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Strong Equivalence of Qualitative Optimization Problems

We introduce the framework of qualitative optimization problems (or, simply, optimization problems) to represent preference theories. The formalism uses separate modules to describe the space of outcomes to be compared (the generator) and the preferences on outcomes (the selector). We consider two types of optimization problems. They differ in the way the generator, which we model by a propositional theory, is interpreted: by the standard propositional logic semantics, and by the equilibrium-model (answer-set) semantics. Under the latter interpretation of generators, optimization problems directly generalize answer-set optimization programs proposed previously. We study strong equivalence of optimization problems, which guarantees their interchangeability within any larger context. We characterize several versions of strong equivalence obtained by restricting the class of optimization problems that can be used as extensions and establish the complexity of associated reasoning tasks. Understanding strong equivalence is essential for modular representation of optimization problems and rewriting techniques to simplify them without changing their inherent properties.


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1 Introduction

We introduce the framework of qualitative optimization problems in which, following the design of answer-set optimization (ASO) programs [Brewka, Niemelä, and Truszczyński2003], we use separate modules to describe the space of outcomes to be compared (the generator) and the preferences on the outcomes (the selector). In all optimization problems we consider, the selector module follows the syntax and the semantics of preference modules in ASO programs, and the generator is given by a propositional theory. If this propositional theory is interpreted according to the standard propositional logic semantics, that is, outcomes to be compared are models of the generator, we speak about the classical optimization problems (CO problems, for short). If the generator theory is interpreted by the semantics of equilibrium models [Pearce1997], also known as answer sets [Ferraris2005], that is, it is the answer sets of the generator that are being compared, we speak about answer-set optimization problems (ASO problems, for short).

Representing and reasoning about preferences in qualitative settings is an important research area for knowledge representation and qualitative decision theory. The main objectives are to design expressive yet intuitive languages to model preferences, and to develop automated methods to reason about formal representations of preferences in these languages. The literature on the subject of preferences is vast. We refer the reader to the special issue of Artificial Intelligence Magazine

[Goldsmith and Junker2008] for a collection of overview articles and references.

Understanding when optimization problems are equivalent, in particular, when one can be interchanged with another within any larger context, is fundamental to any preference formalism. Speaking informally, optimization problems and are interchangeable or strongly equivalent when for every optimization problem (context), and define the same optimal models. Understanding when one optimization problem is equivalent to another in this sense is essential for preference analysis, modular preference representation, and rewriting techniques to simplify optimization problems into forms more amenable to processing, without changing any of their inherent properties. Let us consider a multi-agent setting, in which agents combine their preferences on some set of alternatives with the goal of identifying optimal ones. Can one agent in the ensemble be replaced with another so that the set of optimal alternatives is unaffected not only now, but also under any extension of the ensemble in the future? Strong equivalence of agents’ optimization problems is precisely what is needed to guarantee this full interchangeability property!

The notion of strong equivalence is of general interest, by no means restricted to preference formalisms. In some cases, most notably for classical logic, it coincides with equivalence, the property of having the same models. However, if the semantics is not monotone, that is, extending the theory may introduce new models, not only eliminate some, strong equivalence becomes a strictly stronger concept, and the one to adopt if theories being analyzed are to be placed within a larger context. The nonmonotonicity of the semantics is the salient feature of nonmonotonic logics [Marek and Truszczyński1993]

and strong equivalence of theories in nonmonotonic logics, especially logic programming with the answer-set semantics

[Gelfond and Lifschitz1991], was extensively studied in that setting [Lifschitz, Pearce, and Valverde2001, Turner2003, Eiter, Fink, and Woltran2007]. Preference formalisms also often behave nonmonotonically as adding a new preference may cause a non-optimal outcome (model) to become an optimal one. Thus, in preference formalisms, equivalence and strong equivalence are typically different notions. Accordingly, strong equivalence was studied for logic programs with rule preferences [Faber and Konczak2006], programs with ordered disjunction [Faber, Tompits, and Woltran2008] and programs with weak constraints [Eiter et al.2007].

We extend the study of strong equivalence to the formalism of qualitative optimization problems. The formalism is motivated by the design of answer-set optimization (ASO) programs of bnt03 bnt03. It borrows two key features from ASO programs that make it an attractive alternative to the preference modeling approaches based on logic programming that we mentioned above. First, following ASO programs, optimization problems provide a clear separation of hard constraints, which specify the space of feasible outcomes, and preferences (soft constraints) that impose a preference ordering on feasible outcomes. Second, optimization problems adopt the syntax and the semantics of preference rules of ASO programs that correspond closely to linguistic patterns of simple conditional preferences used by humans.

The separation of preference modules from hard-constraints facilitates eliciting and representing preferences. It is also important for characterizing strong equivalence. When a clear separation is not present, like in logic programs with ordered disjunctions, strong equivalence characterizations are cumbersome as they have to account for complex and mostly implicit interactions between hard constraints and preferences. For optimization problems, which impose the separation, we have “one-dimensional” forms of strong equivalence, in which only hard constraints or only preferences are added. These “one-dimensional” concepts are easier to study yet provide enough information to construct characterizations for the general case.

Main Contributions. (1) We propose a general framework of qualitative optimization problems, extending in several ways the formalism of ASO programs. We focus on two important instantiations of the framework, the classes of classical optimization (CO) problems and answer-set optimization (ASO) problems. The latter one directly generalizes ASO programs. (2) We characterize the concept of strong equivalence of optimization problems relative to changing selector modules. The characterization is independent of the semantics of generators and so, applies both to CO and ASP problems. We also characterize strong equivalence relative to changing generators (with preferences fixed). In this case, not surprisingly, the characterization depends on the semantics of generators. However, we show that the dependence is quite uniform, and involves a characterization of strong equivalence of generators relative to their underlying semantics, when they are considered on their own as propositional theories. Finally, we combine the characterizations of the “one-dimensional” concepts of strong equivalence into a characterization of the general “combined” notion. (3) We develop our results for the case when preferences are ranked. In practice, preferences are commonly ranked due to the hierarchical structure of preference providers. The general case we study allows for additions of preferences of ranks from a specified interval . This covers the case when only some segment in the hierarchy of preference providers is allowed to add preferences (top decision makers, middle management, low-level designers), as well as the case when there is no distinction between the importance of preferences (the non-ranked case). (4) We establish the complexity of deciding whether two optimization problems are strongly equivalent relative to changing selectors, generators, or both. We present only proof sketches and some of the simpler and not overly technical proofs in the main text. Detailed proofs can be found in the Appendix.

2 Optimization Problems

A qualitative optimization problem (an optimization problem

, from now on) is an ordered pair

, where is called the generator and the selector. The role of the generator is to specify the family of outcomes to be compared. The role of the selector is to define a relation on the set of outcomes and, consequently, define the notion of an optimal outcome. The relation induces relations and : we define if and , and if and . For an optimization problem , we write and to refer to its generator and selector, respectively.

Generators. As generators we use propositional theories in the language determined by a fixed countable universe , a boolean constant , and boolean connectives , and , and where we define the constant , and the connectives and in the usual way.111While the choice of primitive connectives is not common for the language of classical propositional logic, it is standard for the of logic here-and-there which underlies the answer-set semantics. Models of the generator, as defined by the semantics used, represent outcomes of the corresponding optimization problem. We consider two quite different semantics for generators: the classical propositional logic semantics and the semantics of equilibrium models [Pearce1997]. Thus, outcomes are either models or equilibrium models, depending on the semantics chosen. The first semantics is of interest due to the fundamental role and widespread use of classical propositional logic, in particular, as a means to describe constraints. Equilibrium models generalize answer sets of logic programs to the case of arbitrary propositional theories [Pearce1997, Ferraris2005] and are often referred to as answer sets. The semantics of equilibrium models is important due to the demonstrated effectiveness of logic programming with the semantics of answer sets for knowledge representation applications. We use the terms equilibrium models and answer sets interchangeably.

Throughout the paper, we represent interpretations of as subsets of . We write to state that an interpretation is a (classical propositional) model of a formula . Furthermore, we denote the set of classical models of a formula or theory by .

Equilibrium models arise in the context of the propositional logic of here-and-there, or the logic HT for short [Heyting1930]. We briefly recall here definitions of concepts, as well as properties of the logic HT that are directly relevant to our work. We refer to the papers by Pearce97 Pearce97 and fer05 fer05 for further details.

The logic HT is a logic located between the intuitionistic and the classical logics. Interpretations in the logic HT are pairs of standard propositional interpretations such that . We write to denote that a formula holds in an interpretation in the logic HT. The relation is defined recursively as follows: for an atom , if and only if . The cases of the boolean connectives and are standard, and if and only if (classical satisfiability) and or . Finally, .

An equilibrium model or answer set of a propositional theory is a standard interpretation such that and for every proper subset of , . Answer sets of a propositional theory are also classical models of . The converse is not true in general.

We denote the set of all answer sets of a theory by , and the set of all HT-models of by , that is, .

For each of the semantics there are two natural concepts of equivalence. Two theories and are equivalent if they have the same models (classical or equilibrium, respectively). They are strongly equivalent if for every theory , and have the same models (again, classical or equilibrium, respectively).

For classical semantics, strong equivalence and equivalence coincide. It is not so for the semantics of equilibrium models. The result by lpv01 lpv01 states that two theories and are strongly equivalent for equilibrium models if and only if and are equivalent in the logic HT, that is .

We call optimization problems under the classical interpretation of generators classical optimization problems or CO problems, for short. When we use the answer-set semantics for generators, we speak about answer-set optimization problems or ASO problems, for short.

Selectors. We follow the definitions of preference modules in ASO programs [Brewka, Niemelä, and Truszczyński2003], adjusting the terminology to our more general setting. A selector is a finite set of ranked preference rules


where and are positive integers, and , , and are propositional formulas over . For a rule of the form (1), the number is the rank of , denoted by , is the head of and is the body of , . Moreover, we write to refer to formula .

If for every preference rule in a selector , then is a simple selector. Otherwise, is ranked. We often omit “” from the notation “” for simple selectors. For a selector , and , we define (where we assume that for every integer , ) and write for the rank interval . We extend this notation to optimization problems. For and a rank interval , we set . For some rank intervals we use shorthands, for example for , for , for , and similar.

For an interpretation , a satisfaction degree of a preference rule is , if and ; otherwise, the rule is irrelevant to , and . We note that bnt03 (bnt03) represented the satisfaction degree of an irrelevant rule by a special non-numeric degree, treated as being equivalent to . The difference is immaterial and the two approaches are equivalent.

Selectors determine a preference relation on interpretations. Given interpretations and and a simple selector , holds precisely when for all , . Therefore, holds if and only if and there exists such that ; if and only if for every , .

Given a ranked selector , we define if for every preference rule , , or if there is a rule such that the following three conditions hold:

  1. for every of the same rank as ,

  2. for every of smaller rank than , .

Moreover, if and only if there is a rule for which the three conditions above hold, and if and only if for every , . Given an optimization problem we often write for (and similarly for and ).

Optimal (preferred) outcomes. For an optimization problem , denotes the set of all outcomes of , that is, the set of all models (under the selected semantics) of the generator of . Thus, stands for all models of in the framework of CO problems and for all answer sets of , when ASO problems are considered. A model is optimal or preferred for if there is no model such that . We denote the set of all preferred models of by .

Relation to ASO programs. Optimization problems extend the formalism of ASO programs [Brewka, Niemelä, and Truszczyński2003] in several ways. First, the generator programs are arbitrary propositional theories. Under the semantics of equilibrium models, our generators properly extend logic programs with the answer-set semantics used as generators in ASO programs. Second, the selectors use arbitrary propositional formulas for options in the heads of preference rule, as well as for conditions in their bodies. Finally, optimization problems explicitly allow for alternative semantics of generators, a possibility mentioned but not pursued by bnt03 bnt03.

Notions of Equivalence. We define the union of optimization problems as expected, that is, for and , we set . Two optimization problems and are strongly equivalent with respect to a class of optimization problems (contexts) if for every optimization problem , .

We consider three general classes of contexts. First and foremost, we are interested in the class of all optimization problems over . We also consider the families and of all optimization problems of the form and , respectively. The first class consists of optimization problems where all models of the generator are equally preferred. We call such optimization problems generator problems. The second class consists of optimization problems in which every interpretation of is an acceptable outcome. We call such optimization problems selector problems. These “one-dimensional” contexts provide essential insights into the general case. For the first class, we simply speak of strong equivalence, denoted . For the latter two classes, we speak of strong gen-equivalence, denoted , and strong sel-equivalence, denoted , respectively.

Constraining ranks of rules in selectors gives rise to two additional classes of contexts parameterized using rank intervals :

The first class of contexts gives rise to strong sel-equivalence with respect to rules of rank in , denoted by . The second class of contexts yields the concept of strong equivalence with respect to rules of rank in . We denote it by . We call problems in the class simple optimization problems.

3 Examples

We present now examples that illustrate key issues relevant to strong equivalence of optimization problems. They point to the necessity of some conditions that appear later in characterizations of strong equivalence and hint at some constructions used in proofs. In all examples except for the last one, we consider simple CO problems. In all problems only atoms explicitly listed matter, so we disregard all others.

Example 1

Let , where and . There are two outcomes here, and , that is, . Let be the only preference rule in . Clearly, and . Thus, and so, .

In addition, let , where and is as above. Then, and, trivially, . It follows that and are equivalent, as they specify the same optimal outcomes. However, they are not strongly gen-equivalent (and so, also not strongly equivalent). Indeed, let . Then and so, . On the other hand, and, therefore, .

Example 1 suggests that we must have if problems and are to be strongly (gen-)equivalent. Otherwise, by properly selecting the context generator, we can eliminate all outcomes in one problem still leaving some in the other.

Example 2

Let , where and . We have . In addition, , , and and are incomparable. Thus, . Let now , where and . Clearly, . Moreover, . Thus, and so, and are equivalent. They are not strongly (gen-)equivalent. Indeed, let . Then, and that .

This example suggests that for two optimization problems to be strongly (gen-)equivalent, they have to define the same preference relation on outcomes.

Example 3

Let , where and . We have and so, . It follows that is equivalent to . Let . Since , . Further, and so, and we get . Thus, and are not strongly (sel-)equivalent.

Informally, this example shows that by modifying the selector part, we can make non-optimal outcomes optimal. Thus, as in the case of strong gen-equivalence (Example 1) the equality of sets of models (i.e., equivalence) is important for strong (sel-)equivalence (a more refined condition will be needed for ranked programs, as we show in Theorem 2).

Example 4

For the next example, let us consider problems and , where is the generator from Example 1, and . We have . Moreover, and . Thus, . Since , we also have (trivially) that . Thus, , too, and the problems and are equivalent. They are not strongly sel-equivalent, though. Let . Then, and so, . On the other hand, . Thus, .

The example suggests that for strong sel-equivalence the equality of the relation induced by the problems considered is important. The equality of the relation is not sufficient. In our example, the relations and are both empty and so – equal. But they are empty for different reasons, absence of preference versus conflicting preferences, which can give rise to different preferred models when extending selectors.

Our last example involves ranked problems. It is meant to hint at issues that arise when ranked problems are considered. In the general case of ranked selectors, the equality of the relation induced by programs being evaluated is not related to strong sel-equivalence in any direct way and an appropriate modification of that requirement has to be used together with yet another condition (cf. Theorem 2).

Example 5

Let , where and , where . Clearly, and . Thus, the two problems are equivalent. They are not strongly sel-equivalent if arbitrary selectors are allowed. For instance, let . Adding this new preference rule to makes and incomparable and so, . On the other hand, since the new rule has rank 2, it dominates the preference rule of , which is of rank 3. Thus, . A similar effect occurs with the problem . Since its only preference rule is dominated by the only preference rule in , . On the other hand, and are incomparable in and, consequently, . Thus, extending and with selectors containing rules of rank 2 or 3 may lead to different optimal outcomes. It is so even though the relations induced by and on the set of all outcomes coincide.

However, adding selectors consisting only of rules of rank greater than 3 cannot have such an effect, since the existing rules would dominate them. Also, adding rules of rank 1 cannot result in differing preferred answer sets, as such rules would dominate the existing ones. Formally, the problems and are strongly sel-equivalent relative to selectors with preference rules of rank greater than 3 or less than 2.

4 Strong sel-equivalence

We start with the case of strong sel-equivalence, the core case for our study. Indeed, characterizations of strong sel-equivalence naturally imply characterizations for the general case thanks to the following simple observation.

Proposition 1

Let and be optimization problems (either under classical or answer-set semantics for the generators) and a rank interval. Then if and only if for every generator , .

Proof.  () Let . Since , and so, .

() Let be any optimization problem in . We have and . By the assumption, it follows that . Thus,

It follows that and, consequently, that .  

Furthermore, the set of outcomes of an optimization problem is unaffected by changes in the selector module. It follows that the choice of the semantics for generators does not matter for characterizations of strong sel-equivalence. Thus, whenever in this section we refer to the set of outcomes of an optimization problem , we use the notation , and not the more specific one, or , that applies to CO and ASO problems, respectively.

Our first main result concerns strong sel-equivalence relative to selectors consisting of preference rules of ranks in a rank interval . Special cases for strong sel-equivalence will follow as corollaries. To state the result, we need some auxiliary notation. For an optimization problem , we define to be the largest such that . If for every we have , then we set . It is clear that is well-defined. Moreover, as , . Furthermore, for a set and a relation over , we write for the restriction of to , that is, .

Theorem 2

For every ranked optimization problems and , and every rank interval , if and only if the following conditions hold:

  1. For every such that or , or both and .

We now comment on this characterization and derive some of its consequences. First, we observe that the conditions (1) and (2) are indeed necessary — differences between and or and can be exploited to construct a selector from whose addition to and results in problems with different sets of optimal outcomes. This is illustrated by Examples 3 and 4 in the case of simple problems and simple contexts (), where and coincide with and , respectively. The condition (3) is necessary, too. Intuitively, if the first ranks where and differentiate between two outcomes and (which are optimal for ranks less than ) are not equal, these first ranks must both be larger than . Otherwise, one can find a selector with rules of ranks in , that will make one of the interpretation optimal in one extended problem but not in the other.

Next, we discuss some special cases of the characterization. First, we consider the case , which allows for a simplification of Theorem 2.

Corollary 3

For every ranked optimization problems and , and every rank interval , if and only if the following conditions hold:

  1. For every , or both and .

Proof.  Starting from Theorem 2, we note that the selector of is empty and hence . Moreover, if the precondition and in condition (3) of Theorem 2 is not satisfied for and a pair , then and and thus the consequent is satisfied in that case as well, which allows for omitting the precondition.  

If in addition , we obtain the case of rank-unrestricted selector contexts, and condition (3) can be simplified once more, since and never hold for .

Corollary 4

For every optimization problems and , (equivalently, or ) if and only if the following conditions hold:

  1. for every , .

Next, we note that if an optimization problem is simple then if and only if , which is equivalent to . This observation leads to the following characterization of strong sel-equivalence of simple optimization problems.

Corollary 5

For every two simple optimization problems and , the following statements are equivalent:

  1. (equivalently, )

  2. (equivalently, )

  3. and .

Proof.  The implication (a)(b) is evident from the definitions.

(b)(c) From Corollary 3 with we directly obtain . The condition follows from conditions (2) and (3) of that corollary. Indeed, let us consider such that and distinguish two cases. If (i) then and by condition (2) of Corollary 3, also , implying . If (ii) then by condition (3) of Corollary 3, . Since are simple, , and consequently . By symmetry, we also have that implies . Thus, .

(c)(a) From (c) it follows that and . Thus, the conditions (1) and (2) of Corollary 4 follow. To prove the condition (3), let us first assume for . It follows that and thus . By our earlier observation also and thus . Hence . For we reason analogously. In the last remaining case, and , so we directly obtain . By Corollary 4, follows.  

Corollary 5 shows, in particular, that for simple problems there is no difference between the relations and . This property reflects the role of preference rules of rank 2 and higher. They allow us to break ties among optimal outcomes, as defined by preference rules of rank 1. Thus, they can eliminate some of these outcomes from the family of optimal ones, but they cannot introduce new optimal outcomes. Therefore, they do not affect strong sel-equivalence of simple problems. This property has the following generalization to ranked optimization problems.

Corollary 6

Let and be ranked optimization problems and let be the maximum rank of a preference rule in . Then the relations and coincide.

Proof.  Clearly, implies . Thus, it is enough to prove that if then . To prove the condition (3), let us consider such that . By the condition (3) of Theorem 2, . Since is the maximum rank of a preference rule in or , and . Thus, (the case is similar).  

Our observation on the role of preference rules with ranks higher than ranks of rules in or also implies that and are strongly sel-equivalent relative to selectors consisting exclusively of such rules if and only if and are equivalent (have the same optimal outcomes), and if optimal outcomes that “tie” in also “tie” in and conversely. Formally, we have the following result.

Corollary 7

Let and be ranked optimization problems and let be the maximum rank of a preference rule in . Then if and only if and .

Proof.  Clearly, and and so, and . Thus, the “only-if” part follows by Theorem 2 (the condition (1) of that theorem reduces to and the condition (3) implies ). To prove the “if” part, we note that the condition (1) of Theorem 2 holds by the assumption. Moreover, the relations and are empty and so, they coincide. Thus, the condition (2) of Theorem 2 holds. Finally, if , and , then and so, . By the assumption, , that is, . The case when is similar. Thus, the condition (3) of Theorem 2 holds, too, and follows.  

Lastly, we give some simple examples illustrating how our results can be used to “safely” modify or simplify optimization problems, that is rewrite one into another strongly sel-equivalent one.

Example 6

Let , where and , and , where . Regarding these problems as CO problems, we have that . Moreover, it is evident that . Thus, by Corollary 5, and are strongly sel-equivalent. In other words, we can faithfully replace rules , in the selector of any optimization problem with generator by the single rule .

Next, for an example of a more general principle, we note that removing preference rules with only one option in the head yields a problem that is strongly sel-equivalent.

Corollary 8

Let and be two CO or ASO problems such that and is obtained from by removing all preference rules with only one option in the head. Then and are strongly sel-equivalent.

Proof.  The conditions (1)-(3) of Theorem 2 all follow from an observation that for every interpretation and every preference rule with just one option in the head, .  

5 Strong gen-equivalence

We now focus on the case of strong gen-equivalence. The semantics of generators makes a difference here but the difference concerns only the fact that under the two semantics we consider, the concepts of strong equivalence are different. Other aspects of the characterizations are the same. Specifically, generators have to be strongly equivalent relative to a selected semantics. Indeed, if the generators are not strongly equivalent, one can extend them uniformly so that after the extension one problem has a single outcome, which is then trivially an optimal one, too, while the other one has no outcomes and so, no optimal ones. Second, the preference relation defined by the selectors of the problems considered must coincide. Thus, a single theorem handles both types of problems.

Theorem 9

For every two CO (ASO, respectively) problems and , if and only if and are strongly equivalent (that is, for CO problems, and for ASO problems) and .

In view of Examples 1 and 2, the result is not unexpected. The two examples demonstrated that the conditions of the characterization cannot, in general, be weakened.

It is clear from Corollary 4 and Theorem 9 that strong sel-equivalence of CO problems is a stronger property than their strong gen-equivalence.

Corollary 10

For every two CO problems and , implies .

In general the implication in Corollary 10 cannot be reversed. The problems and considered in Example 4 are not strongly sel-equivalent. However, based on Theorem 9, they are strongly gen-equivalent. Indeed, and, writing for , the relations and are both empty and so, equal.

The relation between strong sel-equivalence and strong gen-equivalence of ASO problem is more complex. In general, neither property implies the other even if both problems and are assumed to be simple. It is so because if and only if and (Corollary 5), and if and only of and (Theorem 9). Now, (regular equivalence of programs) does not imply (strong equivalence) and does not imply .

6 Strong equivalence — the combined case

Finally, we consider the relation , which results from considering contexts that combine both generators and selectors. Since generators may vary here, as in the previous section, the semantics of generators matters. But, as in the previous section, the difference boils down to different characterizations of strong equivalence of generators.

We start with a result characterizing strong equivalence of CO and ASO problems relative to combined contexts (both generators and selectors possibly non-empty) with selectors consisting of rules of rank at least and at most , respectively.

Theorem 11

For every ranked CO (ASO, respectively) problems and , and every rank interval , if and only if the following conditions hold:

  1. and are strongly equivalent (that is, for CO problems, and for ASO problems)

  2. For every such that or , or both and

  3. .

The corresponding characterizations for CO and ASO problems differ only in their respective conditions (1), which now reflect different conditions guaranteeing strong equivalence of generators under the classical and answer-set semantics. Moreover, the four conditions of Theorem 11 can be obtained by suitably combining and extending the conditions of Theorem 2 and Theorem 9. First, as combined strong equivalence implies strong gen-equivalence, the condition (1) is taken from Theorem 9. Second, we modify the conditions (2) and (3) from Theorem 2 replacing with (and accordingly with ), as each classical model of can give rise to an optimal classical or equilibrium one upon the addition of a context, an aspect also already visible in Theorem 9. Finally, we have to add a new condition that the relations and coincide on the sets of models of and . When generators are allowed to be extended, one can make any two of their models to be the only outcomes after the extension. If the two outcomes, say and , are related differently by the corresponding strict relations induced by rules with ranks less than , then in one extended problem, exactly one of the two outcomes, say , is optimal. In the other extended problem we cannot have both outcomes be optimal nor only be optimal as that would contradict the strong equivalence of problems under considerations. If, however, is the only optimal outcome also in the other extended problem, then must “win” with based on rules that have ranks at least . In such case, there is a way to add new preferences of rank that will “promote” to be optimal too, without making it optimal in the first problem. However, that contradicts strong equivalence.

We conclude this section with observations concerning the relation for both CO and ASO problems. The contexts relevant here may contain preference rules of arbitrary ranks. We start with the case of CO problems, where the results are stronger. While they can be derived from the general theorems above, we will present here arguments relying on results from previous sections, which is possible since for CO problems equivalence and strong-equivalence of generators coincide.

We saw in the last section that for CO problems is a strictly stronger relation than . In fact, for CO problems, coincides with the general relation .

Theorem 12

For every CO problems and , if and only if .

Proof.  The “only-if” implication is evident. To prove the converse implication, we will use Proposition 1 which reduces checking for strong equivalence to checking for strong sel-equivalence. Let be a generator problem. Since , from Corollary 4 we have . Consequently, . Writing for and for we have . Thus, also by Corollary 4, . Finally, the condition (3) of Corollary 4 for and implies the condition (3) of that corollary for and (as has no preference rules and . It follows, again by Corollary 4, that . Thus, by Proposition 1, .  

In particular, Corollary 5 implies that the relations , , , and coincide on simple CO problems.

Corollary 13

For every simple CO problems and all properties , , and are equivalent.

For simple ASO problems we still have that and coincide but in general these notions are different from and .

Corollary 14

For every simple ASO problems and , the following conditions are equivalent