On selfadjoint functors satisfying polynomial relations

We study selfadjoint functors acting on categories of finite dimensional modules over finite dimensional algebras with an emphasis on functors satisfying some polynomial relations. Selfadjoint functors satisfying several easy relations, in particular, idempotents and square roots of a sum of identity functors, are classified. We also describe various natural constructions for new actions using external direct sums, external tensor products, Serre subcategories, quotients and centralizer subalgebras.Comment: 24 page

Non-primitive Recursive Function Definitions

This paper presents an approach to the problem of introducingnon-primitive recursive function definitions in higher order logic. Arecursive specification is translated into a domain theory version, wherethe recursive calls are treated as potentially non-terminating. Once wehave proved termination, the original specification can be derived easily.A collection of algorithms are presented which hide the domain theoryfrom a user. Hence, the derivation of a domain theory specificationhas been automated completely, and for well-founded recursive functionspecifications the process of deriving the original specification from thedomain theory one has been automated as well, though a user mustsupply a well-founded relation and prove certain termination propertiesof the specification. There are constructions for building well-foundedrelations easily

A HOL Basis for Reasoning about Functional Programs

Domain theory is the mathematical theory underlying denotational semantics. This thesis presents a formalization of domain theory in the Higher Order Logic (HOL) theorem proving system along with a mechanization of proof functions and other tools to support reasoning about the denotations of functional programs. By providing a fixed point operator for functions on certain domains which have a special undefined (bottom) element, this extension of HOL supports the definition of recursive functions which are not also primitive recursive. Thus, it provides an approach to the long-standing and important problem of defining non-primitive recursive functions in the HOL system. Our philosophy is that there must be a direct correspondence between elements of complete partial orders (domains) and elements of HOL types, in order to allow the reuse of higher order logic and proof infrastructure already available in the HOL system. Hence, we are able to mix domain theoretic reasoning with reasoning in the set theoretic HOL world to advantage, exploiting HOL types and tools directly. Moreover, by mixing domain and set theoretic reasoning, we are able to eliminate almost all reasoning about the bottom element of complete partial orders that makes the LCF theorem prover, which supports a first order logic of domain theory, difficult and tedious to use. A thorough comparison with LCF is provided. The advantages of combining the best of the domain and set theoretic worlds in the same system are demonstrated in a larger example, showing the correctness of a unification algorithm. A major part of the proof is conducted in the set theoretic setting of higher order logic, and only at a late stage of the proof domain theory is introduced to give a recursive definition of the algorithm, which is not primitive recursive. Furthermore, a total well-founded recursive unification function can be defined easily in pure HOL by proving that the unification algorithm (defined in domain theory) always terminates; this proof is conducted by a non-trivial well-founded induction. In such applications, where non-primitive recursive HOL functions are defined via domain theory and a proof of termination, domain theory constructs only appear temporarily

LCF Examples in HOL

The LCF system provides a logic of fixed point theory and is useful to reason about non-termination, arbitrary recursive definitions and infinite types as lazy lists. It is unsuitable for reasoning about finite types and strict functions. The HOL system provides set theory and supports reasoning about finite types and total functions well. In this paper a number of examples are used to demonstrate that an extension of HOL with domain theory combines the benefits of both systems. The examples illustrate reasoning about infinite values and non-terminating functions and show how mixing domain and set theoretic reasoning eases reasoning about finite LCF types and strict functions. An example presents a proof of the correctness and termination of a recursive unification algorithm using well-founded induction

Familial occurrence of Danish and Dutch cases of the bovine brachyspina syndrome

Abstract Background The bovine brachyspina syndrome is a recently reported malformation in the Holstein breed. The aetiology of this syndrome is unknown, but its occurrence following breeding between genetically related and phenotypically normal cattle may indicate that it is an autosomal recessively inherited disorder. Three cases are reported and compared to the originally reported case. Case presentation Two Danish cases and a Dutch case are described. The calves were delivered following a slightly prolonged gestation period. Gross lesions consisted of growth retardation, significant shortening of the entire spine and long and slender limbs. Additionally, inferior brachygnatism and defects of several internal organs were recorded. The cases were diagnosed as having the brachyspina syndrome based on the presence of essential lesions. The parents of each case were genetically related and linked to the first reported case by a common ancestor. Conclusion The findings support the hypothesis that the brachyspina syndrome in Holstein cattle is inherited autosomal recessively and illustrate some of the assumed phenotypical variation of this syndrome. The brachyspina syndrome may be an emerging disease in the Holstein breed.</p