Coalgebraic Aspects of Bidirectional Computation

We have previously (Bx, 2014; MPC, 2015) shown that several state-based bx formalisms can be captured using monadic functional pro-gramming, using the state monad together with possibly other monadiceffects, giving rise to structures we have called monadic bx (mbx). Inthis paper, we develop a coalgebraic theory of state-based bx, and relatethe resulting coalgebraic structures (cbx) to mbx. We show that cbxsupport a notion of composition coherent with, but conceptually sim-pler than, our previous mbx definition. Coalgebraic bisimulation yieldsa natural notion of behavioural equivalence on cbx, which respects com-position, and essentially includes symmetric lens equivalence as a spe-cial case. Finally, we speculate on the applications of this coalgebraicperspective to other bx constructions and formalisms.