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

An extensional Kleene realizability semantics for the Minimalist Foundation

We build a Kleene realizability semantics for the two-level Minimalist Foundation MF, ideated by Maietti and Sambin in 2005 and completed by Maietti in 2009. Thanks to this semantics we prove that both levels of MF are consistent with the (Extended) formal Church Thesis CT. MF consists of two levels, an intensional one, called mTT and an extensional one, called emTT, based on versions of Martin-L\"of's type theory. Thanks to the link between the two levels, it is enough to build a semantics for the intensional level to get one also for the extensional level. Hence here we just build a realizability semantics for the intensional level mTT. Such a semantics is a modification of the realizability semantics in Beeson 1985 for extensional first order Martin-L\"of's type theory with one universe. So it is formalised in Feferman's classical arithmetic theory of inductive definitions. It is called extensional Kleene realizability semantics since it validates extensional equality of type-theoretic functions extFun, as in Beeson 1985. The main modification we perform on Beeson's semantics is to interpret propositions, which are defined primitively in MF, in a proof-irrelevant way. As a consequence, we gain the validity of CT. Recalling that extFun+ CT+ AC are inconsistent over arithmetics with finite types, we conclude that our semantics does not validate the full Axiom of Choice AC. On the contrary, Beeson's semantics does validate AC, being this a theorem of Martin-L\"of's theory, but it does not validate CT. The semantics we present here appears to be the best Kleene realizability semantics for the extensional level emTT of MF. Indeed Beeson's semantics is not an option for emTT since the full AC added to it entails the excluded middle

Global semantic typing for inductive and coinductive computing

Inductive and coinductive types are commonly construed as ontological (Church-style) types, denoting canonical data-sets such as natural numbers, lists, and streams. For various purposes, notably the study of programs in the context of global semantics, it is preferable to think of types as semantical properties (Curry-style). Intrinsic theories were introduced in the late 1990s to provide a purely logical framework for reasoning about programs and their semantic types. We extend them here to data given by any combination of inductive and coinductive definitions. This approach is of interest because it fits tightly with syntactic, semantic, and proof theoretic fundamentals of formal logic, with potential applications in implicit computational complexity as well as extraction of programs from proofs. We prove a Canonicity Theorem, showing that the global definition of program typing, via the usual (Tarskian) semantics of first-order logic, agrees with their operational semantics in the intended model. Finally, we show that every intrinsic theory is interpretable in a conservative extension of first-order arithmetic. This means that quantification over infinite data objects does not lead, on its own, to proof-theoretic strength beyond that of Peano Arithmetic. Intrinsic theories are perfectly amenable to formulas-as-types Curry-Howard morphisms, and were used to characterize major computational complexity classes Their extensions described here have similar potential which has already been applied

Church's thesis and related axioms in Coq's type theory

"Church's thesis" ($\mathsf{CT}$) as an axiom in constructive logic states that every total function of type $\mathbb{N} \to \mathbb{N}$ is computable, i.e. definable in a model of computation. $\mathsf{CT}$ is inconsistent in both classical mathematics and in Brouwer's intuitionism since it contradicts Weak K\"onig's Lemma and the fan theorem, respectively. Recently, $\mathsf{CT}$ was proved consistent for (univalent) constructive type theory. Since neither Weak K\"onig's Lemma nor the fan theorem are a consequence of just logical axioms or just choice-like axioms assumed in constructive logic, it seems likely that $\mathsf{CT}$ is inconsistent only with a combination of classical logic and choice axioms. We study consequences of $\mathsf{CT}$ and its relation to several classes of axioms in Coq's type theory, a constructive type theory with a universe of propositions which does neither prove classical logical axioms nor strong choice axioms. We thereby provide a partial answer to the question which axioms may preserve computational intuitions inherent to type theory, and which certainly do not. The paper can also be read as a broad survey of axioms in type theory, with all results mechanised in the Coq proof assistant

Virtual Evidence: A Constructive Semantics for Classical Logics

This article presents a computational semantics for classical logic using constructive type theory. Such semantics seems impossible because classical logic allows the Law of Excluded Middle (LEM), not accepted in constructive logic since it does not have computational meaning. However, the apparently oracular powers expressed in the LEM, that for any proposition P either it or its negation, not P, is true can also be explained in terms of constructive evidence that does not refer to "oracles for truth." Types with virtual evidence and the constructive impossibility of negative evidence provide sufficient semantic grounds for classical truth and have a simple computational meaning. This idea is formalized using refinement types, a concept of constructive type theory used since 1984 and explained here. A new axiom creating virtual evidence fully retains the constructive meaning of the logical operators in classical contexts. Key Words: classical logic, constructive logic, intuitionistic logic, propositions-as-types, constructive type theory, refinement types, double negation translation, computational content, virtual evidenc

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

Realizability and recursive mathematics

Section 1: Philosophy, logic and constructivityPhilosophy, formal logic and the theory of computation all bear on problems in the foundations of constructive mathematics. There are few places where these, often competing, disciplines converge more neatly than in the theory of realizability structures. Uealizability applies recursion-theoretic concepts to give interpretations of constructivism along lines suggested originally by Heyting and Kleene. The research reported in the dissertation revives the original insights of Kleene—by which realizability structures are viewed as models rather than proof-theoretic interpretations—to solve a major problem of classification and to draw mathematical consequences from its solution.Section 2: Intuitionism and recursion: the problem of classificationThe internal structure of constructivism presents an interesting problem. Mathematically, it is a problem of classification; for philosophy, it is one of conceptual organization. Within the past seventy years, constructive mathematics has grown into a jungle of fullydeveloped "constructivities," approaches to the mathematics of the calculable which range from strict finitism through hyperarithmetic model theory. The problem we address is taxonomic: to sort through the jungle, set standards for classification and determine those features which run through everything that is properly "constructive."There are two notable approaches to constructivity; these must appear prominently in any proposed classification. The most famous is Brouwer's intuitioniam. Intuitionism relies on a complete constructivization of the basic mathematical objects and logical operations. The other is classical recursive mathematics, as represented by the work of Dekker, Myhill, and Nerode. Classical constructivists use standard logic in a mathematical universe restricted to coded objects and recursive operations.The theorems of the dissertation give a precise answer to the classification problem for intuitionism and classical constructivism. Between these realms arc connected semantically through a model of intuitionistic set theory. The intuitionistic set theory IZF encompasses all of the intuitionistic mathematics that does not involve choice sequences. (This includes all the work of the Bishop school.) IZF has as a model a recursion-theoretic structure, V(A7), based on Kleene realizability. Since realizability takes set variables to range over "effective" objects, large parts of classical constructivism appear over the model as inter¬ preted subsystems of intuitionistic set theory. For example, the entire first-order classical theory of recursive cardinals and ordinals comes out as an intuitionistic theory of cardinals and ordinals under realizability. In brief, we prove that a satisfactory partial solution to the classification problem exists; theories in classical recursive constructivism are identical, under a natural interpretation, to intuitionistic theories. The interpretation is especially satisfactory because it is not a Godel-style translation; the interpretation can be developed so that it leaves the classical logical forms unchanged.Section 3: Mathematical applications of the translation:The solution to the classification problem is a bridge capable of carrying two-way mathematical traffic. In one direction, an identification of classical constructivism with intuitionism yields a certain elimination of recursion theory from the standard mathematical theory of effective structures, leaving pure set theory and a bit of model theory. Not only are the theorems of classical effective mathematics faithfully represented in intuitionistic set theory, but also the arguments that provide proofs of those theorems. Via realizability, one can find set-theoretic proofs of many effective results, and the set-theoretic proofs are often more straightforward than their recursion-theoretic counterparts. The new proofs are also more transparent, because they involve, rather than recursion theory plus set theory, at most the set-theoretic "axioms" of effective mathematics.Working the other way, many of the negative ("cannot be obtained recursively") results of classical constructivism carry over immediately into strong independence results from intuitionism. The theorems of Kalantari and Retzlaff on effective topology, for instance, turn into independence proofs concerning the structure of the usual topology on the intuitionistic reals.The realizability methods that shed so much light over recursive set theory can be applied to "recursive theories" generally. We devote a chapter to verifying that the realizability techniques can be used to good effect in the semantical foundations of computer science. The classical theory of effectively given computational domains a la Scott can be subsumed into the Kleene realizability universe as a species of countable noneffective domains. In this way, the theory of effective domains becomes a chapter (under interpre¬ tation) in an intuitionistic study of denotational semantics. We then show how the "extra information" captured in the logical signs under realizability can be used to give proofs of classical theorems about effective domains.Section 4: Solutions to metamathematical problems:The realizability model for set theory is very tractible; in many ways, it resembles a Boolean-valued universe. The tractibility is apparent in the solutions it offers to a number of open problems in the metamathematics of constructivity. First, there is the perennial problem of finding and delimiting in the wide constructive universe those features that correspond to structures familiar from classical mathematics. In the realizability model, it is easy to locate the collection of classical ordinals and to show that they form, intuitionistically, a set rather than a proper class. Also, one interprets an argument of Dekker and Myhill to prove that the classical powerset of the natural numbers contains at least continuum-many distinct cardinals.Second, a major tenet of Bishop's program for constructivity has been that constructive mathematics is "numerical:" all the properties of constructive objects, including the real numbers, can be represented as properties of the natural numbers. The realizability model shows that Bishop's numericalization of mathematics can, in principle, be accomplished. Every set over the model with decidable equality and every metric space is enumerated by a collection of natural numbers