Preferences constitute a very natural and effective way of resolving indeterminate situations. For example, in scheduling not all deadlines may be simultaneously satisfiable, and in configuration various goals may not be simultaneously met. Preferences among deadlines and goals may allow for an acceptable, non-optimal solution. In legal reasoning, laws may apply in different situations, but laws may also conflict with each other. Conflicts are resolved by appeal to higher-level principles such as authority or recency. So federal laws will have a higher priority than state laws, and newer laws will take priority over old. Further preferences, such as authority holding sway over recency, may also be required. In fact, while logical preference handling constitutes already an indispensable means in legal reasoning systems (cf. [16, 22]), it is also advancing in other application areas such as intelligent agents and e-commerce , information-site selection , and the resolution of grammatical ambiguities .
The increasing practical interest in preferences is also reflected by the large number of proposals for preference handling in logic programming, including [23, 6, 15, 31, 17, 8, 13, 28], and related areas, such as default logic [3, 5, 12]. A common approach in such work has been to employ meta-formalisms for characterizing “preferred answer sets”. This has led to a diversity of approaches that are hardly comparable due to considerably different methods of formal characterization. As a consequence, there is no homogeneous account of preference.
In , we started addressing this shortcoming by proposing a uniform semantical framework for extended logic programming with preferences. To be precise, we develop an (alternating) fixpoint theory for so-called ordered logic programs, building on the basic ideas in . An ordered logic program is an extended logic program whose rules are subject to a strict partial order. In analogy to standard logic programming, such a program is then interpreted by means of an associated fixpoint operator. Different semantics are obtained by distinguishing different subsets of the respective set of alternating fixpoints. As a result, several different approaches to defining preferred answer sets, including [9, 10, 13], can all be captured within our framework and each of these preference strategies is based on an operator, which plays the same role as the consequence operator in the setting of normal logic programs.
In this paper, we show that the preference strategies for defining answer sets turn out to be too weak in the setting of well-founded semantics. For this reason, we propose a new approach to preference handling for logic programs that seems to be more appropriate for well-founded semantics. In fact, we show that for a resulting instance of this approach some attractive properties. We also discuss the relation of our preferred well-founded semantics to other approaches [21, 6, 30].
2 Definitions and notation
An extended logic program is a finite set of rules of the form
where , and each is a literal, ie. either an atom or the negation of . The set of all literals is denoted by . Given a rule as in (1), we let denote the head, , of and the body, , of . Further, let and . A program is called basic if for all its rules; it is called normal if it contains no classical negation symbol .
We define the reduct of a rule as . The reduct, , of a program relative to a set of literals is defined by
A set of literals is closed under a basic program iff for any , whenever . We say that is logically closed iff it is either consistent (ie. it does not contain both a literal and its negation ) or equals . The smallest set of literals which is both logically closed and closed under a basic program is denoted by . With these formalities at hand, we can define answer set semantics for extended logic programs: A set of literals is an answer set of a program iff .
For capturing even more semantics within a similar framework, van Gelder defines in  the operator as . It is important to note that the operator is anti-monotonic, which implies that the operator is monotonic. A fixpoint of is called an alternating fixpoint for . Different semantics are captured by distinguishing different groups of fixpoints of . For instance, given a program , the least alternating fixed point of is known to amount to its well-founded semantics. Answer sets of are simply alternating fixed points of that are also fixed points of .
Alternative inductive characterizations for the operators , , and can be obtained by appeal to immediate consequence operators [26, 19]. Let be a basic program and a set of literals. The immediate consequence operator is defined as follows:
if is consistent, and otherwise. Iterated applications of are written as for , where and for . It is well-known that , for any basic program . Also, for any answer set of program , it holds that .
A reduction from extended to basic programs is avoidable with an extended consequence operator: Let be an extended program and and be sets of literals. The extended immediate consequence operator is defined as follows:
if is consistent, and otherwise. Iterated applications of are written as those of above. Clearly, we have for any basic program and for any extended program . Accordingly, we have for any answer set of program that . Finally, for dealing with the individual rules in (2), we rely on the notion of activeness:111Although activeness is implicitly present in standard logic programming (cf. definition of ), the term as such was (to the best of our knowledge) coined in approaches dealing with preferences in default logic [3, 5]. There, however, activeness additionally stipulated that in order to prevent multiple applications of the same rule. Let be two sets of literals in a program . A rule in is active wrt the pair , if and . Alternatively, we thus have that .
Lastly, an ordered logic program222Also called prioritized logic program by some authors, as eg. in [31, 8]. is simply a pair , where is an extended logic program and is an irreflexive and transitive relation. Given, , the relation is meant to express that has higher priority than .333Some authors, eg. , attribute relation the inverse meaning.
3 Preferred (alternating) fixpoints
We start by describing the semantical framework given in , while concentrating on the formal details needed for capturing the approach introduced in . The formal development of the approach in  and  is analogous and thus omitted here.
The overall idea behind the obtained semantics for ordered logic program is to distinguish the “preferred” answers of a program by means of fixpoint equations. That is, a set of literals constitutes a collection of preferred answers from , if it satisfies the equation for some operator . In view of the classical logic programming approach described in Section 2, this makes us investigate semantics that interpret preferences as inducing selection functions on the set of standard answer sets of the underlying non-ordered program .
Standard answer sets are defined via a reduction of extended logic programs to basic programs. Such a reduction is inappropriate when resolving conflicts among rules by means of preferences since all such conflicts are simultaneously resolved when turning into . Rather conflict resolution must be addressed among the original rules in order to account for blockage between rules. In fact, once the negative body is eliminated there is no way to detect whether holds in case of . Our idea is therefore to characterize preferred answer sets by an inductive development that agrees with the given ordering rather than a simultaneous reduction. In terms of a standard answer set , this means that we favor its formal characterization as over . This leads us to the following definition. 444Fixpoint operators for the approaches in  and  are obtained by appropriate modifications to Condition I and II in Definition 1; cf. .
Let be an ordered logic program and let and be sets of literals.
We define the set of immediate consequences of with respect to and as
if is consistent, and otherwise.
Note that is a refinement of its classical counterpart . To see this, observe that Condition I embodies the standard application condition for rules given in (2)
The actual refinement takes place in Condition II. The idea is to apply a rule only if the “question of applicability” has been settled for all higher-ranked rules . Let us illustrate this in terms of iterated applications of . In these cases, contains the set of conclusions that have been derived so far, while provides the putative answer set (or: provides a set of literals that can be falsified). Then, the “question of applicability” is considered to be settled for a higher ranked rule
if the prerequisites of will never be derivable, viz. , or
if is defeated by what has been derived so far, viz. , or
if or another rule with the same head have already applied, viz. .
The first two conditions show why activeness of is stipulated wrt , as opposed to in Condition I. The last condition serves somehow two purposes: First, it detects whether the higher ranked rule has applied and, second, it suspends the preference whenever the head of the higher ranked has already been derived by another rule. This suspension of preference constitutes a distinguishing feature of the approach at hand; this is discussed in detail in  in connection with other approaches to preference handling.
As with and , iterated applications of are written as for , where and for . This allows us to define the counterpart of fixpoint operator for ordered programs:
Let be an ordered logic program and let be a set of literals.
We define .
In analogy to and , operator is a refinement of its classical counterpart . The major difference of our definition from van Gelder’s is that we directly obtain the consequences from (and ). Unlike this, the usual approach (without preferences) first obtains a basic program from and then the consequences are derived from this basic program .
A preferred answer set is defined as a fixpoint of .
In analogy to van Gelder , we may define the alternating transformation for an ordered logic program as . A fixpoint of is called an alternating fixpoint of . Given that is anti-monotonic , we get that is monotonic. According to results tracing back to Tarski , this implies that possesses a least and a greatest fixpoint, denoted by and , respectively.
Different semantics of ordered logic programs are obtained by distinguishing different subsets of the respective set of alternating fixpoints. In fact, the preferred answer set semantics constitute instances of the overall framework. To see this, observe that each fixpoint of is also a fixpoint of .
4 Preferring least alternating fixpoints?
Let us now investigate the least alternating fixpoint of and with it the comportment of the previous fixpoint operator in the setting of well-founded semantics. As opposed to answer sets semantics, this semantics relies on 3-valued models (or, partial models). Such a model consists of three parts: the set of true literals, the set of false literals, and the set of unknown literals. Given that the union of these three sets is , it is sufficient to specify two of the three sets for determining a 3-valued interpretation. Accordingly, a 3-valued interpretation is a pair where and are sets of literals with . That is, means that is true in , while means that is false in . Otherwise, is considered to be unknown in .
Well-founded semantics constitutes another major semantics for logic programs. In contrast to answers sets semantics, it aims at characterizing skeptical conclusions comprised in a single so-called well-founded model of the underlying program. This model can be characterized within the alternating fixpoint theory in terms of the least fixpoint of operator . That is, the well-founded model of a program is given by the 3-valued interpretation . Hence, it is sufficient to consider the least alternating fixpoint of a program, since it determines its well-founded model. We therefore refer to the least alternating fixpoint of as the well-founded set of . The set is usually referred to as the unfounded set of .
After extending these concepts to preference handling, that is, substituting the classical operators and by and , respectively, one can show that (i) each ordered logic program has a unique preferred well-founded model; (ii) the preferred well-founded set is contained in any preferred answer set (while the unfounded one is not); and (iii) whenever we obtain a two-valued well-founded model, its underlying well-founded set is the unique answer set of the program.555No matter whether we consider the fixpoint operators for the approach in , , or , respectively.
One often criticized deficiency of the standard well-founded model is that it is too skeptical. Unfortunately, this is not remedied by alternating the fixpoint operators of the previous sections, no matter which strategy we consider. To see this, consider the ordered logic program :
The well-founded model of is given by . The same model is obtained by alternating operator . Observe that and . Consequently, is the least alternating fixpoint of .
The question is now why these operators are still too skeptical in defining well-founded semantics (although they work nicely in the setting of answer sets and regular semantics). In fact, the great advantage of a setting like that of answer sets semantics is that we deal with direct fixpoint equations, like , where the context represents the putative answer set. This is different in the setting of well-founded semantics, where we usually start by applying an operator to a rather small context, eg. initially the empty set; this usually results in a larger set, sometimes even , that constitutes then the context of the second application of the operator. Now, looking at the underlying definitions, we see that the actual preference handling condition, eg. Condition II in Definition 1 takes advantage of for deciding applicability. The alternating character in the well-founded setting does not support this sort of analysis since it cannot provide the (putative) final result of the computation.
5 Towards a preferred well-founded semantics
In view of the failure of the above fixpoint operator(s) in the setting of well-founded semantics, the obvious question is now whether an appropriate alternating fixpoint operation is definable that yields a reasonable well-founded semantics for ordered logic programs. As informal guidelines, we would like that the resulting semantics (i) allows for deriving more conclusions than the standard well-founded semantics by appeal to given preferences; (ii) coincides with standard well-founded semantics in the absence of preferences; and finally (iii) approximates the previous preferred answer sets semantics.
The standard well-founded model is defined by means of the least fixpoint of the operator . As above, we aim at integrating preferences by elaborating upon the underlying immediate consequence operator X given in (2). As well, the basic idea is to modify this operator so that more conclusions can be derived by employing preferences. However, as discussed at the end of the previous section, the alternating iterations of face two complementary situations: those with smaller contexts and those with larger ones. Since preferences exploit these contexts, it seems reasonable to distinguish alternating applications or, at least, to concentrate on one such situation while dealing with the other one in the standard way.666Such an approach is also pursued in . For strengthening , we thus have two options: either we make the outer operator derive more literals or we make the inner operator derive less literals.
In what follows, we adopt the former option and elaborate upon the outer operator. The general idea is then to reduce the context considered in the second application of by appeal to preferences in order to make more rules applicable. For this purpose we remove those literals that are derived by means of less preferred, defeated rules.
Let be an ordered logic program and let and be sets of literals.
We define the set of immediate consequences of with respect to and as
if is consistent, and otherwise.
We say that defeats wrt if . The set of removed literals consists thus of those rule heads, all of whose corresponding rules are less preferred than and defeated by or , viz. the literals derived so far. In fact, this condition only removes a literal such as from , if all of its applicable generating rules like are defeated by the preferred rule . Note that is normally different for different rules .
For illustration consider the rules in . For and , we get and . In such a situation, activeness of is checked wrt while that of is checked wrt . When applying , the removal of from context allows us to discard the conclusion of the less preferred rule that is defeated by the preferred rule . This example is continued below.
Notably, the choice of is one among many options. Unfortunately, it leads beyond the scope of this paper to investigate the overall resulting spectrum, so that we concentrate on the above definition and discuss some alternatives at the end of this section. From a general perspective, the above definition offers thus a parameterizable framework for defining well-founded semantics including preferences.
In analogy to the previous sections, we can define a consequence operator as follows.
Let be an ordered logic program and let be a set of literals.
We define .
Of particular interest in view of an alternating fixpoint theory is that enjoys anti-monotonicity:
Let be an ordered logic program and sets of literals.
If , then .
Given this, we may define a new alternating transformation of as
Since both and are anti-monotonic, is monotonic.
Let be an ordered logic program and let be a set of literals.
We define as a preferred well-founded set of iff .
By Tarski’s Theorem , we get that each ordered logic program has a unique preferred well-founded set.
Let be an ordered logic program.
Then, there is a unique preferred well-founded set of .
Given the notion of the preferred well-founded set, we define the preferred well-founded model of an ordered program as follows.
Let be an ordered logic program and let be the well-founded set of .
We define the preferred well-founded model of as .
It is well-known that the standard well-founded semantics for extended logic programs has time complexity [29, 4]. The complexity of the preferred well-founded semantics is still in polynomial time but it is in . The reason is that we have to additionally compute for each .
We first obtain the following corollary to Theorem 2.
Every ordered logic program has a unique preferred well-founded model.
This result shows that our preferred well-founded semantics is as robust as the standard well-founded semantics.
The relationship between the standard well-founded model and the preferred well-founded model can be stated as follows.
Let be the preferred well-founded model of and let be the well-founded model of .
Then, we have
, if .
Let us reconsider . While is the well-founded model of , its ordered counterpart has the preferred well-founded model . To see this, observe that and . Clearly, is a fixpoint of and . Thus, is an alternating fixpoint of . Also, we see that is not an alternating fixpoint. This implies that is the least alternating fixpoint of .
This example along with the last result show that preferences allow us to strengthen the conclusions obtained by the standard well-founded semantics. That is, whenever certain conclusions are not sanctioned in the standard framework one may add appropriate preferences in order to obtain these conclusions within the overall framework of well-founded semantics.
For a complement, consider the following variation of , also discussed in .
Observe that has well-founded model . In contrast to , the preferred well-founded model of is also . As discussed in  this makes sense since preferences should only enrich but not “override” an underlying well-founded model.
Another attractive property of this instance of preferred well-founded semantics is that it provides an approximation of preferred answer sets semantics.
Let be the preferred well-founded model of and let be a preferred answer set of .
Then, we have and .
Finally, let us briefly discuss some alternative choices for . In fact, whenever we express the same preferences among (negative) rules having the same head the previous definition of is equivalent to . However, this conceptually simpler definition is inadequate when it comes to attributing different preferences to rules with the same heads as in the following example.
Consider the ordered program .
The preferred well-founded semantics of gives , while the conceptually simpler one yields , a clearly wrong result! In the simplistic setting would contain the head of the third rule, discarding the fact that already defeats .
Another alternative choice for is indicated by the difference between the strategies employed in  and . In fact, the latter implicitly distinguishes between same literals stemming from different rules. This amounts to distinguishing different occurrences of literals. For this, we may rely on the aforementioned simplistic definition of and suppose that provides us with occurrences of literals, like instead of . Without entering details, let us illustrate this idea by appeal to . An approach distinguishing occurrences of literals would yield and . When considering , we check activeness wrt , viz. . Unlike just above, remains in the reduced context and is inapplicable. An elaboration of this avenue is beyond the scope of this paper, in particular, since it involves an occurrence-based development of well-founded semantics.
In contrast to answer set semantics, the extension of well-founded semantics to ordered logic program has been rarely studied before. In this section we will discuss the relation of our approach to [6, 21, 30].
6.1 Relation to Brewka’s Approach
Brewka defines in  a well-founded semantics for ordered logic programs. Notably, this approach is based on a paraconsistent extension of well-founded semantics that tolerates inconsistencies among the result of the inner operator without trivializing the overall result. Despite this deviation from standard well-founded semantics, the question remains whether Brewka’s semantics can be captured within our semantical framework.
In fact, both approaches are based on quite different intuitions. While the underlying idea of Brewka’s approach is to define a criterion for selecting the intended rules by employing preference, we integrate preferences into the immediate consequence operaor by individually restricting the context of application for each rule.
Nonetheless, it turns out that Brewka’s semantics can be captured through an alternating fixpoint construction. As we show below, Brewka’s modification boils down to using an alternate fixpoint operator of the form “”. To this end, let us first consider the difference among the underlying operators and . Define as the smallest set of literals which is closed under a basic program . Then, given a set of literals, is defined as . Dropping the requirement of logical closure results in a paraconsistent inference operation. For example, given , we get , while . Although the corresponding adaptions are more involved, the surprising result is now that Brewka’s semantics can also be captured within our overall framework, if we use the closure operator instead of .
Moreover, we need the following. Let be an ordered logic program and be a set of literals. We define as the set of rules defeated by wrt and as
Notice that is a set of rules while is a set of literals. is also different from Brewka’s Dom (set of dominated rules) in that is defined wrt a set of literals rather than a set of rules.
Write . Let be the operator obtained from (in Definition 3) by replacing with . This results in a fixpoint operator .
As we show in the full version of this paper, Brewka’s well-founded set corresponds to the least fixpoint of the alternating operator . This means Brewka’s well-founded semantics also enjoys an alternating fixpoint characterization.
6.2 Relation to Other Approaches
In , it is mentioned that a well-founded semantics with preference can be defined in terms of their operator but default negation is not allowed in their syntax. However, even for ordered logic programs without default negation, our basic semantic approach is different from the well-founded semantics in priority logic . The main reason is that they interpret the priority relation in a quite different way: is blocked whenever is applicable. While we attribute to the program
a preferred well-founded model, containing both and , the well-founded model of in priority logic is . That is, cannot be inferred.
Another skeptical semantics for preference is defeasible logic, which was originally introduced by D. Nute  and received extensive studies in recent years [1, 2, 20]. Defeasible logic distinguishes the strict rules from defeasible rules. This already makes its semantics different from our preferred well-founded semantics.
Consider an example from . The following is a theory in defeasible logic:
In defeasible logic, is not derivable, i. e., cannot be defeasibly derived. As pointed out by Brewka, this means a defeasible rule having higher priority can defeat a strict rule.
The above theory can be directly translated into an ordered logic program as follows:
It can be verified that the preferred well-founded model (in our sense) is . Therefore, is derivable under our preferred well-founded semantics.
We have looked into the issue of how van Gelder’s alternating fixpoint theory  for normal logic programs can be suitably extended to define the well-founded semantics for ordered logic programs (extended logic programs with preference). The key of the alternating fixpoint approach is how to specify a suitable consequence relation for ordered logic programs. We argue that the preference strategies for defining answer sets are not suitable for defining preferred well-founded semantics and then some informal criteria for preferred well-founded semantics are proposed. Based on this analysis, we have defined a well-founded semantics for ordered logic programs. This semantics allows an elegant definition and satisfies some attractive properties: (1) Each ordered logic program has a unique preferred well-founded model; (2) The preferred well-founded reasoning is no less skeptical than the standard well-founded reasoning; (3) Any conclusion under the preferred well-founded semantics is also derivable under some major preferred answer sets semantics. Our semantics is different from defeasible logic and the skeptical priority logic. An important result is the equivalence of Brewka’s preferred well-founded semantics and our semantics introduced in Section 5.
This work was supported by DFG under grant FOR 375/1-1, TP C.
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