This paper explores the idea of generic programming in which programs are parameterised by data types. Part of the constructive theory of lists, specically the part dealing with properties of segments, is generalised in two ways: from lists to arbitrary inductive data types, and from functions to relations. The new theory is used to solve a generic problem about segments. 1 Introduction To what extent is it possible to construct programs without knowing exactly what data types are involved? At rst sight this may seem a strange question, but consider the case of pattern matching. Over lists, this problem can be formulated in terms of two strings, a pattern and a text; the object is to determine if and where the pattern occurs as a segment of the text. Now, pattern matching can be generalised to other data types, including arrays and trees of various kinds; the essential step is to be able to dene the notion of `segment' in these types. So the intriguing question arises: can one...

Models of software systems are built in Z and VDM using partial functions between sets and certain operations on these partial functions: extension (⊔), restriction (⊳), removal (⊳−) and override (†). Can these operations be given a categorial semantics? Doing so will show the ‘nature ’ of the operations. The operation of override is found to depend on the ‘shape ’ on X, the poset PX. The operations are developed in an elementary topos E. This is achieved by constructing each operation in the topos Set, of sets and total functions, and then using these constructions as the definition of the operations in an elementary topos. Each of the operations is thus given a categorical semantics. As an example the operation of override is considered in the topos Set · , of total functions and commuting diagrams. Can models of software systems be built in topoi other than Set? 1

Abstract. There is an obvious tension between symbolic and subsymbolic theories, because both show complementary strengths and weaknesses in corresponding applications and underlying methodologies. The resulting gap in the foundations and the applicability of these approaches is theoretically unsatisfactory and practically undesirable. We sketch a theory that bridges this gap between symbolic and subsymbolic approaches by the introduction of a Topos-based semi-symbolic level used for coding logical first-order expressions in a homogeneous framework. This semi-symbolic level can be used for neural learning of logical firstorder theories. Besides a presentation of the general idea of the framework, we sketch some challenges and important open problems for future research with respect to the presented approach and the field of neurosymbolic integration, in general. Keywords. Neuro–Symbolic Integration, Topos Theory, First–Order Logic 1