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

Classifying word problems of finitely generated algebras via computable reducibility

We contribute to a recent research program which aims at revisiting the study of the complexity of word problems, a major area of research in combinatorial algebra, through the lens of the theory of computably enumerable equivalence relations (ceers), which has considerably grown in recent times. To pursue our analysis, we rely on the most popular way of assessing the complexity of ceers, that is via computable reducibility on equivalence relations, and its corresponding degree structure (the c-degrees). On the negative side, building on previous work of Kasymov and Khoussainov, we individuate a collection of c-degrees of ceers which cannot be realized by the word problem of any finitely generated algebra of finite type. On the positive side, we show that word problems of finitely generated semigroups realize a collection of c-degrees which embeds rich structures and is large in several reasonable ways

Synthetic Kolmogorov Complexity in Coq

International audienceWe present a generalised, constructive, and machine-checked approach to Kolmogorov complexity in the constructive type theory underlying the Coq proof assistant. By proving that nonrandom numbers form a simple predicate, we obtain elegant proofs of undecidability for random and nonrandom numbers and a proof of uncomputability of Kolmogorov complexity. We use a general and abstract definition of Kolmogorov complexity and subsequently instantiate it to several definitions frequently found in the literature. Whereas textbook treatments of Kolmogorov complexity usually rely heavily on classical logic and the axiom of choice, we put emphasis on the constructiveness of all our arguments, however without blurring their essence. We first give a high-level proof idea using classical logic, which can be formalised with Markov's principle via folklore techniques we subsequently explain. Lastly, we show a strategy how to eliminate Markov's principle from a certain class of computability proofs, rendering all our results fully constructive. All our results are machine-checked by the Coq proof assistant, which is enabled by using a synthetic approach to computability: rather than formalising a model of computation, which is well-known to introduce a considerable overhead, we abstractly assume a universal function, allowing the proofs to focus on the mathematical essence

On Array Noncomputable Degrees, Maximal Pairs and Simplicity Properties

In this thesis, we give contributions to topics which are related to array noncomputable (a.n.c.) Turing degrees, maximal pairs and to simplicity properties. The outline is as follows. In Chapter 2, we introduce a subclass of the a.n.c. Turing degrees, the so called completely array noncomputable (c.a.n.c. for short) Turing degrees. Here, a computably enumerable (c.e.) Turing degree a is c.a.n.c. if any c.e. set A ∈ a is weak truth-table (wtt) equivalent to an a.n.c. set. We show in Section 2.3 that these degrees exist (indeed, there exist infinitely many low c.a.n.c. degrees) and that they cannot be high. Moreover, we apply some of the ideas used to show the existence of c.a.n.c. Turing degrees to show the stronger result that there exists a c.e. Turing degree whose c.e. members are halves of maximal pairs in the c.e. computably Lipschitz (cl) degrees, thereby solving the first part of the first open problem given in the paper by Ambos-Spies, Ding, Fan and Merkle [ASDFM13]. In Chapter 3, we present an approach to extending the notion of array noncomputability to the setting of almost-c.e. sets (these are the sets which correspond to binary representations of left-c.e. reals). This approach is initiated by the Heidelberg Logic Group and it is worked out in detail in an upcoming paper by Ambos-Spies, Losert and Monath [ASLM18], in the thesis of Losert [Los18] and in [ASFL+]. In [ASLM18], the authors introduce the class of sets with the universal similarity property (u.s.p. for short; throughout this thesis, sets with the u.s.p. will shortly be called u.s.p. sets) which is a strong form of array noncomputability in the setting of almost-c.e. sets and they show that sets with this property exist precisely in the c.e. not totally ω-c.e. degrees. Then it is shown that, using u.s.p. sets, one obtains a simplified method for showing the existence of almost-c.e. sets with a property P (for certain properties P) that are contained in c.e. not totally ω-c.e. degrees, namely by showing that u.s.p. sets have property P. This is demonstrated by showing that u.s.p. sets are computably bounded random (CB-random), thereby extending a result from Brodhead, Downey and Ng [BDN12]. Moreover, it is shown that the c.e. not totally ω-c.e. degrees can be characterized as those c.e. degrees which contain an almost-c.e. set which is not cl-reducible to any complex almost-c.e. set. This affirmatively answers a conjecture by Greenberg. For the if-direction of the latter result, we prove a new result on maximal pairs in the almost-c.e. sets by showing the existence of locally almost-c.e. sets which are halves of maximal pairs in the almost-c.e. sets such that the second half can be chosen to be c.e. and arbitrarily sparse. This extends Yun Fan’s result on maximal pairs [Fan09]. By our result, we also get a new proof of one of the main results in Barmpalias, Downey and Greenberg [BDG10], namely that in any c.e. a.n.c. degree there is a left-c.e. real which is not cl-reducible to any ML-random left-c.e. real. In this thesis, we give an overview of some of the results from [ASLM18] and sketch some of the proofs to illustrate this new methodology and, subsequently, we give a detailed proof of the above maximal pair result. In Chapter 4, we look at the interaction between a.n.c. wtt-degrees and the most commonly known simplicity properties by showing that there exists an a.n.c. wtt-degree which contains an r-maximal set. By this result together with the result by Ambos-Spies [AS18] that no a.n.c. wtt-degree contains a dense simple set, we obtain a complete characterization which of the classical simplicity properties may hold for a.n.c. wtt-degrees. The guiding theme for Chapter 5 is a theorem by Barmpalias, Downey and Greenberg [BDG10] in which they characterize the c.e. not totally ω-c.e. degrees as the c.e. degrees which contain a c.e. set which is not wtt-reducible to any hypersimple set. So Ambos-Spies asked what the above characterization would look like if we replaced hypersimple sets by maximal sets in the above theorem. In other words, what are the c.e. Turing degrees that contain c.e. sets which are not wtt-reducible to any maximal set. We completely solve this question on the set level by introducing the new class of eventually uniformly wtt-array computable (e.u.wtt-a.c.) sets and by showing that the c.e. sets with this property are precisely those c.e. sets which are wtt-reducible to maximal sets. Indeed, this characterization can be extended in that we can replace wtt-reducible by ibT-reducible and maximal sets by dense simple sets. By showing that the c.e. e.u.wtt-a.c. sets are closed downwards under wtt-reductions and under the join operation, it follows that the c.e. wtt-degrees containing e.u.wtt-a.c. sets form an ideal in the upper semilattice of the c.e. wtt-degrees and, further, we obtain a characterization of the c.e. wtt-degrees which contain c.e. sets that are not wtt-reducible to any maximal set. Moreover, we give upper and lower bounds (with respect to ⊆) for the class of the c.e. e.u.wtt-a.c. sets. For the upper bound, we show that any c.e. e.u.wtt-a.c. set has array computable wtt-degree. For the lower bound, we introduce the notion of a wtt-superlow set and show that any wtt-superlow c.e. set is e.u.wtt-a.c. Besides, we show that the wtt-superlow c.e. sets can be characterized as the c.e. sets whose bounded jump is ω-computably approximable (ω-c.a. for short); hence, they are precisely the bounded low sets as introduced in the paper by Anderson, Csima and Lange [ACL17]. Furthermore, we prove a hierarchy theorem for the wtt-superlow c.e. sets and we show that there exists a Turing complete set which lies in the intersection of that hierarchy. Finally, it is shown that the above bounds are strict, i.e., there exist c.e. e.u.wtta. c. sets which are not wtt-superlow and that there exist c.e. sets whose wtt-degree is array computable and which are not e.u.wtt-a.c. (where here, we obtain the separation even on the level of Turing degrees). The results from Chapter 5 will be included in a paper which is in preparation by Ambos-Spies, Downey and Monath [ASDM19]

Multiple Permitting and Bounded Turing Reducibilities

We look at various properties of the computably enumerable (c.e.) not totally ω-c.e. Turing degrees. In particular, we are interested in the variant of multiple permitting given by those degrees. We define a property of left-c.e. sets called universal similarity property which can be viewed as a universal or uniform version of the property of array noncomputable c.e. sets of agreeing with any c.e. set on some component of a very strong array. Using a multiple permitting argument, we prove that the Turing degrees of the left-c.e. sets with the universal similarity property coincide with the c.e. not totally ω-c.e. degrees. We further introduce and look at various notions of socalled universal array noncomputability and show that c.e. sets with those properties can be found exactly in the c.e. not totally ω-c.e. Turing degrees and that they guarantee a special type of multiple permitting called uniform multiple permitting. We apply these properties of the c.e. not totally ω-c.e. degrees to give alternative proofs of well-known results on those degrees as well as to prove new results. E.g., we show that a c.e. Turing degree contains a left-c.e. set which is not cl-reducible to any complex left-c.e. set if and only if it is not totally ω-c.e. Furthermore, we prove that the nondistributive finite lattice S7 can be embedded into the c.e. Turing degrees precisely below any c.e. not totally ω-c.e. degree. We further look at the question of join preservation for bounded Turing reducibilities r and r′ such that r is stronger than r′. We say that join preservation holds for two reducibilities r and r′ if every join in the c.e. r-degrees is also a join in the c.e. r′-degrees. We consider the class of monotone admissible (uniformly) bounded Turing reducibilities, i.e., the reflexive and transitive Turing reducibilities with use bounded by a function that is contained in a (uniformly computable) family of strictly increasing computable functions. This class contains for example identity bounded Turing (ibT-) and computable Lipschitz (cl-) reducibility. Our main result of Chapter 3 is that join preservation fails for cl and any strictly weaker monotone admissible uniformly bounded Turing reducibility. We also look at the dual question of meet preservation and show that for all monotone admissible bounded Turing reducibilities r and r′ such that r is stronger than r′, meet preservation holds. Finally, we completely solve the question of join and meet preservation in the classical reducibilities 1, m, tt, wtt and T

Calibrating the complexity of combinatorics: reverse mathematics and Weihrauch degrees of some principles related to Ramsey’s theorem

In this thesis, we study the proof-theoretical and computational strength of some combinatorial principles related to Ramsey's theorem: this will be accomplished chiefly by analyzing these principles from the points of view of reverse mathematics and Weihrauch complexity. We start by studying a combinatorial principle concerning graphs, introduced by Bill Rival and Ivan Sands as a form of ``inside-outside'' Ramsey's theorem: we will determine its reverse mathematical strength and present the result characterizing its Weihrauch degree. Moreover, we will study a natural restriction of this principle, proving that it is equivalent to Ramsey's theorem. We will then move to a related result, this time concerning countable partial orders, again introduced by Rival and Sands: we will give a thorough reverse mathematical investigation of the strength of this theorem and of its original proof. Moreover, we will be able to generalize it, and this generalization will itself be presented in the reverse mathematical perspective. After this, we will focus on two forms of Ramsey's theorem that can be considered asymmetric. First, we will focus on a restriction of Ramsey's theorem to instances whose solutions have a predetermined color, studying it in reverse mathematics and from the point of view of the complexity of the solutions in a computability theoretic sense. Next, we move to a classical result about partition ordinals, which will undergo the same type of analysis. Finally, we will present some results concerning a recently introduced operator on the Weihrauch degrees, namely the first-order part operator: after presenting an alternative characterization of it, we will embark on the study the result of its applications to jumps of Weak Kőnig's Lemma