Complements witness consistency (short paper)

Much of the existing bx literature, especially that from the PL community on lenses, has described extensional, state-based formalisms. More recently, attention has turned to incorporating intensional information about edits (typically based on monoid actions), or more generally, deltas (typically based on categories), describing how models are updated. Pervasive in both the conceptual modelling, and the mathematics, of varieties of such bx, is the role played by the complement, which generalises the ‘constant complement’ case of the view-update problem in databases. Complements typically reify, or correspond to, data which is abstracted away by passing from a source to a view. In this paper, we present an alternative perspective, which has perhaps been implicit in the lens literature, but not, to our knowledge, previously made explicit anywhere: namely that elements of the complement are witnesses to the consistency relation maintained by the transformation. We illustrate this idea with examples drawn from the bx literature, especially that on lenses.

[1]  H. Läuchli An Abstract Notion of Realizability for Which Intuitionistic Predicate Calculus is Complete , 1970 .

[2]  Lambert Meertens,et al.  Designing Constraint Maintainers for User Interaction , 1998 .

[3]  Saul A. Kripke,et al.  Semantical Analysis of Modal Logic I Normal Modal Propositional Calculi , 1963 .

[4]  Hartmut Ehrig,et al.  From state- to delta-based bidirectional model transformations: the symmetric case , 2011, MODELS'11.

[5]  Martin Hofmann,et al.  Edit lenses , 2012, POPL '12.

[6]  Perdita Stevens,et al.  Bidirectional model transformations in QVT: semantic issues and open questions , 2007, MODELS'07.

[7]  Martin Hofmann,et al.  Symmetric lenses , 2011, POPL '11.

[8]  Benjamin C. Pierce,et al.  Combinators for bi-directional tree transformations: a linguistic approach to the view update problem , 2005, POPL '05.

[9]  James McKinna Bidirectional Transformations with Deltas: A Dependently Typed Approach (Talk Proposal) , 2016, Bx@ETAPS.

[10]  Benjamin C. Pierce,et al.  Matching lenses: alignment and view update , 2010, ICFP '10.

[11]  Stephen Cole Kleene,et al.  On the interpretation of intuitionistic number theory , 1945, Journal of Symbolic Logic.

[12]  James Cheney,et al.  Towards a Principle of Least Surprise for Bidirectional Transformations , 2015, Bx@STAF.

[13]  Krzysztof Czarnecki,et al.  From State- to Delta-Based Bidirectional Model Transformations: the Asymmetric Case , 2011, J. Object Technol..

[14]  J. Lambek,et al.  Introduction to higher order categorical logic , 1986 .

[15]  Tarmo Uustalu,et al.  Coalgebraic Update Lenses , 2014, MFPS.