); G(:(j ^ f )) 6  j lldg G(j ^ f )

The fact that the conjunction rule only holds nonmonotonically means that we have to be careful whether we formalize a goal for p ^ q by G(p) ^ G(q) or by G(p ^ q). Although we have: G(p); G(q) j lldg G(p ^ q) G(p ^ q) j lldgG(p) G(p ^ q) j lldgG(q)

they derive dierent consequences when other goals are added to the goal base, for example: G(p); G(q); G(r) j lldgG(rj:(p ^ q)) G(p ^ q); G(r) 6  j lldgG(rj:(p ^ q))

and G(p); G(q); G(rj:p) 6  j lldgG(rj:p ^ :q) G(p ^ q); G(rj:p) j lldgG(rj:p ^ :q)

Hence, both representations are incomparable, in the sense that in some circumstances the former is more adventurous, and in other circumstances the latter.

3.4 Reasoning by cases The disjunction rule of lldg supports reasoning by cases in a similar way as it does in most conditional logics. The following example shows how the logic reasons by cases to derive a conclusion from con icting goals. Example 3 (Reasoning by cases) As desired, we have G(a ^ cjb); G(a ^ :cj:b) j lldgG(a),

as a consequence of the following derivation. G(a ^ :cj:b)ffa^:c^:bgg G(a ^ cjb)ffa^c^bgg wc wc G(ajb)ffa^c^bgg G(aj:b)ffa^:c^:bgg or G(aj>)ffa^c^bg;fa^:c^:bgg) The example also illustrates why the label has to consist of a set of set of formulas, and not only of a set of formulas. In the latter case, the consistency check would block the intuitive derivation.

4 Goal independence

The logic lldg is cautious in the sense that it, just like System Z, does not have the inheritance of properties to sub-ideal subclasses. For example, we have G(p); G(q) 6  j lldgG(pj:q),

because the strengthening :q of the rst premise makes it impossible to ful ll the premise G(q). Depending on the interpretation of the propositions, this derivation may be desired or not. Solutions for this ambiguity problem proposed in logics of defeasible reasoning use a notion of repair [Benferhat et al., 1998] or they add independence assumptions [Dubois et al., 1997] when this derivation is desired. In lldg we can add independence relations between goals of the goal base. We restrict the consistency check to a subset of all premises, the premises it depends on. As an extreme case, non-defeasible goals only depend on the premises they are derived from. In the following de nition we use the characterization of lldg in Proposition 3. De nition 4 (Independence) Let the extended goal base B = (S; I ) be a set of goals S with a re exive and asymmetric relation I on S S . I (g1 ; g2) should be read as `the goal g1 is independent of the goal g2 .' The ful llment conditions of G(i j i ) in B are F C (G(i j i ) j B ) = ffi ^ i ^ j ^ j j G(j j j ) 2 B and not I (G(i ^ i ); G(j ; j )) and i ^ i ^ j ^ j is consistentg We have B j lldg G( j ) if there is a goal G( j )L that can be derived from the set of goals fG(i j i )ff ^ gg[FC (G( j )jB) j 1  i  ng in llg, i.e. in lldg in which the consistency check R is replaced by the check Rllg of Proposition 2. Goal independence is illustrated in the following example. Example 4 Consider the two extended goal bases B1 = (fG(p); G(q)g; ;) and B1 = (fG(p); G(q)g; f(G(p); G(q)); (G(q); G(p))g). The ful llment conditions of G(p) in B1 and B2 are respectively ffp ^ qgg and ;. We therefore have the following. i

i

i

i

B1 6  j lldg G(pj:q) B2  j lldg G(pj:q)

Hence, we have inheritance to sub-ideal subclasses (:q) only if the two goals are independent.

5 Phased labelled logics of defeasible goals (plldg)

Phased labelled logics of goals [van der Torre, 1998b] extend the label L of a goal G(j )L with the phase (p) in which it is derived. De nition 5 (Language) Let L be a propositional base logic. The language of plldg consists of the labelled dyadic goals G( j )L , with  and sentences of L, and L a pair (F; p) that consists of a set of sets of sentences of L (ful llments) and an integer (the phase). We write j= for entailment in L. Each formula G( j )L occurring as a premise has a label that consists of its own (propositionally consistent) ful llment and phase 0. De nition 6 (Premise) A formula G(j )(ff^ gg;0) , where  ^ is consistent in L, is called a premise of plldg. The phase of a goal is determined by the proof rule used to derive the goal, and the labels are used to check that the phase of reasoning is non-decreasing. De nition 7 (plldg) Let  be a phasing function that associates with each proof rule below an integer called its phase. Moreover, let B be a set of plldg-formulas called the goal base. The phased labelled logic of goals plldg for  consists of the inference rules below, extended with the following two conditions R = RF + Rp . RF : G(j )(F;p) may only be derived if for each F 0 2 F (of the form F1 [ f g, F1 [ F2 or F1 ) and for each goal G(0 j 0 ) 2 B , F 0 [f0 ^ 0 g is only inconsistent if F1 [ f0 ^ 0 g (and F2 [ f0 ^ 0 g) is already inconsistent. Rp : G( j )(F;p) may only be derived if p  p1 (and p  p2 ) for the goal G(1 j 1 )(F1 ;p1 ) (and G(2 j 2 )(F2 ;p2 ) ) it is derived from. The inference rules of plldg are replacements by logical equivalents and the following four rules. G( j 1 )(F;p) ; R saR : G( j 1 ^ 2 )(F f 2 g;(sa)) G( j ^ )(F1 ;p1 ) ; G( j )(F2 ;p2 ) ; R ctaR : G( ^ j )(F1 F2 ;(cta)) G(1 j )(F;p) ; R wcR : G(1 _ 2 j )(F;(wc)) G( j 1 )(F1 ;p1 ) ; G( j 2 )(F2 ;p2 ) ; R orR : G( j 1 _ 2 )(F1 [F2 ;(or))

Clearly plldg is a generalization of lldg, because the logic lldg is the plldg with the phasing function  de ned by (sa) = 1, (cta) = 1, (wc) = 1, (or) = 1. Besides lldg, we are interested in the following fourphase logic. De nition 8 (4lldg) The logic 4lldg is the plldg with the phasing function  de ned by (sa) = 1, (cta) = 2, (wc) = 3, (or) = 4. In Theorem 1 below we show that for each lldg derivation there is an equivalent 4lldg derivation. Consequently, the phasing does not restrict the set of derivable goals. First we repeat a crucial proposition for llg and 4llg from [van der Torre, 1998b]. Proposition 5 We can replace two subsequent steps of an lldg derivation by an equivalent 4lldg derivation.

Proof Follows directly from Proposition 3 in [van der Torre, 1998b], because the goal base is xed. Theorem 1 (Equivalence lldg and 4lldg) Let B be a goal base. We have B j lldg G( j ) if and only if B j 4lldgG(j ). Proof ( Every 4lldg derivation is a lldg derivation. ) We can take any lldg derivation and construct an

equivalent 4lldg derivation, by iteratively replacing two subsequent steps in the wrong order by several steps in the right order, see Proposition 5. If the proof tree is nite, then after a nite number of steps, all derivation steps are ordered, because no set of replacements cycles (and can be used to construct in nite proof trees). Proposition 6 4lldg is the only four-phase logic of defeasible goals in which Theorem 1 holds.

Proof Proposition 5 does not hold for any other four

phase logic. Counterexamples for reversing the order of each two subsequent steps of 4lldg are given below. For sa and cta, the premises G(pj>) and G(qj>) cannot be combined unless one is rst strengthened. For cta and wc, if the premise G(q j>) is weakened then it can no longer be used to detach G(p j q). For wc and or, the latter can only be applied if consequents are equivalent. G(p j >) sa G(p j q) G(q j >) cta G(p ^ q j >) G(p j q) G(q j >) cta G(p ^ q j >) wc G(p j >) G(p1 j q1 ) wc G(p1 _ p2 j q1 ) G(p1 _ p2 j q2 ) or G(p1 _ p2 j q1 _ q2 ) Moreover, Theorem 2 shows that in 4lldg the consistency check on the label can be replaced by a consistency check on the conjunction of the antecedent and consequent of the goal.

Theorem 2 Let RAC be the following condition. RAC : G( j ) F;p may only be derived from G( j ) (and G( j  )) if for each goal G(0 j 0 ) 2 B , f^ ; 0 ^ 0 g is only inconsistent if f ^ ; 0 ^ 0 g (and f ^ ; 0 ^ 0 g) is already inconsistent. (

1

)

2

2

1

2

1

1

2

Consider any potential derivation of 4lldg, satisfying the condition Rp but not necessarily RF . Then the following four conditions are equivalent: 1. The derivation satis es condition RF throughout phase 1 and 2, 2. The derivation satis es RF everywhere, 3. The derivation satis es condition RAC throughout phase 1 and 2, 4. The derivation satis es RAC everywhere. Proof Clearly (2) ) (1) and (4) ) (3). Through phase 1 and 2, for each formula the conjunction of antecedent and consequent is equivalent to the unique element of its label. Hence (1) , (3). In phase 3 and 4 the rules preserve the consistency of the conjunction of antecedent and consequent, and they also preserve the property that each element of the label is consistent. From this we have (3) ) (4) and (1) ) (2). Putting this together gives us (1) , (2) , (3) , (4) and we are done.

6 Conclusions and further research

In this paper we studied restricted applicability of rules in Gabbay's labelled deductive systems. In particular, we introduced a labelled logic of defeasible goals, with several desirable properties. The logic does not have the identity or con ict averse strategies, but it has restricted strengthening of the antecedent, conjunction, transitivity and the disjunction rule. Moreover, we introduced phasing in the logic and we proved two phasing theorems. Phasing makes the proof theory more ecient, because only a single order of rule application has to be considered. Decision-theoretic goals. The use of lldg in decision-theoretic planning is our main present concern. In decision-theoretic planning usually decision-theoretic goals are used, i.e. goals based on (qualitative abstractions of) utilities and probabilities [Pearl, 1993; Boutilier, 1994; Bacchus and Grove, 1996]. To deal with irrelevance problems usually strong independence notions are introduced, but consequently the logic is not able to reason about con icts. Comparison. Moreover, in further research we will contrast lldg with other con ict-tolerant languages. We already observed that the logic of defaults satis es the identity (called supraclassicallity) and the conjunction rule. Another distinction is that in default reasoning usually rst the con ict is repaired, and thereafter classical inference is used to derive conclusions (called the coherence approach in [Benferhat and Garcia, 1997]).

TMS. We have been informed that our labelled logic of

goals has some formal similarities with truth maintenance systems. This formal relationship will be studied in further research.

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