Properties of Stable Model Semantics Extensions

05/15/2014
by   Mário Abrantes, et al.
0

The stable model (SM) semantics lacks the properties of existence, relevance and cumulativity. If we prospectively consider the class of conservative extensions of SM semantics (i.e., semantics that for each normal logic program P retrieve a superset of the set of stable models of P), one may wander how do the semantics of this class behave in what concerns the aforementioned properties. That is the type of issue dealt with in this paper. We define a large class of conservative extensions of the SM semantics, dubbed affix stable model semantics, ASM, and study the above referred properties into two non-disjoint subfamilies of the class ASM, here dubbed ASMh and ASMm. From this study a number of results stem which facilitate the assessment of semantics in the class ASMh U ASMm with respect to the properties of existence, relevance and cumulativity, whilst unveiling relations among these properties. As a result of the approach taken in our work, light is shed on the characterization of the SM semantics, as we show that the properties of (lack of) existence and (lack of) cautious monotony are equivalent, which opposes statements on this issue that may be found in the literature; we also characterize the relevance failure of SM semantics in a more clear way than usually stated in the literature.

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1 Introduction

The semantics [Gelfond and Lifschitz (1988)] is generally accepted by the scientific community working on logic programs semantics as the de facto standard 2-valued semantics. Nevertheless there are some advantageous properties the SM semantics lacks such as (1) model existence for every , (2) relevance, and (3) cumulativity [Pinto and Pereira (2011)]. Model existence guarantees that every has a semantics. This is important to allow arbitrary updates and/or merges involving Knowledge Bases, possibly from different authors or sources [Pinto and Pereira (2011)]. Relevance allows for top-down query solving without the need to always compute complete models, but just the sub-models that sustain the answer to a query, though guaranteed extendable to whole ones [Pinto and Pereira (2011)]. As for cumulativity, it allows the programmer to take advantage of tabling techniques [Swift (1999)] for speeding up computations [Pinto and Pereira (2011)]. Independently of the motivations that underlay the design of a semantics for logic programs, one may ask if it is easy to guarantee some or all of the above properties, or even if it is easy to assess the profile of the resulting semantics in what concerns these properties. In this work we define a family of -valued conservative extensions of the semantics, the affix stable model semantics family, . We then take two subclasses, and , and present a number of results that simplify the task of assessing the semantics in on the properties of existence, relevance and cumulativity. The semantics in these two classes bear resemblance with the already known and semantics (see section 3), and this stands for the motivation to consider them. The following results, obtained in this work, should be emphasized: (1) We present a refined definition of cumulativity for semantics in the class , which turns into an easier job the dismissal of this property by resorting to counter-examples; (2) We divide the sets of rules of into layers, and use the decomposition of models into that layered structure to define three new (structural) properties, defectivity, excessiveness and irregularity, which allow to state a number of relations between the properties of existence, relevance and cumulativity for semantics of the class, and at the same time facilitate the assessment of semantics in this class with respect to those properties; (3) As a result of the approach in our work light is shed on the characterization of semantics, as we show that the properties of (lack of) existence and (lack of) cautious monotony are equivalent, which opposes statements on this issue that may be found in the literature; we also characterize the relevance failure of semantics in a more clear way than usually stated in the literature. It should be stressed that this study is on the properties of a class of 2-valued semantics, under a prospection motivation. The weighing of such semantics rationales under an ‘intuitive’ point of view (or any other equivalently non-objective concept) is beyond the reach of our study. The results presented in this paper are enounced for the universe of finite ground normal logic programs, and are either proved in [Abrantes (2013)], or immediate consequences of results there contained.
The remainder of the paper proceeds as follows. In section 2 we define the language of and the terminology to be used in the sequel. In section 3 the families , and are defined. In section 4 we characterize the property of cumulativity for the families and , whilst in section 5 the properties of defectivity, excessiveness and irregularity are defined. Some relations among existence, relevance and cumulativity, which are revealed by means of those properties, are stated. Section 6 is dedicated to final remarks.

2 Language and Terminology of Logic Programs

A defined over a language is a set of normal rules, each of the form

(1)

where are non-negative integers and are atoms of ; and are generically designated literals, being specifically designated default literal. The operator ‘’ stands for the conjunctive connective, the operator ‘’ stands for negation by default and the operator ‘’ stands for a dependency operator that establishes a dependence of on the conjunction on the right side of ‘’. is the head of the rule and is the body of the rule. A rule is a fact if . A literal (or a program) is ground if it does not contain variables. The set of all ground atoms of a is called Herbrand base of , . A program is finite if it has a finite number of rules111In this work, if nothing to the contrary is said, by ‘logic program’, or simply by ‘program’, we mean a finite set of normal ground rules.. Given a program , program is a subprogram of if , where are envisaged as sets of rules.

For ease of exposition we henceforth use the following abbreviations: , is the set of all atoms that appear in the ground structure , where can be a rule, a set of rules or a set of logic expressions; , is the set of literals in the body of a ground rule ; , is the set of all facts that appear in the set of rules ; , is the set of all atoms that appear in the heads of the set of rules ; if is unitary, we may use ‘’ instead of ‘’ . We may compound some of these abbreviations, as for instance whose meaning is straightforward. Each of the abbreviations may also be taken as the conjunction of the elements contained in the respective sets.

Given a -valued interpretation of a logic program , we represent by (resp. ) the set of its positive literals (resp. atoms whose default negations are true with respect to ). If is -valued, we additionally represent by the set of undefined atoms with respect to .

The following concepts concern the structure of programs. Let be a logic program and any two rules of . Complete rule graph, 222Adapted from [Pinto and Pereira (2011)].: is the directed graph whose vertices are the rules of . Two vertices representing rules and define an arc from to iff . Rule depending on a rule: rule depends on rule iff there is a directed path in from to . Subprogram relevant to an atom333Adapted from [Dix (1995b)]: a rule is relevant to an atom iff there is a rule such that and depends on . The set of all rules of relevant to is represented by , and is named subprogram (of ) relevant to . Loop444Adapted from [Costantini (1995)]: a set of rules forms a loop (or the rules of set are in loop) iff, for any two rules , depends on and depends on . We say that rule is in loop through literal iff there is a rule such that . Rule layering: the rule layering (or just layering, for simplicity) of is the labeling of each rule with the smallest possible natural number, , in the following way: for any two rules and , (1) if rules are in loop, then ; (2) if rule depends on rule but rule does not depend on rule , then . Every integer number in the image of the layer function defines a layer of , meaning the set of rules of labeled with number – we use the expression ‘layer’ to refer both to a set of all rules with that label, and to the label itself. We represent by (resp. ) the set of all rules of whose layer is less than or equal to (resp. greater than) . -segment of a program: we say that is the T-segment of iff . We may also say ‘segment ’ to mean the set of rules corresponding to segment .

Let be a 2-valued semantics and the set of models of a logic program . Let also the set of atoms be dubbed semantic kernel of with respect to (the semantic kernel is not defined if ). The following properties concern semantics of logic programs. We say that a semantics is: Existential iff every has at least one model; Cautious monotonic555 Adapted from [Dix (1995a), Dix (1995b)]. iff for every , and for every set , we have ; Cut iff for every , and for every set , we have ; Cumulative iff it is cautious monotonic and cut; Relevant iff for every we have

(2)

where is the subprogram of relevant to atom ; Global to local relevant iff the logical entailment ’’ stands in formula (2); Local to global relevant iff the logical entailment ’’ stands in formula (2).

3 Conservative Extensions of the Semantics

In this section we define a family of abductive 2-valued semantics666See [Denecker and Kakas (2002)] for abductive semantics., the affix stable model family, , whose members are conservative extensions of the semantics. For that purpose we need the concepts of reduction system and MH semantics.

3.1 Reduction System and Semantics

In [Brass et al. (2001)] the authors propose a set of five operations to reduce a program (i.e., eliminate rules or literals) – positive reduction, PR, negative reduction, NR, success, S, failure, F and loop detection, L (see A for the definitions of these operations). We represent this set of operations as . By non-deterministically applying this set of operations on a program , we obtain the program , the remainder of , which is invariant under a further application of any of the five operations. This transformation is terminating and confluent [Brass et al. (2001)]. We denote the transformation of into as . We also write . It is shown in [Brass et al. (2001)] that , where stands for the well-founded model [Gelder (1993)]. See B for an example of the computation of the remainder of a program.

One way to obtain conservative extensions of the semantics, is to relax some operations of the reduction system , which yields weaker reduction systems, that is, systems that erase less rules or literals than . An example of such a semantics is the minimal hypotheses semantics, [Pinto and Pereira (2011)], whose reduction system is obtained from by replacing the negative reduction operation, , by the layered negative reduction operation, , i.e., . is a weaker version of that instead of eliminating any rule containing say in the body, in the presence of the fact , as does, only eliminates rule if this rule is not in loop through literal . We write , where is the layered remainder of . We also write . See C for an example of the computation of the layered remainder of a program.

3.2 and Families

We define affix stable interpretation and then use this concept to put forward the definition of family.

Affix Stable Interpretation.def:stbintLet P be a , a 2-valued semantics with a corresponding reduction system , and . We say that is an affix stable interpretation of P with respect to set X and semantics (or simply a stable interpretation with affix X) iff and 777Notice that is the set of undefined atoms in the model . that is, is the only stable model of the program . We name an affix (or hypotheses set) of interpretation . We also name assumable hypotheses set of program , , the union of all possible affixes that may be considered to define the stable interpretations (we have ).

Affix Stable Model Semantics Family, .def:afxsmA 2-valued semantics , with a corresponding reduction system , belongs to the affix stable model semantics family, , iff, given any , contains all the models of , in case they exist, plus a subset (possibly empty) of the affix stable interpretations of chosen by resorting to specifically enounced criteria.

Both semantics and belong to the family. The two non-disjoint subfamilies of next defined, and , will be the classes whose formal properties we study in the sequel.

and Families.def:ASMhmA semantics belongs to the or families iff, for any , the models are computed as follows:

  1. For both and the set of assumable hypotheses, , is contained in the set of atoms that appear default negated in 888The purpose of computing the remainder of a program, is to obtain the assumable hypotheses set of the program.;

  2. For semantics in the class , the affixes of the models of are either those non empty minimal with respect to set inclusion, if , or else the empty set if . For semantics in the class , the models in are always minimal models.

We now refer some examples of and members, whose definitions can be found in D. Besides , and others, the following are family members, referred to subsequently:999The first three semantics were suggested by Alexandre Pinto. , , , , . Besides and others, the following are family members, referred to subsequently: , , , .

4 Characterization of Cumulativity for the Class

In this section we lay down a characterization of cumulativity for semantics of the class, via the following theorem.

teo:equalcumLet be a semantics of the class. For every program and for every subset , the following results stand: (1) is cautious monotonic iff ; (2) is cut iff ; (3) is cumulative iff – this is a consequence of statements (1) and (2).101010Notice that represents the set of all models of .

The three items of this theorem correspond to refinements of the classical definitions of cautious monotony, cut and cumulativity (see section 2). The new definitions establish the properties by means of relations among sets of models, as opposed to the relations among sets of atoms that characterize the classical definitions.

The results stated in this theorem are advantageous to spot cumulativity failure in semantics of the class by means of counter-examples (logic programs), when compared with common procedures (e.g., [Dix (1995a), Dix (1995b)]). The reason is that common procedures always need the counter-examples to fail cumulativity111111The general procedure to spot the failure of cumulativity by resorting to counter-examples is as follows: compute all the models of a program ; add to subsets , and compute all the models of the resulting programs , drawing a conclusion about cumulativity failure only in cases where ., whilst the results of theorem LABEL:teo:equalcum allow us to spot failure of cumulativity even in some cases where the counter-examples used do not show any failure of this property. To make this point clear see the examples in E and F.

It should be stressed that there are -valued cumulative semantics to which for some and some (for an example, see the definition of the 2-valued semantics Picky in G). Theorem LABEL:teo:equalcum states this is not the case if .

5 Defectivity, Excessiveness and Irregularity

Theorem LABEL:teo:equalcum application for dismissing the cumulativity property by means of counter-examples, demands computing the set of models of a program , the set , and after this it needs the computation of the sets of models , , to look for a case that eventually makes false. In this section three structural properties are defined, defectivity, excessiveness and irregularity, that will turn the dismissal of existence, relevance or cumulativity spottable by means of one model only. It will be shown that for semantics of the class, defectivity is equivalent to the failure of existence and to the failure of global to local relevance, and also entails the failure of cautious monotony, whilst excessiveness entails the failure of cut, and irregularity is equivalent to the failure of local to global relevance.

5.1 Defectivity

The rationale for the concept of defective semantics is the following: if a segment has a model that is not contained in any whole model of , then we say the semantics is defective, in the sense that it ‘does not use’ all the models of segment in order to get whole models of .

Defective semantics.def:defectiveA 2-valued semantics is called defective iff there is a , , a segment of , and a model of the segment , such that . We also say that is defective with respect to segment of program , and that is a defective model of with respect to segment and semantics .

ex:smnotexistProgram may be used to show that the semantics is defective. In fact, the only model of is with affix . Meanwhile, is a segment that has the stable model , and we have .

The next theorem shows how conclusions about existence, relevance and cumulativity may be immediately taken in the case of a defective semantics.

teo:equivdefectThe following relations are valid for any semantics of the class:

The reader should notice the importance of this theorem: not only defectivity is enough to dismiss existence, relevance and cumulativity, as also these properties appear strongly related for semantics of the class : if existence fails then relevance also fails (through global to local relevance failure); if existence fails then cumulativity also fails (through cautious monotony failure); if relevance fails (through global to local relevance failure), then cumulativity also fails (through cautious monotony failure). Definition LABEL:def:defective above shows the structural nature of defectivity, which allows the verification of the property by wisely constructing a program that satisfies it. This may turn easier the assessment of existence, relevance and cumulativity, when compared to dealing with this issue on the basis of abstract proofs. Even more, the relation between existence and defectivity stated in theorem LABEL:teo:equivdefect, allows the failure of the existence property to be detected by resorting to counter-examples, even in some cases where the program used as counter-example has models. E.g., program in E can be used to detect the failure of existence for semantics, in spite of the existence of stable models for program , since it reveals the defectivity of .121212It should be pointed out that there are 2-valued semantics for which the equivalence fails, e.g., [Apt et al. (1988)] which is not defective in spite of failing the existence property – it is the case that is not a semantics, since it does not conservatively extend the semantics.

The results stated in theorem LABEL:teo:equivdefect also shed some light on the characterization of semantics with respect to the properties of existence and cumulativity. In [Dix (1995b)], section , the author says that the is not cumulative and that this fact does not depend on the non existence of stable models (i.e., the author states that lack of cumulativity is not a consequence of lack of existence). Meanwhile theorem LABEL:teo:equivdefect above shows that is non-existential due to being defective, which in turn makes it not cautious monotonic and thus not cumulative. Thus the failure of cumulativity for the semantics case is indeed a consequence of the failure of existence for this semantics. Moreover, with respect to the semantics a stronger result relating existence and cautious monotony may be enounced: these two properties show up equivalence in the sense stated in proposition LABEL:prop:smexistcm below. To the best of our knowledge, this connection between these two properties had not yet been stated.

prop:smexistcmFor the semantics the following result stands: there is a program that shows existence failure iff there is a program that shows cautious monotony failure.

5.2 Excessiveness and Irregularity

The rationale of the concept of excessive semantics is the following: if a has a model and a layer such that for every model it is the case that , then we say that model (and thus the semantics) is excessive, in the sense that it ‘goes beyond’ the semantics of the segment by not being a ‘consequence’ of it.

Excessive semantics.def:excessiveA 2-valued semantics is called excessive iff there is a logic program , a segment , a model and a model such that:

  1. , where ;

  2. For every model it is the case that ;

  3. There is at last a model of , such that .

We also say that is excessive with respect to segment of program , and that is an excessive model of with respect to segment and semantics .

In the excessiveness example in H it is shown that the semantics are excessive.

The rationale of the concept of irregularity is as follows: given a certain whole model , if the set is not a model of a segment , then we say that is irregular, since ‘is not a consequence’ of the semantics of segment .

Irregular semantics.def:irregularA 2-valued semantics is called irregular iff there is a logic program , a segment and a model of , such that for no model of do we have , where . We also say that is irregular with respect to segment of program , and that is an irregular model of with respect to segment and semantics . A model that is not irregular is called regular, and a semantics that produces only regular models is called regular.131313In comparing excessiveness and irregularity, notice that a whole model can be excessive whilst containing models for all the segments of the program (i.e., be a regular model) - see the excessiveness example in H.

The concepts of excessiveness and irregularity exhibit independence for semantics of the class, meaning there is a semantics in this class for any of the four possible cases of validity or failure of excessiveness and irregularity. As a matter of fact, it can be shown [Abrantes (2013)] that is irregular whilst not excessive (i.e., ); it is also the case that is excessive but not irregular (i.e., ). Also is excessive and irregular, and is not excessive and is not irregular.

The following result states relations between excessiveness and cut, and between irregularity and relevance.

teo:cutexcelgr The following relations stand for any semantics of the class:

As excessiveness and irregularity are structural properties, being thus detectable by construction of adequate programs, they facilitate, via this theorem, the dismissal of cut and relevance. For instance, this result together with the excessiveness example in H, shows that semantics , , , and are excessive, and thus not cut. Also, this result together with the irregularity example in H, shows that semantics , and , , and are irregular, and thus not relevant. As was the case for the relation between the properties of existence and cumulativity for the semantics, our work sheds also some light on the semantics relevance failure, through the following results.

prop:smnotexcLet be a and . Then is neither excessive nor irregular. cor:smvacrel is (vacuously) local to global relevant.

Notice that this corollary, together with the example in E and theorem LABEL:teo:equivdefect, let clear the cause for semantics relevance failure: fails relevance because it fails global to local relevance. This is a more precise characterization than just saying that is not relevant, as usually stated in literature (e.g., [Dix (1995b)]).

If we consider the five formal properties of existence (), global to local relevance (), local to global relevance (), cautious monotony () and cut (), the validity or failure of each of these properties allow, in the general case, the existence of types of semantics. Meanwhile, the study we present in this work shows that only such types of semantics may exist in the class. They are represented in table 1 in I.

6 Final Remarks

In this paper we considered the characterization of 2-valued conservative extensions of the semantics on the properties of existence, relevance and cumulativity. This theoretical endeavor is reasonable under a point of view of prospectively assessing the behavior of such types of semantics with respect to a set of properties that are desirable, both under a computational (relevance and cumulativity) and a semantical (existence) standpoint. For that purpose we focused our study on two subsets of the here defined class of 2-valued conservative extensions of the semantics, the non-disjoint classes and , whose elements maintain a degree of resemblance with already known 2-valued semantics, such as the and the semantics. As a result of this study, refined definitions of cautious monotony, cut and cumulativity were set. This new definitions turn into an easier job the dismissal of the properties of existence, relevance and cumulativity, as shown in section 4. This study also reveals relations among these properties, unveiled by theorems LABEL:teo:equivdefect and LABEL:teo:cutexcelgr, that allow to draw conclusions about some of them on basis of held knowledge about others. This last point builds on top of the new structural properties of defectivity, excessiveness and irregularity, which provide an analytical shortcut to assess existence, relevance and cumulativity. The approach taken in this work (characterizing families of semantics, instead of individual semantics), revealed itself advantageous also in clarifying the profile of the well known and studied semantics, via the results stated in proposition LABEL:prop:smexistcm and corollary LABEL:cor:smvacrel. Our work also states a maximum of 12 types of semantics in the class , with respect to the satisfaction/failure of the properties of existence (), global to local relevance (), local to global relevance (), cautious monotony () and cut ().

Finally, the structural approach put forward in this paper has the potential of being used with semantics other than 2-valued ones, and with other strong and weak properties besides existence, relevance or cumulativity.141414The terms strong and weak applied to formal properties, are here adopted after [Dix (1995a), Dix (1995b)].

Acknowledgments

We thank Alexandre Pinto for some important debates on conservative extensions of the semantics. The work on this paper has been partially supported by Fundação para a Ciência e Tecnologia and Instituto Politécnico de Bragança grant PROTEC : SFHR/49747/2009.

References

  • Abrantes (2013) Abrantes, M. 2013. Revision based total semantics for extended normal logic programs. Ph.D. thesis, Universidade Nova de Lisboa.
  • Apt et al. (1988) Apt, K., Blair, H. A., and Walker, A. 1988. Towards a theory of declarative knowledge. In Foundations of deductive databases and logic programming, J. Minker, Ed. Morgan Kaufmann, Los Altos, CA, 89–142.
  • Brass et al. (2001) Brass, S., Dix, J., Freitag, B., and Zukowski, U. 2001. Transformation-based bottom-up computation of the well-founded model. TPLP, 497–538.
  • Costantini (1995) Costantini, S. 1995. Contributions to the stable model semantics of logic programs with negation. Theoretical Computer Science 149, 2 (2 Oct.), 231–255.
  • Denecker and Kakas (2002) Denecker, M. and Kakas, A. C. 2002. Abduction in logic programming. In Computational Logic: Logic Programming and Beyond’02. 402–436.
  • Dix (1995a) Dix, J. 1995a. A classification theory of semantics of normal logic programs: I. strong properties. Fundam. Inform 22, 3, 227–255.
  • Dix (1995b) Dix, J. 1995b. A classification theory of semantics of normal logic programs: II. weak properties. Fundam. Inform 22, 3, 257–288.
  • Gelder (1993) Gelder, A. V. 1993. The alternating fixpoint of logic programs with negation. J. of Comp. System Sciences 47, 1, 185–221.
  • Gelfond and Lifschitz (1988) Gelfond, M. and Lifschitz, V. 1988. The stable model semantics for logic programming. In ICLP/SLP. MIT Press, 1070–1080.
  • Pinto and Pereira (2011) Pinto, A. M. and Pereira, L. M. 2011. Each normal logic program has a 2-valued minimal hypotheses semantics. INAP 2011, CoRR abs/1108.5766.
  • Swift (1999) Swift, T. 1999. Tabling for non-monotonic programming. Ann. Math. Artif. Intell 25, 3-4, 201–240.

Appendix A Reduction Operations

In the definitions below, and are two ground logic programs.

  1. Positive reduction, PR. Program results from by positive reduction iff there is a rule and a default literal such that , and .

  2. Negative reduction, NR. Program results from by negative reduction iff there is a rule and a default literal such that , and .

  3. Success, S. Program results from by success iff there is a rule and a fact such that , and .

  4. Failure, F. Program results from by failure iff there is a rule and a positive literal such that , and .

  5. Loop Detection, L. Program results from by loop detection iff there is a set of ground atoms such that:

    1. For each rule , if , then ;

    2. .

Appendix B Remainder Computation Example

Let be the set of all rules below. The remainder is the non shadowed part of the program. The labels (i)–(v) indicate the operations used in the corresponding reductions: (i) PR, (ii) NR, (iii) S, (iv) F, (v) L.

Appendix C Minimal Hypotheses Models Computation

Let be the set of rules below, which is equal to the program in B. The layered remainder is the non-shadowed part of the program.

Notice that rule is no longer eliminated by the fact , since this rule and rule are in loop, and in the case of rule the loop is through the literal .
The models of a program are computed as follows: (1) Take as assumable hypotheses set, , the set of all atoms that appear default negated in ; in the case of the previous program we have ; (2) Form all programs , for all possible subsets , (if , then is the only set to consider); take all the interpretations for which is a total model (meaning a model that has no undefined literals); is the hypotheses set of the interpretation ; (3) Take all the interpretations obtained in the previous point, and chose as models the ones that have minimal sets with respect to set inclusion. The models of program in the example above, and the corresponding hypotheses sets, are

Notice that is the only model of . The reduction system keeps some loops intact, which are used as choice devices for generating models, allowing us to have . The sets considered may be taken as abductive explanations [Denecker and Kakas (2002)] for the corresponding models.

Appendix D Definitions of some Elements of and Families

Besides and others, the following are family members.
: the reduction system is obtained by replacing the success operation in by the layered success operation;151515Layered success is an operation proposed by Alexandre Pinto. It weakens the operation of success by allowing it to be performed only in the cases where the rule , whose body contains the positive literal to be erased, is not involved in a loop through literal . models are computed as in the case.
: the reduction system is ; the assumable hypotheses set of a program , , is formed by the atoms that appear default negated in literals involved in loops in the layered remainder ; models are computed as in the case.
: the reduction system is ; models are computed as in the case with the following additional condition: if is a set of hypotheses of a model of , then

that is, no single hypothesis may be defined in the well-founded model if we join to all the other remaining hypotheses.
: the reduction system is ; retrieves the minimal models contained in for any . also belongs to the family, due to the minimality of its models.
: the reduction system is ; retrieves the same models as , except for the irregular ones (cf. Definition LABEL:def:irregular).

Besides , (defined above in this appendix) and others, the following are family members.
: the reduction system is . Given a , contains all the minimal models of .161616See definition of in subsection 3.1.
: the reduction system is . Given a , contains all the models in where is obtained after terminating the following algorithm:171717This algorithm is presented in [Dix (1995a)].
(a) Compute ;
(b) Compute ;
(c) If , then let be the new designation of program ; go to step (a).
Repeat steps (a) – (c) until comes false in (c).
: the reduction system is . Given a , compute through the steps of computation, but taking only the regular models (cf. Definition LABEL:def:irregular) to compute the semantic kernel at steps (a) and (b).
: the reduction system is . Given a , contains all the minimal models of that have the smallest (with respect to set inclusion) subsets of classically unsupported atoms.181818Given a logic program , a model of and an atom , we say that is classically unsupported by iff there is no rule such that and all literals in are true with respect to .

Appendix E Example of Cumulativity Failure Detection

The following -layer program is a counter-example for showing, using theorem LABEL:teo:equalcum, that semantics is not cumulative, due to being not cautious monotonic (program does not allow us to spot the failure of any of these properties by means of the usual definitions of cumulativity and cautious monotony presented in section 2) .

In fact, the models of are and , and thus . Now has the stable models , and , and thus . Hence no negative conclusion can be afforded about cumulativity, by means of the usual definition of this property. Meanwhile, by using the statement (3) of theorem LABEL:teo:equalcum it is straightforward to conclude that semantics does not enjoy the property of cumulativity, because . Moreover, statement (1) of the theorem tells us, via this example, that semantics is not cautious monotonic because .

Appendix F Proof of Cautious Monotony and Cut Failure

The following -layer program is a counter-example for showing, using theorem LABEL:teo:equalcum, that none of the semantics , , , and is either cautious monotonic or cut (program does not allow us to spot the failure of any of these properties by means of the usual definitions of cautious monotony and cut presented in section 2) .

Let represent any of the above semantics. The minimal hypotheses models are the same with respect to any of the four semantics (models are represented considering only positive literals): with affix ; with affix ; with affix . Thus . Now it is the case that the remainder of is the same for any of these semantics:

(as a matter of fact, the remainder for the has the rule instead of ; but this does not change the sequel of this reasoning). The minimal hypotheses models of are the same with respect to any of the four semantics (models are represented considering only positive literals): with affix ; with affix ; with affix . Thus , and no conclusions about cumulativity can be drawn by means of the usual general procedures. Meanwhile, , with affix , is a minimal affix model of but is not a minimal affix model of , which by point (1) of theorem LABEL:teo:equalcum renders any of these semantics not cautious monotonic. Also , with affix , is a minimal affix model of , but not a minimal affix model of , which by point (2) of theorem LABEL:teo:equalcum renders any of these semantics as not cut.

Appendix G Picky, a Special 2-valued Cumulative Semantics

The semantics is defined as follows: for any (1) if , then ; (2) if , then (2a) iff , for every ; (2b) otherwise . This semantics is cumulative, by definition, but it is not always the case that , : for program of the example in E, we have and , which means, by theorem LABEL:teo:equalcum, that is not cumulative. Notice that is not a semantics, because it does not conservatively extend the semantics: for program in the referred example, we have and .

Appendix H Excessiveness and Irregularity

Excessiveness. The following program shows that semantics , , , and are excessive (the dashed lines divide the program into layers; top layer is layer 1, bottom layer is layer 4),

Let represent any of these semantics. It is the case that with affix , is a model of under any of the referred semantics, and for no model , where