Robin Hirsch

Complete representations in algebraic logic

A boolean algebra is shown to be completely representable if and only if it is atomic, whereas it is shown that neither the class of completely representable relation algebras nor the class of completely representable cylindric algebras of any xed dimension are elementary.

Step by step-building representations in algebraic logic

We consider the problem of finding and classifying representations in algebraic logic. This is approached by letting two players build a representation using a game. Homogeneous and universal representations are characterized according to the outcome of certain games. The Lyndon conditions defining representable relation algebras (for the finite case) and a similar schema for cylindric algebras are derived. Finte relation algebras with homogeneous representations are characterized by first order formulas. Equivalence games are defined, and are used to establish whether an algebra is co-categorical. We have a simple proof that the perfect extension of a representable relation algebra is completely representable. An important open problem from algebraic logic is addressed by devising another two-player game, and using it to derive equational axiomatisations for the classes of all representable relation algebras and representable cylindric algebras. Other instances of this approach are looked at, and include the step by step method. ?

Expressive Power and Complexity in Algebraic Logic

Two complexity problems in algebraic logic are surveyed: the satisfaction problem and the network satisfaction problem. Various complexity results are collected here and some new ones are derived. Many examples are given. The network satisfaction problem for most cylindric algebras of dimension four or more is shown to be intractable. Complexity is tied-in with the expressivity of a relation algebra. Expressivity and complexity are analysed in the context of homogeneous representations. The model-theoretic notion of interpretation is used to generalise known complexity results to a range of other algebraic logics. In particular a number of relation algebras are shown to have intractable network satisfaction problems.

Relation algebras of intervals

Abstract Given a representation of a relation algebra we construct relation algebras of pairs and of intervals . If the representation happens to be complete, homogeneous and fully universal then the pair and interval algebras can be constructed direct from the relation algebra. If, further, the original relation algebra is ω-categorical we show that the interval algebra is too. The complexity of relation algebras is studied and it is shown that every pair algebra with infinite representations is intractable. Applications include constructing an interval algebra that combines metric and interval expressivity.

Evolving Lucene search queries for text classification

We describe a method for generating accurate, compact, human understandable text classifiers. Text datasets are indexed using Apache Lucene and Genetic Programs are used to construct Lucene search queries. Genetic programs acquire fitness by producing queries that are effective binary classifiers for a particular category when evaluated against a set of training documents. We describe a set of functions and terminals and provide results from classification tasks.

Relation algebras with n-dimensional relational bases

Abstract We study relation algebras with n -dimensional relational bases in the sense of Maddux. Fix n with 3⩽n⩽ω . Write B n for the class of non-associative algebras with an n -dimensional relational basis, and RA n for the variety generated by B n . We define a notion of relativised representation for algebras in RA n , and use it to give an explicit (hence recursive) equational axiomatisation of RA n , and to reprove Madduxs result that RA n is canonical. We show that the algebras in B n are precisely those that have a complete relativised representation of this type. Then we prove that whenever 4⩽n , RA l is not finitely axiomatisable over RA n . This confirms a conjecture of Maddux. We also prove that B n is elementary for n=3,4 only.

Relation algebras from cylindric algebras, II

Abstract We prove, for each 4⩽n S Ra CA n+1 cannot be defined, using only finitely many first-order axioms, relative to S Ra CA n . The construction also shows that for 5⩽n S Ra CA n is not finitely axiomatisable over RA n , and that for 3⩽m S Nr m CA n+1 is not finitely axiomatisable over S Nr m CA n . In consequence, for a certain standard n-variable first-order proof system ⊢m,n of m-variable formulas, there is no finite set of m-variable schemata whose m-variable instances, when added to ⊢m,n as axioms, yield ⊢m,n+1.

Evolving rules for document classification

We describe a novel method for using Genetic Programming to create compact classification rules based on combinations of N-Grams (character strings). Genetic programs acquire fitness by producing rules that are effective classifiers in terms of precision and recall when evaluated against a set of training documents. We describe a set of functions and terminals and provide results from a classification task using the Reuters 21578 dataset. We also suggest that because the induced rules are meaningful to a human analyst they may have a number of other uses beyond classification and provide a basis for text mining applications.

Strongly representable atom structures of relation algebras

A relation algebra atom structure is said to be strongly rep- resentable if all atomic relation algebras with that atom structure are rep- resentable. This is equivalent to saying that the complex algebra Cm is a representable relation algebra. We show that the class of all strongly repre- sentable relation algebra atom structures is not closed under ultraproducts and is therefore not elementary. This answers a question of Maddux (1982). Our proof is based on the following construction. From an arbitrary undi- rected, loop-free graph , we construct a relation algebra atom structure () and prove, for infinite , that () is strongly representable if and only if the chromatic number of is infinite. A construction of Erdos shows that there are graphs r (r < !) with infinite chromatic number, with a non-principal ul- traproduct Q D r whose chromatic number is just two. It follows that ( r) is strongly representable (each r < !) but Q D ( ( r)) is not.

A finite relation algebra with undecidable network satisfaction problem

We deene a nite relation algebra and show that the network satisfaction problem is undecidable for this algebra 1. be a relation algebra (see JT52] for the original ax-iomatisation or Mad91] for an introduction to relation algebra). An atom a of A is a minimal, non-zero element. At(A) denotes the set of all atoms of A. A is atomic if for all non-zero a 2 A there exists 2 At(A) with a. In the following we assume A is atomic.