Semantics for Possibilistic Disjunctive Programs

06/03/2011 ∙ by Juan Carlos Nieves, et al. ∙ Universitat Politècnica de Catalunya 0

In this paper, a possibilistic disjunctive logic programming approach for modeling uncertain, incomplete and inconsistent information is defined. This approach introduces the use of possibilistic disjunctive clauses which are able to capture incomplete information and incomplete states of a knowledge base at the same time. By considering a possibilistic logic program as a possibilistic logic theory, a construction of a possibilistic logic programming semantic based on answer sets and the proof theory of possibilistic logic is defined. It shows that this possibilistic semantics for disjunctive logic programs can be characterized by a fixed-point operator. It is also shown that the suggested possibilistic semantics can be computed by a resolution algorithm and the consideration of optimal refutations from a possibilistic logic theory. In order to manage inconsistent possibilistic logic programs, a preference criterion between inconsistent possibilistic models is defined; in addition, the approach of cuts for restoring consistency of an inconsistent possibilistic knowledge base is adopted. The approach is illustrated in a medical scenario.

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1 Introduction

Answer Set Programming (ASP) is one of the most successful logic programming approaches in Non-monotonic Reasoning and Artificial Intelligence applications

[Baral (2003), Gelfond (2008)]. In [Nicolas et al. (2006)], a possibilistic framework for reasoning under uncertainty was proposed. This framework is a combination between ASP and possibilistic logic [Dubois et al. (1994)].

Possibilistic Logic is based on possibilistic theory in which, at the mathematical level, degrees of possibility and necessity are closely related to fuzzy sets [Dubois et al. (1994)]. Due to the natural properties of possibilistic logic and ASP, Nicolas et al.’s approach allows us to deal with reasoning that is at the same time non-monotonic and uncertain. Nicolas et al.’s approach is based on the concept of possibilistic stable model which defines a semantics for possibilistic normal logic programs.

An important property of possibilistic logic is that it is axiomatizable in the necessity-valued case [Dubois et al. (1994)]. This means that there is a formal system (a set of axioms and inferences rules) such that from any set of possibilistic fomulæ  and for any possibilistic formula , is a logical consequence of if and only if is derivable from in this formal system. A result of this property is that the inference in possibilistic logic can be managed by both a syntactic approach (axioms and inference rules) and a possibilistic model theory approach (interpretations and possibilistic distributions).

Equally important to consider is that the answer set semantics inference can also be characterized as a logic inference in terms of the proof theory of intuitionistic logic and intermediate logics [Pearce (1999), Osorio et al. (2004)]. This property suggests that one can explore extensions of the answer set semantics by considering the inference of different logics.

Since in [Dubois et al. (1994)] an axiomatization of possibilistic logic has been defined, in this paper we explore the characterization of a possibilistic semantics for capturing possibilistic logic programs in terms of the proof theory of possibilistic logic and the standard answer set semantics. A nice feature of this characterization is that it is applicable to disjunctive as well as normal possibilistic logic programs, and, with minor modification, to possibilistic logic programs containing a strong negation operator.

The use of possibilistic disjunctive logic programs allow us to capture incomplete information and incomplete states of a knowledge base at the same time. In order to illustrate the use of possibilistic disjunctive logic programs, let us consider a scenario in which uncertain and incomplete information is always present. This scenario can be observed in the process of human organ transplanting. There are several factors that make this process sophisticated and complex. For instance:

  • the transplant acceptance criteria vary ostensibly among transplant teams from the same geographical area and substantially between more distant transplant teams [López-Navidad et al. (1997)]. This means that the acceptance criteria applied in one hospital could be invalid or at least questionable in another hospital.

  • there are lots of factors that make the diagnosis of an organ donor’s disease in the organ recipient unpredictable. For instance, if an organ donor has hepatitis, then an organ recipient could be infected by an organ of . According to [López-Navidad and Caballero (2003)], there are cases in which the infection can occur; however, the recipient can spontaneously clear the infection, for example hepatitis. This means that an organ donor’s infection can be present or non-present in the organ recipient. Of course there are infections which can be prevented by treating the organ recipient post-transplant.

  • the clinical state of an organ recipient can be affected by several factors, for example malfunctions of the graft. This means that the clinical state of an organ recipient can be stable or unstable after the graft because the graft can have good graft functions, delayed graft functions and terminal insufficient functions111Usually, when a doctor says that an organ has terminally insufficient functions, it means that there are no clinical treatments for improving the organ’s functions..

It is important to point out that the transplant acceptance criteria rely on the kind of organ (kidney, heart, liver, etc.) considered for transplant and the clinical situation of the potential organ recipients.

Let us consider the particular case of a kind of kidney transplant with organ donors who have a kind of infection, for example: endocarditis, hepatitis. As already stated, the clinical situation of the potential organ recipients is relevant in the organ transplant process. Hence the clinical situation of an organ recipient is denoted by the predicate , such that can be stable, unstable, 0-urgency and

denotes a moment in time. Another important factor, that is considered, is the state of the organ’s functions. This factor is denoted by the predicate

such that can be terminal-insufficient functions, good-graft functions, delayed-graft functions, normal-graft functions and denotes a moment in time. Also, the state of an infection in both the organ recipient and the organ donor are considered, these states are denoted by the predicates and respectively so that denotes a moment in time. The last predicate that is presented is such that can be transplant, wait, post-transplant treatment and denotes a moment in time. This predicate denotes the possible actions of a doctor. In Figure 1222This finite state automata was developed under the supervision of Francisco Caballero M. D. Ph. D. from the Hospital de la Santa Creu I Sant Pau, Barcelona, Spain., a finite state automata is presented. In this automata, each node represents a possible situation where an organ recipient can be found and the arrows represent the doctor’s possible actions. Observe that we are assuming that in the initial state the organ recipient is clinically stable and he does not have an infection; however, he has a kidney whose functions are terminally insufficient. From the initial state, the doctor’s actions would be either to perform a kidney transplantat or just wait333In the automata of Figure 1, we are not considering the possibility that there is a waiting list for organs. This waiting list has different policies for assigning an organ to an organ recipient..

Figure 1: An automata of states and actions for considering infections in kidney organ transplant.

According to Figure 1, an organ recipient could be found in different situations after a graft. The organ recipient may require another graft and the state of the infection could be unpredictable. This situation makes the automata of Figure 1 nondeterministic. Let us consider a couple of extended disjunctive clauses which describe some situations presented in Figure 1.


As syntactic clarification, we want to point out that is regarded as a strong negation which is not exactly the negation in classical logic. In fact, any atom negated by strong negation will be replaced by a new atom as it is done in ASP. This means that cannot be regarded as a logic tautology.

Continuing with our medical scenario, we can see that the intended meaning of the first clause is that if the organ donor has an infection, then the infection can be present or non-present in the organ recipient after the graft, and the intended meaning of the second one is that the graft’s functions can be: good, delayed and terminal after the graft. Observe that these clauses are not capturing the uncertainty that is involved in each statement. For instance, w.r.t. the first clause, one can wish to attach an degree of uncertainty in order to capture the uncertainty that is involved in this statement — keeping in mind that the organ recipient can be infected by the infection of the donor’s organ; however, the infection can be spontaneously cleared by the organ recipient as it is the case of hepatitis [López-Navidad and Caballero (2003)].

In logic programming literature, one can find different approaches for representing uncertain information [Kifer and Subrahmanian (1992), Ng and Subrahmanian (1992), Lukasiewicz (1998), Kern-Isberner and Lukasiewicz (2004), van Emden (1986), Rodríguez-Artalejo and Romero-Díaz (2008), Van-Nieuwenborgh et al. (2007), Fitting (1991), Lakshmanan (1994), Baldwin (1987), Dubois et al. (1991), Alsinet and Godo (2002), Alsinet and Godo (2000), Alsinet et al. (2008), Nicolas et al. (2006)]. Basically, these approaches differ in the underlying notion of uncertainty and how uncertainty values, associated with clauses and facts, are managed. Usually the selection of an approach for representing uncertain information relies on the kind of information which has to be represented. In psychology literature, one can find significant observations related to the presentation of uncertain information. For instance, Tversky and Kahneman have observed in [Tversky and Kahneman (1982)] that people commonly use statements such as “I think that ”, “chances are ”, “

it is probable that

”, “it is plausible that ”, etc., for supporting their decisions. In fact, many times, experts in a domain, such as medicine, appeal to their intuition by using these kinds of statements [Fox and Das (2000), Fox and Modgil (2006)]. One can observe that these statements have adjectives which quantify the information as a common denominator. These adjectives are for example: probable, plausible, etc. This suggests that the consideration of labels for the syntactic representation of uncertain values could help represent uncertain information pervaded by ambiguity.

Since possibilistic logic defines a proof theory in which the strength of a conclusion is the strength of the weakest argument in its proof, the consideration of an ordered set of labels for capturing incomplete states of a knowledge base is feasible. The only formal requirement is that this set of adjectives/labels must be a finite set. For instance, for the given medical scenario, a transplant coordinator444A transplant coordinator is an expert in all of the processes of transplants [López-Navidad et al. (1997)]. can suggest a set of labels in order to quantify a medical knowledge base and, of course, to define an order between those labels. By considering those labels, we can have possibilistic clauses as:

probable:

Informally speaking, the reading of this clause is: it is probable that if the organ donor has an infection, then the organ recipient can be infected or not after a graft.

As we can see, possibilistic programs with negation as failure

represent a rich class of logic programs which are especially adapted to automated reasoning when the available information is pervaded by ambiguity.

In this paper, we extend the work of two earlier papers [Nieves et al. (2007a), Nieves et al. (2007b)] in order to obtain a simple logic characterization of a possibilistic logic programming semantics for capturing possibilistic programs; this semantics is applicable to disjunctive as well as normal logic programs. As we have already mentioned, the construction of the possibilistic semantics is based on the proof theory of possibilistic logic. Following this approach:

  • We define the inference . This inference takes as references the standard definition of the answer set semantics and the inference which corresponds to the inference of possibilistic logic.

  • The possibilistic semantics is defined in terms of a syntactic reduction, and the concept of i-greatest set.

  • Since the inference of possibilistic logic is computable by a generalization of the classical resolution rule, it is shown that the defined possibilistic semantics is computable by inferring optimal refutations.

  • By considering the principle of partial evaluation, it is shown that the given possibilistic semantics can be characterized by a possibilistic partial evaluation operator.

  • Finally, since the possibilistic logic uses -cuts to manage inconsistent possibilistic knowledge bases, an approach of cuts for restoring consistency of an inconsistent possibilistic knowledge base is adopted.

The rest of the paper is divided as follows: In §2 we give all the background and necessary notation. In §3, the syntax of our possibilistic framework is presented. In §4, the semantics for capturing the possibilistic logic programs is defined. Also it is shown that this semantics is computable by considering a possibilistic resolution rule and partial evaluation. In §5, some criteria for managing inconsistent possibilistic logic programs are defined. In §6, we present a small discussion w.r.t. related approaches to our work. Finally, in the last section, we present our conclusions and future work.

2 Background

In this section we introduce the necessary terminology and relevant definitions in order to have a self-contained document. We assume that the reader is familiar with basic concepts of classic logic, logic programming and lattices.

2.1 Lattices and order

We start by defining some fundamental definitions of lattice theory (see [Davey and Priestly (2002)] for more details).

Definition 1

Let be a set. An order (or partial order) on is a binary relation on such that, for all ,

(i)

(ii)

and imply

(iii)

and imply

These conditions are referred to, respectively, as reflexivity, antisymmetry and transitivity.

A set equipped with an order relation is said to be an ordered set (or partial ordered set). It will be denoted by (,).

Definition 2

Let (,) be an ordered set and let . An element is an upper bound of if for all . A lower bound is defined dually. The set of all upper bounds of is denoted by (read as ‘ upper’) and the set of all lower bounds by (read as ‘ lower’).

If has a minimum element , then is called the least upper bound () of . Equivalently, is the least upper bound of if

(i)

is an upper bound of , and

(ii)

for all upper bound of .

The least upper bound of exists if and only if there exists such that

and this characterizes the of . Dually, if has a greatest element, , then is called the greatest lower bound () of . Since the least element and the greatest element are unique, and are unique when they exist.

The least upper bound of is called the supremum of and it is denoted by ; the greatest lower bound of S is called the infimum of S and it is denoted by

Definition 3

Let (,) be a non-empty ordered set.

(i)

If and exist for all , then is called lattice.

(ii)

If and exist for all , then is called a complete lattice.

Example 1

Let us consider the set of labels , 555This set of labels was taken from [Fox and Modgil (2006)]. In that paper, the authors argue that we can construct a set of labels (they call those: modalities) in a way that this set provides a simple scale for ordering the claims of our beliefs. We will use this kind of labels for quantifying the degree of uncertainty of a statement. and let be a partial order such that the following set of relations holds: , , , , , . A graphic representation of according to is showed in Figure 2. It is not difficult to see that is a lattice and further it is a complete lattice.

Figure 2: A graphic representation of a lattice where the following relations holds: , , , , .

2.2 Logic programs: Syntax

The language of a propositional logic has an alphabet consisting of

(i)

proposition symbols:

(ii)

connectives :

(iii)

auxiliary symbols : ( , )

in which are binary-place connectives, , are unary-place connective and is zero-ary connective. The proposition symbols and stand for the indecomposable propositions, which we call atoms, or atomic propositions. Atoms negated by will be called extended atoms.

Remark 1

We will use the concept of atom without paying attention to whether it is an extended atom or not.

The negation sign is regarded as the so called strong negation by the ASP’s literature and the negation as the negation as failure. A literal is an atom, , or the negation of an atom . Given a set of atoms , we write to denote the set of literals An extended disjunctive clause, C, is denoted:

in which , , , each is an atom666Notice that these atoms can be extended atoms.. When and the clause is an abbreviation of ; clauses of these forms are some times written just as . When the clause is an abbreviation of:

Clauses of this form are called constraints (the rest, non-constraint clauses). An extended disjunctive program is a finite set of extended disjunctive clauses. By , we denote the set of atoms in the language of .

Sometimes we denote an extended disjunctive clause C by , contains all the head literals, contains all the positive body literals and contains all the negative body literals. When , the clause is called positive disjunctive clause. A set of positive disjunctive clauses is called a positive disjunctive logic program. When is a singleton set, the clause can be regarded as a normal clause. A normal logic program is a finite set of normal clauses. Finally, when is a singleton set and , the clause can also be regarded as a definite clause. A finite set of definite clauses is called a definite logic program.

We will manage the strong negation (), in our logic programs, as it is done in ASP [Baral (2003)]. Basically, each extended atom is replaced by a new atom symbol which does not appear in the language of the program. For instance, let be the normal program:

. .
. .

Then replacing each extended atom by a new atom symbol, we will have:

. .
. .

In order not to allow models with complementary atoms, that is and , a constraint of the form is usually added to the logic program. In our approach, this constraint can be omitted in order to allow models with complementary atoms. In fact, the user could add/omit this constraint without losing generality.

Formulæ are constructed as usual in classic logic by the connectives: . A theory is a finite set of formulæ. By , we denote the set of atoms that occur in T. When we treat a logic program as a theory,

  • each negative literal is replaced by such that is regarded as the negation in classic logic.

  • each constraint is rewritten according to the formula .

Given a set of proposition symbols and a theory in a logic . If if and only if .

2.3 Interpretations and models

In this section, we define some relevant concepts w.r.t. semantics. The first basic concept that we introduce is interpretation.

Definition 4

Let be a theory, an interpretation is a mapping from to meeting the conditions:

  1. ,

  2. ,

  3. if and only if and ,

  4. ,

  5. .

It is standard to provide interpretations only in terms of a mapping from to . Moreover, it is easy to prove that this mapping is unique by virtue of the definition by recursion [van Dalen (1994)]. Also, it is standard to use sets of atoms to represent interpretations. The set corresponds exactly to those atoms that evaluate to 1.

An interpretation is called a (2-valued) model of the logic program if and only if for each clause , . A theory is consistent if it admits a model, otherwise it is called inconsistent. Given a theory and a formula , we say that is a logical consequence of , denoted by , if every model of holds that . It is a well known result that if and only if is inconsistent [van Dalen (1994)].

We say that a model of a theory is a minimal model if a model of different from such that does not exist. Maximal models are defined in the analogous form.

2.4 Logic programming semantics

In this section, the answer set semantics is presented. This semantics represents a two-valued semantics approach.

2.4.1 Answer set semantics

By using ASP, it is possible to describe a computational problem as a logic program whose answer sets correspond to the solutions of the given problem. It represents one of the most successful approaches of non-monotonic reasoning of the last two decades [Baral (2003)]. The number of applications of this approach have increased due to the efficient implementations of the answer set solvers that exist.

The answer set semantics was first defined in terms of the so called Gelfond-Lifschitz reduction [Gelfond and Lifschitz (1988)] and it is usually studied in the context of syntax dependent transformations on programs. The following definition of an answer set for extended disjunctive logic programs generalizes the definition presented in [Gelfond and Lifschitz (1988)] and it was presented in [Gelfond and Lifschitz (1991)]: Let P be any extended disjunctive logic program. For any set , let be the positive program obtained from P by deleting

(i)

each rule that has a formula in its body with , and then

(ii)

all formulæ of the form in the bodies of the remaining rules.

Clearly does not contain (this means that is either a positive disjunctive logic program or a definite logic program), hence S is called an answer set of P if and only if S is a minimal model of . In order to illustrate this definition, let us consider the following example:

Example 2

Let us consider the set of atoms and the following normal logic program :

. .
. .

We can see that is:

. .

Notice that this program has three models: , and . Since the minimal model among these models is , we can say that is an answer set of .

In the answer set definition, we will normally omit the restriction that if has a pair of complementary literals then . This means that we allow for the possibility that an answer set could have a pair of complementary atoms. For instance, let us consider the program :

. . .

then, the only answer set of this program is : . In Section 5, the inconsistency in possibilistic programs is discussed.

It is worth mentioning that in literature there are several forms for handling an inconsistency program [Baral (2003)]. For instance, by applying the original definition [Gelfond and Lifschitz (1991)] the only answer set of is: . On the other hand, the DLV system [DLV (1996)] returns no models if the program is inconsistent.

2.5 Possibilistic Logic

Since in our approach is based on the proof theory of possibilistic logic, in this section, we present an axiomation of possibilistic logic for the case of necessity-valued formulæ.

Possibilistic logic is a weighted logic introduced and developed in the mid-1980s, in the setting of artificial intelligence, with the goal of developing a simple yet rigorous approach to automated reasoning from uncertain or prioritized incomplete information. Possibilistic logic is especially adapted to automated reasoning when the available information is pervaded by ambiguities. In fact, possibilistic logic is a natural extension of classical logic in which the notion of total order/partial order is embedded in the logic.

Possibilistic Logic is based on possibility theory. Possibilistic theory, as its name implies, deals with the possible rather than probable values of a variable with possibility being a matter of degree. One merit of possibilistic theory is at one and the same time to represent imprecision (in the form of fuzzy sets) and quantity uncertainty (through the pair of numbers that measure possibility and necessity).

Our study in possibilistic logic is devoted to a fragment of possibilistic logic, in which knowledge bases are only necessity-quantified statements. A necessity-valued formula is a pair in which is a classical logic formula and is a positive number. The pair expresses that the formula is certain at least to the level , that is , in which is a necessity measure modeling our possibly incomplete state knowledge [Dubois et al. (1994)].

is not a probability (like it is in probability theory), but it induces a certainty (or confidence) scale. This value is determined by the expert providing the knowledge base. A necessity-valued knowledge base is then defined as a finite set (that is to say a conjunction) of necessity-valued formulæ.

The following properties hold w.r.t. necessity-valued formulæ:

(1)
(2)
(3)

Dubois et al., in [Dubois et al. (1994)] introduced a formal system for necessity-valued logic which is based on the following axioms schemata (propositional case):

(A1)

(A2)

(A3)

Inference rules:

(GMP)

(S)

if

According to Dubois et al., in [Dubois et al. (1994)], basically we need a complete lattice to express the levels of uncertainty in Possibilistic Logic. Dubois et al. extended the axioms schemata and the inference rules for considering partially ordered sets. We shall denote by the inference under Possibilistic Logic without paying attention to whether the necessity-valued formulæ are using a totally ordered set or a partially ordered set for expressing the levels of uncertainty.

The problem of inferring automatically the necessity-value of a classical formula from a possibilistic base was solved by an extended version of resolution for possibilistic logic (see [Dubois et al. (1994)] for details).

One of the main principles of possibilistic logic is that:

Remark 2

The strength of a conclusion is the strength of the weakest argument used in its proof.

According to Dubois and Prade [Dubois and Prade (2004)], the contribution of possibilistic logic setting is to relate this principle (measuring the validity of an inference chain by its weakest link) to fuzzy set-based necessity measures in the framework of Zadeh’s possibilistic theory, since the following pattern then holds:

This interpretive setting provides a semantic justification to the claim that the weight attached to a conclusion should be the weakest among the weights attached to the formulæ involved in the derivation.

3 Syntax

In this section, the general syntax for possibilistic disjunctive logic programs will be presented. This syntax is based on the standard syntax of extended disjunctive logic programs (see Section 2.2).

We start by defining some concepts for managing the possibilistic values of a possibilistic knowledge base777Some concepts presented in this section extend some terms presented in [Nicolas et al. (2006)].. We want to point out that in the whole document only finite lattices are considered. This assumption was made based on the recognition that in real applications we will rarely have an infinite set of labels for expressing the incomplete state of a knowledge base.

A possibilistic atom is a pair , in which is a finite set of atoms and is a lattice. The projection to a possibilistic atom is defined as follows: . Also given a set of possibilistic atoms , over is defined as follows: .

Let be a lattice. A possibilistic disjunctive clause is of the form:

in which and is an extended disjunctive clause as defined in Section 2.2. The projection for a possibilistic clause is . On the other hand, the projection for a possibilistic clause is . This projection denotes the degree of necessity captured by the certainty level of the information described by . A possibilistic constraint is of the form:

in which is the top of the lattice and is a constraint as defined in Section 2.2. The projection for a possibilistic constraint is: . Observe that the possibilistic constraints have the top of the lattice as an uncertain value, this assumption is due to the fact that similar a constraint in standard ASP, the purpose of a possibilistic constraint is to eliminate possibilistic models. Hence, it can be assumed that there is no doubt about the veracity of the information captured by a possibilistic constraint. However, as in standard ASP, one can define possibilistic constraints of the form: such that is an atom which is not used in any other possibilistic clause and . This means that the user can define possibilistic constraints with different levels of certainty.

A possibilistic disjunctive logic program is a tuple of the form , in which is a finite set of possibilistic disjunctive clauses and possibilistic constraints. The generalization of over is as follows: . Notice that is an extended disjunctive program. When is a normal program, is called a possibilistic normal program. Also, when is a positive disjunctive program, is called a possibilistic positive logic program and so on. A given set of possibilistic disjunctive clauses is also represented as to avoid ambiguities with the use of the comma in the body of the clauses.

Given a possibilistic disjunctive logic program , we define the -cut and the strict -cut of , denoted respectively by and , by

such that

such that

Example 3

In order to illustrate a possibilistic program, let us go back to our scenario described in Section 1. Let be the lattice of Figure 2 such that the relation means that is less possible than . The possibilistic program will be the following set of possibilistic clauses:

It is probable that if the organ donor has an infection, then the organ recipient can be infected or not after a graft:

probable:

It is confirmed that the organ’s functions can be: good, delayed and terminal after a graft.

confirmed:

It is confirmed that if the organ’s functions are terminally insufficient then a transplanting is necessary.

confirmed: .

It is plausible that the clinical situation of the organ recipient can be stable if the functions of the graft are good.

plausible: .

It is plausible that the clinical situation of the organ recipient can be unstable if the functions of the graft are delayed.

plausible: .

It is plausible that the clinical situation of the organ recipient can be of 0-urgency if the functions of the graft are terminally insufficient after the graft.

plausible: 0-urgency
.

It is certain that the doctor cannot do two actions at the same time.

certain: .

It is certain that a transplant cannot be done if the organ recipient is dead.

certain: .

The initial state of the automata of Figure 1 is captured by the following possibilistic clauses:

certain: .

certain: .

certain: .

certain: .

4 Semantics

In §3, the syntax for any possibilistic disjunctive program was introduced, Now, in this section, a semantics for capturing these programs is studied. This semantics will be defined in terms of the standard definition of the answer set semantics (§2.4.1) and the proof theory of possibilistic logic (§2.5).

As sets of atoms are considered as interpretations, two basic operations between sets of possibilistic atoms are defined; also a relation of order between them is defined: Given a finite set of atoms and a lattice (,), and

Observe that is the finite set of all the possibilistic atom sets induced by and . Informally speaking, is the subset of such that each set of has no atoms with different uncertain value.

Definition 5

Let be a finite set of atoms and (,) be a lattice. , we define.

.
, and
then .

This definition is almost the same as Definition 7 presented in [Nicolas et al. (2006)]. The main difference is that in Definition 7 from [Nicolas et al. (2006)] the operations and are defined in terms of the operators min and max instead of the operators and . Hence, the following proposition is a direct result of Proposition 6 of [Nicolas et al. (2006)].

Proposition 1

is a complete lattice.

Before moving on, let us define the concept of i-greatest set w.r.t. as follows: Given , is an i-greatest set in iff such that . For instance, let . One can see that has two i-greatest sets: and . The concept of i-greatest set will play a key role in the definition of possibilistic answer sets in order to infer possibilistic answer sets with optimal certainty values.

4.1 Possibilistic answer set semantics

Similar to the definition of answer set semantics, the possibilistic answer set semantics is defined in terms of a syntactic reduction. This reduction is inspired by the Gelfond-Lifschitz reduction.

Definition 6 (Reduction )

Let be a possibilistic disjunctive logic program, M be a set of atoms. reduced by is the positive possibilistic disjunctive logic program:


in which is of the form .

Notice that is not exactly equal to the Gelfond-Lifschitz reduction. For instance, let us consider the following programs:

. . .
. . .
.

The program is obtained from and by applying Definition 6 and the program is obtained from and by applying the Gelfond-Lifschitz reduction. Observe that the reduction of Definition 6 removes from the head of the possibilistic disjunctive clauses any atom which does not belong to . As we will see in Section 4.2, this property will be helpful for characterizing the possibilistic answer set in terms of a fixed-point operator. It is worth mentioning that the reduction also has a different effect from the Gelfond-Lifschitz reduction in the class of normal programs. This difference is illustrated in the following programs:

. . .
. .
. .

Example 4

Continuing with our medical scenario described in the introduction, let be a ground instance of the possibilistic program presented in Example 3:

probable:

confirmed:

confirmed: .
plausible: .
plausible: .
plausible: 0-urgency
.
certain: .
certain: .
certain: .
certain: .
certain: .
certain: .

Observe that the variables of time and were instantiated with the values and respectively; moreover, observe that the atoms and were replaced by and respectively. This change was applied in order to manage the strong negation, .

Now, let be the following possibilistic set:

.

One can see that is:

probable:
confirmed:
confirmed: .
plausible: .
plausible: .
plausible: 0-urgency
.
certain: .
certain: .
certain: .
certain: .
certain: .
certain: .

Once a possibilistic logic program has been reduced by a set of possibilistic atoms , it is possible to test whether is a possibilistic answer set of the program . For this end, we consider a syntactic approach; meaning that it is based on the proof theory of possibilistic logic. Let us remember that the possibilistic logic is axiomatizable [Dubois et al. (1994)]; hence, the inference in possibilistic logic can be managed by both a syntactic approach (axioms and inference rules) and a possibilistic model theory approach (interpretations and possibilistic distributions).

Since the certainty value of a possibilistic disjunctive clause can belong to a partially ordered set, the inference rules of possibilistic logic introduced in Section 2.5 have to be generalized in terms of bounds. The generalization of GMP and S is defined as follows:

(GMP*)