There are now a number of BIDIRECTIONAL PROGRAMMING LANGUAGES, where every program can be read both as a forward transformation mapping one data structure to another and as a reverse transformation mapping an edited output back to a correspondingly edited input. Besides parsimony - the two related transformations are described by just one expression - such languages are attractive because they promise strong behavioral laws about how the two transformations fit together - e.g., their composition is the identity function. It has repeatedly been observed, however, that such laws are actually a bit too strong: in practice, we do not want them "on the nose," but only up to some equivalence, allowing inessential details, such as whitespace, to be modified after a round trip. Some bidirectional languages loosen their laws in this way, but only for specific, baked-in equivalences.

A lens is a bidirectional program. When read from left toright, it denotes an ordinary function that maps inputs to outputs. When read from right to left, it denotes an ''update translator'' that takes an input together with an updated output and produces a new input that reflects the update. Many variants of this idea have been explored in the literature, but none deal fully with ordered data. If, for example, an update changes the order of a list in theoutput, the items in the output list and the chunks of the input that generated them can be misaligned, leading to lost or corrupted data.