Weakest preconditions in fibrations

Abstract

Weakest precondition transformers are useful tools in program verification. One of their key properties is composability, that is, the weakest precondition predicate transformer (wppt for short) associated to program $f;\;g$ should be equal to the composition of the wppts associated to f and g. In this paper, we study the categorical structure behind wppts from a fibrational point of view. We characterize the wppts that satisfy composability as the ones constructed from the Cartesian lifting of a monad. We moreover show that Cartesian liftings of monads along lax slice categories bijectively correspond to Eilenberg–Moore monotone algebras. We then instantiate our techniques by deriving wppts for commonplace effects such as the maybe monad, the nonempty powerset monad, the counter monad, or the distribution monad. We also show how to combine them to derive the wppts appearing in the literature of verification of probabilistic programs.

Higher-Order Weakest Precondition Transformers via a CPS Transformation

“…By choosing appropriate parameters, we can express various properties of effectful programs as a weakest precondition transformer as studied in [3]. Our result in this paper provides a way to syntactically calculate it via the CPS transformation.…”

“…• base types, effect-free constants, and algebraic operations in syntax (and their interpretations in semantics), • an ωCPO-enriched bicartesian closed category C to interpret programs, • a pseudo-lifting strong monad T on C for interpreting computational effects, and • an EM monotone T -algebra on Ω for defining the meaning of the weakest precondition transformer for Ω-valued predicates [3].…”

“…Since category theory provides useful tools for a unified framework for computational effects [2], some extensions explore generalizations via category theory. For example, a recent paper [3] studied various weakest precondition transformers for programs with computational effects using fibrations and monad liftings. Their framework subsumes weakest precondition transformers for total / partial correctness for partiality, may / must correctness for nondeterminism, and expected runtime for probabilistic programs.…”

“…In this paper, we use a general definition of weakest precondition transformers for programs with computational effects [3]. This definition is based on the notion of monad liftings.…”

“…Given a monad lifting, a weakest precondition transformer can be defined by wp[f ](q) = f * ( Ṫ q) where f : I → T J is a morphism in B and q ∈ E J . In [3], it is proved that compositionality of this weakest precondition transformer corresponds to cartesianness of monad lifting. Moreover, if we restrict ourselves to a specific kind of fibrations, then there exists a bijective correspondence between cartesian monad liftings of T and EM monotone T -algebras, which we are going to explain.…”