The Definitional Side of the Forcing

International audienceThis paper studies forcing translations of proofs in dependent type theory, through the Curry-Howard correspondence. Based on a call-by-push-value decomposition, we synthesize two simply-typed translations: i) one call-by-value, corresponding to the translation derived from the presheaf construction as studied in a previous paper ; ii) one call-by-name, whose intuitions already appear in Kriv-ine and Miquel's work. Focusing on the call-by-name translation, we adapt it to the dependent case and prove that it is compatible with the definitional equality of our system, thus avoiding coherence problems. This allows us to use any category as forcing conditions , which is out of reach with the call-by-value translation. Our construction also exploits the notion of storage operators in order to interpret dependent elimination for inductive types. This is a novel example of a dependent theory with side-effects, clarifying how dependent elimination for inductive types must be restricted in a non-pure setting. Being implemented as a Coq plugin, this work gives the possibility to formalize easily consistency results, for instance the consistency of the negation of Voevodsky's univalence axiom

Controlling iterated jumps of solutions to combinatorial problems

Among the Ramsey-type hierarchies, namely, Ramsey's theorem, the free set, the thin set and the rainbow Ramsey theorem, only Ramsey's theorem is known to collapse in reverse mathematics. A promising approach to show the strictness of the hierarchies would be to prove that every computable instance at level n has a low_n solution. In particular, this requires effective control of iterations of the Turing jump. In this paper, we design some variants of Mathias forcing to construct solutions to cohesiveness, the Erdos-Moser theorem and stable Ramsey's theorem for pairs, while controlling their iterated jumps. For this, we define forcing relations which, unlike Mathias forcing, have the same definitional complexity as the formulas they force. This analysis enables us to answer two questions of Wei Wang, namely, whether cohesiveness and the Erdos-Moser theorem admit preservation of the arithmetic hierarchy, and can be seen as a step towards the resolution of the strictness of the Ramsey-type hierarchies.Comment: 32 page

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

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

The weakness of the pigeonhole principle under hyperarithmetical reductions

The infinite pigeonhole principle for 2-partitions ($\mathsf{RT}^1_2$) asserts the existence, for every set $A$, of an infinite subset of $A$ or of its complement. In this paper, we study the infinite pigeonhole principle from a computability-theoretic viewpoint. We prove in particular that $\mathsf{RT}^1_2$ admits strong cone avoidance for arithmetical and hyperarithmetical reductions. We also prove the existence, for every $\Delta^0_n$ set, of an infinite low${}_n$ subset of it or its complement. This answers a question of Wang. For this, we design a new notion of forcing which generalizes the first and second-jump control of Cholak, Jockusch and Slaman.Comment: 29 page

Instrumentalizing Jurors: An Argument Against the Fourth Amendment Exclusionary Rule

In this symposium contribution, I contend that the application of the Fourth Amendment exclusionary rule in cases tried by juries raises troubling moral issues that are not present when a judge adjudicates a case on his or her own. Specifically, I argue that the exclusionary rule infringes upon jurors’ deliberative autonomy by depriving them of available evidence that rationally bears upon their verdict and by instrumentalizing them in service to the Court’s deterrence objectives. After considering ways in which those moral problems could be at least partially mitigated, I contend that the best approach might be to abandon the exclusionary rule entirely. I suggest that the Supreme Court might already be willing to abandon the rule, provided that Congress enacts reforms aimed at making the threat of financial liability for Fourth Amendment violations more robust. I close by identifying several ways in which Congress could help pave the way for the exclusionary rule’s demise

Defining 'Speech': Subtraction, Addition, and Division

In free speech theory ‘speech’ has to be defined as a special term of art. I argue that much free speech discourse comes with a tacit commitment to a ‘Subtractive Approach’ to defining speech. As an initial default, all communicative acts are assumed to qualify as speech, before exceptions are made to ‘subtract’ those acts that don’t warrant the special legal protections owed to ‘speech’. I examine how different versions of the Subtractive Approach operate, and criticise them in terms of their ability to yield a substantive definition of speech which covers all and only those forms of communicative action that – so our arguments for free speech indicate – really do merit special legal protection. In exploring alternative definitional approaches, I argue that what ultimately compromises definitional adequacy in this arena is a theoretical commitment to the significance of a single unified class of privileged communicative acts. I then propose an approach to free speech theory that eschews this theoretical commitment