Irreversible computable functions

International audienceThe strong relationship between topology and computations has played a central role in the development of several branches of theoretical computer science: foundations of functional programming, computational geometry, computability theory, computable analysis. Often it happens that a given function is not computable simply because it is not continuous. In many cases, the function can moreover be proved to be non-computable in the stronger sense that it does not preserve computability: it maps a computable input to a non-computable output. To date, there is no connection between topology and this kind of non-computability, apart from Pour-El and Richards ''First Main Theorem'', applicable to linear operators on Banach spaces only. In the present paper, we establish such a connection. We identify the discontinuity notion, for the inverse of a computable function, that implies non-preservation of computability. Our result is applicable to a wide range of functions, it unifies many existing ad hoc constructions explaining at the same time what makes these constructions possible in particular contexts, sheds light on the relationship between topology and computability and most importantly allows us to solve open problems. In particular it enables us to answer the following open question in the negative: if the sum of two shift-invariant ergodic measures is computable, must these measures be computable as well? We also investigate how generic a point with computable image can be. To this end we introduce a notion of genericity of a point w.r.t. a function, which enables us to unify several finite injury constructions from computability theory

Genericity of weakly computable objects

International audienceIn computability theory many results state the existence of objects that in many respects lack algorithmic structure but at the same time are effective in some sense. Friedberg and Muchnik's answer to Post problem is one of the most celebrated results in this form. The main goal of the paper is to develop a general result that embodies a large number of these particular constructions, capturing the essential idea that is common to all of them, and expressing it in topological terms.To do so, we introduce the effective topological notions of irreversible function and directional genericity and provide two main results that identify situations when such constructions are possible, clarifying the role of topology in many arguments from computability theory. We apply these abstract results to particular situations, illustrating their strength and deriving new results.This paper is an extended version of a conference paper with detailed proofs and new results

Aspects of Computable Analysis

Computable analysis has been well studied ever since Turing famously formalised the computable reals and computable real-valued function in 1936. However, analysis is a broad subject, and there still exist areas that have yet to be explored. For instance, Sierpinski proved that every real-valued function ƒ : ℝ → ℝ is the limit of a sequence of Darboux functions. This is an intriguing result, and the complexity of these sequences has been largely unstudied. Similarly, the Blaschke Selection Theorem, closely related to the Bolzano-Weierstrass Theorem, has great practical importance, but has not been considered from a computability theoretic perspective. The two main contributions of this thesis are: to provide some new, simple proofs of fundamental classical results (highlighting the role of ∏0/1 classes), and to use tools from effective topology to analyse the Darboux property, particularly a result by Sierpinski, and the Blaschke Selection Theorem. This thesis focuses on classical computable analysis. It does not make use of effective measure theory

Computability in constructive type theory

We give a formalised and machine-checked account of computability theory in the Calculus of Inductive Constructions (CIC), the constructive type theory underlying the Coq proof assistant. We first develop synthetic computability theory, pioneered by Richman, Bridges, and Bauer, where one treats all functions as computable, eliminating the need for a model of computation. We assume a novel parametric axiom for synthetic computability and give proofs of results like Rice’s theorem, the Myhill isomorphism theorem, and the existence of Post’s simple and hypersimple predicates relying on no other axioms such as Markov’s principle or choice axioms. As a second step, we introduce models of computation. We give a concise overview of definitions of various standard models and contribute machine-checked simulation proofs, posing a non-trivial engineering effort. We identify a notion of synthetic undecidability relative to a fixed halting problem, allowing axiom-free machine-checked proofs of undecidability. We contribute such undecidability proofs for the historical foundational problems of computability theory which require the identification of invariants left out in the literature and now form the basis of the Coq Library of Undecidability Proofs. We then identify the weak call-by-value λ-calculus L as sweet spot for programming in a model of computation. We introduce a certifying extraction framework and analyse an axiom stating that every function of type ℕ → ℕ is L-computable.Wir behandeln eine formalisierte und maschinengeprüfte Betrachtung von Berechenbarkeitstheorie im Calculus of Inductive Constructions (CIC), der konstruktiven Typtheorie die dem Beweisassistenten Coq zugrunde liegt. Wir entwickeln erst synthetische Berechenbarkeitstheorie, vorbereitet durch die Arbeit von Richman, Bridges und Bauer, wobei alle Funktionen als berechenbar behandelt werden, ohne Notwendigkeit eines Berechnungsmodells. Wir nehmen ein neues, parametrisches Axiom für synthetische Berechenbarkeit an und beweisen Resultate wie das Theorem von Rice, das Isomorphismus Theorem von Myhill und die Existenz von Post’s simplen und hypersimplen Prädikaten ohne Annahme von anderen Axiomen wie Markov’s Prinzip oder Auswahlaxiomen. Als zweiten Schritt führen wir Berechnungsmodelle ein. Wir geben einen kompakten Überblick über die Definition von verschiedenen Berechnungsmodellen und erklären maschinengeprüfte Simulationsbeweise zwischen diesen Modellen, welche einen hohen Konstruktionsaufwand beinhalten. Wir identifizieren einen Begriff von synthetischer Unentscheidbarkeit relativ zu einem fixierten Halteproblem welcher axiomenfreie maschinengeprüfte Unentscheidbarkeitsbeweise erlaubt. Wir erklären solche Beweise für die historisch grundlegenden Probleme der Berechenbarkeitstheorie, die das Identifizieren von Invarianten die normalerweise in der Literatur ausgelassen werden benötigen und nun die Basis der Coq Library of Undecidability Proofs bilden. Wir identifizieren dann den call-by-value λ-Kalkül L als sweet spot für die Programmierung in einem Berechnungsmodell. Wir führen ein zertifizierendes Extraktionsframework ein und analysieren ein Axiom welches postuliert dass jede Funktion vom Typ N→N L-berechenbar ist

Lachlan Non-Splitting Pairs and High Computably Enumerable Turing Degrees

A given c.e. degree a > 0 has a non-trivial splitting into c.e. degrees v and w if a is the join of v and w and v | w. A Lachlan Non-Splitting Pair is a pair of c.e. degrees such that a > d and there is no non-trivial splitting of a into c.e. degrees w and v with w > d and v > d. Lachlan [Lachlan1976] showed that such a pair exists by proving the Lachlan Non-Splitting Theorem. This theorem is remarkable for its discovery of the 0'''-priority method, and became known as the `Monster' due to its significant complexity. Harrington, Shore and Slaman subsequently tried to explain Lachlan's methods in more intuitive and comprehensible terms in a number of unpublished notes. Leonhardi [Leonhardi1997] then published a short account of the Lachlan Non-Splitting Theorem based on these notes and generalised the theorem in a different direction. In their work on the separation of the jump class High from the jump class Low2, Shore and Slaman [SlamanShore1993] also conjectured that every high c.e. degree strictly bounds a Lachlan Non-Splitting Pair, a fact which could be used to separate the two jump classes. While this separation was eventually achieved through the notion of a Slaman Triple, the conjecture itself remained an open question. Cooper, Yi and Li [CooperLiYi2002] also defined the notion of a c.e. Robinson degree as one which does not strictly bound the base d of a Lachlan Non-Splitting Pair , and sought to understand the relationship of this notion to the High/Low Hierarchy. In this dissertation we make the following two contributions. Firstly we show that a counter-example can be found to show that the account of the Lachlan Non-Splitting Theorem given by Leonhardi [Leonhardi1997] fails to satisfy its requirements. By rectifying the construction, we give a complete, correct and intuitive account of the Lachlan Non-Splitting Theorem. Secondly we show that the high permitting method developed by Shore and Slaman [SlamanShore1993] can be combined with the construction of the Lachlan Non-Splitting Theorem just described to prove that every high c.e. degree strictly bounds a Lachlan Non-Splitting Pair. From this it follows that the existence of a Lachlan Non-Splitting Pair can be used to separate the jump classes High and Low2, that the distribution of Lachlan Non-Splitting Pairs with respect to these jump classes mirrors the one for Slaman Triples, and that there is no high c.e. Robinson degree