On the strength of dependent products in the type theory of Martin-L\"of

One may formulate the dependent product types of Martin-L\"of type theory either in terms of abstraction and application operators like those for the lambda-calculus; or in terms of introduction and elimination rules like those for the other constructors of type theory. It is known that the latter rules are at least as strong as the former: we show that they are in fact strictly stronger. We also show, in the presence of the identity types, that the elimination rule for dependent products--which is a "higher-order" inference rule in the sense of Schroeder-Heister--can be reformulated in a first-order manner. Finally, we consider the principle of function extensionality in type theory, which asserts that two elements of a dependent product type which are pointwise propositionally equal, are themselves propositionally equal. We demonstrate that the usual formulation of this principle fails to verify a number of very natural propositional equalities; and suggest an alternative formulation which rectifies this deficiency.Comment: 18 pages; v2: final journal versio

A tutorial on implementing De Morgan cubical type theory

This tutorial explains (one way) how to implement De Morgan cubical type theory to people who know how to implement a dependent type theory. It contains an introduction to basic concepts of cubes, type checking algorithms under a cofibration, the idea of "transportation rules" and cubical operations. This tutorial is a by-product of an experimental implementation of cubical type theory, called Guest0x0.Comment: 27 page

A Cubical Implementation of Homotopical Patch Theory

We consider theoretical models of version control systems based on Homotopy Type Theory (HoTT). The main contribution is an implementation of Angiuli et al.’s Homotopical Patch Theory in Cubical Agda. Additionally the first chapter contains an approachable introduction to HoTT and Cubical Agda aimed at an audience of interested computer science students covering dependent Martin-Löf-style type theory, propositions as types, univalent foundations, higher inductive types and CCHM cubical type theory. Finally, we discuss some other approaches to a theory of version control systems in Darcs’ “algebra of patches” and an unsuccessful attempt to model repositories in type theory as coequalizers.Master's Thesis in InformaticsINF399MAMN-INFMAMN-PRO

Introduction to Q-tensor theory

This paper aims to provide an introduction to a basic form of the ${\bf Q}$-tensor approach to modelling liquid crystals, which has seen increased interest in recent years. The increase in interest in this type of modelling approach has been driven by investigations into the fundamental nature of defects and new applications of liquid crystals such as bistable displays and colloidal systems for which a description of defects and disorder is essential. The work in this paper is not new research, rather it is an introductory guide for anyone wishing to model a system using such a theory. A more complete mathematical description of this theory, including a description of flow effects, can be found in numerous sources but the books by Virga and Sonnet and Virga are recommended. More information can be obtained from the plethora of papers using such approaches, although a general introduction for the novice is lacking. The first few sections of this paper will detail the development of the ${\bf Q}$-tensor approach for nematic liquid crystalline systems and construct the free energy and governing equations for the mesoscopic dependent variables. A number of device surface treatments are considered and theoretical boundary conditions are specified for each instance. Finally, an example of a real device is demonstrated

Multi-kink topological terms and charge-binding domain-wall condensation induced symmetry-protected topological states: Beyond Chern-Simons/BF theory

Quantum-disordering a discrete-symmetry breaking state by condensing domain-walls can lead to a trivial symmetric insulator state. In this work, we show that if we bind a 1D representation of the symmetry (such as a charge) to the intersection point of several domain walls, condensing such modified domain-walls can lead to a non-trivial symmetry-protected topological (SPT) state. This result is obtained by showing that the modified domain-wall condensed state has a non-trivial SPT invariant -- the symmetry-twist dependent partition function. We propose two different kinds of field theories that can describe the above mentioned SPT states. The first one is a Ginzburg-Landau-type non-linear sigma model theory, but with an additional multi-kink domain-wall topological term. Such theory has an anomalous $U^k(1)$ symmetry but an anomaly-free $Z_N^k$ symmetry. The second one is a gauge theory, which is beyond Abelian Chern-Simons/BF gauge theories. We argue that the two field theories are equivalent at low energies. After coupling to the symmetry twists, both theories produce the desired SPT invariant.Comment: 22 pages, 8 figures, symmetry transformaition for the Ginzburg-Landau NL$\sigma$M is added in the introduction part, references adde

Categorical structures for deduction

We begin by introducing categorized judgemental theories and their calculi as a general framework to present and study deductive systems. As an exemplification of their expressivity, we approach dependent type theory and first-order logic as special kinds of categorized judgemental theories. We believe our analysis sheds light on both the topics, providing a new point of view. In the case of type theory, we provide an abstract definition of type constructor featuring the usual formation, introduction, elimination and computation rules. For first-order logic we offer a deep analysis of structural rules, describing some of their properties, and putting them into context. We then put one of the main constructions introduced, namely that of categorized judgemental dependent type theories, to the test: we frame it in the general context of categorical models for dependent types, describe a few examples, study its properties, and use it to model subtyping and as a tool to prove intrinsic properties hidden in other models. Somehow orthogonally, then, we show a different side as to how categories can help the study of deductive systems: we transport a known model from set-based categories to enriched categories, and use the information naturally encoded into it to describe a theory of fuzzy types. We recover structural rules, observe new phenomena, and study different possible enrichments and their interpretation. We open the discussion to include different takes on the topic of definitional equality

On Gaugino Condensation in String Theory

openWe derive the precise form of the low-energy four-dimensional EFT for type IIB string theory compactified on the complex cone over Kähler-Einstein del Pezzo surfaces, including N spacetime-filling D3-branes and assuming Minkowski externally. We explicitly derive the theory for the K\"ahler modulus in the simplest case of a complex cone over the complex projective plane, with a stack of four D7-branes and one O7-plane wrapped around the base of the cone. An effective scalar potential appears in the theory, due to gaugino condensation taking place at low energies over the D7-branes stack, exhibiting a runaway direction and an unstable de Sitter vacuum. We find an explicit cosmological-like solution for the K\"ahler modulus, showing that the warped volume of the internal complex projective plane inflates with time in a runaway fashion. We conclude that type IIB string theory compactified on the complex cone over the complex projective plane, with four D7-branes and one O7-plane wrapped around the complex projective plane, is unstable. We explicitly derive the ten-dimensional equations of motion for maximally symmetric time-dependent metric perturbations by means of an ad hoc procedure, and we exhibit both stationary and time-dependent solutions, whose boundary conditions are imposed in part by the gaugino condensate stress-energy tensor. We partially fix the free parameters of the time-dependent solution using the results from the four-dimensional low-energy EFT. This thesis contains also an introduction to string compactifications, to the KKLT scenario and to the literature about ten-dimensional effects of gaugino condensation.We derive the precise form of the low-energy four-dimensional EFT for type IIB string theory compactified on the complex cone over Kähler-Einstein del Pezzo surfaces, including N spacetime-filling D3-branes and assuming Minkowski externally. We explicitly derive the theory for the K\"ahler modulus in the simplest case of a complex cone over the complex projective plane, with a stack of four D7-branes and one O7-plane wrapped around the base of the cone. An effective scalar potential appears in the theory, due to gaugino condensation taking place at low energies over the D7-branes stack, exhibiting a runaway direction and an unstable de Sitter vacuum. We find an explicit cosmological-like solution for the K\"ahler modulus, showing that the warped volume of the internal complex projective plane inflates with time in a runaway fashion. We conclude that type IIB string theory compactified on the complex cone over the complex projective plane, with four D7-branes and one O7-plane wrapped around the complex projective plane, is unstable. We explicitly derive the ten-dimensional equations of motion for maximally symmetric time-dependent metric perturbations by means of an ad hoc procedure, and we exhibit both stationary and time-dependent solutions, whose boundary conditions are imposed in part by the gaugino condensate stress-energy tensor. We partially fix the free parameters of the time-dependent solution using the results from the four-dimensional low-energy EFT. This thesis contains also an introduction to string compactifications, to the KKLT scenario and to the literature about ten-dimensional effects of gaugino condensation

From Hilbert proofs to consecutions and back

Restall set forth a "consecution" calculus in his An Introduction to Substructural Logics. This is a natural deduction type sequent calculus where the structural rules play an important role.  This paper looks at different ways of extending Restall's calculus. It is shown that Restall's weak soundness and completeness result with regards to a Hilbert calculus can be extended to a strong one so as to encompass what Restall calls proofs from assumptions. It is also shown how to extend the calculus so as to validate the metainferential rule of reasoning by cases, as well as certain theory-dependent rules