# Volume 16, Issue 1

2020

### 1. On the logical complexity of cyclic arithmetic

We study the logical complexity of proofs in cyclic arithmetic ($\mathsf{CA}$), as introduced in Simpson '17, in terms of quantifier alternations of formulae occurring. Writing $C\Sigma_n$ for (the logical consequences of) cyclic proofs containing only $\Sigma_n$ formulae, our main result is that $I\Sigma_{n+1}$ and $C\Sigma_n$ prove the same $\Pi_{n+1}$ theorems, for all $n\geq 0$. Furthermore, due to the 'uniformity' of our method, we also show that $\mathsf{CA}$ and Peano Arithmetic ($\mathsf{PA}$) proofs of the same theorem differ only exponentially in size. The inclusion $I\Sigma_{n+1} \subseteq C\Sigma_n$ is obtained by proof theoretic techniques, relying on normal forms and structural manipulations of $\mathsf{PA}$ proofs. It improves upon the natural result that $I\Sigma_n$ is contained in $C\Sigma_n$. The converse inclusion, $C\Sigma_n \subseteq I\Sigma_{n+1}$, is obtained by calibrating the approach of Simpson '17 with recent results on the reverse mathematics of Büchi's theorem in Ko{\l}odziejczyk, Michalewski, Pradic & Skrzypczak '16 (KMPS'16), and specialising to the case of cyclic proofs. These results improve upon the bounds on proof complexity and logical complexity implicit in Simpson '17 and also an alternative approach due to Berardi & Tatsuta '17. The uniformity of our method also allows us to recover a metamathematical account of fragments of $\mathsf{CA}$; in particular we show that, for $n\geq 0$, the consistency of $C\Sigma_n$ is provable […]

### 2. Modalities in homotopy type theory

Univalent homotopy type theory (HoTT) may be seen as a language for the category of $\infty$-groupoids. It is being developed as a new foundation for mathematics and as an internal language for (elementary) higher toposes. We develop the theory of factorization systems, reflective subuniverses, and modalities in homotopy type theory, including their construction using a "localization" higher inductive type. This produces in particular the ($n$-connected, $n$-truncated) factorization system as well as internal presentations of subtoposes, through lex modalities. We also develop the semantics of these constructions.

### 3. Non-idempotent types for classical calculi in natural deduction style

In the first part of this paper, we define two resource aware typing systems for the {\lambda}{\mu}-calculus based on non-idempotent intersection and union types. The non-idempotent approach provides very simple combinatorial arguments-based on decreasing measures of type derivations-to characterize head and strongly normalizing terms. Moreover, typability provides upper bounds for the lengths of the head reduction and the maximal reduction sequences to normal-form. In the second part of this paper, the {\lambda}{\mu}-calculus is refined to a small-step calculus called {\lambda}{\mu}s, which is inspired by the substitution at a distance paradigm. The {\lambda}{\mu}s-calculus turns out to be compatible with a natural extensionof the non-idempotent interpretations of {\lambda}{\mu}, i.e., {\lambda}{\mu}s-reduction preserves and decreases typing derivations in an extended appropriate typing system. We thus derive a simple arithmetical characterization of strongly {\lambda}{\mu}s-normalizing terms by means of typing.

### 4. A Categorical Reconstruction of Quantum Theory

We reconstruct finite-dimensional quantum theory from categorical principles. That is, we provide properties ensuring that a given physical theory described by a dagger compact category in which one may `discard' objects is equivalent to a generalised finite-dimensional quantum theory over a suitable ring $S$. The principles used resemble those due to Chiribella, D'Ariano and Perinotti. Unlike previous reconstructions, our axioms and proof are fully categorical in nature, in particular not requiring tomography assumptions. Specialising the result to probabilistic theories we obtain either traditional quantum theory with $S$ being the complex numbers, or that over real Hilbert spaces with $S$ being the reals.