Categories Of Partial Maps Research Articles

A uniform model of computational conceptual blending

We present a mathematical model for the cognitive operation of conceptual blending that aims at being uniform across different representation formalisms, while capturing the relevant structure of this operation. The model takes its inspiration from amalgams as applied in case-based reasoning, but lifts them into category theory so as to follow Joseph Goguen’s intuition for a mathematically precise characterisation of conceptual blending at a representation-independent level of abstraction. We prove that our amalgam-based category-theoretical model of conceptual blending is essentially equivalent to the pushout model in the ordered category of partial maps as put forward by Goguen. But unlike Goguen’s approach, our model is more suitable to capture computational realisations of conceptual blending, and we exemplify this by concretising our model to computational conceptual blends for various representation formalisms and application domains.

Partial pushout semantics of generics in DOL

We combine CASL's pushout-style generic specification with DOL's filtering, the latter being a syntactic removal of parts of a specification. The challenge is that now the body of a generic specification can remove parts of the formal parameter. This cannot be handled with usual pushout semantics, but calls for a semantics of “match, delete, glue in” as used in the theory of graph grammars. We hence employ Heindel's theory of MipMap categories as a basis for the use of pushouts in categories of partial maps. We introduce a notion of MipMap institution that can serve as a semantic background for a partial pushout semantics of generics with filtering.

Adhesivity with Partial Maps instead of Spans

The introduction of adhesive categories revived interest in the study of properties of pushouts with respect to pullbacks that started over thirty years ago for the category of graphs. Adhesive categories - of which graphs are the “archetypal” example - are defined by a single property of pushouts along monos that implies essential lemmas and central theorems of double pushout rewriting such as the local Church-Rosser Theorem. The present paper shows that a strictly weaker condition on pushouts suffices to obtain essentially the same results: it suffices to require pushouts to be hereditary, i.e. they have to remain pushouts when they are embedded into the associated category of partial maps. This fact however is not the only reason to introduce partial map adhesive categories as categories with pushouts along monos (of a certain stable class) that are hereditary. There are two equally important motivations: first, there is an application relevant example category that cannot be captured by the more established variations of adhesive categories; second, partial map adhesive categories are “conceptually similar” to adhesive categories as the latter can be characterized as those categories with pushout along monos that remain bi-pushouts when they are embedded into the associated bi-category of spans. Thus, adhesivity with partial maps instead of spans appears to be a natural candidate for a general rewriting framework.

A categorical outlook on relational modalities and simulations

We characterise bicategories of spans, relations and partial maps universally in terms of factorisations involving maps. We apply this characterisation to show that the standard modalities □ and ⋄ arise canonically as extensions of a predicate logic from functions to (abstract) relations. When relations and partial maps are representable, we exhibit logical predicates for the power-object and partial-map-classifier monads. We also show that the □ modality gives the relevant pullbacks of subobjects in the internal logic of categories of partial maps. Organising modal formulae fibrationally, we exhibit an intrinsic relationship between their satisfaction relative to transition systems and the notion of simulation. In this setting, we use the biclosed structure of the bicategory of relations to give a new proof of the standard fact that observational similarity implies similarity.

Boolean and classical restriction categories

A restriction category is an abstract category of partial maps. A Boolean restriction category is a restriction category that supports classical (Boolean) reasoning. Such categories are models of loop-free dynamic logic that is deterministic in the sense that < α > Q ⊂ [α]Q. Classical restriction categories are restriction categories with a locally Boolean structure: it is shown that they are precisely full subcategories of Boolean restriction categories. In particular, a Boolean restriction category may be characterised as a classical restriction category with finite coproducts in which all restriction idempotents split.Every restriction category admits a restriction embedding into a Boolean restriction category. Thus, every abstract category of partial maps admits a conservative extension that supports classical reasoning. An explicit construction of the classical completion of a restriction category is given.

Restriction categories III: colimits, partial limits and extensivity

A restriction category is an abstract formulation for a category of partial maps, defined in terms of certain specified idempotents called the restriction idempotents. All categories of partial maps are restriction categories; conversely, a restriction category is a category of partial maps if and only if the restriction idempotents split. Restriction categories facilitate reasoning about partial maps as they have a purely algebraic formulation.In this paper we consider colimits and limits in restriction categories. As the notion of restriction category is not self-dual, we should not expect colimits and limits in restriction categories to behave in the same manner. The notion of colimit in the restriction context is quite straightforward, but limits are more delicate. The suitable notion of limit turns out to be a kind of lax limit, satisfying certain extra properties.Of particular interest is the behaviour of the coproduct, both by itself and with respect to partial products. We explore various conditions under which the coproducts are ‘extensive’ in the sense that the total category (of the related partial map category) becomes an extensive category. When partial limits are present, they become ordinary limits in the total category. Thus, when the coproducts are extensive we obtain as the total category a lextensive category. This provides, in particular, a description of the extensive completion of a distributive category.

Restriction categories I: categories of partial maps

Given a category with a stable system of monics, one can form the corresponding category of partial maps. To each map in this category there is, on the domain of the map, an associated idempotent, which measures the degree of partiality. This structure is captured abstractly by the notion of a restriction category, in which every arrow is required to have such an associated idempotent. Categories with a stable system of monics, functors preserving this structure, and natural transformations which are cartesian with respect to the chosen monics, form a 2-category which we call M Cat . The construction of categories of partial maps provides a 2-functor Par : M Cat → Cat . We show that Par can be made into an equivalence of 2-categories between M Cat and a 2-category of restriction categories. The underlying ordinary functor Par 0 : M Cat 0 → Cat 0 of the above 2-functor Par turns out to be monadic, and, from this, we deduce the completeness and cocompleteness of the 2-categories of M -categories and of restriction categories. We also consider the problem of how to turn a formal system of subobjects into an actual system of subobjects. A formal system of subobjects is given by a functor into the category sLat of semilattices. This structure gives rise to a restriction category which, via the above equivalence of 2-categories, gives an M -category. This M -category contains the universal realization of the given formal subobjects as actual subobjects.

Order-enrichment for categories of partial maps

Motivated by a desire to treat non-termination directly in the semantics of computation, the notion of approximation between programs is studied in the context of categories of partial maps. In particular, contextual approximation and specialisation are considered and shown to coincide. Moreover, after exhibiting the approximation between total maps as a primitive notion, from an arbitrary (or axiomatic) approximation order on total maps a computationally natural approximation order on partial maps is derived. The main technical contribution is a characterisation of when this approximation order between partial maps is domain-theoretic (in the sense that the category of partial maps Cpo-enriches) provided that the approximation order between total maps is also.

Monads and algebras in the semantics of partial data types

Programming language semantics plays an important role in the development of software technology by providing a coherent approach to the design and implementation of programming languages. Ry avoiding ad hoc techniques, such an approach seeks to provide an environment suitable for the interpretation of powerful yet flexible languages capable of greater modularity and polymorphic behavior. This in turn should lead to greater code reuse, reliability and maintenance so crucial to the future developement of large software systems. Algebraic methods have long provided a plenitude of roles in both defining issues and solving problems in the foundations of programming language semantics. In this paper we focus attention on the use of algebraic structures, particularly categories of algebras and monads, to describe partial data types and their corresponding computations. It has become increasingly evident with the proliferation of semantic categories that no one category is ideal for all purposes. Instead, what is proposed in this paper in the context of partial types is a systematic approach which constructs several categories of algebras and their corresponding adjunctions. The semantic content is then derived not only from the individual categories but from their functorial relationships. Recent attempts at dealing with partial operations on data types have led to the construction of categories of partial maps [ 11, 121. Earlier work had focused on the

``Pathologies'' in two syntactic categories of partial maps.

On developpe une approche algebrique de la theorie de la recursivite initiee par Di Paolo et Heller

Categories of partial maps

This paper attempts to reconcile the various abstract notions of “category of partial maps” which appear in the literature. First a particular algebraic theory (p-categories) is introduced and a representation theorem proved. This gives the authors a coherent framework in which to place the various other definitions. Both algebraic theories and theories which make essential use of the poset-enriched structure of partial maps are discussed. Proofs of equivalence are given where possible and counterexamples where known. The paper concludes with brief sections on the representation of partial maps and on partial algebras.