I certify that this PhD research entitled FORMALIZING THE LOGIC OF SPATIAL QUALIFICATION USING A QUALITATIVE REASONING MEASURE was carried out by Patience Charles BASSEY with student identification number, 130162 in the Department of Computer Science, University of Ibadan, Ibadan, Nigeria under my supervision. SQM - Spatial Qualification Model TAV - Translator along vector TBH - Tafawa Balewa Hall TdH - Teddar Hall. A spatial qualification problem, an aspect of spatial reasoning, concerns the impossibility of knowing an agent's presence at a particular place and time.
Therefore, this work was aimed at formalizing a logical theory for reasoning about the spatial competence of an agent to perform an action based on prior knowledge using a qualitative reasoning approach. The spatial qualification model specified the log of spatial presence and reachability axioms needed to infer an agent's spatial presence. The necessary knowledge to solve the spatial qualification problem belongs to such information necessary for making decisions in reality.
Problem Statement
Spatial qualification is an important condition for any spatially situated agent to participate in an action, and it is seen as lacking in early works in the area of KRR, such as that on temporal reasoning with plans (Allen, 1991). The few cases, where the spatial knowledge is represented for reasoning, include the one used to solve the adversarial geospatial abduction problem (Shakarian et al., 2011). In a stochastic domain (Dean et al., 1993), where space is divided into a grid of locations with four directional conditions assigned to each of the states, the use of reward functions for efficient planning requires the deliberation interval (that is the time interval between the current time and due date).
Although the qualification problem is a well-known problem in AI field, none of these efforts addressed the qualification problem with respect to space. Given a previous antecedent that an intelligent agent was present at a certain location, it is possible that the agent was present at the scene of incidence at the time of the incidence. Is movement of the agent from its last known location to the location of occurrence possible within the time the agent was last seen and the time of occurrence.
Research Questions
Aim and Objectives of the study
An attempt to answer the above question could lead to its refinement to: The specific objectives are:. i) Determine which language is suitable for logical theory. ii). Use the syntax of the language to formalize the domain of spatial qualification. iii)Define the formal semantics of the language. iv) Develop a proof system for the formalized logic. v) Apply the logic model to a planning distribution domain using case studies of spatial qualification problems for research.
Methodology
This follows from the structure of the Kripke model, the fundamental modal logic model (Zalta, 1995). Kripke structure is a threefold M = (W,R,V), where W is the non-empty set of possible worlds (i.e. states in a computation), R W W is the accessibility relation (also called transition relation) and V: (Prop W) → (true, false) is a valuation function (which tells us which properties are true or false in different worlds) (Goldblatt, 2005). Another methodology used in this study is the analytical Tableau Proof method to further prove the formalized logic for soundness and completeness.
The formal axioms in the well-proven logic model are then applied to the domain of product distribution planning to infer schedules with deadlines.
Basic Assumptions
Some of the logical languages relevant to this study are discussed in the following section. Instead of the generalized approach (that is, the potentially infinite models of the real world or possible world) of Montague, Barwise and Perry adopted the finite situations as their basis (Janson, 2012). From the model structure of PWS in (3.1), the availability relation, R, is defined to be.
The expression I[c, w1] denotes the application of the interpretation function I to the constant symbol c in the world w1. In chapter four, a proof system for the formalized spatial qualification logic model (SQM) is developed. Figure 5.3 shows some possible routes along which nodes in the designated routes can be visited.
Axioms were also used in the model with given distances and times of known locations. The hybridization of the two representational languages resulted in a quantified modal logic that was adopted in formalism. On the complexity of qualitative spatial thinking: the largest traceable fragment of the account of regional connections.
Organisation of the rest of the thesis
Qualitative Reasoning (QR)
Impacts of Qualitative Reasoning
Step (i) requires reasoning, as it applies to the logic of spatial qualification, which helps to evaluate the existing plan.
Key principles governing qualitative modelling
Limitations of Qualitative reasoning
- Bouncing ball domain
- Robotic application
Despite the above advantages of qualitative reasoning, there are a number of misconceptions about what qualitative reasoning is not. Cohn's point called for the addition of qualitative non-topological information such as orientation, distance, size and shape to topological relationships (Randell et al, 1992). The application of qualitative theoretical models developed so far introduces their use with imprecise quantitative data.
The qualitative and quantitative approaches have both been integrated with the qualitative theories developed to replace the hypothetical approaches (Escrig, 2005), providing qualitative descriptions of environmental landmarks. The application of qualitative and quantitative approaches in the bouncing ball domain implies the suggestion that the place vocabulary should be embedded in a more quantitative analog representation (metric diagram) with the results of qualitative spatial reasoning (Forbus, 1981). Therefore, in support of the poverty conjecture, this work concludes that qualitative reasoning, as a complement to quantitative reasoning, makes the calculation of common sense properties a reality.
Reasoning with spatial knowledge
Khepera and Pioneer; the automatic construction of mosaic designs using qualitative shape recognition of ceramic pieces; and the navigation in an environment that resembles brain structure with a robot with legs (Escrig, 2005). Spatial reasoning involving spatial concepts such as space and time has led to a lack of consensus, generating a lot of problems over the years. Some of the problems with space and time have been identified to include vagueness, uncertainty and granularity (Galton, 2009; Cohn and Renz, 2008).
One of the lack of consensus was Leibniz's opposition to Newton's view of the universe (Casati, 1999). In other words, to investigate spatial qualification, both space and spatial objects are treated as two inseparable entities, thus following Leibniz's claim, but with a time stamp. Before embarking on the exploration of spatial qualification, it is necessary to examine some defined spatial concepts and the challenges of spatial thinking.
Spatial concepts and challenges of spatial reasoning
This construct is seen in a semi-formal ontological framework, where the semantics of concepts such as habitat and environment, and their relationship with spatial structure of the world are represented (Bennett, 2010). This approach changed the focus of researchers in the field from sticking to the poverty assumption promulgated by Forbus, Nielson and Faltings that: "there is no purely qualitative, general purpose. Because of the transitivity of both Allen's interval logic and numbers, the conjecture of the sparsity of spatial representation in higher dimensions led to the conclusion that for spatial reasoning almost nothing will do worse than numbers.
Therefore, the use of a combined approach, the metric diagram/location vocabulary - MD/PV model (Forbus, Nielsen and Faltings, 1987). The poverty assumption sees common sense reasoning as qualitative reasoning and some level of quantitative knowledge. Based on the impoverishment assumption, qualitative spatial reasoning (QSR) is challenged to provide computations that will allow a machine to represent and infer higher-dimensional spatial entities without resorting to traditional quantitative techniques.
Qualitative Spatial Reasoning (QSR)
Design approaches in QSR
Theories for Spatial reasoning
- Ontology
- Mereology
- Topology
- Mereotopology
- Mereogeometry
With this, one can interpret the situation or the system in relation to the available models. This provides an opportunity to express constraints across worlds or simply to materialize worlds as common objects in the domain. Topology is the theory of how things are connected, that is, how a set of entities can interact with each other.
By considering the values empty () and non-empty () for each of the nine intersections, 29 = 512 binary topological relations can be distinguished. An ontologically well-founded logical language for describing spatial, temporal and material properties of the physical world is also presented (Bennett, 2001a). This gives the three distinct primitives used in describing mereology, topology and morphological properties of the logical theory of space with three-dimensional regions (Borgo et al., 1996).
Aspects of Qualitative Spatial Reasoning
- Temporal reasoning
- Qualitative Spatial calculi
Relations between spatial areas are defined based on the relation C(a. b), read as "a connects with b". Another extension of the RCC-8 relations is the Boolean region connection calculus (BRCC-8), which combines the region variables using the Boolean operators , and. Therefore, most directional relations are ternary due to the introduction of the frame of reference in contrast to the topological relations (Cohn and Renz, 2007) previously discussed.
This uncertainty cannot be handled by some of the movement command models such as M=(CS,TC) (Latombe, 1988), which are quantitative and contain only the control statement that specifies the trajectory along which the controller executing the command must move and the termination condition upon which the controller must terminate the movement (Latombe, 1991). Based on some of the defined spatiotemporal concepts, Muller defined six classes of motion as: LEAVE, REACH, HIT, CROSS, INTERNAL and EXTERNAL. Some of the reviewed qualitative spatial and temporal calculations, their relationships with examples are summarized in table 2.2 with corresponding citations.
Qualitative spatial reasoning: Gaps and way forward
Logical theories
Reasons for using logical theories
This means that logic allows one to distinguish correct reasoning from poor reasoning, thereby aiding one's correct reasoning. A logical system for a language is a set of axioms and rules designed to prove exactly the valid statements in the language.
Logical/Formal languages
First-order logic (FOL)
- Syntax of FOL
The meaning of the classical logical operators is as given in the model semantics of first-order predicate logic. The possibility of the presence of the agent, which is the uncertain knowledge, is therefore the output of the SQ reasoner. Since the logical model will be a first-order modal logic, the model-theoretic nature of the logic must be clarified.
It is especially important to know the variability or otherwise of the domains in the world, as the model moves from world to world. Proceedings of the 2nd International Conference on Principles of Knowledge Representation and Reasoning, Cambridge, MA. Proceedings of the 5th workshop on time, space and motion: meaning and knowledge in the sensible world.
