Enumeration Reducibility in Closure Spaces with Applications to Logic and Algebra

In many instances in first order logic or computable algebra, classical theorems show that many problems are undecidable for general structures, but become decidable if some rigidity is imposed on the structure. For example, the set of theorems in many finitely axiomatisable theories is nonrecursive, but the set of theorems for any finitely axiomatisable complete theory is recursive. Finitely presented groups might have an nonrecursive word problem, but finitely presented simple groups have a recursive word problem. In this article we introduce a topological framework based on closure spaces to show that many of these proofs can be obtained in a similar setting. We will show in particular that these statements can be generalized to cover arbitrary structures, with no finite or recursive presentation/axiomatization. This generalizes in particular work by Kuznetsov and others. Examples from first order logic and symbolic dynamics will be discussed at length

Inductive Inference: An Axiomatic Approach

A predictor is asked to rank eventualities according to their plausibility, based on past cases. We assume that she can form a ranking given any memory that consists of finitely many past cases. Mild consistency requirements on these rankings imply that they have a numerical representation via a matrix assigning numbers to eventuality-case pairs, as follows. Given a memory, each eventuality is ranked according to the sum of the numbers in its row, over cases in memory. The number attached to an eventuality-case pair can be interpreted as the degree of support that the past case lends to the plausibility of the eventuality. Special instances of this result may be viewed as axiomatizing kernel methods for estimation of densities and for classification problems. Interpreting the same result for rankings of theories or hypotheses, rather than of specific eventualities, it is shown that one may ascribe to the predictor subjective conditional probabilities of cases given theories, such that her rankings of theories agree with rankings by the likelihood functions.Inductive inference, case-based reasoning,case-based decision theory, maximum likelihood

Conserved charges and supersymmetry in principal chiral and WZW models

Conserved and commuting charges are investigated in both bosonic and supersymmetric classical chiral models, with and without Wess-Zumino terms. In the bosonic theories, there are conserved currents based on symmetric invariant tensors of the underlying algebra, and the construction of infinitely many commuting charges, with spins equal to the exponents of the algebra modulo its Coxeter number, can be carried out irrespective of the coefficient of the Wess-Zumino term. In the supersymmetric models, a different pattern of conserved quantities emerges, based on antisymmetric invariant tensors. The current algebra is much more complicated than in the bosonic case, and it is analysed in some detail. Two families of commuting charges can be constructed, each with finitely many members whose spins are exactly the exponents of the algebra (with no repetition modulo the Coxeter number). The conserved quantities in the bosonic and supersymmetric theories are only indirectly related, except for the special case of the WZW model and its supersymmetric extension.Comment: LaTeX; 49 pages; v2: minor changes and additions to text and ref

On the geometric theory of local MV-algebras

We investigate the geometric theory of local MV-algebras and its quotients axiomatizing the local MV-algebras in a given proper variety of MV-algebras. We show that, whilst the theory of local MV-algebras is not of presheaf type, each of these quotients is a theory of presheaf type which is Morita-equivalent to an expansion of the theory of lattice-ordered abelian groups. Di Nola-Lettieri's equivalence is recovered from the Morita-equivalence for the quotient axiomatizing the local MV-algebras in Chang's variety, that is, the perfect MV-algebras. We establish along the way a number of results of independent interest, including a constructive treatment of the radical for MV-algebras in a fixed proper variety of MV-algebras and a representation theorem for the finitely presentable algebras in such a variety as finite products of local MV-algebras.Comment: 52 page