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Computability
About: Computability is a research topic. Over the lifetime, 2829 publications have been published within this topic receiving 85162 citations.
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TL;DR: This chapter presents the study of the interplay between recursion in higher types and transfinite recursion and introduces the theory of partial continuous functionals, based on Scott's notion of an information system.
Abstract: This chapter presents the study of the interplay between recursion in higher types and transfinite recursion. The chapter introduces the theory of partial continuous functionals, based on Scott's notion of an information system. The partial continuous functionals are central for any the analysis of higher type computability, which is based on the rather natural assumption that any computation ought to be finite. The reason is simply that they form the mathematically appropriate domain of a computable functional. The chapter also presents the collapsing results and discusses the extended Grzegorczyk hierarchy. An introduction to partial continuous functional and some general material concerning computability in higher types is also presented in the chapter. The chapter discusses bounded fixed point operators and concerns elimination of detours through higher types by transfinite recursion.
17 citations
DOI•
01 Jan 2019
TL;DR: The first full mechanisation of the Davis-Putnam-Robinson-Matiyasevich theorem is given, stating that every recursively enumerable problem -- in this case by a Minsky machine -- is Diophantine.
Abstract: We formalise the undecidability of solvability of Diophantine equations, i.e. polynomial equations over natural numbers, in Coq's constructive type theory. To do so, we give the first full mechanisation of the Davis-Putnam-Robinson-Matiyasevich theorem, stating that every recursively enumerable problem - in our case by a Minsky machine - is Diophantine. We obtain an elegant and comprehensible proof by using a synthetic approach to computability and by introducing Conway's FRACTRAN language as intermediate layer.
17 citations
Journal Article•
TL;DR: A general class of Fine computable functions is introduced and it is proved that the integration operator on the space of Fine continuous functions is lower semi-computable.
Abstract: A metric defined by Fine induces a topology on the unit interval which is strictly stronger than the ordinary Euclidean topology and which has some inter- esting applications in Walsh analysis. We investigate computability properties of a corresponding Fine representation of the real numbers and we construct a structure which characterizes this representation. Moreover, we introduce a general class of Fine computable functions and we compare this class with the class of locally uniformly Fine computable functions defined by Mori. Both classes of functions include all ordinary computable functions and, additionally, some important functions which are discontin- uous with respect to the usual Euclidean metric. Finally, we prove that the integration operator on the space of Fine continuous functions is lower semi-computable.
16 citations
TL;DR: The computability path ordering (CPO) as discussed by the authors is a recursive relation on terms obtained by lifting a precedence on function symbols, which is essentially obtained from the higher-order recursive path ordering by eliminating type checks from some recursive calls and incorporating the treatment of bound variables as in the computability closure.
Abstract: This paper aims at carrying out termination proofs for simply typed higher-order calculi automatically by using ordering comparisons. To this end, we introduce the computability path ordering (CPO), a recursive relation on terms obtained by lifting a precedence on function symbols. A first version, core CPO, is essentially obtained from the higher-order recursive path ordering (HORPO) by eliminating type checks from some recursive calls and by incorporating the treatment of bound variables as in the com-putability closure. The well-foundedness proof shows that core CPO captures the essence of computability arguments ^#224, la Tait and Girard, therefore explaining its name. We further show that no further type check can be eliminated from its recursive calls without loosing well-foundedness, but for one for which we found no counterexample yet. Two extensions of core CPO are then introduced which allow one to consider: the first, higher-order inductive types; the second, a precedence in which some function symbols are smaller than application and abstraction.
16 citations
TL;DR: A series of approximations are described that converge to the $p-radius with a priori computable accuracy for nonnegative matrices, which gives efficient approximation schemes for the p-radius computation.
Abstract: The $p$-radius characterizes the average rate of growth of norms of matrices in a multiplicative semigroup. This quantity has found several applications in recent years. We raise the question of its computability. We prove that the complexity of its approximation increases exponentially with $p$. We then describe a series of approximations that converge to the $p$-radius with a priori computable accuracy. For nonnegative matrices, this gives efficient approximation schemes for the $p$-radius computation.
16 citations