A formalized general theory of syntax with bindings

We present the formalization of a theory of syntax with bindings that has been developed and refined over the last decade to support several large formalization efforts. Terms are defined for an arbitrary number of constructors of varying numbers of inputs, quotiented to alpha-equivalence and sorted according to a binding signature. The theory includes a rich collection of properties of the standard operators on terms, such as substitution and freshness. It also includes induction and recursion principles and support for semantic interpretation, all tailored for smooth interaction with the bindings and the standard operators

Subnormalizers and solvability in finite groups

For a finite group $G$, we study the probability $sp(G)$ that, given two elements $x,y \in G$, the cyclic subgroup $\langle x \rangle$ is subnormal in the subgroup $\langle x, y \rangle$. This can be seen as an intermediate invariant between the probability that two elements generate a nilpotent subgroup and the probability that two elements generate a solvable subgroup. We prove that $sp(G) \leq 1/6$ for every nonsolvable group $G$

Bindings as bounded natural functors

We present a general framework for specifying and reasoning about syntax with bindings. Abstract binder types are modeled using a universe of functors on sets, subject to a number of operations that can be used to construct complex binding patterns and binding-aware datatypes, including non-well-founded and infinitely branching types, in a modular fashion. Despite not committing to any syntactic format, the framework is “concrete” enough to provide definitions of the fundamental operators on terms (free variables, alpha-equivalence, and capture-avoiding substitution) and reasoning and definition principles. This work is compatible with classical higher-order logic and has been formalized in the proof assistant Isabelle/HOL

Synaesthetic colours do not camouflage form in visual search

One of the major issues in synaesthesia research is to identify the level of processing involved in the formation of the subjective colours experienced by synaesthetes: are they perceptual phenomena or are they due to memory and association learning? To address this question, we tested whether the colours reported by a group of grapheme-colour synaesthetes (previously studied in an functional magnetic resonance imaging experiment) influenced them in a visual search task. As well as using a condition where synaesthetic colours should have aided visual search, we introduced a condition where the colours experienced by synaesthetes would be expected to make them worse than controls. We found no evidence for differences between synaesthetes and normal controls, either when colours should have helped them or where they should have hindered. We conclude that the colours reported by our population of synaesthetes are not equivalent to perceptual signals, but arise at a cognitive level where they are unable to affect visual search

‘Paper Seas’. Polar Adventures in the Contemporary German Novel by Sten Nadolny and Christoph Ransmayr

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