Phased Labeled Logics of Conditional Goals Leendert W.N. van der Torre IRIT, Paul Sabatier University 118 Route de Narbonne, Toulouse, France
http://abduction.euridis.fbk.eur.nl/~torre/
[email protected]
Abstract. In this paper we introduce phased labeled logics of condi-
tional goals. Labels are used to impose restrictions on the proof theory of the logic. The restriction discussed in this paper is that a proof rule can be blocked in a derivation due to the fact that another proof rule has been applied earlier in the derivation. We call a set of proof rules that can be applied in any order a phase in the proof theory. We propose a one-phase logic of goals containing four proof rules, and we show that it is equivalent to a four-phase logic of goals in which each phase contains exactly one proof rule. The proof theory of the four-phase logic of goals is much more ecient, because other orderings no longer have to be considered.
1 Introduction In the usual approaches to planning in AI, a planning agent is provided with a description of some state of a airs, a goal state, and charged with the task of discovering (or performing) some sequence of actions to achieve that goal. Recently several logics for conditional or context-sensitive goals and desires have been proposed [3,2,10,1,12,11,8,7] in the context of qualitative decision theory. In [15] we introduced a version of a labeled deductive system [4] to reason about goals. Labeled goals G( j )L can roughly be read as `preferably if , against the background of L.' The label keeps track of the context in which the goal is derived. It has some desirable properties not found in other proposals. First, the logic can reason about con icting goals. This is important, because 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. Second, the labeled logics are stronger than previous proposals in the sense that they validate strengthening of the antecedent and transitivity. It has been shown in [15] that these proof rules can only be combined with the desirable proof rule weakening of the consequent if additional machinery like labels is introduced in the logic. Otherwise counterintuitive conclusions follow. In the phased labeled logics of goals (pllg) introduced in this paper we show how to phase derivations. To impose phasing restrictions on the derivations, the labeled logics of goals are extended in two ways. First, a phase is associated
with each pllg-proof rule by an explicitly given phasing function. Second, the label of a pllg-goal not only contains a set of sets of propositional formulas, the ful llments (F ) of the premises from which the goal is derived, but it also contains an integer, the phase (p) in which the goal is derived. There are two restrictions on the use of proof rules in pllg. First, as in other labeled logics of goals, a consistency check on F is used to restrict derivations such that all premises can be ful lled. The ful llment of a premise G( j ) is the propositional formula ^ [19] and the ful llment of a derived goal depends on the premises from which it is derived. Later in this paper we show how this consistency check restricts derivations to a kind of constructive proofs, in the sense that no goals are derived from con icts. Second, and this is new, the phase p is used to select the proof rules that still may be applied. In this paper we focus on a one-phase labeled logic of goals (llg) and a fourphase labeled logic of goals (4llg). Both contain the four proof rules strengthening of the antecedent, a type of transitivity, weakening of the consequent and the disjunction rule for the antecedent. Moreover, we show that the conjunction rule for the consequent follows from these four rules. In llg these rules can be applied in any order, but in 4llg there is only one order in which they can be applied. Nevertheless, we prove that every formula that can be derived in llg can also be derived in 4llg. Consequently, in the proof theory of llg we can restrict ourselves to one speci c order of the proof rules, which makes the logic much more ecient. Finally, we also prove that in 4llg the consistency check on the label can be replaced by a consistency check on the antecedent and consequent.
2 Phased labeled logic of goals (pllg) Phased labeled logics of goals are versions of a labeled deductive system as it was introduced by Gabbay in [4]. Roughly speaking, the label L of a goal G( j )L consists of a record of the ful llments (F ) of the premises that are used in the derivation of G( j ), and the phase (p) in which it is derived. Where there is no application of reasoning by cases, F can be taken to be a set of boolean formulas, that grows by joining sets as premises are combined. But in general, to cover the parallel tracks created through reasoning by cases, we need to consider sets of sets of boolean formulas [9].
De nition 1 (Language). Let L be a propositional base logic. The language of pllg consists of the labeled 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 2 (Premise). A formula G( j )(ff ^ gg;0), where ^ is consistent in L, is called a premise of pllg.
The phase of a goal is determined by the proof rule used to derive the goal, and the set of ful llments is the union (or) or the product (sa, trans) of the labels of the premises used in this inference rule, where the product is de ned by fS1; : : :; Sn g fT1 ; : : :; Tm g = fS1 [ T1 ; : : :; S1 [ Tm ; : : :; Sn [ Tm g. The labels are used to check that ful llments are consistent and that the phase of reasoning is non-decreasing. In a normative context the consistency check realizes a variant of the Kantian principle that `ought implies can.'
De nition 3 (pllg). Let be a phasing function that associates with each proof rule below an integer called its phase. The phased labeled logic of goals pllg for consists of the inference rules below, extended with the following two conditions R = R F + Rp . RF : G( j )(F;p) may only be derived if each Fi 2 F is consistent: it must always be possible to ful ll a derived goal and each of the goals it is derived from, though not necessarily all of them at the same time. Rp: G( j )(F;p) may only be derived if p pi for all goals G( i j i )(F ;p ) it is derived from. i
i
The inference rules of pllg are replacements by logical equivalents (for antecedent and consequent) and the following four rules. saR transR :
G( j ) ; R : G( j ^ 1) (F;p) 1 2 (F f g;(sa)) 2
G( j ^ )(F1 ;p1 ) ; G( j )(F2 ;p2 ) ; R G( ^ j )(F1 F2 ;(trans))
wcR
G( j ) ; R : G( _ 1 j ()F;p) 1 2 (F;(wc))
G( j 1 )(F1 ;p1 ) ; G( j 2 )(F2 ;p2 ) ; R G( j 1 _ 2 )(F1 [F2 ;(or)) We say fG( i j i ) j 1 i ng `pllg G( j ) if there is a labeled goal G( j )L that can be derived from the set of goals fG( ij i )(ff ^ gg;0) j 1 i ng. orR :
i
i
The unusual transitivity rule (transR ) implies, under certain circumstances, the standard transitivity rule as well as the conjunction rule. First, if we have (sa) (trans) (wc), then we have the following derivation. G( j )(F1 ;p1 ) sa G( j ^ )(F1 f g;(sa)) G( j )(F2 ;p2 ) trans G( ^ j )(F1 f gF2 ;(trans)) wc G( j )(F1 f gF2 ;(wc))
Consequently, the following standard transitivity rule is implied by pllg if we have (sa) = (trans) = (wc), i.e. if sa, trans and wc are in the same phase, because F2 implies (see also Proposition 2). G( j )(F ;p ) ; G( j )(F ;p ) ; R trans0R : G( j )(F F ;(trans)) Second, concerning the conjunction rule, if (sa) (trans), then we can rst strengthen G( 1 j ) to G( 1 j ^ 2), and then apply trans as follows to derive G( 1 ^ 2j ). G( 1j )(F ;p ) sa G( 1j ^ 2 )(F f g;(sa)) G( 2j )(F ;p ) trans G( 1 ^ 2j )(F f gF ;(trans)) Consequently, the following conjunction rule is implied by the logic pllg if we have (sa) = (trans), i.e. if sa and trans are in the same phase, because F2 implies 2 (Proposition 2). G( 1 j )(F ;p ) ; G( 2 j )(F ;p ) ; R andR : G( 1 ^ 2 j )(F F ;(trans)) Labeled logics of goals can reason about con icting goals, and they can combine several proof rules without deriving counterintuitive consequences. It has been shown in [15] that this can be achieved by using the ful llments without using the phases. First, the logics can reason about con icting goals, because we have G(p); G(:p) 6` G(p ^:p) and G(p); G(:p) 6` G(q), where G( ) is short for G( j>), > stands for any tautology like p _ :p and q is not logically implied by p or :p. In particular, the ful llments in the labels are used to block the second derivation step in the following counterintuitive derivation [15], in which a blocked derivation step is represented by a dashed line. Moreover, from the results presented later in this paper follows that this counterintuitive derivation can also be blocked by giving wc a higher phase than and. G(p) wc G(p _ q) G(:p) ? ? ? ? ? ? ? ? ? ? and G(q ^ :p) wc G(q) Second, it is easily checked how the ful llments in the labels are used to combine strengthening of the antecedent (sa) with weakening of the consequent (wc) without validating the following counterintuitive derivation [15]. Moreover, this counterintuitive derivation can also be blocked by giving wc a higher phase than sa, together with a consistency check on the conjunction of antecedent and consequent. G(cj>) wc G(c _ tj>) ? ? ? ? ? sa G(c _ tj:c) 1
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pllg
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pllg
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3 Labeled logic of goals (llg): some examples In the this section we illustrate the use of the consistency check on F in the labeled logics of goals by a variant of the logic proposed in [15], and in the following sections we study the use of di erent phases in the proof theory. The logic llg is the pllg that consists of only one phase.
De nition 4 (llg). The logic llg is the pllg with the phasing function de ned by (sa) = 1, (trans) = 1, (wc) = 1, (or) = 1.
The logic llg derives the proof rules trans0 and and discussed in the previous section, because sa, trans and wc (respectively sa and trans) are in the same phase. The following example illustrates the labeled logic of conditional goals. Example 1. Consider the set S = fG(a _ pj>); G(:aj>)g as premise set, where a can be read as `buying apples' and p as `buying pears' (taken from [13]). We have S 6` G(p j a), as desired. Below it is shown how two derivations of the counterintuitive G(p j a)L are blocked. The non-derived goal is counterintuitive, because when a is true (its antecedent) then the rst premise is ful lled and the second is violated. This pattern holds irrespective of whether p is true or false. Buying pears does not `improve' the situation, once apples are bought. Hence, once a is assumed, there is no longer any reason to infer p. llg
G(a _ pj>)(ffa_pgg;0) G(:aj>)(ff:agg;0) G(:a ^ pj>)(ffa_p;:agg;1) ? ? ? ? ? ? ? ? ? ? ? ? sa G(:a ^ pja)(ffa_p;:a;agg;1) G(pja)(ffa_p;:a;agg;1) wc
and
G(a _ pj>)(ffa_pgg;0) G(:aj>)(ff:agg;0) G(:a ^ pj>)(ffa_p;:agg;1) G(pj>)(ffa_p;:agg;1) wc ? ? ? ? ? ? ? ? ? ? sa G(pja)(ffa_p;:a;agg;1)
and
The following example illustrates how the transitivity rule formalizes that conditional rules can be applied one after the other. Derivations go `as far as possible.' Example 2. Consider the set of goals S = fG(a j b); G(b j c); G(c j>); G(:a j>)g. There is a con ict for a, because we have S ` G(a j>) and S ` G(:a j>). There is not a con ict for b, because we have S ` G(bj>) and S 6` G(:bj>). Hence, derivation chains go as far as possible and there is no weak contraposition as, for example, in conditional entailment [5], see also the discussion in [9]. Moreover, consider the set of goals S 0 = fG(a j b); G(:c j b _ c)g. We have S ` G(ajb _ c), as desired. The following derivation illustrates how this more complex form of transitivity is supported. G(ajb)(ffa^bgg;0) sa G(ajb ^ :c)(ffa^b;:cgg;1) G(:cjb _ c)(ffb^:cgg;0) trans G(a ^ :cjb _ c)(ffa^b;:c;b^:cgg;1) wc G(ajb _ c)(ffa^b;:c;b^:cgg;1) The third example illustrates how the disjunction rule supports reasoning by cases. llg
llg
llg
llg
llg
Example 3. Consider the set S = fG(a ^ cjb); G(a ^:cj:b)g (taken from [9]). We have S ` G(aj>), as desired. The following derivation illustrates this complex llg
type of reasoning by cases. G(a ^ cjb)(ffa^c^bgg;0) G(a ^ :cj:b)(ffa^:c^:bgg;0) wc wc G(ajb)(ffa^c^bgg;1) G(aj:b)(ffa^:c^:bgg;1) or G(aj>)(ffa^c^bg;fa^:c^:bgg);1)
On the other hand, consider the set S = fG(a j b); G(a j:b)g. In [14,21] it is argued that G(aja $ b) is counterintuitive and should therefore not be derived. We have S 6` G(a j a $ b), and we have the blocked derivation below. The non-derived goal is counterintuitive, because when a $ b is true (its antecedent) then the rst premise is ful lled when its antecedent is true and the second is violated when its antecedent is true. This pattern holds irrespective of whether a is true or false. Hence, once a $ b is assumed, there is no longer any reason to infer a. G(ajb)(ffa^bgg;0) G(aj:b)(ffa^:bgg;0) or G(aj>)(ffa^bg;fa^:bgg;1) ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? sa G(aja $ b)(fa^b;a$bg;fa^:b;a$bgg;1) llg
In the following section we study the additional expressive power of pllg over llg by introducing di erent phases in the proof theory.
4 Four-phase labeled logic of goals (4llg) In this section we discuss a labeled deontic logic which completely orders the derivations of llg in the following order: sa, trans, wc, or. We call the resulting logic four-phase labeled logic of goals 4llg.
De nition 5 (4llg). The logic 4llg is the pllg with the phasing function de ned by (sa) = 1, (trans) = 2, (wc) = 3, (or) = 4.
In Theorem 1 below we show that for each llg derivation there is an equivalent 4llg derivation. We rst prove three propositions.
Proposition 1. Consider any potential derivation of pllg, satisfying the condition Rp but not necessarily RF . Then the following two conditions are equivalent: 1. The nal derivation satis es condition RF , 2. The derivation satis es RF everywhere.
Proof The labels are, in a suitable sense, cumulative. Every element of every label in the derivation is classically implied by some element of the label of the nal conclusion.
Proposition 2. For each goal G( j )(F;p) derived in pllg we have for each Fi 2 F that Fi j= ^ . 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. Proposition 3. We can replace two subsequent steps of an llg derivation by an equivalent 4llg derivation.
Proof The replacements are given below. For each replacement the original llg derivation as well as its 4llg replacement are given. From Proposition 2 follows that the replacement does not violate the consistency check. For example, consider the reversing e.1. of or4 and trans2 . Call the ful llments of the three premises G( j 1 ^ ), G( j 2 ^ ) and G( 1 _ 2 j ) respectively F1, F2 and F3 . From the llg-derivation follows that each element of (F1 [ F2) F3 is consistent, and therefore all elements of F1 F3 and F2 F3 are consistent. Moreover, from Proposition 2 follows for each F1;i 2 F1 that F1;i j= 1 and for each F2;i 2 F2 that F2;i j= 2 . Consequently, for the 4llg-derivation we have that the labels of the replacements F1 f 1 _ : 2 g F3 and F2 f 2 _ : 1 g F3 are equivalent to F1 F3 and F2 F3, and all elements of them are therefore consistent. The other proofs are analogous. G( j ^ 1 ) G( j 1 ) trans G( ^ j 1 ) sa G( ^ j 1 ^ 2 )
G( j 1) G( j ^ 1 ) sa sa G( j ^ 1 ^ 2 ) G( j 1 ^ 2 ) trans G( ^ j 1 ^ 2 )
a. Reversing the order of trans2 and sa1 G( 1j 1) G( 1j 1) wc sa G( 1 _ 2j 1 ) G( 1j 1 ^ 2 ) sa wc G( 1 _ 2j 1 ^ 2 ) G( 1 _ 2j 1 ^ 2 ) b. Reversing the order of wc3 and sa1 G( 1j ^ ) G( j ) G( 1j ^ ) wc trans G( 1 _ 2j ^ ) G( j ) G( 1 ^ j ) trans wc G(( 1 _ 2) ^ j ) G(( 1 _ 2) ^ j ) c.1. Reversing the order of wc3 and trans2 G( j( 1 _ 2 ) ^ ) sa G( 1 j ) G( j 1 ^ ) G( 1 j ) wc trans G( j( 1 _ 2 ) ^ ) G( 1 _ 2 j ) G( ^ 1j ) trans wc G(( ^ ( 1 _ 2 )j ) G(( ^ ( 1 _ 2 )j ) c.2. Reversing the order of wc3 and trans2
G( j 1) G( j 2) or G( j 1 _ 2 ) sa G( j( 1 _ 2 ) ^ 3 )
G( j 2) G( j 1) sa sa G( j 1 ^ 3 ) G( j 2 ^ 3 ) or G( j( 1 ^ 3 ) _ ( 2 ^ 3 ))
d. Reversing the order of or4 and sa1 G( j 1 ^ ) G( j 2 ^ ) or G( j( 1 _ 2 ) ^ ) G( 1 _ 2 j ) G( ^ ( 1 _ 2 )j )
trans
G( 1 _ 2j ) G( 1 _ 2j ) G ( j 2 ^ ) G( 1 _ 2j ^ ( 2 _ : 1 )) G( j 1 ^ ) G( 1 _ 2j ^ ( 1 _ : 2 )) sa trans G( ^ ( 1 _ 2 )j ^ ( 1 _ : 2 )) G( ^ ( 1 _ 2 )j ^ ( 2 _ : 1 )) or G( ^ ( 1 _ 2 )j )
sa trans
e.1. Reversing the order of or4 and trans2 G( j 1) G( j 2 ) or G( j ^ ( 1 _ 2 )) G( j 1 _ 2 ) trans G( ^ j 1 _ 2 ) G( j ^ ( 1 _ 2 )) G( j ^ ( 1 _ 2 )) sa sa G( j ^ 1 ) G( j 1 ) G( j ^ 2 ) G( j 2 ) trans trans G( ^ j 1 ) G( ^ j 2 ) or G( ^ j 1 _ 2 )
e.2. Reversing the order of or4 and trans2 G( 1j 1 ) G( 1j 2) G( 1j 2) G( 1j 1) or wc G( 1j 1 _ 2 ) G( 1 _ 2j 1) G( 1 _ 2j 2) wc G( 1 _ 2j 1 _ 2 ) G( 1 _ 2j 1 _ 2 )
wc or
f. Reversing the order of or4 and wc3
Theorem 1 (Equivalence llg and 4llg). Let S be a set of conditional goals. We have S ` G( j ) if and only if S ` G( j ). llg
4llg
Proof ( Every 4llg derivation is a llg derivation. ) We can take any llg
derivation and construct an equivalent 4llg derivation, by iteratively replacing two subsequent steps in the wrong order by several steps in the right order, see Proposition 3. 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 4. 4llg is the only four-phase logic of goals for which Theorem 1
holds.
Proof Proposition 3 does not hold for any other four phase logic. Counterexamples for reversing the order of each two subsequent steps of 4llg are given below. For sa and trans, the premises G(pj>) and G(qj>) cannot be combined unless one is rst strengthened. For trans 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 >) G(p j q) sa G(q j >) G(p ^ q j >)
trans
G(p j q) G(q j >) G(p1 j q1 ) G(p ^ q j >) trans G(p1 _ p2 j q1 ) wc G(p1 _ p2 j q2 ) G(p j >) wc G(p1 _ p2 j q1 _ q2 )
or
It is easy to see that the following proof rule or0 is implied by or and wc in llg. If we replace or in llg by the more general or0 , then we can use it as a phase-3 rule. G( 1 j 1 )(F ;p ) ; G( 2 j 2 )(F ;p ) ; R or0R : G( 1 _ 2 j 1 _ 2 )(F [F ;(or)) Moreover, Theorem 2 shows that in 4llg the consistency check on the label can be replaced by a consistency check on the conjunction of the antecedent and consequent of the goal. 1
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Theorem 2. Consider any potential derivation of 4llg, 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, 2. The derivation satis es RF everywhere, 3. Each consequent is consistent with its antecedent throughout phase 1, 4. Each consequent is consistent with its antecedent everywhere. Proof Clearly (2) ) (1) and (4) ) (3). Through phase 1, for each formula the conjunction of antecedent and consequent is equivalent to the unique element of its label. Hence (1) , (3). In phase 2 the conjunction of antecedent and consequent is also equivalent to the unique element of its label, which is equivalent to the label of the rst premise of each derivation step. 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.
In the following section we illustrate that the rst two phases of the fourphase logic 4llg, i.e. sa and trans, can be combined in one phase without invalidating Theorem 2. Moreover, the latter two phases of 4llg, i.e. wc and or, can be combined similarly. The two-phase logic 2llg rst combines goals in derivation chains or arguments (sa and trans), and then combines arguments (wc and or) with reasoning by cases.
5 Two-phase labeled logic of goals 2llg The logic 2llg is the pllg that combines the rst two phases and the last two phases of 4llg.
De nition 6 (2llg). 2llg is the pllg with the phasing function de ned by (sa) = 1, (trans) = 1, (wc) = 2, (or) = 2.
It is easy to see that 2llg is equivalent to llg and 4llg in the sense of Theorem 1, because its phasing is in between the phasing of llg and 4llg. Moreover, Theorem 3 below is analogous to Theorem 2 for the two-phase logic 2llg.
Theorem 3. Consider any potential derivation of 2llg, 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, 2. The derivation satis es RF everywhere, 3. Each consequent is consistent with its antecedent throughout phase 1, 4. Each consequent is consistent with its antecedent everywhere. Proof Analogous to the proof of Theorem 2, because in phase 1 the conjunction of antecedent and consequent is also equivalent to the unique element of its label, and in phase 2 the rules preserve the consistency of the conjunction of antecedent and consequent, as well as the property that each element of the label is consistent. We can construct other two-phase logics of goals, for example the one in which the rst phase consists of only sa, but the following proposition shows that in those logics we cannot restrict ourselves to a consistency check on the conjunction of the antecedent and conjunction.
Proposition 5. The logic 2llg is the only two-phase pllg that validates versions of Theorem 1 and 2, in which 4llg is replaced by the two-phase logic.
Proof From the proof of Proposition 4 follows that 2llg is the only two-phase
for which Theorem 1 holds and in which wc cannot occur before trans. The following derivation shows that if wc can occur before trans, then we need the labels for the consistency check.
pllg
G(:p ^ qjr)(ff:p^q^rgg;0) wc G(pjq ^ r)(ffp^q^rgg;0) G(qjr)(ff:p^q^rgg;(wc)) trans G(p ^ qjr)(ffp^q^r;:p^q^rgg;(trans ))
The surprising theorems follow from the consistency check in pllg, which is stronger than it seems at rst sight. In the following section we illustrate that the consistency check restricts derivations to a type of constructive proofs in the sense that no conclusions may be drawn from con icts.
6 Con icting goals In this section we discuss three derivations related to con icting goals. They are not valid in pllg, because their usefulness depends on the application area of the logic of goals. In natural language di erent types of goals are used through each other, with possible confusion. An agent can either commit himself to a goal, because it maximizes his expected utility, or the goal may be imposed upon him by some authority, e.g. his boss or owner. In the latter case, the agent can simply adopt the goal as a desirable state, or he may interpret it as a re ection of the authority's maximal expected utility. Most formalisms for goals are developed to be used by intelligent robots, whose goals are the commands given by his owner. In the robot case, the logical properties of goals resemble the logical properties of obligations. As a consequence of the analogy between goals and obligations, the following unrestricted conjunction rule may be accepted in the logic of goals. G( 1 j )(F ;p ) ; G( 2 j )(F ;p ) and0 : G( 1 ^ 2 j )(F F ;max(p ;p )) Moreover, the deontic `ought implies can' axiom :G(?j ) may be accepted. For a discussion on the unrestricted conjunction rule and the associated problems, as well as a development of logics satisfying this proof rule, we refer to the deontic logic literature (see e.g. [14,20] for a discussion and references). However, even if the unrestricted conjunction rule and the deontic `ought implies can' axiom are not accepted in the logic of goals, then there is another interesting issue of con icting goals. It concerns the following derivation, which we call Forbidden Con ict (fc) [17]. In this rule, as well as in the derivations in the remainder of this section, we leave the labels unspeci ed. G(:aj>) G(ajb) fc G(:bj>) 1
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In contrast to the constructive proofs of pllg, this proof rule formalizes a con ict averse strategy. If b is the case then a con ict arises; therefore the agent desires that b is false. The following continuation of Example 2 illustrates that derivations no longer go as far as possible, but instead we have weak contraposition. Example 4. Assume the proof rules trans, wc and fc together with the four goals S = fG(a j b); G(b j c); G(c j>); G(:a j>)g. The goal G(b j>) can be derived from G(bjc) and G(cj>) by trans and wc, and the goal G(:bj>) from G(ajb) and G(:a j>) by fc. Hence, there is a con ict for b, because G(b j>) as well as G(:bj>) can be derived. At rst sight, this seems undesirable (see also [9]). The following derivation is the third and last one we discuss related to con icting goals. G(ajb) G(aj:b) G(a ! bj>) G(a ^ bj>)
It formalizes a more complex con ict averse strategy. From the two goals G(aj:b) and G(a ! bj>) follows that if :b is the case then a con ict arises: a as well as a ! b cannot both be the case as well. The goal G(a ^ bj>) can thus be derived as follows. G(aj:b) G(a ! bj>) fc G(ajb) G(bj>) trans G(a ^ bj>) It follows from Proposition 2 that in pllg this derivation is blocked, because all consistent extensions of the ful llment of the premise G(aj:b) do not derive a ^ b. The following derivation illustrates that the goal can be derived in 4llg without ful llment check with the unrestricted conjunction rule and0 . G(ajb) G(aj:b) or G(aj>) G(a ! bj>) and G(a ^ bj>) The consequences of con ict averse strategies for pllg are yet unclear, and left for further research.
7 Related research Several authors have observed the relation between goals and desires in qualitative decision theory and obligations in deontic logic [10,1,8] and we have discussed the relation between decision theory, diagnosis theory and deontic logic in [19]. Labeled deontic logic was introduced in [16,17], though its properties have not been studied. It does not make a dilemma like O(p j>) ^ O(:p j >) inconsistent, in contrast to traditional deontic logics like for example so-called standard deontic logic (SDL), and we therefore now interpret it as a logic of goals. Makinson [9] extended labeled deontic logic with the disjunction rule to cover reasoning by cases and an unrestricted conjunction rule. Two-phase deontic logic has been proposed in [13], but phased deontic logic has not been related to labeled deontic logic (although it has been suggested in [9]). In [15] we introduced the one-phase labeled logic of goals llg0 based on labeled formulas G( j )(F;V ) and two consistency checks. In llg0 , a formula G( j )(f ^ g;f: ^ g) is called a premise, and the label of a goal derived by an inference rule is the union of the labels of the premises used in this inference rule. llg0 based on a violation check and a ful llment check consists of inference rules, extended with a condition R = RV + RF . RV : G( j )(F;V ) may only be derived if ^ 6j= for all 2 V : ful lling a derived goal should not imply a violation of one of the goals it is derived from; RF : G( j )(F;V ) may only be derived if ^ 6j= : for all 2 F : it must always be possible to ful ll a derived goal and each of the goals it is derived from, though not necessarily all of them at the same time.
This logic is stronger than the logics proposed in this paper, because it validates the following counterintuitive derivation. Notice that we cannot strengthen the condition of this logic, such that for example it is always possible that all the premises can be ful lled at the same time, because then the intuitive rst derivation of Example 3 is blocked. G(pjr) G(qjr) and G(p ^ qjr) G(p ^ qj:r) or G(p ^ qj>) wc G((p ^ q) _ sj>) sa G((p ^ q) _ sj:(p ^ q ^ r)) Alternatively, and perhaps more intuitively, we take the logic pllg and replace its sa rule by the following rule sa, and replace RF by RF with a consistency check on the conjunction of the ful llments in the label and the antecedent of the goal, as in llg0 . We call the resulting logics pllg. G( j 1 )(F;p) ; R sa*R : G( j 1 ^ 2 )(F;(sa)) RF : G( j )(F;p) may only be derived if f g [ Fi is consistent for each Fi 2 F . It is easy to see that 4llg* is equivalent to 4llg. Moreover, we conjecture that Theorem 1 still holds, i.e. that llg* is equivalent to 4llg*, and we conjecture that pllg* is equivalent to pllg. However, the following example illustrates that the proof of Theorem 1 no longer holds, because some llg derivations have intermediate results (here G(pjq1)) which cannot be used in 4llg*. G(pj>) sa G(pjq1) G(pjq2) or G(pjq1 _ q2) sa G(pj:q1 ^ q2) Makinson [9] proposes a one-phase labeled logic based on RF in which the premises are represented by G( j )ff gg, i.e. in which only the consequents are represented in the label. Obviously, Theorem 1 no longer holds. Typical properties are that the disjunction rule always holds, such that the second derivation in Example 3 is not blocked, and the last derivation of Section 6 is also valid. Moreover, the logic has been extended with an unrestricted conjunction rule. Most other logics of goals that have been proposed have a built in mechanism such that more speci c and con icting goals override more general ones, see e.g. [1,12,11]. However, speci city is only one possible rule to decide con icts, which may be overridden itself (as in legal reasoning, where more general later rules override more speci c earlier rules). Moreover, these logics make con icts like G(p) ^ G(:p) inconsistent, which is not in line with the idea that goals can refer to di erent objectives. Finally, the non-monotonic mechanisms do not formalize the deliberating robber satisfactorily [18].
8 Conclusions and further research Labeled logic is a powerful formalism to analyze and construct proofs. In this paper we discussed four proof rules in the framework of labeled deduction and we showed how labeled logics can combine the proof rules without deriving counterintuitive consequences. In particular, we showed how derivations can be partitioned into several phases. Surprisingly, the phasing of 4llg does not restrict the set of derivable conclusions of llg. Consequently, the phasing of 4llg can be considered as a useful heuristic to make the proof theory of llg more ecient, because only a small subset of all proofs of llg have to be considered to proof the (in)validity of a formula. Presently we are looking for ways to de ne a semantics for pllg, such that also negations and disjunctions of goals are de ned. One approach has been given in [9]. A possible worlds semantics can be de ned along the lines of the two-phase deontic logic in [13,21]. Theorem 2 and 3 show that in 4llg and 2llg we can get rid of the ful llments in the label by checking the consistency of the conjunction of the antecedent and consequent of the dyadic operators. Moreover, we can also get rid of the integer in the label by introducing di erent operators for each phase. For example, for 2llg we can de ne the two operators G1 (with proof rules sa and trans) and G2 (with proof rules wc and or). The premises are goals G1 ( i j i ), the conclusion is a goal G2( j ), and the two phases are linked to each other with the following new proof rule. G1( j ) G2( j ) To fully grasp the logical properties of goals, we think it is important to not only consider the proof theory, but also to consider the semantic relation between goals, desires, utilities and preferences in a decision-theoretic setting (see e.g. [10,8, 22]). In particular, goals serve as computationally useful heuristic approximations of the relative preferences over the possible results of a plan [3], and goals are used to communicate desires in a compact and ecient way [6]. We think this is not a rivaling approach to the approach taken in this paper, but a complementary one.
Acknowledgement Thanks to Patrick van der Laag, David Makinson and Emil Weydert for discussions on the issues raised in this paper, and comments on earlier versions of it. Many of the issues and examples discussed in this paper were raised during a discussion on labeled deontic logic with David Makinson.
References 1. C. Boutilier. Toward a logic for qualitative decision theory. In Proceedings of the KR'94, pages 75{86, 1994.
2. J. Doyle, Y. Shoham, and M.P. Wellman. The logic of relative desires. In Sixth International Symposium on Methodologies for Intelligent Systems, Charlotte, North Carolina, 1991. 3. J. Doyle and M.P. Wellman. Preferential semantics for goals. In Proceedings of the AAAI'91, pages 698{703, 1991. 4. D. Gabbay. Labelled Deductive Systems, volume 1. Oxford University Press, 1996. 5. H. Ge ner and J. Pearl. Conditional entailment: bridging two approaches to default reasoning. Arti cial Intelligence, 53:209{244, 1992. 6. P. Haddawy and S. Hanks. Representations for decision-theoretic planning: Utility functions for dead-line goals. In Proceedings of the KR'92, pages 71{82, 1992. 7. Z. Huang and J. Bell. Dynamic goal hierarchies. In Intelligent Agent Systems: Theoretical and Practical Issues, LNAI 1027, pages 88{103. Springer, 1997. 8. J. Lang. Conditional desires and utilities - an alternative approach to qualitative decision theory. In Proceedings of the ECAI'96, pages 318{322, 1996. 9. D. Makinson. On a fundamental problem of deontic reasoning. In P. McNamara and H. Prakken, editors, Norms, Logics and Information Systems. New Studies on Deontic Logic and Computer Science. IOS Press, 1998. 10. J. Pearl. From conditional oughts to qualitative decision theory. In Proceedings of the UAI'93, pages 12{20, 1993. 11. S.-W. Tan and J. Pearl. Qualitative decision theory. In Proceedings of the AAAI'94, 1994. 12. S.-W. Tan and J. Pearl. Speci cation and evaluation of preferences under uncertainty. In Proceedings of the KR'94, pages 530{539, 1994. 13. Y.-H. Tan and L.W.N. van der Torre. How to combine ordering and minimizing in a deontic logic based on preferences. In Deontic Logic, Agency and Normative Systems. Proceedings of the EON'96, Workshops in Computing, pages 216{232. Springer, 1996. 14. L.W.N. van der Torre. Reasoning about Obligations: Defeasibility in Preferencebased Deontic Logic. PhD thesis, Erasmus University Rotterdam, 1997. 15. L.W.N. van der Torre. Labeled logics of conditional goals. In Proceedings of the ECAI'98, pages 368{369. John Wiley & Sons, 1998. 16. L.W.N. van der Torre and Y.-H. Tan. Cancelling and overshadowing: two types of defeasibility in defeasible deontic logic. In Proceedings of the IJCAI'95, pages 1525{1532. Morgan Kaufman, 1995. 17. L.W.N. van der Torre and Y.-H. Tan. The many faces of defeasibility in defeasible deontic logic. In D. Nute, editor, Defeasible Deontic Logic, pages 79{121. Kluwer, 1997. 18. L.W.N. van der Torre and Y.-H. Tan. Deliberate robbery, or the calculating Samaritan. In Proceedings of the ECAI'98 Workshop on Practical Reasoning and Rationality (PRR'98), 1998. 19. L.W.N. van der Torre and Y.-H. Tan. Diagnosis and decision making in normative reasoning. Arti cial Intelligence and Law, 1998. 20. L.W.N. van der Torre and Y.-H. Tan. Prohairetic Deontic Logic (PDL). In this volume, 1998. 21. L.W.N. van der Torre and Y.-H. Tan. Two-phase deontic logic. 1998. Submitted. 22. L.W.N. van der Torre and E. Weydert. Goals, desires, utilities and preferences. In Proceedings of the ECAI'98 Workshop on Decision Theory meets Arti cial Intelligence, 1998.
