1 Introduction
LP^{MLN} [10], a newly developed knowledge representation and reasoning language, is designed to handle nonmonotonic and uncertain knowledge by combining the methods of Answer Set Programming (ASP) [3, 7] and Markov Logic Networks (MLN) [16]. Specifically, an LP^{MLN} program can be viewed as a weighted ASP program, where each ASP rule is assigned a weight denoting its certainty degree, and each weighted rule is allowed to be violated by a set of beliefs associated with the program. For example, LP^{MLN} rule “” is a weighted constraint denoting facts and are contrary, is the weight of the constraint. In the view of ASP, the set is impossible to be a belief set of any ASP programs containing the constraint, while in the context of LP^{MLN}, is a valid belief set. Since violates the constraint, the weight is regarded as the certainty degree of . It is easy to observe that the example can also be encoded by weak constraints in ASP. From this perspective, LP^{MLN} can be viewed as an extension of ASP with weak constraints, that is, ASP with weak rules. Besides, several inference tasks are introduced to LP^{MLN}
such as computing marginal probability distribution of beliefs, computing most probable belief sets etc., which makes LP
^{MLN} suitable for knowledge reasoning in the context that contains uncertain and inconsistent data. For example, Eiter and Kaminski [6] used LP^{MLN}in the tasks of classifying visual objects, and some unpublished work tried to use LP
^{MLN} as the bridge between text and logical knowledge bases.Recent results on LP^{MLN}aim to establish the relationships among LP^{MLN} and other logic formalisms [2, 12], develop LP^{MLN} solvers [9, 18, 20], acquire the weights of rules automatically [11], explore the properties of LP^{MLN} [19] etc. All these results lay the foundation for the problems solving via LP^{MLN}, however, many theoretical problems of LP^{MLN} are still unsolved, which prevents the wider applications of LP^{MLN}. In this paper, we investigate the strong equivalences between LP^{MLN} programs, which is regarded as an important property in the field of logic programming. For two ASP programs and , they are strongly equivalent, iff for any ASP program , the programs and have the same stable models [13]. In other words, an ASP program can be replaced by one of its strong equivalent without considering its context, which helps us to simplify logic programs, enhance inference engines, construct humanfriendly knowledge bases etc. For example, an ASP rule such that its positive and negative body have common atoms is strongly equivalent to [8, 14, 15], therefore, such kinds of rules can be eliminated from any context, which leads to a more concise knowledge base and makes the reasoning easier. By investigating the strong equivalences in LP^{MLN}, it is expected to improve the knowledge base constructing and knowledge reasoning in LP^{MLN}, furthermore, help us to facilitate the applications and extend the understandings of LP^{MLN}.
Our contributions are as follows. Firstly, we define the notions of strong equivalences in LP^{MLN}, that is, the pstrong and wstrong equivalences. As we showed in above example, a stable model defined in LP^{MLN} is associated with a certainty degree, therefore, the notions of strong equivalences in LP^{MLN} are also relevant to the certainty degree. Secondly, we present a modeltheoretical approach to characterizing the defined notions, which can be viewed as a generalization of the strongequivalence models (SEmodel) approach in ASP [17]. Finally, we show the use of the strong equivalences in simplifying LP^{MLN} programs, and present a sufficient and necessary syntactic condition that guarantees the strong equivalences between a single LP^{MLN} rule and the empty program.
2 Preliminaries
In this section, we review the knowledge representation and reasoning language LP^{MLN} presented in [10]. An LP^{MLN} program is a finite set of weighted rules , where is the weight of rule , and is an ASP rule of the form
(1) 
where s are literals, is epistemic disjunction, and is default negation. The weight of an LP^{MLN} rule is either a real number or a symbol “” denoting “infinite weight”, and if is a real number, the rule is called soft, otherwise, it is called hard. For convenient description, we introduce some notations. By we denote the set of unweighted ASP counterpart of an LP^{MLN} program , i.e. . For an ASP rule of the form (1), the literals occurred in head, positive body, and negative body of are denoted by , , and respectively. Therefore, an ASP rule of the form (1) can also be abbreviated as “”. By we denote the set of literals occurred in rule , and by we denote the set of literals occurred in an ASP program .
An LP^{MLN} program is called ground if its rules contain no variables. Usually, a nonground LP^{MLN} program is considered as a shorthand for the corresponding ground program, therefore, we limited our attention to the strong equivalences between ground LP^{MLN} programs in this paper. For a ground LP^{MLN} program , we use to denote the weight degree of , i.e. . A ground LP^{MLN} rule is satisfied by a consistent set of ground literals, denoted by , if by the notion of satisfiability in ASP. An LP^{MLN} program is satisfied by , denoted by , if satisfies all rules in . By we denote the LP^{MLN} reduct of an LP^{MLN} program w.r.t. , i.e. . A consistent set of literals is a stable model of an ASP program , if satisfies all rules in and is minimal in the sense of set inclusion, where is the GelfondLifschitz reduct (GLreduct) of w.r.t. , i.e. . The set is a stable model of an LP^{MLN} program if is a stable model of the ASP program . And by we denote the set of all stable models of an LP^{MLN} program . For a stable model of an LP^{MLN} program , the weight degree of w.r.t. is defined as , and the probability degree of w.r.t. is defined as
(2) 
For a literal , the probability degree of w.r.t. is defined as
(3) 
A stable model of an LP^{MLN} program is called a probabilistic stable model of if . By we denote the set of all probabilistic stable models of . It is easy to check that is a probabilistic stable model of , iff is stable model of that satisfies the most hard rules. Based on above definitions, there are two kinds of main inference tasks for an LP^{MLN} program [9]:

Maximum A Posteriori (MAP) inference: compute the stable models with the highest weight or probability degree of the program , i.e. the most probable stable model;

Marginal Probability Distribution (MPD) inference: compute the probability degrees of a set of literals w.r.t. the program .
3 Strong Equivalences for LP^{Mln}
In this section, we investigate the strong equivalences in LP^{MLN}. Firstly, we define the notions of strong equivalences based on two different certainty degrees in LP^{MLN}. Secondly, we present a modeltheoretical approach to characterizing the notions. Finally, we present the relationships among these notions.
3.1 Notions of Strong Equivalences
The notion of strong equivalence is built on the notion of ordinary equivalence, in this section, we define two notions of ordinary equivalences between LP^{MLN} programs, which is relevant to the weight and probability defined for stable models in LP^{MLN}.
Definition 1 (wordinary equivalence).
Two LP^{MLN} programs and are wordinarily equivalent, denoted by , if their stable models coincide, and for each stable model of the programs, .
Definition 2 (pordinary equivalence).
Two LP^{MLN} programs and are pordinarily equivalent, denoted by , if their stable models coincide, and for each stable model of the programs, .
From Definition 1 and Definition 2, it can be observed that both of the wordinary and pordinary equivalences can guarantee two LP^{MLN} programs have the same MAP and MPD inference results, and the pordinary equivalence is a little weaker, i.e. if two LP^{MLN} programs are pordinarily equivalent, then they are wordinarily equivalent, but the inverse dose not hold generally. Based on the definitions of ordinary equivalences, we can define two kinds of strong equivalences between LP^{MLN} programs.
Definition 3 (strong equivalences for LP^{Mln}).
For two LP^{MLN} programs and ,

they are wstrongly equivalent, denoted by , if for any LP^{MLN} program , ;

they are pstrongly equivalent, denoted by , if for any LP^{MLN} program , .
The notions of wstrong and pstrong equivalences can guarantee the faithful replacement of an LP^{MLN} program in any context. Here, we introduce a new notion of strong equivalence, semistrong equivalence, that does not guarantee the faithful replacement, but helps us to simplify the characterizations of other strong equivalences.
Definition 4 (semistrong equivalence).
Two LP^{MLN} programs and are semistrongly equivalent, denoted by , if for any LP^{MLN} program , the programs and have the same stable models.
3.2 Characterizations of Strong Equivalences
In this section, we present the characterizations for wstrong and pstrong equivalences. From Definition 3 and Definition 4, the notions of wstrong and pstrong equivalences can be viewed as the strengthened semistrong equivalence by introducing the certainty evaluations. Therefore, we present the characterization of semistrong equivalence firstly, which severs as the basis of characterizing wstrong and pstrong equivalences.
3.2.1 Characterizing SemiStrong Equivalence
Here, we characterize the semistrong equivalence between LP^{MLN} programs by generalizing the strongequivalence models (SEmodels) approach presented in [17]. For the convenient description, we introduce following notions.
Definition 5 (SEinterpretation).
A strong equivalence interpretation (SEinterpretation) is a pair of consistent sets of literals such that . An SEinterpretation is called total if , and nontotal if .
Definition 6 (SEmodels for LP^{Mln}).
For an LP^{MLN} program , an SEinterpretation is an SEmodel of , if and , where .
In Definition 6, is an ASP program obtained from by a threestep transformation. In the first step, is obtained from by removing all rules that cannot be satisfied by , which is the LP^{MLN} reduct of w.r.t. . In the second step, is obtained by dropping weight of each rule in . In the third step, is obtained by the GLreduct. Clearly, an SEmodel for the LP^{MLN} program is an SEmodel of a consistent unweighted subset of that is obtained by LP^{MLN} reduct, which means the definition of SEmodels for LP^{MLN} programs is built on the definition of SEmodels for ASP programs. In what follows, we use to denote the set of all SEmodels of an LP^{MLN} program .
Definition 7.
For an LP^{MLN} program and an SEmodel of , the weight degree of w.r.t. the program is defined as
(4) 
Example 1.
Consider an LP^{MLN} program . For the set , it is easy to check that , therefore, the LP^{MLN} reduct is itself. By the definitions of GLreduct, , therefore, both and are SEmodels of , and .
Now, we show some useful properties of the SEmodels for LP^{MLN} programs. Proposition 1 is an immediate result according to the definition of SEmodels.
Proposition 1.
Let be an LP^{MLN} program and an SEinterpretation,

if , then is an SEmodel of ;

is not an SEmodel of , iff .
Proposition 2 shows the relationships between the SEmodels and the stable models of an LP^{MLN} program.
Proposition 2.
For an LP^{MLN} program and a total SEmodel of ,

there must be an LP^{MLN} program such that is a stable model of , for example, ;

is a stable model of , iff for any proper subset of .
Based on above results, a characterization of semistrong equivalence between LP^{MLN} programs is presented in Lemma 1.
Lemma 1.
Let and be two LP^{MLN} programs, they are semistrongly equivalent, iff they have the same SEmodels, i.e. .
Proof.
The proof proceeds basically along the lines of the corresponding proof by Turner [17].
For the if direction, suppose , we need to prove that for any LP^{MLN} program , the programs and have the same stable models. We use proof by contradiction. Assume is a set of literals such that . By the definition, we have . By Proposition 1, we have is an SEmodel of . Hence, is also an SEmodel of . Then, we have and . By the assumption , there exists a consistent set of literals such that , then we have and , hence, is an SEmodel of , which means is also an SEmodel of . By the definition of stable model, cannot be a stable model of , which contradicts with the assumption . Therefore, the programs and have the same stable models, and the if direction of Lemma 1 is proven.
For the onlyif direction, suppose , we need to prove that . We use proof by contradiction. Assume is an SEinterpretation such that . By Proposition 1, we have . Let . We have . Let be a set of literals such that and . By the construction of , we have . Since , we have . Hence, there must exist a literal such that . By the construction of , we have , which means . By the definition of stable models, is a stable model of , which means should also be a stable model of . By the definition of stable model, cannot be an SEmodel of , which contradicts with the assumption . Therefore, and have the same SEmodels, and the onlyif direction of Lemma 1 is proven. ∎
3.2.2 Characterizing WStrong and PStrong Equivalences
Now we present a main result of the paper, that is, the characterizations of wstrong and pstrong equivalences. Based on Lemma 1, Lemma 2 provides a sufficient condition to characterize the pstrong equivalence for LP^{MLN} programs.
Lemma 2.
Two LP^{MLN} programs and are pstrongly equivalent, if , and there exist two constants and such that for each SEmodel , .
Proof.
For two LP^{MLN} programs and , by Lemma 1, if , and are semistrongly equivalent, i.e. for any LP^{MLN} program , . Suppose there exist two constants and such that for each SEmodel , , we need to show that and are pstrongly equivalent. Let be an LP^{MLN} program, it is easy to check that is a probabilistic stable model of iff is a probabilistic stable model of , i.e. . For a stable model , the probability degree of can be reformulated as
(5) 
By the definition of pstrong equivalence, we have . ∎
The condition in Lemma 2, called PSEcondition, is sufficient to characterize the pstrong equivalence. One may ask that whether the PSEcondition is also necessary. To answer the question, we need to consider the hard rules of LP^{MLN} particularly. For LP^{MLN} programs containing no hard rules, it is easy to check that the PSEcondition is necessary. But for arbitrary LP^{MLN} programs, this is not an immediate result, which is shown as follows. Firstly, we introduce some notations. For a set of literals, we use to denote the power set of , and use to denote the maximal consistent part of the power set of , i.e. .
Lemma 3.
For two pstrongly equivalent LP^{MLN} programs and , let and be arbitrary LP^{MLN} programs such that . There exist two constants and such that for any SEmodels of , if , then .
By Lemma 3, for two pstrongly equivalent LP^{MLN} programs and , to prove the necessity of the PSEcondition, we need to find a set of LP^{MLN} programs satisfying

, ; and

, where is the set of literals occurred in and , i.e. .
Above set is called a set of necessary extensions w.r.t. LP^{MLN} programs and . As shown in Proposition 1, an arbitrary total SEinterpretation is an SEmodel of an LP^{MLN} program, therefore, if there exists a set of necessary extensions of two pstrongly equivalent programs and , then the necessity of the PSEcondition can be proven. In what follows, we present a method to construct a set of necessary extensions.
Definition 8.
For two consistent sets and of literals, and an atom such that , by we denote an LP^{MLN} program as follows
(6)  
(7) 
Definition 9 (flattening extension).
For an LP^{MLN} program and a set of literals such that , a flattening extension of w.r.t. is defined as

;

,
where is a set of weighted facts constructed from , i.e. , is a probabilistic stable model of , i.e. , and .
According to the splitting set theorem of LP^{MLN} [19], the flattening extension has following properties.
Proposition 3.
For an LP^{MLN} program and a set of literals, if is constructed from by adding , then we have

;

; and

the weight degrees of stable models have following relationships
(8) and for two stable models and of , if , then .
Example 2.
Let be the LP^{MLN} program in Example 1, and a set of literals . By Definition 9, , it is easy to check that all subsets of are the stable models of , is the unique probabilistic stable model. By Definition 8, is as follows
(9)  
(10) 
and we have . The stable models and their weight degrees of , , and are shown in Table 1. From the table, we can observe that the flattening extension can be used to adjust the sets of literals that satisfy the most hard rules.
Weight  

    
  
Lemma 4.
Let and be two pstrongly equivalent LP^{MLN} programs, and . For two consistent subsets and of , there exists a flattening extension such that and are probabilistic stable models of .
Lemma 4 provides a method to construct a set of necessary extensions of two pstrongly equivalent LP^{MLN} programs by constructing a set of flattening extensions, which means the PSEcondition is necessary to characterize the pstrong equivalence for LP^{MLN} programs.
Theorem 1.
Let and be two LP^{MLN} programs,

and are pstrongly equivalent iff , and there exist two constants and such that for each SEmodel , ;

and are wstrongly equivalent iff they are pstrongly equivalent and the constants .
Example 3.
Consider LP^{MLN} programs and , where is a variable denoting the weight of corresponding rule. It is easy to check that is the unique nontotal SEmodel of and , therefore, and are semistrongly equivalent. If the programs are also pstrongly equivalent, we have following system of linear equations, where and .
(11) 
Solve the system of equations, we have and are pstrongly equivalent iff and ; and they are wstrongly equivalent iff and .
4 Simplifying LP^{Mln} Programs
The notions of strong equivalences can be used to study the simplifications of logic programs. Specifically, if LP^{MLN} program and are strongly equivalent, and the program is easier to solve or more friendly for human, then can be replaced by . In this section, we investigate the simplifications of LP^{MLN} programs via using the notions of strong equivalences. In particular, we present an algorithm to simplify and solve LP^{MLN} programs based on strong equivalences firstly. Then, we present some syntactic conditions that guarantee the strong equivalence between a single LP^{MLN} rule and the empty set , which can be used to check the strong equivalences efficiently.
Definition 10.
An LP^{MLN} rule is called semivalid, if is semistrongly equivalent to ; the rule is called valid, if is pstrong equivalent to .
In Definition 10, we specify two kinds of LP^{MLN} rules w.r.t semistrong and pstrong equivalences. Obviously, a valid LP^{MLN} rule can be eliminated from any LP^{MLN} programs, while a semivalid LP^{MLN} rule cannot. By the definition, eliminating a semivalid LP^{MLN} rule does not change the stable models of original programs, but changes the probability distributions of the stable models, which means it may change the probabilistic stable models of original programs.
Example 4.
Consider three LP^{MLN} programs , , and . It is easy to check that rules in and are valid and semivalid, respectively. Table 2 shows the stable models and their probability degrees of LP^{MLN} programs , , and . It can be observed that eliminating the rule of from makes all stable models of probabilistic, which means semivalid rules cannot be eliminated directly.
Stable Model  

Algorithm 1 provides a framework to simplify and solve LP^{MLN} programs based on the notions of semivalid and valid LP^{MLN} rules. Firstly, simplify an LP^{MLN} program by removing all semivalid and valid rules (line 2  8). Then, compute the stable models of the simplified LP^{MLN} program via using some existing LP^{MLN} solvers, such as LPMLN2ASP, LPMLN2MLN [9], and LPMLNModels [20] ect. Finally, compute the probability degrees of the stable models w.r.t. the simplified program and all semivalid rules (line 9  12). The correctness of the algorithm can be proved by corresponding definitions.
Name  Definition  Strong Equivalence 

TAUT  p, semi  
CONTRA  p, semi  
CONSTR1  semi  
CONSTR2  semi  
CONSTR3  , , and  p, semi 
In Algorithm 1, a crucial problem is to decide whether an LP^{MLN} rule is valid or semivalid. Theoretically, it can be done by checking the SEmodels of a rule, however, the approach is highly complex in computation. Therefore, we investigate the syntactic conditions for the problem. Table 3 shows five syntactic conditions for a rule , where TAUT and CONTRA have been introduced to investigate the program simplification of ASP [15, 5], CONSTR1 means the rule is a constraint, and CONSTR3 is a special case of CONSTR1. Rules satisfying CONSTR2 is usually used to eliminate constraints in ASP, for example, rule “” is equivalent to rule “”, if the atom does not occur in other rules. Based on these conditions, we present the characterization of semivalid and valid LP^{MLN} rules.
Theorem 2.
An LP^{MLN} rule is semivalid, iff the rule satisfies one of TAUT, CONTRA, CONSTR1 and CONSTR2.
Theorem 3.
An LP^{MLN} rule is valid, iff one of following condition is satisfied

rule satisfies one of TAUT, CONTRA, and CONSTR3; or

rule satisfies CONSTR1 or CONSTR2, and .
Theorem 2 and Theorem 3 can be proven by Lemma 1 and Theorem 1. It is worthy noting that conditions CONSTR1 and CONSTR2 means the only effect of constraints in LP^{MLN} is to change the probability distribution of inference results, which can also be observed in Example 2. In this sense, the constraints in LP^{MLN} can be regarded as the weak constraints in ASP, and Algorithm 1 is similar to the algorithm of solving ASP containing weak constraints. In both of algorithms, stable models are computed by removing (weak) constraints, and the certainty evaluations of the stable models are computed by combining these constraints.
Combining Theorem 2 and Theorem 3, Algorithm 1 is an alternative approach to enhance LP^{MLN} solvers. In addition, Theorem 2 and Theorem 3
also contribute to the field of knowledge acquiring. On the one hand, although it is impossible that rules of the form TAUT, CONTRA, and CONSTR3 are constructed by a skillful knowledge engineer, these rules may be obtained from data via rule learning. Therefore, we can use TAUT, CONTRA, and CONSTR3 as the heuristic information to improve the results of rule learning. On the other hand, CONSTR1 and CONSTR2 imply a kind of methodology of problem modeling in LP
^{MLN}, that is, we can encode objects and relations by LP^{MLN} rules and facts, and adjust the certainty degrees of inference results by LP^{MLN} constraints. In fact, this is the core idea of ASP with weak constraints, LP^{MLN} is more flexible by contrast, since LP^{MLN} provides weak facts and rules besides weak constraints.5 Conclusion and Future Work
In this paper, we present four kinds of notions of strong equivalences between LP^{MLN} programs by comparing the certainty degrees of stable models in different ways, i.e. semistrong, wstrong and pstrong equivalences, where wstrong equivalence is the strongest notion, and semistrong equivalence is the weakest notion. For each notion, we present a sufficient and necessary condition to characterize it, which can be viewed as a generalization of SEmodel approach in ASP. After that, we present a sufficient and necessary condition that guarantees the strong equivalence between a single LP^{MLN} rule and the empty set, and we present an algorithm to simplify and solve LP^{MLN} programs by using the condition. The condition can also be used to improve the knowledge acquiring and increase the understanding of the methodology of problems modeling in LP^{MLN}.
As we showed in the paper, there is a close relationship between LP^{MLN} and ASP, especially, the constraints in LP^{MLN} can be regarded as the weak constraints in ASP. Concerning related work, the strong equivalence for ASP programs with weak constraints (abbreviated to ASP^{wc}) has been investigated [4]. It is easy to observe that the strong equivalence and corresponding characterizations of ASP^{wc} can be viewed as a special case of the pstrong equivalence in ASP.
For the future, we plan to improve the equivalences checking in the paper, and use these technologies to enhance LP^{MLN} solvers. And we also plan to extend the strong equivalence discovering method introduced in [14] to LP^{MLN}, which would help us to decide strong equivalence via some syntactic conditions.
6 Acknowledgments
We are grateful to the anonymous referees for their useful comments. The work was supported by the National Key Research and Development Plan of China (Grant No.2017YFB1002801).
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