Generating Relation Algebras for Qualitative Spatial Reasoning

Basic relationships between certain regions of space are formulated in natural language in everyday situations. For example, a customer specifies the outline of his future home to the architect by indicating which rooms should be close to each other. Qualitative spatial reasoning as an area of artificial intelligence tries to develop a theory of space based on similar notions. In formal ontology and in ontological computer science, mereotopology is a first-order theory, embodying mereological and topological concepts, of the relations among wholes, parts, parts of parts, and the boundaries between parts. We shall introduce abstract relation algebras and present their structural properties as well as their connection to algebras of binary relations. This will be followed by details of the expressiveness of algebras of relations for region based models. Mereotopology has been the main basis for most region based theories of space. Since its earliest inception many theories have been proposed for mereotopology in artificial intelligence among which Region Connection Calculus is most prominent. The expressiveness of the region connection calculus in relational logic is far greater than its original eight base relations might suggest. In the thesis we formulate ways to automatically generate representable relation algebras using spatial data based on region connection calculus. The generation of new algebras is a two pronged approach involving splitting of existing relations to form new algebras and refinement of such newly generated algebras. We present an implementation of a system for automating aforementioned steps and provide an effective and convenient interface to define new spatial relations and generate representable relational algebras

There is no finite-variable equational axiomatization of representable relation algebras over weakly representable relation algebras

We prove that any equational basis that defines RRA over wRRA must contain infinitely many variables. The proof uses a construction of arbitrarily large finite weakly representable but not representable relation algebras whose "small" subalgebras are representable.Comment: To appear in Review of Symbolic Logi

Introducing Boolean Semilattices

We present and discuss a variety of Boolean algebras with operators that is closely related to the variety generated by all complex algebras of semilattices. We consider the problem of finding a generating set for the variety, representation questions, and axiomatizability. Several interesting subvarieties are presented. We contrast our results with those obtained for a number of other varieties generated by complex algebras of groupoids

Complex algebras of semigroups

The notion of a Boolean algebra with operators (BAO) was first defined by Jonsson and Tarski in 1951. Since that time, many varieties of BAOs, including modal algebras, closure algebras, monadic algebras, and of course relation algebras, have been studied. With the exception of relation algebras, these varieties each have one unary operator. This paper investigates a variety of BAOs with one binary operator. Begin with a semigroup S = (S,·). The complex algebra of S, denoted S+, is a Boolean algebra whose underlying set is the power set of S with set union, intersection and complementation as the Boolean algebra operations. The multiplication operation defined on the semigroup induces a normal, associative binary operation, \u27*\u27, on the complex algebra as follows: for all subsets A and B contained in S, A * B= a· b:a ϵ A, b ϵ B . Hence, the complex algebra of a semigroup is a BAO with one normal binary associative operator;Let S+ be the class of all complex algebras of semigroups and consider the variety generated by the class S+, denoted V(S+). The tools required to study this variety are developed, including the duality between BAOs and relational structures as it applies to V(S+). A closure operator is defined which is used to determine homomorphic images of members of V(S+). Theorems on the subdirectly irreducible and simples algebras in V(S+) are proved;Next, the structure of V(S+) is analyzed. The general problem of representing a BAO with one binary, normal, associative operator as a member of V(S+) is discussed. Several examples and theorems concerning representation are presented. It is shown that the quasivariety generated by S+ is strictly contained in V(S+). Lastly, the structure of the lattice of subvarieties of V(S+) is investigated. There are precisely two atoms in this lattice; each atom is generated by a two element algebra. Two infinite chains of varieties exist, with the smallest element in each chain a cover of exactly one atom. This leads to a discussion of certain splitting algebras and conjugate varieties. Finally, equations characterizing some important subvarieties are developed