Computability and analysis: the legacy of Alan Turing

We discuss the legacy of Alan Turing and his impact on computability and analysis.Comment: 49 page

Computation with Advice

Computation with advice is suggested as generalization of both computation with discrete advice and Type-2 Nondeterminism. Several embodiments of the generic concept are discussed, and the close connection to Weihrauch reducibility is pointed out. As a novel concept, computability with random advice is studied; which corresponds to correct solutions being guessable with positive probability. In the framework of computation with advice, it is possible to define computational complexity for certain concepts of hypercomputation. Finally, some examples are given which illuminate the interplay of uniform and non-uniform techniques in order to investigate both computability with advice and the Weihrauch lattice

Banach Spaces as Data Types

We introduce the operators "modified limit" and "accumulation" on a Banach space, and we use this to define what we mean by being internally computable over the space. We prove that any externally computable function from a computable metric space to a computable Banach space is internally computable. We motivate the need for internal concepts of computability by observing that the complexity of the set of finite sets of closed balls with a nonempty intersection is not uniformly hyperarithmetical, and thus that approximating an externally computable function is highly complex.Comment: 20 page

Complexity Hierarchies and Higher-order Cons-free Term Rewriting

Constructor rewriting systems are said to be cons-free if, roughly, constructor terms in the right-hand sides of rules are subterms of the left-hand sides; the computational intuition is that rules cannot build new data structures. In programming language research, cons-free languages have been used to characterize hierarchies of computational complexity classes; in term rewriting, cons-free first-order TRSs have been used to characterize the class PTIME. We investigate cons-free higher-order term rewriting systems, the complexity classes they characterize, and how these depend on the type order of the systems. We prove that, for every K $\geq$ 1, left-linear cons-free systems with type order K characterize E$^K$TIME if unrestricted evaluation is used (i.e., the system does not have a fixed reduction strategy). The main difference with prior work in implicit complexity is that (i) our results hold for non-orthogonal term rewriting systems with no assumptions on reduction strategy, (ii) we consequently obtain much larger classes for each type order (E$^K$TIME versus EXP$^{K-1}$TIME), and (iii) results for cons-free term rewriting systems have previously only been obtained for K = 1, and with additional syntactic restrictions besides cons-freeness and left-linearity. Our results are among the first implicit characterizations of the hierarchy E = E$^1$TIME $\subsetneq$ E$^2$TIME $\subsetneq$ ... Our work confirms prior results that having full non-determinism (via overlapping rules) does not directly allow for characterization of non-deterministic complexity classes like NE. We also show that non-determinism makes the classes characterized highly sensitive to minor syntactic changes like admitting product types or non-left-linear rules.Comment: extended version of a paper submitted to FSCD 2016. arXiv admin note: substantial text overlap with arXiv:1604.0893