Standpoint logics offer unified modal logic-based formalisms for representing multiple heterogeneous viewpoints. At the same time, many non-monotonic reasoning frameworks can be naturally captured using modal logics — in particular using the modal logic S4F. In this work, we propose a novel formalism called S4F Standpoint Logic, which generalises both S4F and standpoint propositional logic and is therefore capable of expressing multi-viewpoint, non-monotonic semantic commitments. We define its syntax and semantics and analyze its computational complexity, obtaining the result that S4F Standpoint Logic is not computationally harder than its constituent logics, whether in monotonic or non-monotonic form. We also outline mechanisms for credulous and sceptical acceptance and illustrate the framework with an example.

Fuzzy logic programming is an established approach for reasoning under uncertainty. Several semantics from classical, two-valued logic programming have been generalized to the case of fuzzy logic programs. In this paper, we show that two of the most prominent classical semantics, namely the stable model and the well-founded semantics, can be reconstructed within the general framework of approximation fixpoint theory (AFT). This not only widens the scope of AFT from two- to many-valued logics, but allows a wide range of existing AFT results to be applied to fuzzy logic programming. As first examples of such applications, we clarify the formal relationship between existing semantics, generalize the notion of stratification from classical to fuzzy logic programs, and devise “more precise” variants of the semantics.

The tractability of the lightweight description logic EL has allowed for the construction of large and widely used ontologies that support semantic interoperability. However, comprehensive domains with a broad user base are often at odds with strong axiomatisations otherwise useful for inferencing, since these are usually context dependent and subject to diverging perspectives. In this paper we introduce Standpoint EL, a multi-modal extension of EL that allows for the integrated representation of domain knowledge relative to diverse, possibly conflicting standpoints (or contexts), which can be hierarchically organised and put in relation to each other. We establish that Standpoint EL still exhibits EL's favourable PTime standard reasoning, whereas introducing additional features like empty standpoints, rigid roles, and nominals makes standard reasoning tasks intractable.

Standpoint EL is a multi-modal extension of the popular description logic EL that allows for the integrated representation of domain knowledge relative to diverse standpoints or perspectives. Advantageously, its satisfiability problem has recently been shown to be in PTime, making it a promising framework for large-scale knowledge integration. In this paper, we show that we can further push the expressivity of this formalism, arriving at an extended logic, called Standpoint EL+, which allows for axiom negation, role chain axioms, self-loops, and other features, while maintaining tractability. This is achieved by designing a satisfiability-checking deduction calculus, which at the same time addresses the need for practical algorithms. We demonstrate the feasibility of our calculus by presenting a prototypical Datalog implementation of its deduction rules.

Abstract The importance of taking individual, potentially conflicting perspectives into account when dealing with knowledge has been widely recognised. Many existing ontology management approaches fully merge knowledge perspectives, which may require weakening in order to maintain consistency; others represent the distinct views in an entirely detached way. As an alternative, we proposeStandpoint Logic, a simple, yet versatile multi-modal logic “add-on” for existing KR languages intended for the integrated representation of domain knowledge relative to diverse, possibly conflictingstandpoints, which can be hierarchically organised, combined, and put in relation with each other. Starting from the generic framework ofFirst-Order Standpoint Logic(FOSL), we subsequently focus our attention on the fragment ofsententialformulas, for which we provide a polytime translation into the standpoint-free version. This result yields decidability and favourable complexities for a variety of highly expressive decidable fragments of first-order logic. Using some elaborate encoding tricks, we then establish a similar translation for the very expressive description logic $$\mathcal {SROIQ}b_s$$ SROIQbs underlying the OWL 2 DL ontology language. By virtue of this result, existing highly optimised OWL reasoners can be used to provide practical reasoning support for ontology languages extended by standpoint modelling.

Abstract dialectical frameworks (ADFs) are a recently introduced powerful generalization of Dung’s popular abstract argumentation frameworks (AFs). Inspired by similar work for AFs, we introduce a decomposition scheme for ADFs, which proceeds along the ADF’s strongly connected components. We find that, for several semantics, the decompositionbased version coincides with the original semantics, whereas for others, it gives rise to a new semantics. These new semantics allow us to deal with pertinent problems such as odd-length negative cycles in a more general setting, that for instance also encompasses logic programs. We perform an exhaustive analysis of the computational complexity of these new, so-called naive-based semantics. The results are quite interesting, for some of them involve little-known classes of the so-called Boolean hierarchy (another hierarchy in between classes of the polynomial hierarchy). Furthermore, in credulous and sceptical entailment, the complexity can be different depending on whether we check for truth or falsity of a specific statement.

Abstract Dialectical Frameworks (ADFs) generalize Dung's argumentation frameworks allowing various relationships among arguments to be expressed in a systematic way. We further generalize ADFs so as to accommodate arbitrary acceptance degrees for the arguments. This makes ADFs applicable in domains where both the initial status of arguments and their relationship are only insufficiently specified by Boolean functions. We define all standard ADF semantics for the weighted case, including grounded, preferred and stable semantics. We illustrate our approach using acceptance degrees from the unit interval and show how other valuation structures can be integrated. In each case it is sufficient to specify how the generalized acceptance conditions are represented by formulas, and to specify the information ordering underlying the characteristic ADF operator. We also present complexity results for problems related to weighted ADFs.

We present a translation from defeasible theory bases to abstract dialectical frameworks, a recent generalization of abstract argumentation frameworks. Using several problematic examples from the literature, we first show how our translation addresses important issues of existing approaches. We then prove that the translated frameworks satisfy the rationality postulates closure and direct/indirect consistency. Furthermore, the frameworks can detect inconsistencies in the set of strict inference rules and cyclic (strict and defeasible) supports among literals. We also show that the translation involves at most a quadratic blowup and is, therefore, effectively and efficiently computable. In the last part of the article, we also define a direct, possible-worlds semantics for defeasible theory bases, which illustrates the technical difficulties arising in this process. In particular, the possible-worlds semantics is eager to apply defeasible rules, which is in contrast to the previously studied translation-based approaches.

A Boolean function is bipolar iff it is monotone or anti-monotone in each of its arguments. We investigate the number |$b(n)$| of |$n$|-ary bipolar Boolean functions. We present an (almost) closed-form expression for |$b(n)$| that uses the number |$a(n)$| of antichain covers of an |$n$|-element set. This is closely related to Dedekind’s problem, which can be rephrased as determining the number |$d(n)$| of Boolean functions that are monotone in all arguments. Indeed, a closed-form solution of |$a(n)$| would directly yield a closed-form solution of |$d(n)$|⁠, suggesting that determining |$a(n)$| is a non-trivial problem of itself.

Boolean functions in Answer Set Programming have proven a useful modelling tool. They are usually specified by means of aggregates or external atoms. A crucial step in computing answer sets for logic programs containing Boolean functions is verifying whether partial interpretations satisfy a Boolean function for all possible values of its undefined atoms. In this paper, we develop a new methodology for showing when such checks can be done in deterministic polynomial time. This provides a unifying view on all currently known polynomial-time decidability results, and furthermore identifies promising new classes that go well beyond the state of the art. Our main technique consists of using an ordering on the atoms to significantly reduce the necessary number of model checks. For many standard aggregates, we show how this ordering can be automatically obtained.

We present a reimplementation of the DIAMOND system for computing with abstract dialectical frameworks. The original DIAMOND was a script-based tool that called an external ASP solver. This reimplementation uses the clingo library in a native C++ environment and thus avoids communication overhead.

We introduce a general framework for analyzing realizability in abstract dialectical frameworks (ADFs) and various of its subclasses. In particular, the framework applies to Dung argumentation frameworks, SETAFs by Nielsen and Parsons, and bipolar ADFs. We present a uniform characterization method for the admissible, complete, preferred and model/stable semantics. We employ this method to devise an algorithm that decides realizability for the mentioned formalisms and semantics; moreover the algorithm allows for constructing a desired knowledge base whenever one exists. The algorithm is built in a modular way and thus easily extensible to new formalisms and semantics. We have implemented our approach in answer set programming, and used the implementation to obtain several novel results on the relative expressiveness of the abovemen-tioned formalisms.

We analyze the relative expressiveness of the two-valued semantics of abstract argumentation frameworks, normal logic programs and abstract dialectical frameworks. By expressiveness we mean the ability to encode a desired set of two-valued interpretations over a given propositional vocabulary A using only atoms from A. While the computational complexity of the two-valued model existence problem for all these languages is (almost) the same, we show that the languages form a neat hierarchy with respect to their expressiveness. We then demonstrate that this hierarchy collapses once we allow to introduce a linear number of new vocabulary elements.

We analyse the expressiveness of Brewka and Woltran's abstract dialectical frameworks for two-valued semantics. By expressiveness we mean the ability to encode a desired set of two-valued interpretations over a given propositional vocabulary A using only atoms from A. We also compare ADFs' expressiveness with that of (the two-valued semantics of) abstract argumentation frameworks, normal logic programs and propositional logic. While the computational complexity of the two-valued model existence problem for all these languages is (almost) the same, we show that the languages form a neat hierarchy with respect to their expressiveness. We then demonstrate that this hierarchy collapses once we allow to introduce a linear number of new vocabulary elements. We finally also analyse and compare the representational succinctness of ADFs (for two-valued model semantics), that is, their capability to represent two-valued interpretation sets in a space-efficient manner.

We introduce probo, a general benchmark framework for comparing abstract argumentation solvers. probo is intended to act as the core of an argumentation competition intended to run in 2015.

We present DIAMOND, an implementation of Brewka and Woltran's abstract dialectical frameworks (ADFs). The system uses answer set programming encodings to compute interpretations of ADFs according to various semantics. We evaluate the performance of the system using an actual reasoning problem as opposed to using randomly generated frameworks.

We introduce a decomposition scheme for abstract dialectical frameworks (ADFs). The decomposition proceeds along the ADF's strongly connected components. For several semantics, the decomposition-based version coincides with the original semantics. For others, the scheme defines new semantics. These new semantics allow us to deal with pertinent problems such as odd-length negative cycles in a more general setting, that for instance also encompasses logic programs.

We present an implementation of Wyner, Bench-Capon and Dunne's [2013] approach to instantiate knowledge bases in argumentation frameworks. The translation is encoded into answer set programming (ASP); the encoding can be used with ASP-based implementations of argumentation frameworks, such as aspartix or diamond.

Abstract argumentation frameworks (AFs) are one of the most studied formalisms in AI. In this work, we introduce a certain subclass of AFs which we call compact. Given an extension-based semantics, the corresponding compact AFs are characterized by the feature that each argument of the AF occurs in at least one extension. This not only guarantees a certain notion of fairness; compact AFs are thus also minimal in the sense that no argument can be removed without changing the outcome. We address the following questions in the paper: (1) How are the classes of compact AFs related for different semantics? (2) Under which circumstances can AFs be transformed into equivalent compact ones? (3) Finally, we show that compact AFs are indeed a non-trivial subclass, since the verification problem remains coNP-hard for certain semantics.

We show how the pure logic of GK can be embedded into disjunctive logic programming. The translation we present is polynomial, but not modular, and introduces new variables. The result can then be used to compute the extension/expansion semantics of default and autoepistemic logics using disjunctive ASP solvers.

Reasoning about actions is a subfield of artificial intelligence that is concerned with representing and reasoning about dynamic domains. We propose to employ abstract argumentation for this purpose. Specifically, we present a translation of action domains from a specification language into Dung-style argumentation frameworks (AFs). As the key advantage of our approach, we use existing semantics for argumentation to make predictions about the domain in various manners and utilise existing results about argumentation to show that the approach can be efficiently implemented. This demonstrates the practical value not only of its theoretical results, but also abstract argumentation itself.