## Institut für Informatik und Computational Science

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We address the problem of Finite Model Computation (FMC) of first-order theories and show that FMC can efficiently and transparently be solved by taking advantage of a recent extension of Answer Set Programming (ASP), called incremental Answer Set Programming (iASP). The idea is to use the incremental parameter in iASP programs to account for the domain size of a model. The FMC problem is then successively addressed for increasing domain sizes until an answer set, representing a finite model of the original first-order theory, is found. We implemented a system based on the iASP solver iClingo and demonstrate its competitiveness by showing that it slightly outperforms the winner of the FNT division of CADE's 2009 Automated Theorem Proving (ATP) competition on the respective benchmark collection.

Preference handling and optimization are indispensable means for addressing nontrivial applications in Answer Set Programming (ASP). However, their implementation becomes difficult whenever they bring about a significant increase in computational complexity. As a consequence, existing ASP systems do not offer complex optimization capacities, supporting, for instance, inclusion-based minimization or Pareto efficiency. Rather, such complex criteria are typically addressed by resorting to dedicated modeling techniques, like saturation. Unlike the ease of common ASP modeling, however, these techniques are rather involved and hardly usable by ASP laymen. We address this problem by developing a general implementation technique by means of meta-prpogramming, thus reusing existing ASP systems to capture various forms of qualitative preferences among answer sets. In this way, complex preferences and optimization capacities become readily available for ASP applications.

We introduce an approach to computing answer sets of logic programs, based on concepts successfully applied in Satisfiability (SAT) checking. The idea is to view inferences in Answer Set Programming (ASP) as unit propagation on nogoods. This provides us with a uniform constraint-based framework capturing diverse inferences encountered in ASP solving. Moreover, our approach allows us to apply advanced solving techniques from the area of SAT. As a result, we present the first full-fledged algorithmic framework for native conflict-driven ASP solving. Our approach is implemented in the ASP solver clasp that has demonstrated its competitiveness and versatility by winning first places at various solver contests.

We introduce an approach to detecting inconsistencies in large biological networks by using answer set programming. To this end, we build upon a recently proposed notion of consistency between biochemical/genetic reactions and high-throughput profiles of cell activity. We then present an approach based on answer set programming to check the consistency of large-scale data sets. Moreover, we extend this methodology to provide explanations for inconsistencies by determining minimal representations of conflicts. In practice, this can be used to identify unreliable data or to indicate missing reactions.

Boolean networks provide a simple yet powerful qualitative modeling approach in systems biology. However, manual identification of logic rules underlying the system being studied is in most cases out of reach. Therefore, automated inference of Boolean logical networks from experimental data is a fundamental question in this field. This paper addresses the problem consisting of learning from a prior knowledge network describing causal interactions and phosphorylation activities at a pseudo-steady state, Boolean logic models of immediate-early response in signaling transduction networks. The underlying optimization problem has been so far addressed through mathematical programming approaches and the use of dedicated genetic algorithms. In a recent work we have shown severe limitations of stochastic approaches in this domain and proposed to use Answer Set Programming (ASP), considering a simpler problem setting. Herein, we extend our previous work in order to consider more realistic biological conditions including numerical datasets, the presence of feedback-loops in the prior knowledge network and the necessity of multi-objective optimization. In order to cope with such extensions, we propose several discretization schemes and elaborate upon our previous ASP encoding. Towards real-world biological data, we evaluate the performance of our approach over in silico numerical datasets based on a real and large-scale prior knowledge network. The correctness of our encoding and discretization schemes are dealt with in Appendices A-B. (C) 2014 Elsevier B.V. All rights reserved.

Answer set programming (ASP) does not allow for incrementally constructing answer sets or locally validating constructions like proofs by only looking at a part of the given program. In this article, we elaborate upon an alternative approach to ASP that allows for incremental constructions. Our approach draws its basic intuitions from the area of default logics. We investigate the feasibility of the concept of semi-monotonicity known from default logics as a basis of incrementality. On the one hand, every logic program has at least one answer set in our alternative setting, which moreover can be constructed incrementally based on generating rules. On the other hand, the approach may produce answer sets lacking characteristic properties of standard answer sets, such as being a model of the given program. We show how integrity constraints can be used to re-establish such properties, even up to correspondence with standard answer sets. Furthermore, we develop an SLD-like proof procedure for our incremental approach to ASP, which allows for query-oriented computations. Also, we provide a characterization of our definition of answer sets via a modification of Clarks completion. Based on this notion of program completion, we present an algorithm for computing the answer sets of a logic program in our approach.

We present the new multi-threaded version of the state-of-the-art answer set solver clasp. We detail its component and communication architecture and illustrate how they support the principal functionalities of clasp. Also, we provide some insights into the data representation used for different constraint types handled by clasp. All this is accompanied by an extensive experimental analysis of the major features related to multi-threading in clasp.

Using the notion of an elementary loop, Gebser and Schaub (2005. Proceedings of the Eighth International Conference on Logic Programming and Nonmonotonic Reasoning (LPNMR'05), 53-65) refined the theorem on loop formulas attributable to Lin and Zhao (2004) by considering loop formulas of elementary loops only. In this paper, we reformulate the definition of an elementary loop, extend it to disjunctive programs, and study several properties of elementary loops, including how maximal elementary loops are related to minimal unfounded sets. The results provide useful insights into the stable model semantics in terms of elementary loops. For a nondisjunctive program, using a graph-theoretic characterization of an elementary loop, we show that the problem of recognizing an elementary loop is tractable. On the other hand, we also show that the corresponding problem is coNP-complete for a disjunctive program. Based on the notion of an elementary loop, we present the class of Head-Elementary-loop-Free (HEF) programs, which strictly generalizes the class of Head-Cycle-Free (HCF) programs attributable to Ben-Eliyahu and Dechter (1994. Annals of Mathematics and Artificial Intelligence 12, 53-87). Like an Ha: program, an HEF program can be turned into an equivalent nondisjunctive program in polynomial time by shifting head atoms into the body.

Answer Set Programming (ASP) is an emerging paradigm for declarative programming, in which a computational problem is specified by a logic program such that particular models, called answer sets, match solutions. ASP faces a growing range of applications, demanding for high-performance tools able to solve complex problems. ASP integrates ideas from a variety of neighboring fields. In particular, automated techniques to search for answer sets are inspired by Boolean Satisfiability (SAT) solving approaches. While the latter have firm proof-theoretic foundations, ASP lacks formal frameworks for characterizing and comparing solving methods. Furthermore, sophisticated search patterns of modern SAT solvers, successfully applied in areas like, e.g., model checking and verification, are not yet established in ASP solving. We address these deficiencies by, for one, providing proof-theoretic frameworks that allow for characterizing, comparing, and analyzing approaches to answer set computation. For another, we devise modern ASP solving algorithms that integrate and extend state-of-the-art techniques for Boolean constraint solving. We thus contribute to the understanding of existing ASP solving approaches and their interconnections as well as to their enhancement by incorporating sophisticated search patterns. The central idea of our approach is to identify atomic as well as composite constituents of a propositional logic program with Boolean variables. This enables us to describe fundamental inference steps, and to selectively combine them in proof-theoretic characterizations of various ASP solving methods. In particular, we show that different concepts of case analyses applied by existing ASP solvers implicate mutual exponential separations regarding their best-case complexities. We also develop a generic proof-theoretic framework amenable to language extensions, and we point out that exponential separations can likewise be obtained due to case analyses on them. We further exploit fundamental inference steps to derive Boolean constraints characterizing answer sets. They enable the conception of ASP solving algorithms including search patterns of modern SAT solvers, while also allowing for direct technology transfers between the areas of ASP and SAT solving. Beyond the search for one answer set of a logic program, we address the enumeration of answer sets and their projections to a subvocabulary, respectively. The algorithms we develop enable repetition-free enumeration in polynomial space without being intrusive, i.e., they do not necessitate any modifications of computations before an answer set is found. Our approach to ASP solving is implemented in clasp, a state-of-the-art Boolean constraint solver that has successfully participated in recent solver competitions. Although we do here not address the implementation techniques of clasp or all of its features, we present the principles of its success in the context of ASP solving.

We introduce formal proof systems based on tableau methods for analyzing computations in Answer Set Programming (ASP). Our approach furnishes fine-grained instruments for characterizing operations as well as strategies of ASP solvers. The granularity is detailed enough to capture a variety of propagation and choice methods of algorithms used for ASP solving, also incorporating SAT-based and conflict-driven learning approaches to some extent. This provides us with a uniform setting for identifying and comparing fundamental properties of ASP solving approaches. In particular, we investigate their proof complexities and show that the run-times of best-case computations can vary exponentially between different existing ASP solvers. Apart from providing a framework for comparing ASP solving approaches, our characterizations also contribute to their understanding by pinning down the constitutive atomic operations. Furthermore, our framework is flexible enough to integrate new inference patterns, and so to study their relation to existing ones. To this end, we generalize our approach and provide an extensible basis aiming at a modular incorporation of additional language constructs. This is exemplified by augmenting our basic tableau methods with cardinality constraints and disjunctions.

We introduce a simple approach extending the input language of Answer Set Programming (ASP) systems by multi-valued propositions. Our approach is implemented as a (prototypical) preprocessor translating logic programs with multi-valued propositions into logic programs with Boolean propositions only. Our translation is modular and heavily benefits from the expressive input language of ASP. The resulting approach, along with its implementation, allows for solving interesting constraint satisfaction problems in ASP, showing a good performance.