Classical Mathematics for a Constructive World

Interactive theorem provers based on dependent type theory have the flexibility to support both constructive and classical reasoning. Constructive reasoning is supported natively by dependent type theory and classical reasoning is typically supported by adding additional non-constructive axioms. However, there is another perspective that views constructive logic as an extension of classical logic. This paper will illustrate how classical reasoning can be supported in a practical manner inside dependent type theory without additional axioms. We will see several examples of how classical results can be applied to constructive mathematics. Finally, we will see how to extend this perspective from logic to mathematics by representing classical function spaces using a weak value monad.Comment: v2: Final copy for publicatio

Dynamic Separation Logic

This paper introduces a dynamic logic extension of separation logic. The assertion language of separation logic is extended with modalities for the five types of the basic instructions of separation logic: simple assignment, look-up, mutation, allocation, and de-allocation. The main novelty of the resulting dynamic logic is that it allows to combine different approaches to resolving these modalities. One such approach is based on the standard weakest precondition calculus of separation logic. The other approach introduced in this paper provides a novel alternative formalization in the proposed dynamic logic extension of separation logic. The soundness and completeness of this axiomatization has been formalized in the Coq theorem prover

Reviews for the American Mathematical Society (AMS): Russell O’Connor. Classical mathematics for a constructive world. MSCS (21): 861–882, Cambridge University Press. 2010.

Russell O'Connor presents an alternative method of bridging the gap between constructive and classical logics to facilitate integrated reasoning in particular for interactive theorem provers. Instead of adding classical axioms (like the principle of the excluded middle x∨¬x) to a constructive logic, the author proposes the opposite. He interprets constructive logics as an extension to classical logics by adding two new logical connectives + and Σ for constructive disjunction and constructive existential quantification