Symmetry Breaking for Answer Set Programming

In the context of answer set programming, this work investigates symmetry detection and symmetry breaking to eliminate symmetric parts of the search space and, thereby, simplify the solution process. We contribute a reduction of symmetry detection to a graph automorphism problem which allows to extract symmetries of a logic program from the symmetries of the constructed coloured graph. We also propose an encoding of symmetry-breaking constraints in terms of permutation cycles and use only generators in this process which implicitly represent symmetries and always with exponential compression. These ideas are formulated as preprocessing and implemented in a completely automated flow that first detects symmetries from a given answer set program, adds symmetry-breaking constraints, and can be applied to any existing answer set solver. We demonstrate computational impact on benchmarks versus direct application of the solver. Furthermore, we explore symmetry breaking for answer set programming in two domains: first, constraint answer set programming as a novel approach to represent and solve constraint satisfaction problems, and second, distributed nonmonotonic multi-context systems. In particular, we formulate a translation-based approach to constraint answer set solving which allows for the application of our symmetry detection and symmetry breaking methods. To compare their performance with a-priori symmetry breaking techniques, we also contribute a decomposition of the global value precedence constraint that enforces domain consistency on the original constraint via the unit-propagation of an answer set solver. We evaluate both options in an empirical analysis. In the context of distributed nonmonotonic multi-context system, we develop an algorithm for distributed symmetry detection and also carry over symmetry-breaking constraints for distributed answer set programming.Comment: Diploma thesis. Vienna University of Technology, August 201

Towards a typology of spatial relations and properties for urban applications

Relations that occur between features located in space–like the fact that a street is surrounded by very high buildings, that an airport is close to a city- as well as spatial properties of features–like the height and width of a door- play an important role for many urban applications. Digital models of cities can assist in the evaluation of these relations and properties either through visualisation or through computation, mainly based on geometrical information. Hence, considering the objective of explaining to potential users of these city models what useful information they can derive from these data and how, a possible way to address this objective lies in the usage of a pivot model composed of relevant spatial properties and relations, connected to information meaningful to the user and connected to the possible computation of them on available data. This paper firstly sets the ground for a typology of such relevant relations and properties that are shared by different applications and that can be derived/approximated from existing data. It then proposes a model to describe these properties and relations and connect them to their possible computation based on data (2D or 3D). An important aspect of this model is to distinguish between a conceptual layer where relations occur between “real world” features and an implementation layer where they are calculated based on database features and geometries

Tensor Network Methods for Quantum Phases

The physics that emerges when large numbers of particles interact can be complex and exotic. The collective behaviour may not re ect the underlying constituents, for example fermionic quasiparticles can emerge from models of interacting bosons. Due to this emergent complexity, manybody phenomena can be very challenging to study, but also very useful. A theoretical understanding of such systems is important for robust quantum information storage and processing. The emergent, macroscopic physics can be classi ed using the idea of a quantum phase. All models within a given phase exhibit similar low-energy emergent physics, which is distinct from that displayed by models in di erent phases. In this thesis, we utilise tensor networks to study many-body systems in a range of quantum phases. These include topologically ordered phases, gapless symmetry-protected phases, and symmetry-enriched topological phases

Complex Networks and Symmetry I: A Review

In this review we establish various connections between complex networks and symmetry. While special types of symmetries (e.g., automorphisms) are studied in detail within discrete mathematics for particular classes of deterministic graphs, the analysis of more general symmetries in real complex networks is far less developed. We argue that real networks, as any entity characterized by imperfections or errors, necessarily require a stochastic notion of invariance. We therefore propose a definition of stochastic symmetry based on graph ensembles and use it to review the main results of network theory from an unusual perspective. The results discussed here and in a companion paper show that stochastic symmetry highlights the most informative topological properties of real networks, even in noisy situations unaccessible to exact techniques.Comment: Final accepted versio

Bioinspired symmetry detection on resource limited embedded platforms

This work is inspired by the vision of flying insects which enables them to detect and locate a set of relevant objects with remarkable effectiveness despite very limited brainpower. The bioinspired approach worked out here focuses on detection of symmetric objects to be performed by resource-limited embedded platforms such as micro air vehicles. Symmetry detection is posed as a pattern matching problem which is solved by an approach based on the use of composite correlation filters. Two variants of the approach are proposed, analysed and tested in which symmetry detection is cast as 1) static and 2) dynamic pattern matching problems. In the static variant, images of objects are input to two dimentional spatial composite correlation filters. In the dynamic variant, a video (resulting from platform motion) is input to a composite correlation filter of which its peak response is used to define symmetry. In both cases, a novel method is used for designing the composite filter templates for symmetry detection. This method significantly reduces the level of detail which needs to be matched to achieve good detection performance. The resulting performance is systematically quantified using the ROC analysis; it is demonstrated that the bioinspired detection approach is better and with a lower computational cost compared to the best state-of-the-art solution hitherto available

Optimisation in ‘Self-modelling’ Complex Adaptive Systems

When a dynamical system with multiple point attractors is released from an arbitrary initial condition it will relax into a configuration that locally resolves the constraints or opposing forces between interdependent state variables. However, when there are many conflicting interdependencies between variables, finding a configuration that globally optimises these constraints by this method is unlikely, or may take many attempts. Here we show that a simple distributed mechanism can incrementally alter a dynamical system such that it finds lower energy configurations, more reliably and more quickly. Specifically, when Hebbian learning is applied to the connections of a simple dynamical system undergoing repeated relaxation, the system will develop an associative memory that amplifies a subset of its own attractor states. This modifies the dynamics of the system such that its ability to find configurations that minimise total system energy, and globally resolve conflicts between interdependent variables, is enhanced. Moreover, we show that the system is not merely ‘recalling’ low energy states that have been previously visited but ‘predicting’ their location by generalising over local attractor states that have already been visited. This ‘self-modelling’ framework, i.e. a system that augments its behaviour with an associative memory of its own attractors, helps us better-understand the conditions under which a simple locally-mediated mechanism of self-organisation can promote significantly enhanced global resolution of conflicts between the components of a complex adaptive system. We illustrate this process in random and modular network constraint problems equivalent to graph colouring and distributed task allocation problems