Abstract

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 ɛ. 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 ?

Highlights

  • An example the operation of override is considered in the topos Set·, of total functions and commuting diagrams

  • The VDM (Jones 1990) and Z (Spivey 1992) notations have been used widely in the specification and development of software systems. These methods share with the Irish School of Constructive Mathematics (M♣C ), of which the Irish School of the VDM (VDM♣) (Mac an Airchinnigh 1990; Mac an Airchinnigh 1991; Mac an Airchinnigh 1993) is a part, a collection of mathematical operations: extend, restriction, removal and override

  • We would like to give a categorical semantics to these operations

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Summary

Prologue

The VDM (Jones 1990) and Z (Spivey 1992) notations have been used widely in the specification and development of software systems These methods share with the Irish School of Constructive Mathematics (M♣C ), of which the Irish School of the VDM (VDM♣) (Mac an Airchinnigh 1990; Mac an Airchinnigh 1991; Mac an Airchinnigh 1993) is a part, a collection of mathematical operations: extend, restriction, removal and override. This choice problem will only have a solution if f −1({y}) = ∅ When this is true h = 1X † [x → x ], where 1X denotes the identity function on the set X and x ∈ f −1({y}). We are led deeper into category theory to find a solution

Categorial Preliminaries
Monic Arrows
Initial Object
Terminal Object
Products
Co-products
Pullback
Exponentiation
2.11 Subobjects
2.12 Subobject Classifier
Towards Override on Partial Arrows between Objects in a Topos
Partial Functions and Partial Arrows
Extend in Set and E
Restrict in Set and E
Removal in Set and E
Override in Set and E
Epilogue
Full Text
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