1 Introduction
Probabilistic reasoning is inevitable in almost all realworld applications. Therefore, developing welldefined frameworks for representing and reasoning in the presence of probabilistic knowledge and under probabilistic environments is vital. Thus, many frameworks have been developed for representing and reasoning in the presence of probabilistic knowledge and under probabilistic environments. Among these frameworks are probability answer set programming which are probability logic programs with probability answer set semantics
[Saad and Pontelli2006, Saad2006, Saad2007a].The importance of the probability answer set programming frameworks of [Saad and Pontelli2006, Saad2006, Saad2007a] lies in the fact that the probability answer set programming frameworks of [Saad and Pontelli2006, Saad2006, Saad2007a] have been shown applicable to a variety of fundamental probabilistic reasoning tasks. These probabilistic reasoning tasks include, but are not limited to, probabilistic planning [Saad2007b], probabilistic planning with imperfect sensing actions [Saad2009]
, reinforcement learning in MDP environments
[Saad2008a], reinforcement learning in POMDP environments [Saad2011], and Bayes reasoning [Saad2006]. Moreover, in [Saad2008b] it has been proved that stochastic satisfiability (SSAT) can be modularly encoded as probability answer set programs with probability answer set semantics, therefore, the applicability of SSAT to variety of fundamental probabilistic reasoning tasks also carry over to probability answer set programming [Saad and Pontelli2006, Saad2006, Saad2007a].In addition, the probability answer set programming frameworks of [Saad and Pontelli2006, Saad2006, Saad2007a] are strictly expressive. This is because the way how a rule fires in [Saad and Pontelli2006, Saad2006, Saad2007a] is close to the way how it fires in classical answer set programming [Gelfond and Lifschitz1988, Gelfond and Lifschitz1991], which makes any possible extension to [Saad and Pontelli2006, Saad2006, Saad2007a] to more expressive forms of probability answer set programming is more flexible and more intuitive.
Extended and normal disjunctive hybrid probability logic programs with probability answer set semantics is an expressive probability answer set programming framework [Saad2007a] that generalize and subsume extended hybrid probability logic programs [Saad2006] and normal hybrid probability logic programs [Saad and Pontelli2006] with probability answer set semantics as well as classical extended and classical normal disjunctive logic programs with classical answer set semantics [Gelfond and Lifschitz1991] in a unified logical framework to allow nonmonotonic negation, classical negation, and disjunctions under probabilistic uncertainty.
The probability answer set programming framework of [Saad2007a] allows directly and intuitively to represent and reason in the presence of both probabilistic uncertainty and qualitative uncertainty in a unified logical framework. This is necessary to provide the ability to assign probabilistic uncertainly over the possible outcomes of qualitative uncertainty, which is required in most real life applications, e.g., representing and reasoning about probability quantitative preferences. However, the probability answer set programming framework of [Saad2007a] is insufficient for representing and reasoning about probability quantitative preferences. This is because any probability answer set program encoding of a probability quantitative preferences reasoning problem provides all possible solutions to the problem that satisfy the probability quantitative preferences represented in the probability answer set program encoding of the problem, rather than ranking all the possible solutions that satisfy these probability quantitative preferences from the top preferred solution to the least preferred solution.
For example, consider the following simple instance of the wellknown Nurse Restoring Problem from Operation Research [Bard and Purnomo2005]. Consider that a nurses, , in a hospital, need to be assigned to one shift among two shifts in a given day, , such that nurse is assigned exactly one shift. If nurse is neutral regarding servicing at either shifts in that given day, then classical disjunctive logic program can be used to model this problem as a classical disjunctive logic program of the form
with and are the possible classical answer sets, according to the classical answer set semantics of classical disjunctive logic programs [Gelfond and Lifschitz1991]. Consider that nurse, , prefers to service at shift over shift in day,
, due to some circumstances, where this preference relation is specified as a probability distribution over the shifts
in the day . Consider also that the probability nurse prefers to service at shift in day is characterized by the probability value and the probability nurse prefers to service at shift in day is characterized by the probability value . In this case, classical disjunctive logic programs cannot represent the nurse’s preferences over the shifts in the day , since classical disjunctive logic programs are incapable in general of reasoning in the presence of probabilistic uncertainty. However, this variant of the nurse restoring problem can be intuitively represented as disjunctive hybrid probability logic program with probability answer set semantics of the formWe call this variant of the nurse restoring problem Nurse Restoring with probability Preferences problem.
The probability answer set program encoding of the nurse restoring with probability preferences problem instance described above has two probability answer sets namely and , according to the probability answer set semantics of probability answer set programming of [Saad2007a]. It is clear that the probability answer set represents nurse ’s top servicing preferences, which means that the probability answer set is the most preferred probability answer set according to the probability quantitative preferences represented by the probability answer set program. Furthermore, assume that nurse is neutral regarding servicing at shifts and , where this servicing preference of nurse is characterized by the probability value for both shifts. In this case, this nurse restoring with probability preferences problem instance can be represented as a probability answer set program of the form
with and are the probability answer sets, according to the probability answer set semantics of probability answer set programming of [Saad2007a]. Although nurse is neutral regarding servicing at either shifts with probability preference each, however, it can be the case that nurse has more appeal in servicing at shift over shift (qualitative preferences). This makes is the most preferred probability answer set in this case.
The existing probability answer set programs semantics [Saad and Pontelli2006, Saad2006, Saad2007a] does not have the ability to rank probability answer sets either according to probability quantitative preferences or according to qualitative preferences. Rather, probability answer set programs semantics is capable of finding probability answer sets that satisfy probability quantitative preferences represented by the probability answer set program and considers all the resulting probability answer sets as equally preferred. Although, in many applications, it is necessary to rank the probability answer sets generated by the probability answer set programs from the top (most) preferred probability answer set to the least preferred probability answer set, where the top (most) preferred probability answer set is the one that is most desirable. This requires probability answer set programs to be capable of representing both probability quantitative and qualitative preferences and to be capable of reasoning in the presence of both probability quantitative and qualitative preferences across probability answer sets.
In this paper we develop a unified logical framework that is capable of representing and reasoning about both probability quantitative and qualitative preferences. This is accomplished by defining the notion of probability answer set optimization programs. A probability answer set optimization program is a set of probability logic rules under the probability answer set semantics whose probability answer sets are ranked according to probability preferences relations specified by the user. Probability answer set optimization programs modify and generalize the classical answer set optimization programs described in [Brewka et al.2003]. We show the application of probability answer set optimization programs to a variant of the wellknown nurse restoring problem, called the nurse restoring with probability preferences problem, where a probability answer set program (disjunctive hybrid probability logic program with probability answer set semantics) [Saad2007a] is used as probability answer sets generator rules. To the best of our knowledge, this development is the first to consider a logical framework for reasoning about probability quantitative preferences, in general, and reasoning about both probability quantitative and qualitative preferences in particular.
2 Probability Answer Sets
We use probability logic rules under the probability answer set semantics to generate probability answer sets, that are ultimately ranked by probability preference rules. Therefore, in this section we review the probability answer set semantics of disjunctive hybrid probability logic sets of rules, a form of probability answer set programming, as described in [Saad2007a].
2.1 Syntax
Let denotes an arbitrary firstorder language with finitely many predicate symbols, function symbols, constants, and infinitely many variables. A standard atom is a predicate in , where is the Herbrand base of . Nonmonotonic negation or the negation as failure is denoted by . In disjunctive hybrid probability logic rules, probabilities are assigned to primitive events (atoms) and compound events (conjunctions or disjunctions of atoms) as intervals in , where denotes the set of all closed intervals in . For , the truth order on is defined as iff and .
The type of dependency among the primitive events within a compound event is described by a probabilistic strategy, which can be a conjunctive pstrategy or a disjunctive pstrategy. Conjunctive (disjunctive) pstrategies are used to combine events belonging to a conjunctive (disjunctive) formula [Saad and Pontelli2006]. The probabilistic composition function, , of a probabilistic strategy (pstrategy), , is a mapping , where the probabilistic composition function, , computes the probability interval of a conjunction (disjunction) of two events from the probability of its components. Let be a multiset of probability intervals. For convenience, we use to denote .
A probability annotation is a probability interval of the form , where are called probability annotation items. A probability annotation item is either a constant in (called probability annotation constant), a variable ranging over (called probability annotation variable), or (called probability annotation function), where is a representation of a computable function and are probability annotation items.
Let be an arbitrary set of pstrategies, where () is the set of all conjunctive (disjunctive) pstrategies in . A hybrid basic formula is an expression of the form or , where are atoms and and are pstrategies. Let be the set of all ground hybrid basic formulae formed using distinct atoms from and pstrategies from . If is a hybrid basic formula and is a probability annotation then is called a probability annotated hybrid basic formula. A disjunctive hybrid probability logic rule is an expression of the form
(1) 
where () are atoms, () are hybrid basic formulae, and () are probability annotations. A disjunctive hybrid probability logic rule says that if for each , where , the probability interval of is at least and for each , where , it is not believable that the probability interval of is at least , then there exist at least , where , such that the probability interval of is at least . Associated with every set of disjunctive hybrid probability logic rules is a mapping, , where . The mapping associates to each atom in a disjunctive pstrategy that is used to combine the probability intervals obtained from different disjunctive hybrid probability logic rules with the atom, , appearing in their heads.
A disjunctive hybrid probability logic rule is ground if it does not contain any variables. For the simplicity of the presentation, hybrid basic formulae that appearing in a disjunctive hybrid probability logic rule without probability annotations are assumed to be associated with the annotation . In addition, annotated hybrid basic formulae of the form are simply presented as .
2.2 Probability Answer Set Semantics
A probabilistic interpretation (pinterpretation), , for a set of disjunctive hybrid probability logic rules is a mapping . Let be a disjunctive hybrid probability logic rule of form (1) and and .
Definition 1
Let be a set of ground disjunctive hybrid probability logic rules, be a mapping associated to , be a pinterpretation for , and be a disjunctive hybrid probability logic rule of the form (1). Then:

satisfies in iff .

satisfies in iff .

satisfies in iff .

satisfies iff satisfies and satisfies .

satisfies iff such that satisfies .

satisfies iff satisfies whenever satisfies or does not satisfy .

satisfies iff satisfies every disjunctive hybrid probability logic rule in and

such that satisfies and satisfies in the .

such that are atoms in and , where .

A probabilistic model (pmodel) of a set of disjunctive hybrid probability logic rules, , associated with a mapping , is a pinterpretation for that satisfies . A pmodel of is minimal w.r.t. iff there does not exist a pmodel of such that . Let be a set of ground disjunctive hybrid probability logic rules, be a mapping associated to , and be a pinterpretation for . Then, the probabilistic reduct, , of w.r.t. is the set of ground nonmonotonicnegationfree disjunctive hybrid probability logic rules associated to and
is in iff
is in and .
Definition 2
A pinterpretation, , for a set of ground disjunctive hybrid probability logic rules, , associated to a mapping , is a probabilistic answer set for if is minimal pmodel for .
3 Probability Answer Set Optimization Programs
Probability answer set optimization programs are probability logic programs under the probability answer set semantics whose probability answer sets are ranked according to probability preference rules represented in the programs. A probability answer set optimization program, , is a pair of the form
, where is a union of two sets of probability logic rules and is a mapping, , associated to the set of probability logic rules . The first set of probability logic rules, , is called the generator rules that generate the probability answer sets that satisfy every probability logic rule in and the mapping associates to each atom, , appearing in , a disjunctive pstrategy that is used to combine the probability intervals obtained from different probability logic rules in with an atom appearing in their heads. is any set of probability logic rules with welldefined probability answer set semantics including normal, extended, and disjunctive hybrid probability logic rules [Saad and Pontelli2006, Saad2006, Saad2007a], as well as hybrid probability logic rules with probability aggregates (all are forms of probability answer set programming).
The second set of probability logic rules, , is called the probability preference rules, which are probability logic rules that represent the user’s probability quantitative and qualitative preferences over the probability answer sets generated by . The probability preference rules in are used to rank the generated probability answer sets from from the top preferred probability answer set to the least preferred probability answer set. Similar to [Brewka et al.2003], an advantage of probability answer set optimization programs is that and are independent. This makes probability preference elicitation easier and the whole approach is more intuitive and easy to use in practice.
In our introduction of probability answer set optimization programs, we focus on the syntax and semantics of the probability preference rules, , of the probability answer set optimization programs, since the syntax and semantics of the probability answer sets generator rules, , are the same as syntax and semantics of any set of probability logic rules with welldefined probability answer set semantics as described in [Saad and Pontelli2006, Saad2006, Saad2007a].
3.1 Probability Preference Rules Syntax
Let be a firstorder language with finitely many predicate symbols, function symbols, constants, and infinitely many variables. A literal is either an atom in or the negation of an atom (), where is the Herbrand base of and is the classical negation. Nonmonotonic negation or the negation as failure is denoted by . Let be the set of all literals in , where . A probability annotation is a probability interval of the form , where are called probability annotation items. A probability annotation item is either a constant in (called probability annotation constant), a variable ranging over (called probability annotation variable), or (called probability annotation function) where is a representation of a computable function and are probability annotation items.
Let be an arbitrary set of pstrategies, where () is the set of all conjunctive (disjunctive) pstrategies in . A hybrid literals is an expression of the form or , where are literals and and are pstrategies from . is the set of all ground hybrid literals formed using distinct literals from and pstrategies from . If is a hybrid literal is a probability annotation then is called a probability annotated hybrid literal. Let be a set of probability annotated hybrid literals. A boolean combination over is a boolean formula over probability annotated hybrid literals in constructed by conjunction, disjunction, and nonmonotonic negation (), where nonmonotonic negation is combined only with probability annotated hybrid literals.
Definition 3
A probability preference rule, , over a set of probability annotated hybrid literals, , is an expression of the form
(2) 
where are probability annotated hybrid literals and are boolean combinations over .
Let and , where is a probability preference rule of the form (2). Intuitively, a probability preference rule, , of the form (2) means that any probability answer set that satisfies and is preferred over the probability answer sets that satisfy , some , but not , and any probability answer set that satisfies and is preferred over probability answer sets that satisfy , some , but neither nor , etc.
Definition 4
A probability answer set optimization program, , is a pair of the form , where is a set of probability logic rules with welldefined probability answer set semantics, the generator rules, is a set of probability preference rules, and is the mapping that associates to each atom, , appearing in a disjunctive pstrategy.
3.2 Probability Preference Rules Semantics
In this section, we define the satisfaction of probability preference rules and the ranking of the probability answer sets with respect to a probability preference rule and with respect to a set of probability preference rules. We say that a probability preference rule is ground if it does not contain any variables. A probability answer set optimization program, , is ground if no variables appearing in any of the probability logic rules in or in any of the preference rules in
Definition 5
Let be a ground probability answer set optimization program, be a probability answer set of (possibly partial), and be a probability preference rule in of the form (2). Then the satisfaction of a boolean combination, , appearing in the by , denoted by , is defined inductively as follows:

iff .

iff or is undefined in .

iff and .

iff or .
Given and appearing in , the satisfaction of by , denoted by , is defined inductively as follows:

iff

iff or is undefined in .

iff , and , .
The satisfaction of probability preference rules is defined as follows.
Definition 6
Let be a ground probability answer set optimization program, be a probability answer set for , and be a probability preference rule in , and be a boolean combination in . Then, we define the following notions of satisfaction of by :

iff and .

iff and does not satisfy any in .

iff does not satisfy .
means that the body of and the boolean combination that appearing in the head of is satisfied by . However, means that is irrelevant (denoted by ) to , or, in other words, the probability preference rule is not satisfied by , because either one of two reasons. Either because the body of and non of the boolean combinations that appearing in the head of are satisfied by . Or because the body of is not satisfied by .
Definition 7
Let be a ground probability answer set optimization program, be two probability answer sets of , be a probability preference rule in , and be boolean combination appearing in . Then, is strictly preferred over w.r.t. , denoted by , iff and or and and one of the following holds:

implies iff .

implies iff or is undefined in but defined in .

implies iff there exists such that and for all other , we have .

implies iff there exists such that and for all other , we have .
We say, and are equally preferred w.r.t. , denoted by , iff and or and and one of the following holds:

implies iff .

implies iff or is undefined in both and .

implies iff

implies iff
We say, is at least as preferred as w.r.t. , denoted by , iff or .
Definition 8
Let be a ground probability answer set optimization program, be two probability answer sets of , be a probability preference rule in , and be boolean combination appearing in . Then, is strictly preferred over w.r.t. , denoted by , iff one of the following holds:

and and ,
where and . 
and and ,
where . 
and .
We say, and are equally preferred w.r.t. , denoted by , iff one of the following holds:

and and ,
where . 
and .
We say, is at least as preferred as w.r.t. , denoted by , iff or .
The previous two definitions characterize how probability answer sets are ranked with respect to a boolean combination and with respect to a probability preference rule. Definition 7 presents the ranking of probability answer sets with respect to a boolean combination. But, Definition 8 presents the ranking of probability answer sets with respect to a probability preference rule. The following definitions specify the ranking of probability answer sets according to a set of probability preference rules.
Definition 9 (Pareto Preference)
Let be a probability answer set optimization program and be probability answer sets of . Then, is (Pareto) preferred over w.r.t. , denoted by , iff there exists at least one probability preference rule such that and for every other rule , . We say, and are equally (Pareto) preferred w.r.t. , denoted by , iff for all , .
Definition 10 (Maximal Preference)
Let be a probability answer set optimization program and be probability answer sets of . Then, is (Maximal) preferred over w.r.t. , denoted by , iff
We say, and are equally (Maximal) preferred w.r.t. , denoted by , iff
It is worth noting that the Maximal preference definition is more general than the Pareto preference definition, since the Maximal preference relation subsumes the Pareto preference relation.
4 Nurse Restoring with Probability Preferences Problem
Nurse restoring problem is wellknown scheduling problem in Operation Research [Bard and Purnomo2005]. In this section, we extend the nurse restoring problem to allow nurses to express their quantitative and qualitative preferences in terms of probability values over their choices, creating a new version of the nurse restoring problem called Nurse Restoring with Probability Preference Problem. We show that nurse restoring with probability preferences problem can be easily and intuitively represented and solved in the probability answer set optimization framework.
Nurse restoring with probability preferences problem is a multiobjective scheduling problem with several conflicting factors, like the hospitals views of the continuing insurance of sufficient nursing service at minimum cost and the nurses quantitative and qualitative preferences over working hours and days off, where the hospital management must resolve that conflict, since in any hospital’s budget, the nursing service is one of its largest components. To accomplish the scheduling process, the nurse manger must collect information regarding the nursing service demands and the nurses quantitative and qualitative preferences over the available working hours. Hospitals typically employ nurses to work in shifts that cover the twenty four hours of the day, namely early, day, late, and night shifts with their obvious meanings.
The aim is to assign nurses to shifts over days, weeks, or months in order to provide a certain level of care in terms of nursing service whereas taking into consideration each individual nurse quantitative and qualitative preferences over shifts so that fairness and transparency are assured. Nurse preferences over shifts on a given day is given as a probability distribution over shifts on that day. Nurse restoring with probability preferences problem is formalized as given in the following example.
Example 1
Assume that we have different nurses (denoted by ) that need to be assigned to shifts among different shifts per a day (denoted by ) for different days (denoted by ) with the nurse manger demanding that each nurse is assigned exactly one shift per day and no two nurses are assigned the same shift on the same day. Each nurse prefers to work at certain shifts at certain days over other shifts in these certain days. Each nurse preferences over shifts per a day is represented as a probability distribution over shifts per that day. This nurse restoring with probability preferences problem can be represented as a probability answer set optimization program, , where is any arbitrary assignments of probabilistic pstrategies and is a set of disjunctive hybrid probability logic rules with probability answer set semantics of the form:
and , where are probability annotation variables act as place holders and, for any (), represents that nurse prefers to service at shift in day with probability (which is the nurse preference in servicing at the shift in the day ). The first disjunctive hybrid probability logic rule represents a nurse preferences over shifts per day while the second disjunctive hybrid probability logic rule represents the constraints that a nurse is assigned exactly one shift per day and one shift in a given day cannot be assigned to more than one nurse.
The set of probability preference rules, , of the probability answer set optimization program representation of the nurse restoring with probability preferences problem consists and of the probability preference rule
where and , we have .
However, the probability preference rules, , of the probability answer set optimization program representation of the nurse restoring with probability preferences problem can be easily and intuitively modified according to the nurses preferences in many and very flexible ways. For example, it can be the case that nurse is neutral regarding servicing at shifts and in a day with probability value each. This means that shifts and in day are equally preferred to nurse . Hence, this situation can be represented in nurse probability preference rule in as
Moreover, although nurse is neutral regarding servicing at shifts and in day with probability value each, it can be the case that nurse has more appeal in servicing at shift over shift in day . Therefore, this situation can be intuitively represented in nurse probability preference rule in as
Furthermore, it can be the case that each nurse has the preference of servicing at several shifts per a day with varying degrees of probability values. This also can be easily and intuitively accomplished by replacing the disjunctive hybrid probability logic rules in , of the probability answer set optimization program, , representation of the nurse restoring with probability preferences problem by the following set of disjunctive hybrid probability logic rules:
and and for all possible values of , where the variable,
, is a dummy variable, where the number of values that the dummy variable,
, takes is equal to the number of shifts that a nurse is allowed to service per day. For example if the maximum number of shifts for a nurse to service per day is two, then the variable can be assigned to any two dummy values, e.g., and . For all possible values of , the first disjunctive hybrid probability logic rule assigns multiple shifts per day, , to a nurse . The last two disjunctive hybrid probability logic rules ensure that a shift per day is not assigned more than once to the same nurse.Moreover, the disjunctive hybrid probability logic rules in, , allow multiple nurses to be assigned to the same shifts per a day, which can be necessary in situations where large number of patients are required to be serviced at given shifts per a day. In this case the hospital management may need to bound the number of allowable nurses per a shift per day. This also can be represented in the probability answer set optimization framework as a constraint using aggregate atoms in the style of the aggregate atoms presented in [Faber et al.2010].
In addition to replacing the probability preference rules in , of the probability answer set optimization program, , representation of the nurse restoring with probability preferences problem by the following probability preference rule and and for all possible values of :
where and , we have . This shows in general that probability answer set optimization programs can be intuitively and flexibly used to represent and reason in the presence of both probability quantitative preferences and qualitative preferences. This is illuminated by the following instance of the nurse restoring with probability preferences problem described below.
Example 2
Assume that the nurse manger wants to schedule the nursing service for the Saturday and Sunday of this week. However, three nurses, Jeen, Lily, and Lucci are available over the weekends of this week. Jeen, Lily, and Lucci probability quantitative and qualitative preferences over shifts per this Saturday and Sunday are given as described below. In addition, each of the nurses requires to be assigned exactly one shift per day and the nurse manger requires that no two nurses are assigned the same shift on the same day.
This instance of the nurse restoring with probability preferences problem can be represented as an instance of the probability answer set optimization program, , presented in Example 1, as a probability answer set optimization program, , where in addition to the last disjunctive hybrid probability logic rule in of described in Example 1, also contains the following disjunctive hybrid probability logic rules:
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