Robin Hirsch

Laurence Hirsch, Robin Hirsch, Bayode Ogunleye

Text clustering holds significant value across various domains due to its ability to identify patterns and group related information. Current approaches which rely heavily on a computed similarity measure between documents are often limited in accuracy and interpretability. We present a novel approach to the problem based on a set of evolved search queries. Clusters are formed as the set of documents matched by a single search query in the set of queries. The queries are optimized to maximize the number of documents returned and to minimize the overlap between clusters (documents returned by more than one query). Where queries contain more than one word they are interpreted disjunctively. We have found it useful to assign one word to be the root and constrain the query construction such that the set of documents returned by any additional query words intersect with the set returned by the root word. Not all documents in a collection are returned by any of the search queries in a set, so once the search query evolution is completed a second stage is performed whereby a KNN algorithm is applied to assign all unassigned documents to their nearest cluster. We describe the method and present results using 8 text datasets comparing effectiveness with well-known existing algorithms. We note that as well as achieving the highest accuracy on these datasets the search query format provides the qualitative benefits of being interpretable and modifiable whilst providing a causal explanation of cluster construction.

Robin Hirsch, Szabolcs Mikulás, Tim Stokes

Demonic composition, demonic refinement and demonic union are alternatives to the usual "angelic" composition, angelic refinement (inclusion) and angelic (usual) union defined on binary relations. We first motivate both the angelic and demonic via an analysis of the behaviour of non-deterministic programs, with the angelic associated with partial correctness and demonic with total correctness, both cases emerging from a richer algebraic model of non-deterministic programs incorporating both aspects. Zareckii has shown that the isomorphism class of algebras of binary relations under angelic composition and inclusion is finitely axiomatised as the class of ordered semigroups. The proof can be used to establish that the same axiomatisation applies to binary relations under demonic composition and refinement, and a further modification of the proof can be used to incorporate a zero element representing the empty relation in the angelic case and the full relation in the demonic case. For the signature of angelic composition and union, it is known that no finite axiomatisation exists, and we show the analogous result for demonic composition and demonic union by showing that the same axiomatisation holds for both. We show that the isomorphism class of algebras of binary relations with the "mixed" signature of demonic composition and angelic inclusion has no finite axiomatisation. As a contrast, we show that the isomorphism class of partial algebras of binary relations with the partial operation of constellation product and inclusion (also a "mixed" signature) is finitely axiomatisable.

Robin Hirsch, Marcel Jackson, Tomasz Kowalski

A qualitative representation $φ$ is like an ordinary representation of a relation algebra, but instead of requiring $(a; b)^φ= a^φ| b^φ$, as we do for ordinary representations, we only require that $c^φ\supseteq a^φ| b^φ\iff c\geq a ; b$, for each $c$ in the algebra. A constraint network is qualitatively satisfiable if its nodes can be mapped to elements of a qualitative representation, preserving the constraints. If a constraint network is satisfiable then it is clearly qualitatively satisfiable, but the converse can fail. However, for a wide range of relation algebras including the point algebra, the Allen Interval Algebra, RCC8 and many others, a network is satisfiable if and only if it is qualitatively satisfiable. Unlike ordinary composition, the weak composition arising from qualitative representations need not be associative, so we can generalise by considering network satisfaction problems over non-associative algebras. We prove that computationally, qualitative representations have many advantages over ordinary representations: whereas many finite relation algebras have only infinite representations, every finite qualitatively representable algebra has a finite qualitative representation; the representability problem for (the atom structures of) finite non-associative algebras is NP-complete; the network satisfaction problem over a finite qualitatively representable algebra is always in NP; the validity of equations over qualitative representations is co-NP-complete. On the other hand we prove that there is no finite axiomatisation of the class of qualitatively representable algebras.