On the Relation between Weak Completion Semantics and Answer Set Semantics
aa r X i v : . [ c s . A I] O c t On the Relation between Weak CompletionSemantics and Answer Set Semantics
Emmanuelle-Anna Dietz Saldanha and Jorge Fandinno Technische Universitat Dresden, Technische Universitat Potsdam,Germany
Abstract.
The Weak Completion Semantics (WCS) is a computationalcognitive theory that has shown to be successful in modeling episodesof human reasoning. As the WCS is a recently developed logic program-ming approach, this paper investigates the correspondence of the WCSwith respect to the well-established Answer Set Semantics (ASP). Theunderlying three-valued logic of both semantics is different and theirmodels are evaluated with respect to different program transformations.We first illustrate these differences by the formal representation of someexamples of a well-known psychological experiment, the suppression task.After that, we will provide a translation from logic programs understoodunder the WCS into logic programs understood under the ASP. In partic-ular, we will show that logic programs under the WCS can be representedas logic programs under the ASP by means of a definition completion ,where all defined atoms in a program must be false when their definitionsare false.
Keywords:
Answer Set Programming, Weak Completion Semantics,Strong Negation, Human Reasoning
The Weak Completion Semantics (WCS), originally presented in [9], has beensuggested as a computational cognitive theory, and demonstrated to be adequatefor modeling various episodes of human reasoning summarized in [8]. Considera well-known psychological experiment, the suppression task [2], which showedthat participants’ answer systematically diverged from classical logic correct an-swers. Participants were asked to derive conclusions given variations of a set ofpremises. The first group was given the following two premises: If she has an essay to finish, then she will study late in the library. ( e ℓ ) She does not have an essay to finish. ( not e ) The participants received only the natural language sentences, not the abbreviations. Emmanuelle-Anna Dietz Saldanha and Jorge Fandinno Then, they were asked what necessarily follows assuming that the above premiseswere true and given three possible answer from where they could choose:
She will study late in the library. ( ℓ ) She will not study late in the library. ( not ℓ ) She may or may not study late in the library. ( ℓ or not ℓ )54% of the participants answered that She will not study late in the library . Thesecond group received, additionally to ( e ℓ ) and ( not e ), the following premise: If she has a textbook to read, then she will study late in the library. ( t ℓ )Now, only 4% of the participants answered that She will not study late in thelibrary . With these results, Byrne showed that humans seem to reason non-monotonically, i.e. they suppressed previously drawn conclusions. The above ex-amples demonstrates that humans do not always apply the close world assump-tion in their inferences. In particular, if they are made aware of alternatives,they might rather apply the open world assumption . Stenning and van Lam-balgen [15] suggested a formal representation of these premises by licenses forinferences. For the first group, the following logic program rules were suggested: ℓ ← e ∧ not ab e ← ⊥ ab ← ⊥ (1) ab is an abnormality predicate. ab ← ⊥ and e ← ⊥ are (negative) assumptions which assume e and ab being false. In the designated model under the WCS [9],which is the least model of the weak completion of that program under the three-valued Lukasiewicz logic [11], e , ℓ and ab are false. On the other hand, the logicprogram rules for the second group was suggested to be as follows: ℓ ← e ∧ not ab ℓ ← t ∧ not ab e ← ⊥ ab ← ⊥ ab ← ⊥ (2)Here, in the designated model under the WCS, e , ab and ab are false, and ℓ and t unknown . The differences between both models, where in the first case ℓ isfalse, and in the second case ℓ is unknown, seems to represent well the suppressioneffect occurring in the second group: In the first group, 46% concluded that Shewill not study late in the library , whereas in the second group, only 4% concludedthat
She will not study late in the library . The overall results of all the twelvecases of the suppression tasks seem to be adequately modeled under the WCS [3].In this paper we will investigate how the above two cases of the suppressiontask can be modeled under the Answer Set Semantics [7] (ASP), in particular howboth semantics correspond to each other. For this purpose, we first introduce thenotions and notation used throughout the paper and the underlying three-valuedlogics. Section 3 introduces ASP and WCS and shows some intermediate results.The main result is presented in Section 4, where the formal correspondencebetween both semantics is shown, and the above two cases will be discussedagain. The rules will be understood under their weak completion .n the Relation between WCS and ASP 3
In this section, we present the general notation and terminology that will be usedthroughout the paper together with the semantics for classical logic with strongnegation [16] and three-valued Lukasiewicz logic [11]. In the sequel, definitionsare specified in the running text, except if we intend to emphasize them.
We assume a fixed non-empty and (possibly infinite) set of ground atoms, de-noted by At . The set of (strongly) negated atoms for the atoms in S ⊆ At , isdefined as ¬ S def = {¬ A | A ∈ S } . A literal L is either an atom or its (strong)negation, that is L ∈ ( At ∪ ¬ At ). Given a set of atoms At , a formula is definedaccording to the following grammar: ϕ ::= A | ⊥ | ⊤ | U | ϕ ◦ ψ | ¬ ϕ | not ϕ ⊤ , ⊥ and U denote the truth constants true , false and unknown , respectively.The connective “ not ” stands for weak or default negation , whereas “ ¬ ” stands for strong negation . The connectives “ ← CL ”, “ ← L ” and “ ← ” stand for classical (ormaterial) implication, Lukasiewicz implication and logic programming implica-tion, respectively. The logic programming implication sign ← is purely syntacticand, different to ← CL and ← L , will not be assigned a fixed underlying semantics.We will study two different logic programming semantics and depending on thesemantics in consideration, the meaning for ← is then specified accordingly. Weuse ↔ X as an abbreviation defined by ϕ ↔ X ψ def = ( ϕ ← X ψ ) ∧ ( ψ ← X ϕ ) , (3)where X ∈ { C , L } . A formula ϕ is regular if its only occurrences of implicationsstrong negation ¬ only occurs in front of atoms. A formula ϕ is called implication-free if there are no occurrences of the implication connectives ← CL , ← L or ← ,i.e. its set of connectives is {∧ , ∨ , ¬ , not } . A formula ϕ is called basic if it isimplication-free, and in addiction, it has no occurrences of weak negation, i.e.its only connective are {∧ , ∨ , ¬ } . Basic formulas are implication-free formulas,but not vice versa. Note that, in general, regular formulas that are implication-free need not to be basic nor basic formulas need to be regular. Example 1.
Consider the following three formulas: ϕ = ( ¬ ( ¬ p ∧ q ) ∨ not r ) ϕ = ( ¬ ( ¬ p ∧ q )) ϕ = ( not ( ¬ p ∧ q )) ϕ ϕ ϕ ϕ ϕ ϕ Emmanuelle-Anna Dietz Saldanha and Jorge Fandinno F not F ⊤ ⊥ U ⊤⊥ ⊤ ∧ ⊤ U ⊥⊤ ⊤ U ⊥ U U U ⊥⊥ ⊥ ⊥ ⊥ ← CL ⊤ U ⊥⊤ ⊤ ⊤ ⊤ U U ⊤ ⊤⊥ ⊥ ⊤ ⊤ ↔ CL ⊤ U ⊥⊤ ⊤ U ⊥ U U ⊤ ⊤⊥ ⊥ ⊤ ⊤ F ¬ F ⊤ ⊥ U U ⊥ ⊤ ∨ ⊤ U ⊥⊤ ⊤ ⊤ ⊤ U ⊤ U U ⊥ ⊤ U ⊥ ← L ⊤ U ⊥⊤ ⊤ ⊤ ⊤ U U ⊤ ⊤⊥ ⊥ U ⊤ ↔ L ⊤ U ⊥⊤ ⊤ U ⊥ U U ⊤ U ⊥ ⊥ U ⊤ Table 1.
Truth tables for three-valued Lukasiewicz logic {¬ , ∧ , ∨ , ← L , ↔ L } , and Clas-sical Logic extended with strong negation {¬ , not , ∧ , ∨ , ← CL , ↔ CL } . A (three-valued) interpretation I : At −→ {⊤ , ⊥ , U } is a function mapping eachatom to a truth constant. We introduce now two different alternative representa-tions of interpretations that are equivalent and common in the literature: An in-terpretation I can be represented as a pair of set of atoms, I = h I ⊤ , I ⊥ i such that I ⊤ ∩ I ⊥ = ∅ where the correspondence is given as follows: I ⊤ = { A | I ( A ) = ⊤} and I ⊥ = { A | I ( A ) = ⊥} . Note that A / ∈ ( I ⊤ ∪ I ⊥ ) holds iff I ( A ) = U . Alterna-tively, I can be represented as a set of literals I ⊤ ∪ ¬ I ⊥ . The first representationis usual in the context of the WCS while the second is usual in the context ofAnswer Set Programming (ASP) [1]. We will use them interchangeably.Three-valued interpretations can be ordered either by knowledge or by truth :Given two interpretations I = h I ⊤ , I ⊥ i and J = h J ⊤ , J ⊥ i , we say that I containsless knowledge than J , in symbols I ⊆ k J , iff I ⊤ ⊆ J ⊤ and I ⊥ ⊆ J ⊥ iff ( I ⊤ ∪¬ I ⊥ ) ⊆ ( J ⊤ ∪ ¬ J ⊥ ). In other words, I and J agree in all atoms which are knownin I , but I can have more unknown atoms. On the other hand, when the truthorder is applied, i.e. ⊥ ≤ U ≤ ⊤ , then, given two interpretations I = h I ⊤ , I ⊥ i and J = h J ⊤ , J ⊥ i , we say that I contains less truth than J , in symbols I ⊆ t J , iff I ⊤ ⊆ J ⊤ and J ⊥ ⊆ I ⊥ iff ( I ⊤ ∪ ¬ J ⊥ ) ⊆ ( J ⊤ ∪ ¬ I ⊥ ). As we are only interestedin the knowledge ordering, we will omit the subscript k in the following, andsimply write I ⊆ J when we refer to I ⊆ k J .In this paper, we will consider two different three-valued logics, so we in-troduce some general definitions parametrized by the logic. Given a (three-valued) logic L , a three-valued interpretation I satisfies a formula ϕ , in symbols I | = L ϕ , iff I evaluates ϕ as true, that is I ( ϕ ) = ⊤ . Furthermore, I is calleda (three-valued) model of a theory Γ (where Γ is a set of formulas) under L ,denoted by I | = L Γ , iff I | = L ϕ for all ϕ ∈ Γ . I is a ⊆ -minimal model of Γ ifffor no other model J of Γ , J ⊂ I (ordered according to the knowledge). I isthe ⊆ -least model of Γ iff it is the unique minimal model of Γ . A formula ϕ is valid in L , denoted by | = L ϕ , iff I | = L ϕ for every interpretation I . Furthermore,we write Γ | = L ϕ iff I | = L Γ implies I | = L ϕ for every interpretation I . Fortheories Γ and Γ ′ , we write Γ ≡ L Γ ′ iff every interpretation I satisfies: I | = L Γ n the Relation between WCS and ASP 5 iff I | = L Γ ′ . We will omit the brackets { and } , in case ≡ L is applied to formulas,i.e. we write ϕ ≡ L ϕ ′ iff { ϕ } ≡ L { ϕ ′ } . The distinction between strong and weak negation was first noticed by Nel-son [13] in the context of Intuitionistic Logic and later studied by Vakarelov [16]in the context of Classical Logic. The syntax is obtained from the syntax of Clas-sical Logic extended with a the connective “ ¬ ” standing for strong negation,that is, formulas are built from the set of connectives {∧ , ∨ , ← CL , not , ¬ } . Notethat here classical negation is dentoed by “ not ”. We call them the N -formulas.Accordingly, a N -theory is a set of N -formulas. Evaluation of its connectives, ∧ , ∨ , ← CL , not and ¬ , is given by the corresponding truth tables in Table 1. Inthe sequel, we refer to N -logic if N -formulas or N -theories are considered andevaluated with respect to these truth tables. Note that N -logic is a conservativeextension of classical logic in the sense that, if we restrict ourselves to formulaswithout strong negation (formulas without the ¬ connective), the valid formulasin N -logic and Classical logic are the same. We use | = CL and ≡ CL to denoteentailment and equivalence according to N -logic. Weak negation “ not ” can bedefined in terms of “ ← CL ” by the following equivalence: not ϕ ≡ CL ⊥ ← CL ϕ (4)We will consider weak negation here as connective in its own right because ofits importance for logic programming.It is interesting to note that, every (possibly non-regular) N -formula can berewritten as as an equivalent regular N -formula applying the following equiva-lences taken from Vorob’ev calculus (see Section 2.1 in [14] for more details): ¬¬ ϕ ≡ CL ϕ (5) ¬ not ϕ ≡ CL ϕ (6) ¬ ( ϕ ∧ ψ ) ≡ CL ¬ ϕ ∨ ¬ ψ (7) ¬ ( ϕ ∨ ψ ) ≡ CL ¬ ϕ ∧ ¬ ψ (8) ¬ ( ϕ ← CL ψ ) ≡ CL ¬ ϕ ∧ ψ (9)For any N -formula ϕ , we write reg( ϕ ) for the regular formula obtained from ϕ byapplying the above equivalences. For a theory Γ , by reg( Γ ) def = { reg( ϕ ) (cid:12)(cid:12) ϕ ∈ Γ } we denote the regular theory obtained in the same way. The syntax of the three-valued Lukasiewicz logic introduced in [11] is restrictedto the set of connectives {⊥ , ∧ , ∨ , ← L , ¬ } , that is, by replacing in Classical Logicconnectives “ ← CL ” and “ not ” by “ ← L ” and “ ¬ ”, respectively. We call formulasbuild from these connectives L -formulas. Accordingly, a L -theory is a set of Emmanuelle-Anna Dietz Saldanha and Jorge Fandinno L -formulas. Evaluation of its connectives, ⊥ , ∧ , ∨ , ← L and ¬ , is given by thecorresponding truth tables in Table 1. In the sequel, we refer to the L -logic,if L -formulas or a L -theory are considered and evaluated with respect to thesetruth tables. We use | = L and ≡ L to denote entailment and equivalence accordingto L -logic. A rule r is an expression of the form: ϕ ← ψ (10)where ϕ and ψ are implication-free formulas respectively called the head the body of the rule and L is a literal. Head ( r ) and Body ( r ) denote the set of literalsthat occur in the head and the body of the rule r , respectively. A rule is basic (resp. regular ) iff its body and head are basic (resp. regular). A normal nestedrule iff is a rule whose head ϕ ∈ ( At ∪ ¬ At ) is a literal, an extended rule is anormal nested rules such that its body is either ⊤ , (this rule is called a positivefact), or a conjunction of literals or literals preced by weak negation. A normalrule is an extended nested rules such that its head is a positive literal.A (logic) program P is a set of rules and, for any program P , by at ( P ),we denote the set of all atoms occurring in program P . If it is clear from thecontext, then we assume that At = at ( P ). We also denote by Head ( P ) = { L | L ∈ Head ( r ) and r ∈ P} and Body ( P ) = { L | L ∈ Body ( r ) and r ∈ P} the setof literals occurring in the head and the body of a program P , respectively.A program is called basic (resp. regular , normal , extended or normal nested )iff all its rules are basic (resp. regular, normal, extended or normal nested). Incase that ψ = ⊤ , we will usually write just ϕ instead of ϕ ← ⊤ . We will use ϕ ↔ ψ as an abbreviation for a pair of rules ϕ ← ψ and ψ ← ϕ . For instance,the program { a ↔ b } is an abbreviation for the program { a ← b, b ← a } . We review now and extend the definition of Answer Set Semantics from [10] topossibly non-regular programs. The definition of the Answer Set Semantics tonon-regular programs was first introduced by Pearce [14] using an equilibriumcondition over the models of the logic of here-and-there with strong negation.The formulation we present here is equivalent, but using classical logic withstrong negation and a reduct instead.An interpretation I is said to be closed under some regular program P iffevery ( ϕ ← ψ ) ∈ P satisfies that I | = CL ϕ whenever I | = CL ψ holds. The reduct of a regular formula ϕ with respect to an interpretation I , in symbols ϕ I , is n the Relation between WCS and ASP 7 recursively defined as follows: ϕ I def = ϕ if ϕ is a literal (11)( ϕ ∧ ψ ) I def = ϕ I ∧ ψ I (12)( ϕ ∨ ψ ) I def = ϕ I ∨ ψ I (13)( not ϕ ) I def = ( ⊥ if I | = CL ϕ I ⊤ otherwise (14)The reduct of a rule ( ϕ ← ψ ) I def = ϕ I ← ψ I is obtained by applying the reductto its head and body. The reduct of a logic program is obtained by applyingthe reduct to all its rules, that is P I def = { r I (cid:12)(cid:12) r ∈ P } . Furthermore, asdone for N -theories, we also assign to any program P an equivalent regularprogram reg( P ) def = { reg( r ) (cid:12)(cid:12) r ∈ P } with reg( ϕ ← ψ ) def = reg( ϕ ) ← reg( ψ ) forany rule of the form r = ( ϕ ← ψ ). For any N -formula ϕ , we obtain the regularformula reg( ϕ ) by applying the equivalences (5-8) specified in Section 2.3. Then,answer sets are defined in terms of the regular counterpart of any program. Definition 1.
Given an interpretation I and a program P , I is an answer set of P iff I is an ⊆ -minimal closed interpretation under reg ( P ) I . It is easy to see that, for regular programs, Definition 1 precisely coincidewith the definition of answer set from [10].It is well-known that every answer set of a logic program without strongnegation is also a model in classical logic of the propositional theory obtainedby replacing the logic programming implication ← by classical implication ← CL .We extend this result to the case of logic programs with strong negation byreplacing classical logic by its extension with strong negation. Formally, given alogic program P , by N ( P ) we denote the N -theory resulting of replacing in P each occurrence of ← by ← CL . Proposition 1.
Given an interpretation I and a program P , I is closed under P if and only if I is a model of N ( P ) under N -logic.Proof. Note that I is closed under P iff all (rules) ϕ ← ψ ∈ P satisfy I | = CL ϕ whenever I | = CL ψ iff all ϕ ← ψ ∈ P satisfy either I ( ϕ ) = ⊤ or I ( ψ ) = ⊤ iff all ϕ ← ψ ∈ P satisfy I ( ϕ ← CL ψ ) = ⊤ iff all formulas of the form ϕ ← CL ψ ∈ N ( P )satisfy I ( ϕ ← CL ψ ) = ⊤ iff I is a model of N ( P ) under N -logic. Corollary 1.
Given an interpretation I and a regular program P , I is an answerset of P if and only if I is a ⊆ -minimal model of N ( P I ) under N -logic.Proof. From Definition 1 and Proposition 1, it follows that I is an answer set of P if and only if I is a ⊆ -minimal model of N ( P I ) under N -logic. Note that, since P is regular reg( P ) = P . Furthermore, we also have N (reg( P ′ )) = reg( N ( P ′ )) ≡ CL N ( P ′ ) for every program P ′ . Hence, the statement holds. Emmanuelle-Anna Dietz Saldanha and Jorge Fandinno Proposition 2.
Given an interpretation I and a regular program P , if I is ananswer set of P , then I is a model of N ( P ) under N -logic.Proof (sketch). The proof can be carried out by induction in the structure of theformula by noting that ϕ I is just the result of partially evaluating subformulasfor the form of not ψ with respect to I . Hence, from Proposition 1, it follows I being an answer set of P implies that I is a ( ⊆ -minimal) model of N (reg( P ) I ).Take any formula of the form of ϕ ← CL ψ in P . Then, the formula reg( ϕ ) I ← CL reg( ψ ) I belongs to N (reg( P ) I ) and, since I is a model of N (reg( P ) I ), we getthat I (reg( ϕ ) I ← CL reg( ψ ) I ) = ⊤ . This implies that either I (reg( ϕ ) I ) = ⊤ or I (reg( ψ ) I ) = ⊤ , which in its turn implies that either I ( ϕ ) = ⊤ or I ( ψ ) = ⊤ holds. Consequently, I ( ϕ ← CL ψ ) = ⊤ and I is a model of N ( P ).As may be expected and as the following example shows, the other directionof Proposition 2 does not hold. Example 2. h∅ , ∅i = ∅ is the unique answer set of P = {¬ p ← q } while itscorresponding N -theory N ( P ) = {¬ p ← CL q } has several other models, as forinstance h{ p } , { q }i = { p, ¬ q } .It is also well known that normal nested programs without weak nor strongnegation (usually called positive ) have a unique answer set which coincides withthe ⊆ -minimal classical model (usually called the ⊆ -least model) of the corre-sponding propositional theory (see Proposition 3 in [1]). Similarly, normal nestedprograms without weak negation (i.e. basic) have at most one ⊆ -minimal model(and possibly no model) in classical logic with strong negation. For instance, inthe case of { a, ¬ a } is inconsistent, so it has no ⊆ -least model, Proposition 3.
Given any basic normal nested program P , one of the followingtwo statements holds: P has a unique answer set which is also the ⊆ -least model of N ( P ) , or2. P has no answer set and N ( P ) has no model.Proof. First, note that if N ( P ) has no model, from Proposition 2, we immedi-ately get that P has no answer set. Let us show now that if N ( P ) has a model,then it has a ⊆ -least model. Obviously, since At is finite, if N ( P ) has a model,then it has at least some ⊆ -minimal model. Suppose, for the sake of contradic-tion, that N ( P ) has two different ⊆ -minimal models I and I . Then, there areliterals L and L such that I ( L ) = I ( L ) = ⊤ and I ( L ) = I ( L ) = ⊥ . Let J be an interpretation such that J ( L ) = I ( L ) if I ( L ) = I ( L ), and J ( L ) = U ,otherwise. Then, we have that J ⊂ I and J ⊂ I so J = CL N ( P ). Hence, thereis a formula in N ( P ) is of the form L ← CL ϕ with L a literal and ϕ a basicformula such that J ( L ) = ⊤ and J ( ϕ ) = ⊤ . Furthermore, it can be shown byinduction in the structure of the formula that, for every basic formula ϕ and pairof interpretations J ⊆ I , we have that J ( ϕ ) = ⊤ implies I ( ϕ ) = ⊤ . Hence, we Recall, basic programs only consist of rules, whose head and body are implication-free and have no occurences of weak negation.n the Relation between WCS and ASP 9 have that I ( ϕ ) = ⊤ and I ( ϕ ) = ⊤ which, since I and I are models of N ( P ),implies that I ( L ) = I ( L ) = ⊤ . Hence, by construction we have that J ( L ) = ⊤ ,which is a contradiction with the fact that J ( L ) = ⊤ . Consequently, there is aunique ⊆ -minimal model I . Finally, note that, since P has no weak negation, P I = P and, thus, I is also the unique answer set of P . Formally, given a program P , by L ( P ) we denote the L -theory resulting ofreplacing in P each occurrence of ← by ← L . Furthermore, for a normal nestedprogram P , we say that a literal L is defined in P iff P contains a rule whosehead is L ; otherwise we say that L is undefined in P . The set of rules defining aliteral L (those with L in the head) is denoted as def ( P , L ). The set of all literalsthat are undefined in P is denoted by undef ( P ). We specify the set of definedliterals in P as Head ( P ) = ( At ∪ ¬ At ) \ undef ( P ). The set srf ( P ) of a normalnested program P , is defined as follows: srf ( P ) def = { L ← ϕ ∨ · · · ∨ ϕ n | L ∈ ( At ∪ ¬ At ) and def ( P , L ) = { L ← ϕ, . . . , L ← ϕ n } 6 ∅} . Below we straightforwardly extend the definition of the weak completion [9] tonormal nested programs, that is, to programs that may contain rules wherethe head is a strongly negated literal. The weak completion of a normal nestedprogram P , denoted wc ( P ), is defined as follows: wc ( P ) def = { L ↔ ϕ | ( L ← ϕ ) ∈ srf ( P ) } Note that wc ( P ) is also a normal nested program because we consider that ϕ ↔ ψ in a program is a shorthand for the two rules ϕ ← ψ and ψ ← ϕ . Definition 2.
An interpretation I is called wc-model of a normal nested pro-gram P iff I is a ⊆ -minimal model of L ( wc ( P )) . Originally WCS was only defined for basic normal programs, extended withrules of the form A ← ⊥ , called (negative) assumption [9]. Here, we will callthese programs, wc-normal programs. Hence, we are only considering programswith one type of negation, which we will show, corresponds to strong negation inthe ASP. Note that, as opposed to the ASP, the WCS is defined in terms of thethree-valued Lukasiewicz logic instead of classical logic with strong negation.[9] showed that wc-normal programs always have a unique wc-model whichcan be computed by the following consequence operator [15]: Given an interpre-tation I and a wc-normal program P , the application of Φ to I and P , denotedby Φ P ( I ), is an interpretation J = h J ⊤ , J ⊥ i defined as follows: J ⊤ = { A | there is A ← Body ∈ P such that I ( Body ) = ⊤} ,J ⊥ = { A | there is A ← Body ∈ P andall A ← Body ∈ P satisfy I ( Body ) = ⊥} . The correspondence of this unique wc-model and the well-founded model [17] forwc-normal programs without positive cycles, has been shown in [4].0 Emmanuelle-Anna Dietz Saldanha and Jorge Fandinno The following example illustrates the WCS by means of two cases of Byrne’ssuppression task from the introduction.
Example 3.
Let P be the wc-normal program consisting of the rules in (1) inthe introduction. L ( wc ( P L -theory: ℓ ↔ L e ∧ ¬ ab e ↔ L ⊥ ab ↔ L ⊥ whose unique wc-model is h∅ , { e, ℓ, ab }i = {¬ e, ¬ ℓ, ¬ ab } . This program illus-trates why assumptions such as e ← ⊥ , though being tautologies in Lukasiewiczlogic, are not tautologies under the WCS: After the the weak completion transfor-mation they become equivalences, e ↔ L ⊥ , and, thus, e has to be false. Note thatassumptions can also be overwritten by facts. Let for instance P = P ∪{ e ← ⊤} be the program obtained by adding the fact e to the above program. Then, itsweak completion L ( wc ( P ℓ ↔ L e ∧ ¬ ab e ↔ L ⊥ ∨ ⊤ ab ↔ L ⊥ As e ↔ L ⊥ ∨ ⊤ ≡ L e ↔ L ⊤ , the unique wc-model of P h{ e, ℓ } , { ab }i = { e, ℓ, ¬ ab } , where e and ℓ are true. Let P be the wc-normal program consistingof the rules in (2) in the introduction. L ( wc ( P L -theory: ℓ ↔ L ( e ∧ ¬ ab ) ∨ ( t ∧ ¬ ab ) e ↔ L ⊥ ab ↔ L ⊥ ab ↔ L ⊥ whose unique wc-model is h∅ , { e, ab , ab }i = {¬ e, ¬ ab , ¬ ab } . That is, e , ab and ab are false, while ℓ and t are unknown. Let us first discuss the main differences between both semantics according tothe two examples of the suppression task in the introduction.
Example 4 (Ex. 3 continued).
Consider the wc-normal program P
1: Its corre-sponding normal program can be obtained by replacing every assumption of theform A ← ⊥ in P ¬ A . The resulting program, P consists of the following rules: ℓ ← e ∧ ¬ ab ¬ e ¬ ab (15)Its unique answer set is h∅ , { e, ab }i = {¬ e, ¬ ab } , which does not coincide withthe wc-model of P
1, as ℓ is false under the WCS, but unknown under the ASP.The above example illustrates that replacing assumptions by strong negationfacts is not enough to obtain the same results between WCS and ASP.As mentioned previously the ASP and WCS can be respectively definedin terms of classical logic with strong negation and three-valued Lukasiewiczlogic. Interestingly, Vakarelov [16] showed that there is a correspondence between L -logic and N -logic, in the sense that all connectives of one logic are definable n the Relation between WCS and ASP 11 in terms of the other one. In particular, here we are interested in translatingfrom the WCS to the ASP and, thus, that implies a translation from L -logic to N -logic. Formally, given a L -theory Γ , by ˜ N ( Γ ) we denote the result of replacingin Γ every occurrence of ϕ ← L ψ by ( ϕ ← CL ψ ) ∧ ( ¬ ψ ← CL ¬ ϕ ). Theorem 1 (Theorem 11 in [16]).
Given any L -theory Γ , an interpretation I is a model of Γ under L -logic iff I is a model of N ( Γ ) under N -logic. Based on this result, we can establish the correspondence between the ASPand the WCS.We need rules that negatively complete the information of the given program.Let us now formalize this idea by defining the definition completion of a program.
Definition 3.
Given a normal nested program P , its definition completion isdefined as follows: ˜ P def = P ∪ {¬ L ← ¬ ϕ | ( L ← ϕ ) ∈ srf ( P ) } (16)Let us apply the suggested characterization for the programs in Example 4. Example 5 (Ex. 4 continued).
Given P
4, its definition completion is as follows:˜ P P ∪ { ¬ ℓ ← ¬ ( e ∧ ¬ ab ) , ¬¬ e ← ¬⊤ , ¬¬ ab ← ¬⊤ } Note that ¬ ℓ ← ( ¬ e ∧ ¬ ab ) is equivalent to ¬ ℓ ← ¬ e ∨ ab , while the last tworules are tautologies under the ASP. The unique answer set of ˜ P P We will now introduce some auxiliary results that will help us to show the corre-spondence between WCS and ASP. Let us start by showing that the answer setsof any program coincide with the answer sets of its weak completion. The proofof this statement relies on the the following lemma which is a straightforwardlifting of the Completion Lemma from [6, p. 23] to the class of programs withstrong negation.
Lemma 1.
Let P be any program, let At be any set of atoms (not necessar-ily equal to at ( P ) ) and let Q ⊆ ( At ∪ ¬ At ) be any set of literals such that Q ∩ Head( P ) = ∅ . Let ϕ L be some implication-free formula for each literal L ∈ Q and I be an interpretation. Then, the following two statements are equivalent:1. I is an answer set of P ∪ { L ← ϕ L (cid:12)(cid:12) L ∈ Q } .2. I is an answer set of P ∪ { L ↔ ϕ L (cid:12)(cid:12) L ∈ Q } . Proposition 4.
Given any normal nested program P , an interpretation I is ananswer set of P if and only if I is an answer set of wc ( P ) . and Jorge Fandinno Proof.
Assume that P is regular. From item (ii) of Proposition 6 in [10], P and srf ( P ) have the same answer sets. Note that there is a unique rule with head L for each literal L ∈ ( At ∪ ¬ At ) in srf ( P ). Hence, wc ( P ) is obtained by replacingall occurrences of ← in srf ( P ) by ↔ and, from Lemma 1 (by taking P = ∅ ), srf ( P ) and wc ( P ) have the same answer sets. In case that P is not regular, wehave that I is an answer set of P iff I is an answer set of reg( P ) iff I is an answerset of wc (reg( P )) = reg( wc ( P )) iff I is an answer set of wc ( P ). Lemma 2.
Given any normal nested logic program P , an interpretation I is amodel of N ( wc ( ˜ P )) under N -logic if and only if I is a model of L ( wc ( P )) .Proof. Note that N ( wc ( ˜ P )) has a pair of equivalences of the form A ↔ CL ϕ ∨ · · · ∨ ϕ n (17) ¬ A ↔ CL ¬ ϕ ∧ · · · ∧ ¬ ϕ n (18)for each A ∈ At with def ( P , L ) = { A ← ϕ , . . . , A ← ϕ n } 6 = ∅ . On the otherhand, we have that L ( wc ( P )) has an equivalences of the form A ↔ L ϕ ∨ · · · ∨ ϕ n (19)for each atom A ∈ At with def ( P , L ) = { A ← ϕ , . . . , A ← ϕ n } 6 = ∅ . By Theo-rem 1, the models of L ( wc ( P )) under L -logic and the models of N ( L ( wc ( P )))under N -logic coincide. Note now that, by definition, we have that (19) is equiv-alent to the following formula( A ← L ϕ ∨ . . . ∨ ϕ n ) ∧ ( ϕ ∨ . . . ∨ ϕ n ← L A ) (20)Hence, N ( L ( wc ( P ))) contains a formula of the form ψ A ∧ ψ A ∧ ψ A ∧ ψ A for eachatom A ∈ At with def ( P , A ) = { A ← ϕ , . . . , A ← ϕ n } 6 = ∅ where ψ A def = A ← CL ϕ ∨ · · · ∨ ϕ n (21) ψ A def = ¬ A ← CL ¬ ( ϕ ∧ · · · ∧ ϕ n ) (22) ψ A def = ϕ ∨ · · · ∨ ϕ n ← CL A (23) ψ A def = ¬ ( ϕ ∧ · · · ∧ ϕ n ) ← CL ¬ A (24)Note that, by definition, ψ A ∧ ψ A is equivalent to (17). Besides ψ A ∧ ψ A can beequivalently rewritten as ¬ A ↔ CL ¬ ( ϕ ∨ . . . ∨ ϕ n ) (25)that is equivalent to (18), i.e. L ( wc ( P )) and N ( wc ( ˜ P )) have the same models. Proposition 5.
Given any normal nested program P and interpretation I , if I is an answer set of ˜ P , then I is a model of L ( wc ( P )) .Proof. From Lemma 2, I is a model of N ( wc ( ˜ P )) iff I is a model of L ( wc ( P )).From Proposition 4, the answer sets of ˜ P and wc ( ˜ P ) are the same. Furthermore,from Proposition 2, the answer sets of wc ( ˜ P ) are models of N ( wc ( ˜ P )) under N -logic. Hence, the answer sets of ˜ P are wc-models of P . n the Relation between WCS and ASP 13 Given Lemma 2, Proposition 3,4 and 5, we can now show how the definitioncompletion of a program precisely characterizes the WCS in terms of the ASP.
Theorem 2.
Given any wc-normal program P and interpretation I , the follow-ing two statements are equivalent:1. I is the unique wc-model of P ,2. I is the unique answer set of ˜ P .The following two statements are also equivalent:1. P has no wc-model,2. ˜ P has no answer set.Proof. Assume that N ( wc ( ˜ P )) has no model. From Lemma 2, it follows that L ( wc ( P )) has no model either and, thus, there is no wc-model of P . Besides,from Proposition 5, the lack of model of L ( wc ( P )) also implies that ˜ P has noanswer set. Otherwise, N ( wc ( ˜ P )) has a model and, since is a wc-normal programand thus basic, from Proposition 3, we get that there is an interpretation I whichthe ⊆ -least model of N ( wc ( ˜ P )) and, thus, the unique answer set of wc ( ˜ P ). FromProposition 4 this implies that I is the unique answer set of ˜ P . Furthermore,from Lemma 2, this also implies that it is the ⊆ -least model of L ( wc ( P )) and,thus, the unique wc-model of P . We have shown how logic programs under the Weak Completion Semantics canbe translated into logic programs under the Answer Set Semantics by using the definition completion . This completion adds rules supporting the strong negationof a defined atom whenever all the bodies of all rules defining it are false. Thistransformation has been illustrated by two examples of Byrne’s suppression task.This result allows us to use all the knowledge representation features of An-swer Set Programming, including default negation , in combination with this com-pletion and opens two interesting future possibilities: On the one hand, in [5],logic programs under the Weak Completion Semantics were extended with acontext operator to capture negation as failure. Hence, an immediate questionis whether these contextual logic programs can also be translated into logic pro-grams under the Answer Set Semantics by using default negation. On the otherhand, it would be interesting to investigate how the twelve cases of the sup-pression task could be represented by means of default negation or the contextoperator in order to ensure elaboration tolerance [12].Another interesting observation is that the proof of the correspondence be-tween these two semantics relies on the use of strong negation as a connectivein its own right, as opposed to the usual convention of considering that strongnegation can only be applied to atoms. This extension was first considered byDavid Pearce in [14]. It would be worth to investigate how the usual propertiesof the Answer Set Semantics can be extended to this class of programs and howits use can ease other knowledge representation problems. and Jorge Fandinno Acknowledgements.
We are thankful to David Pearce for pointing to Vakarelov’swork on the relation between Lukasiewicz logic and classical logic with strongnegation.