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

Axioms and Decidability for Type Isomorphism in the Presence of Sums

We consider the problem of characterizing isomorphisms of types, or, equivalently, constructive cardinality of sets, in the simultaneous presence of disjoint unions, Cartesian products, and exponentials. Mostly relying on results about polynomials with exponentiation that have not been used in our context, we derive: that the usual finite axiomatization known as High-School Identities (HSI) is complete for a significant subclass of types; that it is decidable for that subclass when two types are isomorphic; that, for the whole of the set of types, a recursive extension of the axioms of HSI exists that is complete; and that, for the whole of the set of types, the question as to whether two types are isomorphic is decidable when base types are to be interpreted as finite sets. We also point out certain related open problems

An interpretation of the Sigma-2 fragment of classical Analysis in System T

We show that it is possible to define a realizability interpretation for the $\Sigma_2$-fragment of classical Analysis using G\"odel's System T only. This supplements a previous result of Schwichtenberg regarding bar recursion at types 0 and 1 by showing how to avoid using bar recursion altogether. Our result is proved via a conservative extension of System T with an operator for composable continuations from the theory of programming languages due to Danvy and Filinski. The fragment of Analysis is therefore essentially constructive, even in presence of the full Axiom of Choice schema: Weak Church's Rule holds of it in spite of the fact that it is strong enough to refute the formal arithmetical version of Church's Thesis

A Direct Version of Veldman's Proof of Open Induction on Cantor Space via Delimited Control Operators

First, we reconstruct Wim Veldman's result that Open Induction on Cantor space can be derived from Double-negation Shift and Markov's Principle. In doing this, we notice that one has to use a countable choice axiom in the proof and that Markov's Principle is replaceable by slightly strengthening the Double-negation Shift schema. We show that this strengthened version of Double-negation Shift can nonetheless be derived in a constructive intermediate logic based on delimited control operators, extended with axioms for higher-type Heyting Arithmetic. We formalize the argument and thus obtain a proof term that directly derives Open Induction on Cantor space by the shift and reset delimited control operators of Danvy and Filinski

FLASH: ultra-fast protocol to identify RNA-protein interactions in cells

Determination of the in vivo binding sites of RNA-binding proteins (RBPs) is paramount to understanding their function and how they affect different aspects of gene regulation. With hundreds of RNA-binding proteins identified in human cells, a flexible, high-resolution, high-throughput, highly multiplexible and radioactivity-free method to determine their binding sites has not been described to date. Here we report FLASH (Fast Ligation of RNA after some sort of Affinity Purification for High-throughput Sequencing), which uses a special adapter design and an optimized protocol to determine protein-RNA interactions in living cells. The entire FLASH protocol, starting from cells on plates to a sequencing library, takes 1.5 days. We demonstrate the flexibility, speed and versatility of FLASH by using it to determine RNA targets of both tagged and endogenously expressed proteins under diverse conditions in vivo

Classical polarizations yield double-negation translations

Double-negation translations map formulas to formulas in such a way that if a formula is a classical theorem then its translation is an intuitionistic theorem. We shall go beyond just examining provability by looking at correspondences between inference rules in classical proofs and in intuitionistic proofs of translated formulas. In order to make this comparison interesting and precise, we will examine focused versions of proofs in classical and intuitionistic logics using the LKF and LJF proof systems. We shall show that for a number of known double-negation translations, one can get essentially identical (focused) intuitionistic proofs as (focused) classical proofs. Thus the choice of a common double-negation translation is really the same choice as a polarization of classical logic (of which there are many)

Colour change of twig-mimicking peppered moth larvae is a continuous reaction norm that increases camouflage against avian predators

Camouflage, and in particular background-matching, is one of the most commonanti-predator strategies observed in nature. Animals can improve their match to thecolour/pattern of their surroundings through background selection, and/or by plasticcolour change. Colour change can occur rapidly (a few seconds), or it may be slow,taking hours to days. Many studies have explored the cues and mechanisms behindrapid colour change, but there is a considerable lack of information about slow colourchange in the context of predation: the cues that initiate it, and the range of phenotypesthat are produced. Here we show that peppered moth (Biston betularia) larvae respondto colour and luminance of the twigs they rest on, and exhibit a continuous reactionnorm of phenotypes. When presented with a heterogeneous environment of mixed twigcolours, individual larvae specialise crypsis towards one colour rather than developingan intermediate colour. Flexible colour change in this species has likely evolved inassociation with wind dispersal and polyphagy, which result in caterpillars settling andfeeding in a diverse range of visual environments. This is the first example of visuallyinduced slow colour change in Lepidoptera that has been objectively quantified andmeasured from the visual perspective of natural predators

Ecumenical modal logic

The discussion about how to put together Gentzen's systems for classical and intuitionistic logic in a single unified system is back in fashion. Indeed, recently Prawitz and others have been discussing the so called Ecumenical Systems, where connectives from these logics can co-exist in peace. In Prawitz' system, the classical logician and the intuitionistic logician would share the universal quantifier, conjunction, negation, and the constant for the absurd, but they would each have their own existential quantifier, disjunction, and implication, with different meanings. Prawitz' main idea is that these different meanings are given by a semantical framework that can be accepted by both parties. In a recent work, Ecumenical sequent calculi and a nested system were presented, and some very interesting proof theoretical properties of the systems were established. In this work we extend Prawitz' Ecumenical idea to alethic K-modalities