The Power of Non-determinism in Higher-Order Implicit Complexity
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We investigate the power of non-determinism in purely functional programming languages with higher-order types. Specifically, we consider cons-free programs of varying data orders, equipped with explicit non-deterministic choice. Cons-freeness roughly means that data constructors cannot occur in function bodies and all manipulation of storage space thus has to happen indirectly using the call stack.
While cons-free programs have previously been used by several authors to characterise complexity classes, the work on non-deterministic programs has almost exclusively considered programs of data order 0. Previous work has shown that adding explicit non-determinism to cons-free programs taking data of order 0 does not increase expressivity; we prove that this—dramatically—is not the case for higher data orders: adding non-determinism to programs with data order at least 1 allows for a characterisation of the entire class of elementary-time decidable sets.
Finally we show how, even with non-deterministic choice, the original hierarchy of characterisations is restored by imposing different restrictions.
KeywordsImplicit computational complexity Cons-free programming EXPTIME hierarchy Non-deterministic programming Unitary variables
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