Symmetric delta lenses and spans of asymmetric delta lenses.

Abstract

Bidirectional Transformations provide mechanisms for maintaining synchronization between updatable data sources. Lenses are certain mathematically specified bidirectional transformations. As part of a project to unify the treatment of symmetric lenses (of various kinds) as equivalence classes of spans of asymmetric lenses (of corresponding kinds), we relate symmetric delta lenses with spans of asymmetric delta lenses. Because delta lenses are based on state spaces which are categories rather than sets, there is further structure that needs to be accounted for. One of the main findings in this paper is that the required equivalence relation among spans is compatible with, but coarser than, the one expected. The main result is an isomorphism of categories between a category whose morphisms are equivalence classes of symmetric delta lenses (here called fb-lenses) and the category of spans of delta lenses modulo the new equivalence.

Multiple model synchronization with multiary delta lenses with amendment andK-Putput

“…It is easy to see that the chain (5) symmetric lens → wide span → symmetric lens results in a lens isomorphic to the initial one, but the chain (6) wide span → symmetric lens → wide span ends with a different span as the first transformation above loses information about updates in the head of the span. As shown by Johnson and Rosebrugh in [22] for the binary case, the two spans can only be equivalent modulo certain equivalence relation, and an equivalence relation between lenses is also needed to align all constructs together. These equivalences may be different for our multiary lenses with amendments, and we leave this important question for future research.…”

Multiple Model Synchronization with Multiary Delta Lenses

“…where An open question is whether the span-to-lens transformation in Theorem 2 and the lens-to-span transformation described above are mutually inverse. The results for the binary case in [8] show that this is only the case modulo certain equivalence relations. These equivalences may be different for our reflective multiary lenses, and we leave this important question for future research.…”

“…Compositionality as a fundamental principle for building synchronization tools was proposed by Pierce and his coauthors, and realized for several types of binary lenses in [4,6,7]. In the delta-lens world, a fundamental theory of equivalence of symmetric lenses and spans of a-lenses (for the binary case) is developed by Johnson and Rosebrugh [8], but they do not consider reflective updates. The PutGetPut law has been discussed (in a different context of state-based asymmetric injective editing) in several early bx work from Tokyo, e.g., [13].…”

Multi-Model Evolution through Model Repair.

“…Compositionality (Foster et al 2007;Diskin et al 2019), i.e. preservation of properties from elementary model repair functions under composition is another aspect of model repair that has been a focus of many theoretical studies in this area, such as (Johnson & Rosebrugh 2017, 2016. Remark 1 (State-Based vs. Delta-Based).…”