The complexity of reasoning with relative directions

© 2014 The Authors and IOS Press. Whether reasoning with relative directions can be performed in NP has been an open problem in qualitative spatial reasoning. Efficient reasoning with relative directions is essential, for example, in rule-compliant agent navigation. In this paper, we prove that reasoning with relative directions is ∃ℝ-complete. As a consequence, reasoning with relative directions is not in NP, unless NP=∃ℝ

Algebraic foundations for qualitative calculi and networks

A qualitative representation $\phi$ is like an ordinary representation of a relation algebra, but instead of requiring $(a; b)^\phi = a^\phi | b^\phi$, as we do for ordinary representations, we only require that $c^\phi\supseteq a^\phi | b^\phi \iff c\geq a ; b$, for each $c$ in the algebra. A constraint network is qualitatively satisfiable if its nodes can be mapped to elements of a qualitative representation, preserving the constraints. If a constraint network is satisfiable then it is clearly qualitatively satisfiable, but the converse can fail. However, for a wide range of relation algebras including the point algebra, the Allen Interval Algebra, RCC8 and many others, a network is satisfiable if and only if it is qualitatively satisfiable. Unlike ordinary composition, the weak composition arising from qualitative representations need not be associative, so we can generalise by considering network satisfaction problems over non-associative algebras. We prove that computationally, qualitative representations have many advantages over ordinary representations: whereas many finite relation algebras have only infinite representations, every finite qualitatively representable algebra has a finite qualitative representation; the representability problem for (the atom structures of) finite non-associative algebras is NP-complete; the network satisfaction problem over a finite qualitatively representable algebra is always in NP; the validity of equations over qualitative representations is co-NP-complete. On the other hand we prove that there is no finite axiomatisation of the class of qualitatively representable algebras.Comment: 22 page

Qualitative Spatial and Temporal Reasoning based on And/Or Linear Programming An approach to partially grounded qualitative spatial reasoning

Acting intelligently in dynamic environments involves anticipating surrounding processes, for example to foresee a dangerous situation or acceptable social behavior. Knowledge about spatial configurations and how they develop over time enables intelligent robots to safely navigate by reasoning about possible actions. The seamless connection of high-level deliberative processes to perception and action selection remains a challenge though. Moreover, an integration should allow the robot to build awareness of these processes as in reality there will be misunderstandings a robot should be able to respond to. My aim is to verify that actions selected by the robot do not violate navigation or safety regulations and thereby endanger the robot or others. Navigation rules specified qualitatively allow an autonomous agent to consistently combine all rules applicable in a context. Within this thesis, I develop a formal, symbolic representation of right-of-way-rules based on a qualitative spatial representation. This cumulative dissertation consists of 5 peer-reviewed papers and 1 manuscript under review. The contribution of this thesis is an approach to represent navigation patterns based on qualitative spatio-temporal representation and the development of corresponding effective sound reasoning techniques. The approach is based on a spatial logic in the sense of Aiello, Pratt-Hartmann, and van Benthem. This logic has clear spatial and temporal semantics and I demonstrate how it allows various navigation rules and social conventions to be represented. I demonstrate the applicability of the developed method in three different areas, an autonomous robotic system in an industrial setting, an autonomous sailing boat, and a robot that should act politely by adhering to social conventions. In all three settings, the navigation behavior is specified by logic formulas. Temporal reasoning is performed via model checking. An important aspect is that a logic symbol, such as \emph{turn left}, comprises a family of movement behaviors rather than a single pre-specified movement command. This enables to incorporate the current spatial context, the possible changing kinematics of the robotic system, and so on without changing a single formula. Additionally, I show that the developed approach can be integrated into various robotic software architectures. Further, an answer to three long standing questions in the field of qualitative spatial reasoning is presented. Using generalized linear programming as a unifying basis for reasoning, one can jointly reason about relations from different qualitative calculi. Also, concrete entities (fixed points, regions fixed in shape and/or position, etc.) can be mixed with free variables. In addition, a realization of qualitative spatial description can be calculated, i.e., a specific instance/example. All three features are important for applications but cannot be handled by other techniques. I advocate the use of And/Or trees to facilitate efficient reasoning and I show the feasibility of my approach. Last but not least, I investigate a fourth question, how to integrate And/Or trees with linear temporal logic, to enable spatio-temporal reasoning