Qualitative Calculus Research Articles

A Logical–Algebraic Approach to Revising Formal Ontologies: Application in Mereotopology

In ontology engineering, reusing (or extending) ontologies poses a significant challenge, requiring revising their ontological commitments and ensuring accurate representation and coherent reasoning. This study aims to address two main objectives. Firstly, it seeks to develop a methodological approach supporting ontology extension practices. Secondly, it aims to demonstrate its feasibility by applying the approach to the case of extending qualitative spatial reasoning (QSR) theories. Key questions involve effectively interpreting spatial extensions while maintaining consistency. The framework systematically analyzes extensions of formal ontologies, providing a reconstruction of a qualitative calculus. Reconstructed qualitative calculus demonstrates improved interpretative capabilities and reasoning accuracy. The research underscores the importance of methodological approaches when extending formal ontologies, with spatial interpretation serving as a valuable case study.

GEQCA: Generic Qualitative Constraint Acquisition

Many planning, scheduling or multi-dimensional packing problems involve the design of subtle logical combinations of temporal or spatial constraints. On the one hand, the precise modelling of these constraints, which are formulated in various relation algebras, entails a number of possible logical combinations and requires expertise in constraint-based modelling. On the other hand, active constraint acquisition (CA) has been used successfully to support non-experienced users in learning conjunctive constraint networks through the generation of a sequence of queries. In this paper, we propose GEACQ, which stands for Generic Qualitative Constraint Acquisition, an active CA method that learns qualitative constraints via the concept of qualitative queries. GEACQ combines qualitative queries with time-bounded path consistency (PC) and background knowledge propagation to acquire the qualitative constraints of any scheduling or packing problem. We prove soundness, completeness and termination of GEACQ by exploiting the jointly exhaustive and pairwise disjoint property of qualitative calculus and we give an experimental evaluation that shows (i) the efficiency of our approach in learning temporal constraints and, (ii) the use of GEACQ on real scheduling instances.

So, what exactly is a qualitative calculus?

The paradigm of algebraic constraint-based reasoning, embodied in the notion of a qualitative calculus, is studied within two alternative frameworks. One framework defines a qualitative calculus as “a non-associative relation algebra (NA) with a qualitative representation”, the other as “an algebra generated by jointly exhaustive and pairwise disjoint (JEPD) relations”. These frameworks provide complementary perspectives: the first is intensional (axiom-based), whereas the second one is extensional (based on semantic structures). However, each definition admits calculi that lie beyond the scope of the other. Thus, a qualitatively representable NA may be incomplete or non-atomic, whereas an algebra generated by JEPD relations may have non-involutive converse and no identity element. The divergence of definitions creates a confusion around the notion of a qualitative calculus and makes the “what” question posed by Ligozat and Renz actual once again. Here we define the relation-type qualitative calculus unifying the intensional and extensional approaches. By introducing the notions of weak identity, inference completeness and Q-homomorphism, we give equivalent definitions of qualitative calculi both intensionally and extensionally. We show that “algebras generated by JEPD relations” and “qualitatively representable NAs” are embedded into the class of relation-type qualitative algebras.

A Generalised Approach for Encoding and Reasoning with Qualitative Theories in Answer Set Programming

AbstractQualitative reasoning involves expressing and deriving knowledge based on qualitative terms such as natural language expressions, rather than strict mathematical quantities. Well over 40 qualitative calculi have been proposed so far, mostly in the spatial and temporal domains, with several practical applications such as naval traffic monitoring, warehouse process optimisation and robot manipulation. Even if a number of specialised qualitative reasoning tools have been developed so far, an important barrier to the wider adoption of these tools is that only qualitative reasoning is supported natively, when real-world problems most often require a combination of qualitative and other forms of reasoning. In this work, we propose to overcome this barrier by using ASP as a unifying formalism to tackle problems that require qualitative reasoning in addition to non-qualitative reasoning. A family of ASP encodings is proposed which can handle any qualitative calculus with binary relations. These encodings are experimentally evaluated using a real-world dataset based on a case study of determining optimal coverage of telecommunication antennas, and compared with the performance of two well-known dedicated reasoners. Experimental results show that the proposed encodings outperform one of the two reasoners, but fall behind the other, an acceptable trade-off given the added benefits of handling any type of reasoning as well as the interpretability of logic programs.

Spatial movement pattern recognition in soccer based on relative player movements.

Knowledge of spatial movement patterns in soccer occurring on a regular basis can give a soccer coach, analyst or reporter insights in the playing style or tactics of a group of players or team. Furthermore, it can support a coach to better prepare for a soccer match by analysing (trained) movement patterns of both his own as well as opponent players. We explore the use of the Qualitative Trajectory Calculus (QTC), a spatiotemporal qualitative calculus describing the relative movement between objects, for spatial movement pattern recognition of players movements in soccer. The proposed method allows for the recognition of spatial movement patterns that occur on different parts of the field and/or at different spatial scales. Furthermore, the Levenshtein distance metric supports the recognition of similar movements that occur at different speeds and enables the comparison of movements that have different temporal lengths. We first present the basics of the calculus, and subsequently illustrate its applicability with a real soccer case. To that end, we present a situation where a user chooses the movements of two players during 20 seconds of a real soccer match of a 2016–2017 professional soccer competition as a reference fragment. Following a pattern matching procedure, we describe all other fragments with QTC and calculate their distance with the QTC representation of the reference fragment. The top-k most similar fragments of the same match are presented and validated by means of a duo-trio test. The analyses show the potential of QTC for spatial movement pattern recognition in soccer.

Eleventh Graders\u2019 Increasingly Elaborate Language Use for Disentangling Amount and Change: A\xa0Case Study on the Epistemic Role of Syntactic Language Complexity

The syntactic dimension of academic language has often been studied with respect to students’ difficulties with syntactic features in mathematical textbooks and test items, and these studies have contributed to understanding the communicative role of language. In contrast, the epistemic role of students’ language use has mainly been explored in lexical and discourse dimensions. This research has shown that higher order cognitive demands require more elaborate language means. The aim of this article is to contribute to theorizing the epistemic role of syntactic language complexity by means of a topic-specific investigation using the mathematical topic of qualitative calculus, i.e., the informal meanings of amount and change. In order to do this, the learning process study presented in this article investigates 18 eleventh graders’ conceptual pathways while dealing with challenging tasks on amount and change. The identification of different syntactic complexities in students’ utterances provides an overview of the variance of possible phrase structures. Further, it shows that successive conceptual conciseness requires either increasing syntactic complexity or conceptual condensation. So increasing elaborateness in the lexical and syntactic dimensions seem to compensate each other.

A Process Algebra With Qualitative Calculus for Modeling Spatio-Temporal Behaviour

Distributive real-time autonomous systems like cyber-physical systems (CPS) and biology systems have become a hot topic nowadays. The high-level design of these systems shares a common characteristic that the behavior of agents is always space-related: i.e., agents are involved in an environment where the space relations between agents count. Agents are always sensitive to the discrete change of space relations among them at explicit time. We call such behavior the spatio-temporal behavior, which has become a critical research point recently. In this paper, we propose a process algebra called Communicating Sequential Process with Qualitative calculus (QCSP) as a formal language for modeling the spatio-temporal behavior. The key feature of our algebra is that it can reason about the space relations between agents by a finite set of binary relations that obey certain mathematical conditions - a qualitative calculus. In QCSP, each agent is confined in a space region. The space region (with an agent in it) can move as a whole and interact with other space regions. The topological structure of space regions forms the outer environment of agents, and the agents react timely according to their environment from time to time. We built our calculi by conservatively extending Timed CSP with qualitative calculus embedded in it, and illustrate how QCSP can be used to model the spatio-temporal behavior by analyzing three examples. We propose a special process algebra as a formal model for behavior that is sensitive to qualitative changing space and time. The key feature of our algebra is that it can reason about the relative location between agents by using qualitative calculus. Each agent is confined in a space region. The space region can move as a whole along with the agent confined in it and interact with other space regions. The behavior of agent depends on the interaction of space regions at an explicit time, shown by that certain actions can be triggered (or allowed) under certain space region relations and time condition. Such spatio-temporal behavior is a critical research point in nowadays distributive real-time autonomous systems like CPS or intelligent transportation systems (ITS). We propose our calculi by conservatively extending Timed CSP with qualitative calculus embedded in it to capture space region relations among agents.

Assessing the potential of the qualitative trajectory calculus to detect gait pathologies: a case study of children with developmental coordination disorder

The Qualitative Trajectory Calculus (QTC) is a qualitative spatio-temporal calculus for describing interactions between moving point objects. So far, it remained unclear whether QTC is useful for describing subtle differences, such as between the movements of different parts of a human body. We tested the applicability of QTC to detect differences in the gait patterns of children with or without Developmental Coordination Disorder (DCD). We found that using a combination of three markers (i.e. ankle, toe and trochanter), QTC can achieve a high classification accuracy (i.e. 83.3%) of classifying subjects correctly to either the DCD group or the control group.

A Trajectory Calculus for Qualitative Spatial Reasoning Using Answer Set Programming

AbstractSpatial information is often expressed using qualitative terms such as natural language expressions instead of coordinates; reasoning over such terms has several practical applications, such as bus routes planning. Representing and reasoning on trajectories is a specific case of qualitative spatial reasoning that focuses on moving objects and their paths. In this work, we propose two versions of a trajectory calculus based on the allowed properties over trajectories, where trajectories are defined as a sequence of non-overlapping regions of a partitioned map. More specifically, if a given trajectory is allowed to start and finish at the same region, 6 base relations are defined (TC-6). If a given trajectory should have different start and finish regions but cycles are allowed within, 10 base relations are defined (TC-10). Both versions of the calculus are implemented as ASP programs; we propose several different encodings, including a generalised program capable of encoding any qualitative calculus in ASP. All proposed encodings are experimentally evaluated using a real-world dataset. Experiment results show that the best performing implementation can scale up to an input of 250 trajectories for TC-6 and 150 trajectories for TC-10 for the problem of discovering a consistent configuration, a significant improvement compared to previous ASP implementations for similar qualitative spatial and temporal calculi.

A Survey of Qualitative Spatial and Temporal Calculi

Qualitative spatial and temporal reasoning (QSTR) is concerned with symbolic knowledge representation, typically over infinite domains. The motivations for employing QSTR techniques include exploiting computational properties that allow efficient reasoning to capture human cognitive concepts in a computational framework. The notion of a qualitative calculus is one of the most prominent QSTR formalisms. This article presents the first overview of all qualitative calculi developed to date and their computational properties, together with generalized definitions of the fundamental concepts and methods that now encompass all existing calculi. Moreover, we provide a classification of calculi according to their algebraic properties.

Movement Pattern Analysis Based on Sequence Signatures

Increased affordability and deployment of advanced tracking technologies have led researchers from various domains to analyze the resulting spatio-temporal movement data sets for the purpose of knowledge discovery. Two different approaches can be considered in the analysis of moving objects: quantitative analysis and qualitative analysis. This research focuses on the latter and uses the qualitative trajectory calculus (QTC), a type of calculus that represents qualitative data on moving point objects (MPOs), and establishes a framework to analyze the relative movement of multiple MPOs. A visualization technique called sequence signature (SESI) is used, which enables to map QTC patterns in a 2D indexed rasterized space in order to evaluate the similarity of relative movement patterns of multiple MPOs. The applicability of the proposed methodology is illustrated by means of two practical examples of interacting MPOs: cars on a highway and body parts of a samba dancer. The results show that the proposed method can be effectively used to analyze interactions of multiple MPOs in different domains.

On redundant topological constraints

Redundancy checking is an important task in the research of knowledge representation and reasoning. In this paper, we consider redundant qualitative constraints. For a set Γ of qualitative constraints, we say a constraint (xRy) in Γ is redundant if it is entailed by the rest of Γ. A prime subnetwork of Γ is a subset of Γ which contains no redundant constraints and has the same solution set as Γ. It is natural to ask how to compute such a prime subnetwork, and when it is unique. We show that this problem is in general intractable, but becomes tractable if Γ is over a tractable subalgebra S of a qualitative calculus. Furthermore, if S is a subalgebra of the Region Connection Calculus RCC8 in which weak composition distributes over nonempty intersections, then Γ has a unique prime subnetwork, which can be obtained in cubic time by removing all redundant constraints simultaneously from Γ. As a by-product, we show that any path-consistent network over such a distributive subalgebra is minimal and globally consistent in a qualitative sense. A thorough empirical analysis of the prime subnetwork upon real geographical data sets demonstrates the approach is able to identify significantly more redundant constraints than previously proposed algorithms, especially in constraint networks with larger proportions of partial overlap relations.

The cognition of change: scaling deformations in mind and spatial theories

The dynamics of the earth and its inhabitants have become a core topic and focus in research in the spatial sciences. The spatio-temporal data avalanche challenges researchers to provide efficient and effective means to process spatio-temporal data. It is of vital importance to develop mechanisms that allow for the transition of data not only into information but also into knowledge. Knowledge representation techniques from artificial intelligence play an important role in laying the foundations for theories dealing with spatio-temporal data. Specifically, the advances in the area of qualitative spatial representation and reasoning (QSTR) have led to promising results. Categorical distinctions of spatio-temporal information identified by QSTR calculi potentially correspond to those relevant to humans. This article presents the first behavioral evaluation of qualitative calculi modeling geographic events associated with scaling deformations of entities, that is, changes in size by either expansion or contraction. Examples of such dynamics include a lake flooding its surroundings or an expanding oil spill in the ocean. We compare four experiments using four different semantic domains. Each domain consists of two spatially extended entities: one entity is undergoing scaling deformations while the other is static. We kept the formal QSTR characterization, which are paths through a topologically defined conceptual neighborhood graph, identical across all semantic domains. Our results show that for geographic events associated with scaling deformations (a) topological relations are not equally salient cognitively; (b) domain semantics has an influence on the conceptual salience of topological relations.

Reasoning about topological and cardinal direction relations between 2-dimensional spatial objects

Increasing the expressiveness of qualitative spatial calculi is an essential step towards meeting the requirements of applications. This can be achieved by combining existing calculi in a way that ...

Local-to-Global Consistency Implies Tractability of Abduction

Abduction is a form of nonmonotonic reasoning that looks for an explanation, built from a given set of hypotheses, for an observed manifestation according to some knowledge base. Following the concept behind the Schaefer's parametrization CSP(Gamma) of the Constraint Satisfaction Problem (CSP), we study here the complexity of the abduction problem Abduction(Gamma, Hyp, M) parametrized by certain (omega-categorical) infinite relational structures Gamma, Hyp, and M from which a knowledge base, hypotheses and a manifestation are built, respectively. We say that Gamma has local-to-global consistency if there is k such that establishing strong k-consistency on an instance of CSP(Gamma) yields a globally consistent (whose every solution may be obtained straightforwardly from partial solutions) set of constraints. In this case CSP(Gamma) is solvable in polynomial time. Our main contribution is an algorithm that under some natural conditions decides Abduction(Gamma, Hyp, M) in P when Gamma has local-to-global consistency. As we show in the number of examples, our approach offers an opportunity to consider abduction in the context of spatial and temporal reasoning (qualitative calculi such as Allen's interval algebra or RCC-5) and that our procedure solves some related abduction problems in polynomial time.

A Qualitative Calculus for Three-Dimensional Rotations

We have developed a qualitative calculus for three-dimensional directions and rotations. A direction is characterized in terms of the signs of its components relative to an absolute coordinate system. A rotation is characterized in terms of the signs of the components of the associated 3 × 3 rotation matrix. A system has been implemented that can solve the following problems: 1. Given the signs of direction and rotation matrix P, find the possible signs of the image of under P. Moreover, for each possible sign vector of · P, generate numerical instantiations of and P that yields that result. 2. Given the signs of rotation matrices P and Q, find the possible signs of the composition P · Q. Moreover, for each possible sign matrix for the composition, generate numerical instantiations of P and Q that yield that result. We have also proved some related complexity and expressivity results. The satisfiability problem for a qualitative rotation constraint network is NP-complete in two dimensions and NP-hard in three dimensions. In three dimensions, any two directions are distinguishable by a qualitative rotation constraint network.

Spatial reasoning with augmented points: Extending cardinal directions with local distances

We present an approach for supplying existing qualitative direction calculi with a distance component to support fully fledged positional reasoning. The general underlying idea of augmenting points with local reference properties has already been applied in the OPRAm calculus. In this existing calculus, point objects are attached with a local reference direction to obtain oriented points and able to express relative direction using binary relations. We show how this approach can be extended to attach a granular distance concept to direction calculi such as the cardinal direction calculus or adjustable granularity calculi such as OPRAm or the Star calculus. We focus on the cardinal direction calculus and extend it to a multi-granular positional calculus called EPRAm. We provide a formal specification of EPRAm including a composition table for EPRA2 automatically determined using real algebraic geometry. We also report on an experimental performance analysis of EPRA2 in the context of a topological map-learning task proposed for benchmarking qualitative calculi. Our results confirm that our approach of adding a relative distance component to existing calculi improves the performance in realistic tasks when using algebraic closure for consistency checking.

Formally grounding spatio-temporal thinking

To navigate through daily life, humans use their ability to conceptualize spatio-temporal information, which ultimately leads to a system of categories. Likewise, the spatial sciences rely heavily on conceptualization and categorization as means to create knowledge when they process spatio-temporal data. In the spatial sciences and in related branches of artificial intelligence, an approach has been developed for processing spatio-temporal data on the level of coarse categories: qualitative spatio-temporal representation and reasoning (QSTR). Calculi developed in QSTR allow for the meaningful processing of and reasoning with spatio-temporal information. While qualitative calculi are widely acknowledged in the cognitive sciences, there is little behavioral assessment whether these calculi are indeed cognitively adequate. This is an astonishing conundrum given that these calculi are ubiquitous, are often intended to improve processes at the human-machine interface, and are on several occasions claimed to be cognitively adequate. We have systematically evaluated several approaches to formally characterize spatial relations from a cognitive-behavioral perspective for both static and dynamically changing spatial relations. This contribution will detail our framework, which is addressing the question how formal characterization of space can help us understand how people think with, in, and about space.

Policy override in practice: model, evaluation, and decision support

The predominant strategy in restricting permissions in information systems is to limit users on the basis of the 'need-to-know' principle. Although appropriate in highly security-sensitive contexts, this culture of protection will, in other contexts, often reduce users' productivity and is seen as a hassle because the everyday exceptions to the routine tasks can be severely hindered. This paper proposes a more flexible authorization model, policy override, which allows end users to override authorization in a controlled manner. In this article, I describe the authorization model and its implementation in a medium enterprise's business application. I evaluated policy override use over a period of 1year through quantitative and qualitative analysis to identify challenges and offer advice on the implementation of policy override in practice. One important challenge is the setting of adequate bounds for policy override. To overcome this obstacle, I propose and evaluate a qualitative risk-based calculus that offers decision support to balance additional risks of policy override with the benefits of more flexible authorization. Copyright © 2012 John Wiley & Sons, Ltd.

Analyzing Qualitative Spatio-Temporal Calculi using Algebraic Geometry

Qualitative spatial reasoning is based on calculi which comprise relations and operation tables that encode operations like relation composition. Designing a calculus involves determining these tables and analyzing reasoning properties—a demanding task that is susceptible to errors if performed manually. This paper is concerned with automating computation of operation tables and analysis of qualitative calculi over real-valued domains like the plane 2. We present an approach to specify qualitative relations using polynomial equations that allows methods from algebraic geometry to be applied. This paper shows how reasoning with qualitative relations can be posed algebraically and demonstrates algebraic reasoning using Gröbner base analysis. We evaluate this approach and describe our implementation, which is freely available as part of the spatial reasoning toolbox SparQ.