Recursive Definitions of Monadic Functions

Abstract

Using standard domain-theoretic fixed-points, we present an approach for defining recursive functions that are formulated in monadic style. The method works both in the simple option monad and the state-exception monad of Isabelle/HOL's imperative programming extension, which results in a convenient definition principle for imperative programs, which were previously hard to define.For such monadic functions, the recursion equation can always be derived without preconditions, even if the function is partial. The construction is easy to automate, and convenient induction principles can be derived automatically.

Concrete Semantics

“…One can hide the lfp in denotational semantics too and allow direct 18 11 Denotational Semantics 2 recursive definitions. This is what Isabelle's partial _ function command does [50,92].…”

Refinement to Imperative HOL

“…We do not provide explicit rules for recursion combinators, but rely on the standard Isabelle infrastructure, in particular on the partial function package [16]. However, in Section 4, we will provide rules to refine the recursion combinators of the IRF to corresponding recursion combinators of the heap monad.…”

A formalization of the Berlekamp-Zassenhaus factorization algorithm

“…The prime number generator lazily generates all primes and aborts as soon as the first suitable prime is detected. This is easy to model in Isabelle by defining the generator (suitable prime bz) via partial function [11].…”