Augmenting Types with Unbounded Demonic and Angelic Nondeterminacy

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Abstract

We show how to introduce demonic and angelic nondeterminacy into the term language of each type in typical programming or specification language. For each type we introduce (binary infix) operators ⊓ and ⊔ on terms of the type, corresponding to demonic and angelic nondeterminacy, respectively. We generalise these operators to accommodate unbounded nondeterminacy. We axiomatise the operators and derive their important properties. We show that a suitable model for nondeterminacy is the free completely distributive complete lattice over a poset, and we use this to show that our axiomatisation is sound. In the process, we exhibit a strong relationship between nondeterminacy and free lattices that has not hitherto been evident. Although nondeterminacy arises naturally in specification and programming languages, we speculate that it combines fruitfully with function theory to the extent that it can play an important role in facilitating proofs of programs that have no apparent connection with nondeterminacy.