Brandon Bennett

Qualitative Spatial Representation and Reasoning with the Region Connection Calculus

This paper surveys the work of the qualitative spatial reasoning group at the University of Leeds. The group has developed a number of logical calculi for representing and reasoning with qualitative spatial relations over regions. We motivate the use of regions as the primary spatial entity and show how a rich language can be built up from surprisingly few primitives. This language can distinguish between convex and a variety of concave shapes and there is also an extension which handles regions with uncertain boundaries. We also present a variety of reasoning techniques, both for static and dynamic situations. A number of possible application areas are briefly mentioned.

Spatial Reasoning with Propositional Logics

I present a method for reasoning about spatial relationships on the basis of entailments in propositional logic. Formalisms for representing topological and other spatial information (e.g. [2] [10] [11]) have generally employed the 1st-order predicate calculus. Whilst this language is much more expressive than 0-order (propositional) calculi it is correspondingly harder to reason with. Hence, by encoding spatial relationships in a propositional representation automated reasoning becomes more effective. I specify representations in both classical and intuitionistic propositional logic, which — together with well-defined meta-level reasoning algorithms — provide for efficient reasoning about a large class of spatial relations.

Multi-Dimensional Modal Logic as a Framework for Spatio-Temporal Reasoning

In this paper we advocate the use of multi-dimensional modal logics as a framework for knowledge representation and, in particular, for representing spatio-temporal information. We construct a two-dimensional logic capable of describing topological relationships that change over time. This logic, called PSTL (Propositional Spatio-Temporal Logic) is the Cartesian product of the well-known temporal logic PTL and the modal logic S4u, which is the Lewis system S4 augmented with the universal modality. Although it is an open problem whether the full PSTL is decidable, we show that it contains decidable fragments into which various temporal extensions (both point-based and interval based) of the spatial logic RCC-8 can be embedded. We consider known decidability and complexity results that are relevant to computation with multi-dimensional formalisms and discuss possible directions for further research.

Modal Logics for Qualitative Spatial Reasoning

Spatial reasoning is essential for many AI applications. In most existing systems the representation is primarily numerical, so the information that can be handled is limited to precise quantitative data. However, for many purposes the ability to manipulate high-level qualitative spatial information in a exible way would be extremely useful. Such capabilities can be proveded by logical calculi; and indeed 1st-order theories of certain spatial relations have been given 20]. But computing inferences in 1st-order logic is generally intractable unless special (domain dependent) methods are known. 0-order modal logics provide an alternative representation which is more expressive than classical 0-order logic and yet often more amenable to automated deduction than 1st-order formalisms. These calculi are usually interpreted as propositional logics: non-logical constants are taken as denoting propositions. However, they can also be given a nominal interpretation in which the constants stand for some kind of object. I show how 0-order logics can be given a spatial interpretation: constants denote regions and logical operators correspond to operations on regions which are important for characterising spatial situations. Representing certain spatial concepts requires the introduction of modal operators, interpreted as functions generating regions related in speciic ways to those denoted by their arguments. A signiicant example is the convex-hull operator whose value is the smallest convex region containing its argument. I investigate how this this operator can be captured in a multi-modal logic.

Representing and Reasoning with Qualitative Spatial Relations About Regions

Qualitative Reasoning (QR) has now become a mature subfield of AI as its tenth annual international workshop, several books (e.g. (Weld and de Kleer, 1990; Faltings and Struss, 1992)) and a wealth of conference and journal publications testify. QR tries to make explicit our everyday commonsense knowledge about the physical world and also the underlying abstractions used by scientists and engineers when they create models. Given this kind of knowledge and appropriate reasoning methods, a computer could make predictions and diagnoses and explain the behavior of physical systems in a qualitative manner, even when a precise quantitative description is not available or is computationally intractable. Note that a representation is not normally deemed to be qualitative by the QR community simply because it is symbolic and utilizes discrete quantity spaces but because the distinctions made in these discretizations are relevant to high-level descriptions of the system or behavior being modeled.

A unifying semantics for time and events

We give a formal semantics for a highly expressive language for representing temporal relationships and events. This language, which we call Versatile Event Logic (VEL), provides a general temporal ontology and semantics encompassing many other representations. The system incorporates a number of features that have not been widely employed in AI formalisms. It has the ability to describe alternative histories using a modal operator. It provides a semantics for individuals that explicitly models their identity through time and across alternative possible histories; and enables one to distinguish between necessary and extensional identity of individuals. In virtue of its treatment of individuals and count nouns, the formalism offers a solution to certain puzzles of identity, which arise when individuals are described in different ways. We propose that VEL can be used as a foundational interlingua for comparing and interfacing different AI languages and illustrate this by considering how Situation Calculus and Event Calculus can be represented within VEL.

A Proximity Approach to Some Region-Based Theories of Space

This paper is a continuation of [VAK 01]. The notion of local connection algebra, based on the primitive notions of connection and boundedness, is introduced. It is slightly different but equivalent to Roepers notion of region-based topology [ROE 97]. The similarity between the local proximity spaces of Leader [LEA 67] and local connection algebras is emphasized. Machinery, analogous to that introduced by Efremovi?c [EFR 51],[EFR 52], Smirnov [SMI 52] and Leader [LEA 67] for proximity and local proximity spaces, is developed. This permits us to give new proximity-type models of local connection algebras, to obtain a representation theorem for such algebras and to give a new shorter proof of the main theorem of Roepers paper [ROE 97]. Finally, the notion of MVD-algebra is introduced. It is similar to Mormanns notion of enriched Boolean algebra [MOR 98], based on a single mereological relation of interior parthood. It is shown that MVD-algebras are equivalent to local connection algebras. This means that the connection relation and boundedness can be incorporated into one, mereological in nature relation. In this way a formalization of the Whiteheadian theory of space based on a single mereological relation is obtained.

Combinations of Modal Logics

There is increasing use of combinations of modal logics in bothfoundational and applied research areas. This article provides anintroduction to both the principles of such combinations and to thevariety of techniques that have been developed for them. In addition,the article outlines many key research problems yet to be tackledwithin this callenging area of work.

Space, time, matter and things

I present a logical language for describing spatial, temporal and material properties of the physical world. The formalism is ontologically well-founded in the sense that it is interpreted with respect to model structures that have a specific physical interpretation in terms of the distribution of matter in space and time.

A note on proximity spaces and connection based mereology

Representation theorems for systems of regions have been ofinterest for some time, and various contexts have been used forthis purpose: Mormann [17] has demonstrated the fruitfulness of themethods of continuous lattices to obtain a topologicalrepresentation theorem for his formalisation of Whiteheadianontological theory of space; similar results have been obtained byRoeper [20]. In this note, we prove a topological representationtheorem for a connection based class of systems, using methods andtools from the theory of proximity spaces. The key novelty is a newproximity semantics for connection relations.