A propositional logic program P may be identified with a PfPf-coalgebra on the set of atomic propositions in the program. The corresponding C(PfPf)-coalgebra, where C(PfPf) is the cofree comonad on PfPf, describes derivations by resolution. That correspondence has been developed to model first-order programs in two ways, with lax semantics and saturated semantics, based on locally ordered categories and right Kan extensions respectively. We unify the two approaches, exhibiting them as complementary rather than competing, reflecting the theorem-proving and proof-search aspects of logic programming. While maintaining that unity, we further refine lax semantics to give finitary models of logic programs with existential variables, and to develop a precise semantic relationship between variables in logic programming and worlds in local state.

Coinductive definitions, such as that of an infinite stream, may often be described by elegant logic programs, but ones for which SLD-refutation is of no value as SLD-derivations fall into infinite loops. Such definitions give rise to questions of lazy corecursive derivations and parallelism, as execution of such logic programs can have both recursive and corecursive features at once. Observational and coalgebraic semantics have been used to study them abstractly. The programming developments have often occurred separately and have usually been implementation-led. Here, we give a coherent semantics-led account of the issues, starting with abstract category theoretic semantics, developing coalgebra to characterize naturally arising trees and proceeding towards implementation of a new dialect, CoALP, of logic programming, characterised by guarded lazy corecursion and parallelism.

A propositional logic program P may be identified with a PfPf -coalgebra on the set of atomic propositions in the program. The corresponding C(PfPf)-coalgebra, where C(PfPf) is the cofree comonad on PfPf, describes derivations by resolution. Using lax semantics, that correspondence may be extended to a class of first-order logic programs without existential variables. The resulting extension captures the proofs by term-matching resolution in logic programming. Refining the lax approach, we further extend it to arbitrary logic programs. We also exhibit a refinement of Bonchi and Zanasi's saturation semantics for logic programming that complements lax semantics.

In 2007, Harmer, Hyland and Melliès gave a formal mathematical foundation for game semantics using a notion they called a ⊸-schedule, and the similar notion of ⊗-schedule, both structures describing interleavings of plays in games. Their definition was combinatorial in nature, but researchers often draw pictures when describing schedules in practice. Moreover, several proofs of key properties, such as that the composition of ⊸-schedules is associative, involve cumbersome combinatorial detail, whereas in terms of pictures the proof is straightforward, reflecting the geometry of the plane. Here, we give a geometric formulation of ⊸-schedules and ⊗-schedules, prove that they are isomorphic to Harmer et al.'s definitions, and illustrate their value by giving such geometric proofs. Harmer et al.'s notions may be combined to describe plays in multi-component games, and researchers have similarly developed intuitive graphical representations of plays in these games. We give a characterisation of these diagrams and explicitly describe how they relate to the underlying schedules, finally using this relation to provide new, intuitive proofs of key categorical properties.

The proliferation of semi-presidential regimes is the most important constitutional development of the modern age, and they now rival in numbers the more familiar presidential and parliamentary sys...

A longitudinal study conducted over a three-year period established six learning networks consisting of 105 entrepreneurs and SME owner/managers. The principal objective of the study was to uncover the critical elements that ensure networks operate effectively in order to advise and inform on practice for the creation of sustainable self-facilitated learning networks. The methodological approach adopted for this study was primarily direct participant observation. Findings provide a guide to action for the development of self-facilitated sustainable learning networks. This paper makes a significant contribution to practice providing knowledge on how to create sustainable learning networks thus having implications for entrepreneurs, government, policy-makers and academics interested in understanding how to effectively guide the development of sustainable learning networks for entrepreneurs to solve their own problems. This paper concludes with a discussion of theoretical and applied significance.

A setoid is a set together with a constructive representation of an equivalence relation on it. Here, we give category theoretic support to the notion. We first define a category Setoid and prove it is Cartesian closed with coproducts. We then enrich it in the Cartesian closed category Equiv of sets and classical equivalence relations, extend the above results, and prove that Setoid as an Equiv-enriched category has a relaxed form of equalisers. We then recall the definition of E-category, generalising that of Equiv-enriched category, and show that Setoid as an E-category has a relaxed form of coequalisers. In doing all this, we carefully compare our category theoretic constructs with Agda code for type-theoretic constructs on setoids.

In 2007, Harmer, Hyland and Melliès gave a formal mathematical foundation for game semantics using a notion they called a schedule. Their definition was combinatorial in nature, but researchers often draw pictures when describing schedules in practice. Moreover, a proof that the composition of schedules is associative involves cumbersome combinatorial detail, whereas in terms of pictures the proof is straightforward, reflecting the geometry of the plane. Here, we give a geometric formulation of schedule, prove that it is equivalent to Harmer et al.ʼs definition, and illustrate its value by giving a proof of associativity of composition.

We compare approaches to implicit and explicit resource handling in coinductive and concurrent logic programming. We show various effects that implicit and explicit handling of resources have on implementation and semantics. In particular, we show that recently introduced coalgebraic logic programming [17] is a paradigm in which, in contrast to many other alternative systems, the aspects of logic and control are intertwined, and computational resources are handled implicitly.

Enrichment of ordinary monads over Cat or Gpd is fundamental to Max Kelly’s unified theory of coherence for categories with structure. So here, we investigate existence and unicity of enrichments of ordinary functors, natural transformations, and hence also monads, over Cat and Gpd. We show that every ordinary natural transformation between 2-functors whose domain 2-category has either tensors or cotensors with the arrow category is 2-natural. We use that to prove that an ordinary monad, or endofunctor, on such a 2-category has at most one enrichment over Cat or Gpd. We also describe a monad on Cat that has no enrichment. So enrichment over Cat is a non-trivial property of a monad rather than a structure that is additional to it. Finally, we present an example, due to Kelly, of V other than Cat or Gpd and an ordinary monad for which more than one enrichment over V exists, showing that our main theorem is specific to Cat and Gpd.

Every variable-free logic program induces a P f P f -coalgebra on the set of atomic formulae in the program. The coalgebra p sends an atomic formula A to the set of the sets of atomic formulae in the antecedent of each clause for which A is the head. In an earlier paper, we identified a variable-free logic program with a P f P f -coalgebra on Set and showed that, if C(P f P f ) is the cofree comonad on P f P f , then given a logic program P qua P f P f -coalgebra, the corresponding C(P f P f )-coalgebra structure describes the parallel and-or derivation trees of P. In this paper, we extend that analysis to arbitrary logic programs. That requires a subtle analysis of lax natural transformations between Poset-valued functors on a Lawvere theory, of locally ordered endofunctors and comonads on locally ordered categories, and of coalgebras, oplax maps of coalgebras, and the relationships between such for locally ordered endofunctors and the cofree comonads on them.

Local variables in imperative languages have been given denotational semantics in at least two fundamentally different ways. One is by use of functor categories, focusing on the idea of possible worlds. The other might be termed event-based, exemplified by Reddy's object spaces and models based on game semantics. O'Hearn and Reddy have related the two approaches by giving functor category models whose worlds are object spaces, then showing that their model is fully abstract for Idealised Algol programs up to order two. But the category of object spaces is not small, and so in order to construct a functor category that is locally small, and hence Cartesian closed, they need to work with a restricted collection of object spaces. This weakens the connection between the object spaces model and the functor-category model: the Yoneda embedding no longer provides a full embedding of the original category of object spaces into the functor-category. Moreoever the choice of the restricted collection of object spaces is ad hoc. In this paper, we refine the approach by proving that the finite objects form a small dense subcategory of a simplified object-spaces model. The functor category over these finite objects is therefore locally small and Cartesian closed, and contains the object-spaces category as a full subcategory. All this work is necessarily enriched in Cpo. We further refine their full abstraction result by showing that full abstraction fails at order three.

Australian Journal of Public AdministrationVolume 68, Issue 2 p. 236-237 The Queen's Other Realms: The Crown and its Legacy in Australia, Canada and New Zealand by Peter Boyce John Power, John Power University of MelbourneSearch for more papers by this author John Power, John Power University of MelbourneSearch for more papers by this author First published: 19 May 2009 https://doi.org/10.1111/j.1467-8500.2009.00634_5.xRead the full textAboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Volume68, Issue2June 2009Pages 236-237 RelatedInformation

We give a general construct that extends denotational semantics for a global computational eect to yield denotational semantics for a corresponding local computational eect. Our leading example yields a construction of the usual denotational semantics for local state from that for global state. Given any Lawvere theory L, possibly countable and possibly enriched, modelling a specic computational eect, we rst give a universal construction that extends L, hence the global operations

We introduce a fibrational semantics for many-valued logic programming, use it to define an SLD-resolution for annotation-free many valued logic programs as defined by Fitting, and prove a soundness and completeness result relating the two. We show that fibrational semantics corresponds with the traditional declarative (ground) semantics and deduce a soundness and completeness result for our SLD-resolution algorithm with respect to the ground semantics.