Dr. Stefan Wölfl - Publications

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2012

Julien Hué, Matthias Westphal and Stefan Wölfl.
An automatic Decomposition Method for Qualitative Spatial and Temporal Reasoning.
In Internation Conference on Tools for Artificial Intelligence (ICTAI), pp. 588-595. 2012.
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Qualitative spatial and temporal reasoning is a research field that studies relational, constraint-based formalisms for representing, and reasoning about, spatial and temporal information. The standard approach for checking consistency is based on an exhaustive representation of possible configurations between three entities, the so-called composition tables. These tables, however, encode semantic background knowledge in a redundant way, which becomes a size and efficiency issue, when the composition table needs to be grounded as done in SAT encodings of problem instances. In this paper, we present a new framework that allows for decomposing composition tables into logically simpler parts, while preserving logical equivalence, e.g., the decomposition in start- and end-points for Allen’s Interval Calculus. We show that finding such decompositions is an NP-complete problem and present a SAT-based method to generate decompositions. Finally, we discuss the impact of our decomposition method on SAT encodings of problem instances, and present a reasoning system built on decompositions that compares favorably with state-of-the-art solvers.
Stefan Wölfl (ed.).
Poster and Demo Track of the 35th German Conference on Artificial Intelligence (KI-2012), September 24-27, 2012, Saarbrücken, Germany.
2012.
(PDF)

2011

Anthony G. Cohn, Jochen Renz and Stefan Wölfl (eds.).
Proceedings of the IJCAI-2011 Workshop on Benchmarks and Applications of Spatial Reasoning, Barcelona, Spain, July 17, 2011.
2011.
(PDF)
Manuel Bodirsky and Stefan Wölfl.
RCC8 is Polynomial on Networks of Bounded Treewidth.
In Proceedings of the 22nd International Joint Conference on Artificial Intelligence (IJCAI 2011), pp. 756-761. AAAI Press 2011.
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We construct an homogeneous (and ω-categorical) representation of the relation algebra RCC8, which is one of the fundamental formalisms for spatial reasoning. As a consequence we obtain that the network consistency problem for RCC8 can be solved in polynomial time for networks of bounded treewidth.
Matthias Westphal, Stefan Wölfl, Bernhard Nebel and Jochen Renz.
On Qualitative Route Descriptions: Representation and Computational Complexity.
In Proceedings of the 22nd International Joint Conference on Artificial Intelligence (IJCAI 2011), pp. 1120-1125. AAAI Press 2011.
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The generation of route descriptions is a fundamental task of navigation systems. A particular problem in this context is to identify routes that can easily be described and processed by users. In this work, we present a framework for representing route networks with the qualitative information necessary to evaluate and optimize route descriptions with regard to ambiguities in them. We identify different agent models that differ in how agents are assumed to process route descriptions while navigating through route networks. Further, we analyze the computational complexity of matching route descriptions and paths in route networks in dependency of the agent model. Finally we empirically evaluate the influence of the agent model on the optimization and the processing of route instructions.
Mehul Bhatt, Hans Guesgen, Stefan Wölfl and Shyamanta Hazarika.
Qualitative Spatial and Temporal Reasoning: Emerging Applications, Trends, and Directions.
Spatial Cognition & Computation: An Interdisciplinary Journal 11 (1), pp. 1-14. 2011.
(DOI)
Matthias Westphal, Christian Dornhege, Stefan Wölfl, Marc Gissler and Bernhard Nebel.
Guiding the Generation of Manipulation Plans by Qualitative Spatial Reasoning.
Spatial Cognition & Computation: An Interdisciplinary Journal 11 (1), pp. 75-102. 2011.
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Manipulation planning is a complex task for robots with a manipulator arm that need to grasp objects in the environment, specifically under narrow spatial conditions restricting the workspace of the robot. A popular approach for generating motion plans is probabilistic roadmap planning. However, the sampling strategy of such planners is usually unguided, and hence may lead to motion plans that seem counterintuitive for a human observer. In this article we present an approach that generates heuristics for the probabilistic sampling strategy from spatial plans that abstract from concrete metric data. These spatial plans describe a free trajectory in the workspace of the robot on a purely qualitative level, i.e., by employing spatial relations from formalisms considered in the domain of Qualitative Spatial and Tem- poral Reasoning. We discuss how such formalisms and constraint-based reasoning methods can be applied to approximate geometrically feasible motions. The paper is completed by an evaluation of a hybrid planning system in different spatial settings showing that run-times are notably improved when an abstract plan is considered as a guidance heuristic.

2010

Jochen Renz and Stefan Wölfl.
A Qualitative Representation of Route Networks.
In Proceedings of the 19th European Conference on Artificial Intelligence (ECAI 2010), pp. 1091-1092. IOS Press 2010.
(DBLP)
Matthias Westphal, Stefan Wölfl and Jason Jingshi Li.
Restarts and Nogood Recording in Qualitative Constraint-based Reasoning.
In Proceedings of the 19th European Conference on Artificial Intelligence (ECAI 2010), pp. 1093-1094. IOS Press 2010.
Also, see the follow up paper at KI 2012: Nogoods in Qualitative Constaint-based Reasoning.
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This paper introduces restart and nogood recording techniques in the domain of qualitative spatial and temporal reasoning. Nogoods and restarts can be applied orthogonally to usual methods for solving qualitative constraint satisfaction problems. In particular, we propose a more general definition of nogoods that allows for exploiting information about nogoods and tractable subclasses during backtracking search. First evaluations of the proposed techniques show promising results.

2009

Florian Pommerening, Stefan Wölfl and Matthias Westphal.
Right-of-Way Rules as Use Case for Integrating GOLOG and Qualitative Reasoning.
In Bärbel Mertsching, Marcus Hund and Muhammad Zaheer Aziz (eds.), Proceedings of the 32nd Annual Conference on Artificial Intelligence (KI 2009), pp. 468-475. Springer-Verlag 2009.
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Agents interacting in a dynamically changing spatial environment often need to access the same spatial resources. A typical example is given by moving vehicles that meet at an intersection in a street network. In such situations right-of-way rules regulate the actions the vehicles involved may perform. For this application scenario we show how the Golog framework for reasoning about action and change can be enhanced by external reasoning services that implement techniques known from the domain of Qualitative Spatial Reasoning.
Matthias Westphal and Stefan Wölfl.
Qualitative CSP, finite CSP, and SAT: Comparing methods for qualitative constraint-based reasoning.
In Proceedings of the 21th International Joint Conference on Artificial Intelligence (IJCAI 2009). 2009.
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Qualitative Spatial and Temporal Reasoning (QSR) is concerned with constraint-based formalisms for representing, and reasoning with, spatial and temporal information over infinite domains. Within the community it has been a widely accepted assumption that genuine qualitative reasoning methods outperform other reasoning methods that are applicable to encodings of qualitative CSP instances. Recently this assumption has been tackled by several authors, who proposed to encode qualitative CSP instances as finite CSP or SAT instances. In this paper we report on the results of a broad empirical study in which we compared the performance of several reasoners on instances from different qualitative formalisms. Our results show that for small-sized qualitative calculi (e.g. Allen's interval algebra and RCC-8) a state-of-the-art implementation of QSR methods currently gives the most efficient performance. However, on recently suggested large-size calculi, e.g. OPRA_4, finite CSP encodings provide a considerable performance gain. These results confirm a conjecture by Bessière stating that support-based constraint propagation algorithms provide better performance for large-sized qualitative calculi.
Stefan Wölfl and Matthias Westphal.
On combinations of binary qualitative constraint calculi.
In Proceedings of the 21th International Joint Conference on Artificial Intelligence (IJCAI 2009). 2009.
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Qualitative constraint calculi are representation formalisms that allow for efficient reasoning about spatial and temporal information. Many of the calculi discussed in the field of Qualitative Spatial and Temporal Reasoning can be defined as combinations of other, simpler and more compact formalisms. On the other hand, existing calculi can be combined to a new formalism in which one can represent, and reason about, different aspects of a domain at the same time. For example, Gerevini and Renz presented a loose combination of the region connection calculus RCC-8 and the point algebra: the resulting formalism integrates topological and qualitative size relations between spatially extended objects. In this paper we compare the approach by Gerevini and Renz to a method that generates a new qualitative calculus by exploiting the semantic interdependencies between the component calculi. We will compare these two methods and analyze some formal relationships between a combined calculus and its components. The paper is completed by an empirical case study in which the reasoning performance of the suggested methods is compared on random test instances.
Bernhard Nebel and Stefan Wölfl.
Benchmarking of Qualitative Spatial and Temporal Reasoning Systems.
2009.
AAAI Technical Report SS-09-02.
(AAAI)
Matthias Westphal and Stefan Wölfl.
Confirming the QSR Promise.
In AAAI Spring Symposium on Benchmarking of Qualitative Spatial and Temporal Reasoning Systems. 2009.
AAAI Technical Report SS-09-02.
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Within the qualitative spatial reasoning community it has been a widely accepted commonplace that reasoning in qualitative constraint calculi outperforms reasoning in other more general and expressive formalisms. To check the correctness of this assumption we conducted some empirical case studies in which we compared the performance of a qualitative constraint solver with different automated reasoning systems, namely first-order and description logic reasoners. We also report on some first results from comparing the performance of qualitative and finite constraint solvers. Our empirical tests are based on randomly generated instances of qualitative constraint satisfaction problems, which have been encoded as reasoning problems for first-order reasoners, description logic reasoners, and finite CSP solvers, respectively. Given our currently used encodings, these studies show that first-order and description logic reasoners are far from being feasible for problem sizes that can easily be solved by a qualitative reasoner. In contrast, finite CSP solvers are competitive, but still outperformed by a qualitative reasoner on the problem instances considered here.
Matthias Westphal, Stefan Wölfl and Zeno Gantner.
GQR: A Fast Solver for Binary Qualitative Constraint Networks.
In AAAI Spring Symposium on Benchmarking of Qualitative Spatial and Temporal Reasoning Systems. 2009.
AAAI Technical Report SS-09-02.
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Qualitative calculi are constraint-based representation formalisms that allow for efficient reasoning about continuous aspects of the world with inherently infinite domains, such as time and space. GQR (Generic Qualitative Reasoner) is a tool that provides reasoning services for arbitrary binary qualitative calculi. Given qualitative information expressible in a qualitative calculus, GQR checks whether this information is consistent w.r.t. the calculus definition. GQR employs state-of-the-art techniques in both qualitative and constraint reasoning, such as heuristic search and usage of known tractable subclasses. In contrast to specialized reasoners, it offers reasoning services for a variety of calculi known in the literature, which can be defined in a simple specification language. The main focus in the design and implementation of GQR is to provide a generic and extensible solver that preserves efficiency and scalability.

2008

Christian Freksa, Nora Newcombe, Peter Gärdenfors and Stefan Wölfl (eds.).
Spatial Cognition VI: Learning, Reasoning, and Talking about Space, International Conference Spatial Cognition 2008 (SC '08), Freiburg, Germany, September 15-19, 2008.
Volume 5248 of Lecture Notes in Artificial Intelligence.
Springer 2008.
(Springer) (DBLP)
Diedrich Wolter, Frank Dylla, Stefan Wölfl, Jan Oliver Wallgrün, Lutz Frommberger, Bernhard Nebel and Christian Freksa.
SailAway: Spatial Cognition in Sea Navigation.
Künstliche Intelligenz 08 (1), pp. 28-30. 2008.
(DBLP)
Frank Dylla, Diedrich Wolter, Lutz Frommberger, Christian Freksa, Stefan Wölfl and Bernhard Nebel.
Qualitative Methoden zur Steuerung von Agenten - SailAway: Raumkognition zur Steuerung von Schiffen.
Industrie Management 4. 2008.
(BIB)
Marco Ragni and Stefan Wölfl.
Reasoning about topological and positional information in dynamic settings.
In Proceedings of the Twenty-First International FLAIRS Conference (2008), pp. 606-611. 2008.
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Typical application fields of spatial and spatio-temporal representation formalisms and reasoning techniques include geographic information systems (GIS), mobile assistance systems for route finding, layout descriptions, and navigation tasks of robots interacting with humans. Often these formalisms focus on specific spatial aspects in that they use either topological or positional relations. In this paper we propose a formalism which allows for representing topological and positional relations between disks in the plane. Although this formalism employs a rather abstract representation of objects as disks, it provides an interesting test-bed for investigating typical problems that arise when topological and positional information about objects are combined, or when such combined formalisms are used to represent continuous changes in the considered application scenario.
Matthias Westphal and Stefan Wölfl.
Bipath Consistency Revisited.
In Proceedings of the ECAI'08 Workshop on Spatial and Temporal Reasoning. Patras, Greece 2008.
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In the field of qualitative spatial and temporal reasoning combinations of constraint calculi have attracted considerable research interest in recent years. Beside combinations of spatial and temporal calculi, it is an important research topic to develop generic methods for combining calculi dealing with different spatial aspects. The prototypical example is the combination of the region connection calculus RCC8 and the point algebra first discussed by Gerevini and Renz, which allows to represent, and reason about, topological and size information about spatially extended objects. To solve constraints in this calculus, Gerevini and Renz also proposed an algorithm, the bipath consistency algorithm, which allows for deciding consistency of a given constraint network for specific sets of relations combining topology and size. In this article we will compare the "bipath consistency"-based combination method to the standard method, which is to combine calculi by generating a new calculus and applying the standard path consistency method. Gerevini and Renz's calculus combining topological and size information will serve as the running example of such combinations and also as a test case for an empirical analysis.
Zeno Gantner, Matthias Westphal and Stefan Wölfl.
GQR - A Fast Reasoner for Binary Qualitative Constraint Calculi.
In Proceedings of the AAAI'08 Workshop on Spatial and Temporal Reasoning. Chicago, USA 2008.
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GQR (Generic Qualitative Reasoner) is a solver for binary qualitative constraint networks. GQR takes a calculus description and one or more constraint networks as input, and tries to solve the networks using the path consistency method and (heuristic) backtracking. In contrast to specialized reasoners, it offers reasoning services for different qualitative calculi, which means that these calculi are not hard-coded into the reasoner. Currently, GQR supports arbitrary binary constraint calculi developed for spatial and temporal reasoning, such as calculi from the RCC family, the intersection calculi, Allen's interval algebra, cardinal direction calculi, and calculi from the OPRA family. New calculi can be added to the system by specifications in a simple text format or in an XML file format. The tool is designed and implemented with genericity and extensibility in mind, while preserving efficiency and scalability. The user can choose between different data structures and heuristics, and new ones can be easily added to the object-oriented framework. GQR is free software distributed under the terms of the GNU General Public License.

2007

Stefan Wölfl, Till Mossakowski and Lutz Schröder.
Qualitative constraint calculi: Heterogeneous verification of composition tables.
In Proceedings of the Twentieth International Florida Artificial Intelligence Research Society Conference (FLAIRS 2007), pp. 665-670. AAAI Press 2007.
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In the domain of qualitative constraint reasoning, a subfield of AI which has evolved in the past 25 years, a large number of calculi for efficient reasoning about spatial and temporal entities has been developed. Reasoning techniques developed for these constraint calculi typically rely on so-called composition tables of the calculus at hand, which allow for replacing semantic reasoning by symbolic operations. Often these composition tables are developed in a quite informal, pictorial manner--a method which seems to be error-prone. In view of possible safety critical applications of qualitative calculi, however, it is desirable to formally verify these composition tables. In general, the verification of composition tables is a tedious task, in particular in cases where the semantics of the calculus depends on higher-order constructs such as sets. In this paper we address this problem by presenting a heterogeneous proof method that allows for combining a higher-order proof assistance system (such as Isabelle) with an automatic (first order) reasoner (such as SPASS or VAMPIRE). The benefit of this method is that the number of proof obligations that is to be proven interactively with a semi-automatic reasoner can be minimized to an acceptable level.
Diedrich Wolter, Frank Dylla, Lutz Frommberger, Jan Oliver Wallgrün, Bernhard Nebel and Stefan Wölfl.
Qualitative Spatial Reasoning for Rule Compliant Agent Navigation.
In Proceedings of the Twentieth International Florida Artificial Intelligence Research Society Conference (FLAIRS 2007), pp. 673-674. AAAI Press 2007.
(DBLP)
Frank Dylla, Lutz Frommberger, Jan Oliver Wallgrün, Diedrich Wolter, Berhard Nebel and Stefan Wölfl.
SailAway: Formalizing navigation rules.
In Proceedings of the Artificial and Ambient Intelligence Symposium on Spatial Reasoning and Communication (AISB 2007). 2007.
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Agents that have to solve navigational tasks need to consider aspects that go far beyond single-agent goal-directed deliberation: What an agent does in a specific situation often interferes with what other agents do at the same time. In order to avoid conflicts or even collisions, situations in space are governed by laws, rules, and agreements between the involved agents. For this reason, artificial agents interacting with humans must be able to process such rule sets, which are usually formulated in natural language. In this paper we present a case study on how to formalize navigation rules in the domain of sea navigation. We present an approach that uses qualitative representations of navigation rules. Qualitative spatial reasoning methods can be applied to distinguish permissible actions in the set of all possible actions. We argue that an agent’s spatial representation can be modeled on a qualitative level in a natural way and that this also empowers sophisticated high-level agent control.

2006

Stefan Wölfl and Till Mossakowski.
Qualitative Constraint Calculi - Application and Integration, Workshop at KI 2006, Bremen, Germany, June 14, 2006, Workshop Proceedings.
2006.
(PDF)
Till Mossakowski, Lutz Schroeder and Stefan Wölfl.
A categorical perspective on qualitative constraint calculi.
In Qualitative Constraint Calculi - Application and Integration, Workshop at KI 2006. 2006.
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In the domain of qualitative constraint reasoning, a subfield of AI which has evolved in the last 25 years, a large number of calculi for efficient reasoning about space and time has been developed. Reasoning problems in such calculi are usually formulated as constraint satisfaction problems. For temporal and spatial reasoning, these problems often have infinite domains, which need to be abstracted to (finite) algebras in order to become computationally feasible. Ligozat has argued that the notion of weak representation plays a crucial role: it not only captures the correspondence between abstract relations (in a relation algebra or non-associative algebra) and relations in a concrete domain, but also corresponds to algebraically closed constraint networks. In this work, we examine properties of the category of weak representations and treat the relations between partition schemes, non-associative algebras and concrete domains in a systematic way. This leads to the notion of semi-strong representation, which captures the correspondence between abstract and concrete relations better than the notion of weak representation does. The slogan is that semi-strong representations avoid unnecessary loss of information. Furthermore, we hope that the categorical perspective will help in the future to provide new insights on the important problem of determining whether algebraic closedness decides consistency of constraint networks.
Marco Ragni and Stefan Wölfl.
Temporalizing Cardinal Directions: From Constraint Satisfaction to Planning.
In Proceedings of the Knowledge Representation Conference (KR 2006). 2006.
(PDF)

2005

Marco Ragni and Stefan Wölfl.
Temporalizing Spatial Calculi: On Generalized Neighborhood Graphs.
In Proceedings of the 28th Annual German Conference on AI (KI 2005), pp. 64-78. 2005.
(PDF)
Stefan Wölfl and Till Mossakowski.
CASL specifications of qualitative calculi.
In Spatial Information Theory: Cognitive and Computational Foundations, Proceedings of COSIT'05, pp. 200-217. 2005.
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In AI a large number of calculi for efficient reasoning about spatial and temporal entities have been developed. The most prominent temporal calculi are the point algebra of linear time and Allen's interval calculus. Examples of spatial calculi include mereotopological calculi, Frank's cardinal direction calculus, Freksa's double cross calculus, Egenhofer and Franzosa's intersection calculi, and Randell, Cui, and Cohn's region connection calculi. These calculi are designed for modeling specific aspects of space or time, respectively, to the effect that the class of intended models may vary widely with the calculus at hand. But from a formal point of view these calculi are often closely related to each other. For example, the spatial region connection calculus RCC5 may be considered a coarsening of Allen's (temporal) interval calculus. And vice versa, intervals can be used to represent spatial objects that feature an internal direction. The central question of this paper is how these calculi as well as their mutual dependencies can be axiomatized by algebraic specifications. This question will be investigated within the framework of the Common Algebraic Specification Language (CASL), a specification language developed by the Common Framework Initiative for algebraic specification and development (COFI). We explain scope and expressiveness of CASL by discussing the specifications of some of the calculi mentioned before.
Stefan Wölfl.
Events in branching time.
Studia Logica 79 (2), pp. 255-282. 2005.
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The concept of event is one of the key notions of many theories dealing with causality or agency. In this paper we study different approaches to events that share the basic assumption that events can be analyzed fruitfully in branching-time structures. The terminological framework developed thereby may be helpful for further analyses in the fields of causality and agency and also in those fields of computational semantics, where similar concepts are considered.

2004

Marco Ragni and Stefan Wölfl.
Branching Allen: Reasoning with Intervals in Branching Time.
In Spatial Cognition IV: Reasoning, Action, Interaction, International Conference Spatial Cognition 2004, 2004. Proceedings. Springer-Verlag 2004.
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Allen’s interval calculus is one of the most prominent formalisms in the domain of qualitative spatial and temporal reasoning. Applications of this calculus, however, are restricted to domains that deal with linear flows of time. But how the fundamental ideas of Allen’s calculus can be extended to other, weaker structures than linear orders has gained only little attention in the literature. In this paper we will investigate intervals in branching flows of time, which are of special interest for temporal reasoning, since they allow for representing indeterministic aspects of systems, scenarios, planning tasks, etc. As well, branching time models, i. e., treelike non-linear structures, do have interesting applications in the field of spatial reasoning, for example, for modeling traffic networks. In a first step we discuss interval relations for branching time, thereby comprising various sources from the literature. Then, in a second step, we present some new complexity results concerning constraint satisfaction problems of interval relations in branching time.
Stefan Wölfl.
Qualitative action theory: A comparison of the semantics of Alternating-time Temporal Logic and the Kutschera-Belnap approach to agency.
In J. J. Alferes and J. Leite (eds.), Proceedings of the 9th European Conference on Logics in Artificial Intelligence (JELIA 2004). Springer-Verlag 2004.
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Qualitative action theory deals with purely qualitative descriptions and formal representations of agency, i.e., agents and their possibilities for intervening in the causal flow of events. This means that, contrary to game theory, qualitative action theory abstains from any metric evaluation of the outcomes of actions. In this paper we present and compare two qualitative approaches to action theory that have been discussed in the literature. The first one coming from philosophical action theory is the Kutschera-Belnap approach, which is the semantic basis of so-called Stit-logics. The second approach is the semantics of Alur, Henzinger, and Kupferman's Alternating-time Temporal Logic (ATL). In computer science, ATL has been introduced as an extension of Computational Tree Logic (CTL) to allow for modeling systems that interact with their environment. Surprisingly, although both approaches are very close in spirit, a systematic analysis of the mutual dependencies between these approaches does not exist. The paper aims at bringing together these two research streams, which seem to have been developed independently in philosophy and computer science. In particular, we will investigate the assumptions with which both approaches may be considered equivalent. Finally, further research on this topic promises interesting results that translate between the approaches presented here.

2002

Stefan Wölfl.
Propositional Q-Logic.
Journal of Philosophical Logic 31, pp. 387-414. 2002.
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Topic of the paper is Q-logic a logic of agency in its temporal and modal context. Q-logic may be considered as a basal logic of agency since the most important stit-operators discussed in the literature can be defined or axiomatized easily within its semantical and syntactical framework. Its basic agent dependent operator, the Q-operator (also known as Delta- or cstit-operator), which has been discussed independently by F. v. Kutschera and B. F. Chellas, is investigated here in respect of its relation to other temporal and modal operators. The main result of the paper, then, is a completeness result for a calculus of Q-logic with respect to a semantics defined on the tree-approach to agency as introduced and developed by, among others, F. v. Kutschera and N. D. Belnap.

1999

Stefan Wölfl.
Combinations of tense and modality for predicate logic.
Journal of Philosophical Logic 28, pp. 371-398. 1999.
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In recent years combinations of tense and modality have moved into the focus of logical research. From a philosophical point of view, logical systems combining tense and modality are of interest because these logics have a wide field of application in original philosophical issues, for example in the theory of causation, of action, etc. But until now only methods yielding completeness results for propositional languages have been developed. In view of philosophical applications, analogous results with respect to languages of predicate logic are desirable, and in this paper I present two such results. The main developments in this area can be split into two directions, differing in the question whether the ordering of time is world-independent or not. Semantically, this difference appears in the discussion whether T×W-frames or Kamp-frames (resp. Ockham-frames) provide a suitable semantics for combinations of tense and modality. Here, two calculi are presented, the first adequate with respect to Kamp-semantics, the second to T×W-semantics. (Both calculi contain an appropriate version of Gabbay's irreflexivity rule.) Furthermore, the proposed constructions of canonical frames simplify some of those which have hitherto been discussed.
Stefan Wölfl.
Kombinierte Zeit- und Modallogik: Vollständigkeitsresultate für prädikatenlogische Sprachen.
Volume 5 of Logische Philosophie.
Logos Verlag, Berlin, Germany 1999.
Dissertation Thesis.

My Erdös number is presumably equal to 4: Stefan Wölfl, Till Mossakowski, George E. Strecker, Marcel Erne, Paul Erdös.