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

Diller-Nahm Bar Recursion

International audienceWe present a generalization of Spector's bar recursion to the Diller-Nahm variant of Gödel's Dialectica interpretation. This generalized bar recursion collects witnesses of universal formulas in sets of approximation sequences to provide an interpretation to the double-negation shift principle. The interpretation is presented in a fully computational way, implementing sets via lists. We also present a demand-driven version of this extended bar recursion manipulating partial sequences rather than initial segments. We explain why in a Diller-Nahm context there seems to be several versions of this demand-driven bar recursion, but no canonical one

Stratified Negation in Limit Datalog Programs

There has recently been an increasing interest in declarative data analysis, where analytic tasks are specified using a logical language, and their implementation and optimisation are delegated to a general-purpose query engine. Existing declarative languages for data analysis can be formalised as variants of logic programming equipped with arithmetic function symbols and/or aggregation, and are typically undecidable. In prior work, the language of $\mathit{limit\ programs}$ was proposed, which is sufficiently powerful to capture many analysis tasks and has decidable entailment problem. Rules in this language, however, do not allow for negation. In this paper, we study an extension of limit programs with stratified negation-as-failure. We show that the additional expressive power makes reasoning computationally more demanding, and provide tight data complexity bounds. We also identify a fragment with tractable data complexity and sufficient expressivity to capture many relevant tasks.Comment: 14 pages; full version of a paper accepted at IJCAI-1

A Logical Foundation for Environment Classifiers

Taha and Nielsen have developed a multi-stage calculus {\lambda}{\alpha} with a sound type system using the notion of environment classifiers. They are special identifiers, with which code fragments and variable declarations are annotated, and their scoping mechanism is used to ensure statically that certain code fragments are closed and safely runnable. In this paper, we investigate the Curry-Howard isomorphism for environment classifiers by developing a typed {\lambda}-calculus {\lambda}|>. It corresponds to multi-modal logic that allows quantification by transition variables---a counterpart of classifiers---which range over (possibly empty) sequences of labeled transitions between possible worlds. This interpretation will reduce the "run" construct---which has a special typing rule in {\lambda}{\alpha}---and embedding of closed code into other code fragments of different stages---which would be only realized by the cross-stage persistence operator in {\lambda}{\alpha}---to merely a special case of classifier application. {\lambda}|> enjoys not only basic properties including subject reduction, confluence, and strong normalization but also an important property as a multi-stage calculus: time-ordered normalization of full reduction. Then, we develop a big-step evaluation semantics for an ML-like language based on {\lambda}|> with its type system and prove that the evaluation of a well-typed {\lambda}|> program is properly staged. We also identify a fragment of the language, where erasure evaluation is possible. Finally, we show that the proof system augmented with a classical axiom is sound and complete with respect to a Kripke semantics of the logic

The strength of countable saturation

We determine the proof-theoretic strength of the principle of countable saturation in the context of the systems for nonstandard arithmetic introduced in our earlier work.Comment: Corrected typos in Lemma 3.4 and the final paragraph of the conclusio

Mathematical Logic: Proof theory, Constructive Mathematics

The workshop “Mathematical Logic: Proof Theory, Constructive Mathematics” was centered around proof-theoretic aspects of current mathematics, constructive mathematics and logical aspects of computational complexit

The Computer Modelling of Mathematical Reasoning

xv, 403 p.; 23 cm