Reducibility has been used to prove a number of properties in the λ-calculus and is well known to offer on one hand very general proofs which can be applied to a number of instantiations, and on the other hand, to be quite mysterious and inflexible. In this paper, we look at two related but different results in λ-calculi with intersection types. We show that one such result (which aims at givin...

We introduce the polynomial-time tree reducibility (ptt-reducibility). Our main result states that for languages B and C it holds that B ptt-reduces to C if and only if the unbalanced leaf-language class of B is robustly contained in the unbalanced leaf-language class of C. This is the unbalanced analogue of the well-known result by Bovet, Crescenzi, Silvestri, and Vereshchagin which connects p...

In this note we determine the values of parameters c for which the polynomial representation of the degenerate double affine Hecke algebra (DAHA), i.e. the trigonometric Cherednik algebra, is reducible. Namely, we show that c is a reducibility point for the polynomial representation of the trigonometric Cherednik algebra for a root system R if and only if it is a reducibility point for the rati...

Frege’s Grundgesetze der Arithmetik is formally inconsistent. This system is, except for minor differences, second-order logic together with an abstraction operator governed by Frege’s Axiom V. A few years ago, Richard Heck showed that the ramified predicative second-order fragment of the Grundgesetze is consistent. In this paper, we show that the above fragment augmented with the axiom of redu...

In this work we investigate the hardness of a computational problem introduced in the recent work of Baumslag et al. [BFNSS11]. In particular, we study the Bn-LHN problem, which is a generalized version of the learning with errors (LWE) problem, instantiated with a particular family of non-abelian groups (free Burnside groups of exponent 3). In our main result, we demonstrate a random self-redu...

We say that a theory T is intermediate under effective reducibility if the isomorphism problems among its computable models is neither hyperarithmetic nor on top under effective reducibility. We prove that if an infinitary sentence T is uniformly effectively dense, a property we define in the paper, then no extension of it is intermediate, at least when relativized to every oracle on a cone. As...