Hans Tompits

Tobias Geibinger, Hans Tompits

Paraconsistent logics constitute an important class of formalisms dealing with non-trivial reasoning from inconsistent premisses. In this paper, we introduce uniform axiomatisations for a family of nonmonotonic paraconsistent logics based on minimal inconsistency in terms of sequent-type proof systems. The latter are prominent and widely-used forms of calculi well-suited for analysing proof search. In particular, we provide sequent-type calculi for Priest's three-valued minimally inconsistent logic of paradox, and for four-valued paraconsistent inference relations due to Arieli and Avron. Our calculi follow the sequent method first introduced in the context of nonmonotonic reasoning by Bonatti and Olivetti, whose distinguishing feature is the use of a so-called rejection calculus for axiomatising invalid formulas. In fact, we present a general method to obtain sequent systems for any many-valued logic based on minimal inconsistency, yielding the calculi for the logics of Priest and of Arieli and Avron as special instances.

Johannes Oetsch, Jörg Pührer, Hans Tompits

We introduce a stepping methodology for answer-set programming (ASP) that allows for debugging answer-set programs and is based on the stepwise application of rules. Similar to debugging in imperative languages, where the behaviour of a program is observed during a step-by-step execution, stepping for ASP allows for observing the effects that rule applications have in the computation of an answer set. While the approach is inspired from debugging in imperative programming, it is conceptually different to stepping in other paradigms due to non-determinism and declarativity that are inherent to ASP. In particular, unlike statements in an imperative program that are executed following a strict control flow, there is no predetermined order in which to consider rules in ASP during a computation. In our approach, the user is free to decide which rule to consider active in the next step following his or her intuition. This way, one can focus on interesting parts of the debugging search space. Bugs are detected during stepping by revealing differences between the actual semantics of the program and the expectations of the user. As a solid formal basis for stepping, we develop a framework of computations for answer-set programs. For fully supporting different solver languages, we build our framework on an abstract ASP language that is sufficiently general to capture different solver languages. To this end, we make use of abstract constraints as an established abstraction for popular language constructs such as aggregates. Stepping has been implemented in SeaLion, an integrated development environment for ASP. We illustrate stepping using an example scenario and discuss the stepping plugin of SeaLion. Moreover, we elaborate on methodological aspects and the embedding of stepping in the ASP development process.

Tomi Janhunen, Emilia Oikarinen, Hans Tompits et al.

Practically all programming languages allow the programmer to split a program into several modules which brings along several advantages in software development. In this paper, we are interested in the area of answer-set programming where fully declarative and nonmonotonic languages are applied. In this context, obtaining a modular structure for programs is by no means straightforward since the output of an entire program cannot in general be composed from the output of its components. To better understand the effects of disjunctive information on modularity we restrict the scope of analysis to the case of disjunctive logic programs (DLPs) subject to stable-model semantics. We define the notion of a DLP-function, where a well-defined input/output interface is provided, and establish a novel module theorem which indicates the compositionality of stable-model semantics for DLP-functions. The module theorem extends the well-known splitting-set theorem and enables the decomposition of DLP-functions given their strongly connected components based on positive dependencies induced by rules. In this setting, it is also possible to split shared disjunctive rules among components using a generalized shifting technique. The concept of modular equivalence is introduced for the mutual comparison of DLP-functions using a generalization of a translation-based verification method.

Marina De Vos, Doğa Gizem Kıza, Johannes Oetsch et al.

While past research in answer-set programming (ASP) mainly focused on theory, ASP solver technology, and applications, the present work situates itself in the context of a quite recent research trend: development support for ASP. In particular, we propose to augment answer-set programs with additional meta-information formulated in a dedicated annotation language, called LANA. This language allows the grouping of rules into coherent blocks and to specify language signatures, types, pre- and postconditions, as well as unit tests for such blocks. While these annotations are invisible to an ASP solver, as they take the form of program comments, they can be interpreted by tools for documentation, testing, and verification purposes, as well as to eliminate sources of common programming errors by realising syntax checking or code completion features. To demonstrate its versatility, we introduce two such tools, viz. (i) ASPDOC, for generating an HTML documentation for a program based on the annotated information, and (ii) ASPUNIT, for running and monitoring unit tests on program blocks. LANA is also exploited in the SeaLion system, an integrated development environment for ASP based on Eclipse. To appear in Theory and Practice of Logic Programming