Stable Semantics

Let us now turn to our only example for collapsing semantics, namely σ = stb. If we are interested in credulous reasoning, we face a similar situation (and a similar proof which can be found in Appendix A) as in Theorem 6.4.5.

Theorem 6.4.8. VER-MIN-REPAIR_stb,credisDp₁-complete.

We turn to skeptical reasoning. Since finding a stable extension is NP-complete, it is not hard to see that there is a coNP lower bound for skeptical reasoning. However, as the framework in question might collapse, we also need to verify that there is at least one stable extension of a given framework. The result is a Dp₁lower bound (see [93]). Interestingly, this observation is not relevant in our case. The coNP lower bound is already responsible for VER-MIN-REPAIRstb,skepto have a Πp2 lower bound: Given H ⊆ K the decision problem

VER-MIN-REPAIR_stb,skepinvolves checking whether all sets H0with H ⊆ H0 ⊆ K do not possess any skeptically accepted argument. Since the latter test has a NP lower bound, we have a Πp₂lower bound for VER-MIN-REPAIRstb,skep. More precisely:

Theorem 6.4.9. VER-MIN-REPAIRstb,skepisΠp2-complete.

The following table summarizes the results for the decision problem VER-MIN-REPAIRσ,

we obtained. Except for gr semantics, credulous reasoning yields Dp₁-completeness for all considered semantics. Due to special properties of semantics, the upper bound from Propo- sition 6.4.2 is not always a lower bound for skeptical reasoning.

credulous skeptical gr P P ad Dp₁-c trivial pr Dp₁-c Πp₃-c co Dp₁-c P stb Dp₁-c Πp₂-c

Table 6.1: Complexity of verifying maximal consistency in AFs

6.5

Conclusion and Related Work

In this chapter, we investigated the computational complexity of decision and function prob- lems related to strong inconsistency. We placed value on a comparison between this refined notion and ordinary inconsistency in monotonic logics. We argued that the this comparison is not meaningful if the satisfiability check for the logic is in PSPACE (or more demand- ing) and thus restricted the investigation to logics where this problem is in the polynomial hierarchy. For this, we extended a result [91] about the problem MU to our generalized version QBF-MU(Q1, ..., Qm). We then gave general upper bounds for the decision prob-

lems of verifying (minimal) strong inconsistency. Although they appear rather generic, we showed that they cannot be improved in general. We also demonstrated how to infer the corresponding lower bounds for ASP. In a similar fashion, we investigated the problem of computing |IMSI(K)|, i. e., the number of minimal strongly inconsistent subsets of a given

knowledge base. Our last step was a similar investigation of inconsistency in AFs, where we focused on maximal consistency rather than minimal (strong) inconsistency.

The results of this chapter suggest that in terms of a worst case analysis, strong incon- sistency is in many cases not more demanding than mere inconsistency. This result does not transfer to logics where the satisfiability check is in P since this class does not allow for a non-deterministic guess of supersets of a given H ⊆ K. Similar observations can be made for the corresponding counting problems. We did not consider the notions concerned with adding formulas to a knowledge base, for example bidirectional non-repairs. It is easy to see that similar results transfer to these notions, which we did not state explicitly here in order to keep our investigation concise.

We already mentioned that Reiter [96] was also concerned about computing hitting sets. Many algorithms and systems for enumerating minimal inconsistent sets –[8; 78; 79]– build on the duality results. Hitting sets are also utilized in computation of causes and responsibilities of inconsistency in databases [98]. Our discussion regarding the complexity of minimal unsatisfiability for QBFs was inspired by [91]. A rather thorough discussion about the complexity of inconsistency measures has been done in [107; 108], which also discusses various problems about minimal inconsistent and maximal consistent sets of a given knowledge base. The paper [46] discusses computational complexity as well.

Conclusion

7.1

Summary

In this work we studied inconsistency in an abstract setting covering arbitrary logics, in par- ticular non-monotonic ones. We demonstrated that in the general case the standard notion of inconsistency is unable to play the same role it does in monotonic reasoning. One of our main contributions is the identification and investigation of an adequate strengthening of inconsistency. Our main results can be summarized as follows:

In Chapter 3 we focused on structural properties of knowledge bases, especially the con- nection between consistent and inconsistent subsets. We gave a generalization of Reiter’s well-known hitting set duality to non-monotonic logics (Theorem 3.1.12). Reiter’s duality – tailored for a monotonic setting– is only concerned about removing formulas. We also gave a duality characterization for repairs which can be obtained by adding (Theorem 3.2.9) or adding and removing formulas (Theorem 3.2.32). We demonstrated structural properties of knowledge bases, which are themselves interesting, not just as tools for proving the main theorems (cf. Propositions 3.2.25 and 3.2.27 as well as Propositions 3.2.37 and 3.2.40). We also considered more fine-grained modifications like strengthening and weakening instead of adding and removing formulas (Corollary 3.3.13). Using several examples we illustrated that infinite knowledge bases are in general not as well-behaved as finite ones. We identi- fied sufficient conditions in order to overcome some of the arising issues, most notably the so-called compactness property (see Theorem 3.4.15).

We devoted Chapter 4 to measuring inconsistency in non-monotonic logics. We in- troduced inconsistency measures for this setting –IMSI, IMSIC and I_p– which are natural

generalizations of measures from the literature. In order to help assessing the quality of our measures, we refined existing rationality postulates to obtain meaningful ones for non- monotonic logics. Thereby, the most important ones were the four postulates for a basic inconsistency measure(see [68]). As a result, we proposed strong monotony which is sim- ilar to the monotony postulate, but requires an additional premise. Based on two refined notions of free formulas we considered SI-free and Independence, and we argued that dom- inance is not meaningful in non-monotonic logics. We analyzed the compliance of our generalized measures with the refined rationality postulates. Our investigation continued with the question how to assess inconsistencies of a knowledge base within the context of a larger one. Interestingly, a well-behaved approach was based on our notion of bidirectional non-repairs (see Definition 3.2.19). We also discussed measuring inconsistency in ASP as a special case of our definition of a general logic.

The results of this chapter suggest that the existing work on measuring inconsistency in propositional knowledge bases can be extended to non-monotonic logics, when considering appropriate adjustments to the established approaches. Interesting novel questions arise, because additional information may resolve conflicts. However, one needs to accept that not all aspects can be covered by a general definition of a logic. Consideration of particu- lar frameworks cannot easily be subsumed if a thorough understanding of inconsistency is desired.

In Chapter 5 we investigated inconsistency in AFs, with respect to various semantics and the two standard reasoning modes, credulous and skeptical. We pointed out that ex- istence of repairs is guaranteed in most cases (see Theorem 5.2.16, Theorem 5.2.19 and Fact 5.2.21). We investigated the relations between repair notions (see Theorems 5.2.4 and 5.2.5 and in particular Conjecture 5.2.5, Theorem 5.2.8 and the examples we gave). We demonstrated how our previous results yield duality characterizations for repairs of AFs (Propositions 5.3.3 and 5.3.4). In order to refine our analysis, we considered specific sit- uations, e.g., symmetry of the attack relation (see Proposition 5.4.6) or splitting (Proposi- tions 5.4.13 and 5.4.14). In a brief discussion on infinite AFs our main result was proving the co-compactness property (introduced in Section 3.4) for finitary AFs (see Theorem 5.4.26). We also performed a short case study, illustrating how to repair some given AFs.

This chapter shall demonstrate how to apply our techniques to tackle inconsistency, when given a specific logic. We found connections between the repair notions and were able to apply our duality results to characterize repairs of AFs, independent of the underlying semantics or reasoning mode. We gave encouraging connections to subclasses of AFs and splitting results from the literature.

In Chapter 6 we performed an analysis of the computational complexity of related de- cision and function problems. For this, we established results for the generic monotonic framework of QBFs (Theorem 6.2.2). We found that in many cases, strong inconsistency is not more demanding than the ordinary one, from a computational point of view (The- orem 6.2.5). In addition to general upper bounds, we gave corresponding lower bounds for ASP (Theorems 6.2.7 and 6.2.10). We extended the investigation to the corresponding problem of counting strongly inconsistent subsets. Since the results can be derived from the corresponding decision problems, strong inconsistency is again comparable to mere in- consistency in many cases (Theorems 6.3.1 and 6.3.2). In Section 6.4 we gave complexity results for verifying minimal repairs for AFs, covering various semantics and the two rea- soning modes considered in Chapter 5.