Strong Bisimulation for Control Operators

The purpose of this paper is to identify programs with control operators whose reduction semantics are in exact correspondence. This is achieved by introducing a relation $\simeq$, defined over a revised presentation of Parigot's $λμ$-calculus we dub $ΛM$. Our result builds on two fundamental ingredients: (1) factorization of $λμ$-reduction into multiplicative and exponential steps by means of explicit term operators of $ΛM$, and (2) translation of $ΛM$-terms into Laurent's polarized proof-nets (PPN) such that cut-elimination in PPN simulates our calculus. Our proposed relation $\simeq$ is shown to characterize structural equivalence in PPN. Most notably, $\simeq$ is shown to be a strong bisimulation with respect to reduction in $ΛM$, i.e. two $\simeq$-equivalent terms have the exact same reduction semantics, a result which fails for Regnier's $σ$-equivalence in $λ$-calculus as well as for Laurent's $σ$-equivalence in $λμ$.