Perspectives for proof unwinding by programming languages techniques

In this chapter, we propose some future directions of work, potentially beneficial to Mathematics and its foundations, based on the recent import of methodology from the theory of programming languages into proof theory. This scientific essay, written for the audience of proof theorists as well as the working mathematician, is not a survey of the field, but rather a personal view of the author who hopes that it may inspire future and fellow researchers

Isomorphisms of types in the presence of higher-order references (extended version)

We investigate the problem of type isomorphisms in the presence of higher-order references. We first introduce a finitary programming language with sum types and higher-order references, for which we build a fully abstract games model following the work of Abramsky, Honda and McCusker. Solving an open problem by Laurent, we show that two finitely branching arenas are isomorphic if and only if they are geometrically the same, up to renaming of moves (Laurent's forest isomorphism). We deduce from this an equational theory characterizing isomorphisms of types in our language. We show however that Laurent's conjecture does not hold on infinitely branching arenas, yielding new non-trivial type isomorphisms in a variant of our language with natural numbers

Isomorphisms of types in the presence of higher-order references

We investigate the problem of type isomorphisms in a programming language with higher-order references. We first recall the game-theoretic model of higher-order references by Abramsky, Honda and McCusker. Solving an open problem by Laurent, we show that two finitely branching arenas are isomorphic if and only if they are geometrically the same, up to renaming of moves (Laurent's forest isomorphism). We deduce from this an equational theory characterizing isomorphisms of types in a finitary language with higher order references. We show however that Laurent's conjecture does not hold on infinitely branching arenas, yielding a non-trivial type isomorphism in the extension of this language with natural numbers.Comment: Twenty-Sixth Annual IEEE Symposium on Logic In Computer Science (LICS 2011), Toronto : Canada (2011

Call-By-Push-Value

Submitted for the degree of Doctor of PhilosophyCall-by-push-value (CBPV) is a new programming language paradigm, based on the slogan “a value is, a computation does”. We claim that CBPV provides the semantic primitives from which the call-by-value and call-by-name paradigms are built. The primary goal of the thesis is to present the evidence for this claim, which is found in a remarkably wide range of semantics: from operational semantics, in big-step form and in machine form, to denotational models using domains, possible worlds, continuations and games. In the first part of the thesis, we come to CBPV and its equational theory by looking critically at the call-by-value and call-by-name paradigms in the presence of general computational effects. We give a Felleisen/Friedman-style CK-machine semantics, which explains how CBPV can be understood in terms of push/pop instructions. In the second part we give simple CBPV models for printing, divergence, global store, errors, erratic choice and control effects, as well as for various combinations of these effects. We develop the store model into a possible world model for cell generation, and (following Steele) we develop the control model into a “jumping implementation” using a continuation language called Jump-With-Argument (JWA). We present a pointer game model for CBPV, in the style of Hyland and Ong. We see that the game concepts of questioning and answering correspond to the CBPV concepts of forcing and producing respectively. We observe that this game semantics is closely related to the jumping implementation. In the third part of the thesis, we study the categorical semantics for the CBPV equational theory. We present and compare 3 approaches: models using strong monads, in the style of Moggi; models using value/producer structures, in the style of Power and Robinson; models using (strong) adjunctions. All the concrete models in the thesis are seen to be adjunction models

Type-Theoretic Signatures for Algebraic Theories and Inductive Types

We develop the usage of certain type theories as specification languages for algebraic theories and inductive types. We observe that the expressive power of dependent type theories proves useful in the specification of more complicated algebraic theories. We describe syntax and semantics for three classes of algebraic theories: finitary quotient inductive-inductive theories, their infinitary generalization, and finally higher inductive-inductive theories. In each case, an algebraic signature is a typing context or a closed type in a specific type theory

Parametricity and Local Variables

We propose that the phenomenon of local state may be understood in terms of Strachey\u27s concept of parametric (i.e., uniform) polymorphism. The intuitive basis for our proposal is the following analogy: a non-local procedure is independent of locally-declared variables in the same way that a parametrically polymorphic function is independent of types to which it is instantiated. A connection between parametricity and representational abstraction was first suggested by J. C. Reynolds. Reynolds used logical relations to formalize this connection in languages with type variables and user-defined types. We use relational parametricity to construct a model for an Algol-like language in which interactions between local and non-local entities satisfy certain relational criteria. Reasoning about local variables essentially involves proving properties of polymorphic functions. The new model supports straightforward validations of all the test equivalences that have been proposed in the literature for local-variable semantics, and encompasses standard methods of reasoning about data representations. It is not known whether our techniques yield fully abstract semantics. A model based on partial equivalence relations on the natural numbers is also briefly examined

Relational Parametricity and Control

We study the equational theory of Parigot's second-order λμ-calculus in connection with a call-by-name continuation-passing style (CPS) translation into a fragment of the second-order λ-calculus. It is observed that the relational parametricity on the target calculus induces a natural notion of equivalence on the λμ-terms. On the other hand, the unconstrained relational parametricity on the λμ-calculus turns out to be inconsistent with this CPS semantics. Following these facts, we propose to formulate the relational parametricity on the λμ-calculus in a constrained way, which might be called ``focal parametricity''.Comment: 22 pages, for Logical Methods in Computer Scienc