A dependent nominal type theory

Nominal abstract syntax is an approach to representing names and binding pioneered by Gabbay and Pitts. So far nominal techniques have mostly been studied using classical logic or model theory, not type theory. Nominal extensions to simple, dependent and ML-like polymorphic languages have been studied, but decidability and normalization results have only been established for simple nominal type theories. We present a LF-style dependent type theory extended with name-abstraction types, prove soundness and decidability of beta-eta-equivalence checking, discuss adequacy and canonical forms via an example, and discuss extensions such as dependently-typed recursion and induction principles

A Simple Nominal Type Theory

Abstract. Nominal logic is an extension of first-order logic with features useful for reasoning about abstract syntax with bound names. For computational applications such as programming and formal reasoning, it is desirable to develop constructive type theories for nominal logic which extend standard type theories for propositional, first- or higher-order logic. This has proven difficult, largely because of complex interactions between nominal logic’s name-abstraction operation and ordinary functional abstraction. This difficulty already arises in the case of propositional logic and simple type theory. In this paper we show how this difficulty can be overcome, and present a simple nominal type theory which enjoys properties such as type soundness and strong normalization, and which can be soundly interpreted using existing nominal set models of nominal logic. We also sketch how recursion combinators for languages with binding structure can be provided. This is an important first step towards understanding the constructive content of nominal logic and incorporating it into existing logics and type theories.

Escapist policy rules

We study a simple, microfounded macroeconomic system in which the monetary authority employs a Taylor-type policy rule. We analyze situations in which the self-confirming equilibrium is unique and learnable according to Bullard and Mitra (2002). We explore the prospects for the use of 'large deviation' theory in this context, as employed by Sargent (1999) and Cho, Williams, and Sargent (2002). We show that our system can sometimes depart from the self-confirming equilibrium towards a non-equilibrium outcome characterized by persistently low nominal interest rates and persistently low inflation. Thus we generate events that have some of the properties of "liquidity traps" observed in the data, even though the policymaker remains committed to a Taylor-type policy rule which otherwise has desirable stabilization properties

Escapist Policy Rules

We study a simple, microfounded macroeconomic system in which the monetary authority employs a Taylor-type policy rule. We analyze situations in which the self-confirming equilibrium is unique and learnable according to Bullard and Mitra (2002). We explore the prospects for the use of ‘large deviation’ theory in this context, as employed by Sargent (1999) and Cho, Williams, and Sargent (2002). We show that our system can sometimes depart from the self-confirming equilibrium towards a non-equilibrium outcome characterized by persistently low nominal interest rates and persistently low inflation. Thus we generate events that have some of the properties of “liquidity traps” observed in the data, even though the policymaker remains committed to a Taylor-type policy rule which otherwise has desirable stabilization properties.Learning, monetary policy rules, escape dynamics

Escapist policy rules

We study a simple, microfounded macroeconomic system in which the monetary authority employs a Taylor-type policy rule. We analyze situations in which the self-confirming equilibrium is unique and learnable according to Bullard and Mitra (2002). We explore the prospects for the use of ‘large deviation’ theory in this context, as employed by Sargent (1999) and Cho, Williams, and Sargent (2002). We show that our system can sometimes depart from the self-confirming equilibrium towards a non-equilibrium outcome characterized by persistently low nominal interest rates and persistently low in- flation. Thus we generate events that have some of the properties of “liquidity traps” observed in the data, even though the policymaker remains committed to a Taylor-type policy rule which otherwise has desirable stabilization properties.Monetary policy ; Inflation (Finance) ; Interest rates

Resonance-state-induced superconductivity at high Indium contents in In-doped SnTe

We report a reinvestigation of superconducting Sn$_{1-x}$In$_{x}$Te at both low and high In doping levels. Analysis of the superconductivity reveals a fundamental change as a function of \textit{x}: the system evolves from a weakly coupled to a strongly coupled superconductor with increasing indium content. Hall Effect measurements further show that the carrier density does not vary linearly with Indium content; indeed at high Indium content, the samples are overall \textit{n}-type, which is contrary to expectations of the standard picture of In$^{1+}$ replacing Sn$^{2+}$ in this material. Density functional theory calculations probing the electronic state of In in SnTe show that it does not act as a trivial hole dopant, but instead forms a distinct, partly filled In 5\textit{s} - Te 5\textit{p} hybridized state centered around E$_F$, very different from what is seen for other nominal hole dopants such as Na, Ag, and vacant Sn sites. We conclude that superconducting In-doped SnTe therefore cannot be considered as a simple hole doped semiconductor.Comment: 12 pages and 7 figure

A Normalizing Computation Rule for Propositional Extensionality in Higher-Order Minimal Logic

The univalence axiom expresses the principle of extensionality for dependent type theory. However, if we simply add the univalence axiom to type theory, then we lose the property of canonicity - that every closed term computes to a canonical form. A computation becomes "stuck" when it reaches the point that it needs to evaluate a proof term that is an application of the univalence axiom. So we wish to find a way to compute with the univalence axiom. While this problem has been solved with the formulation of cubical type theory, where the computations are expressed using a nominal extension of lambda-calculus, it may be interesting to explore alternative solutions, which do not require such an extension. As a first step, we present here a system of propositional higher-order minimal logic (PHOML). There are three kinds of typing judgement in PHOML. There are terms which inhabit types, which are the simple types over Omega. There are proofs which inhabit propositions, which are the terms of type Omega. The canonical propositions are those constructed from false by implication. Thirdly, there are paths which inhabit equations M =_A N, where M and N are terms of type A. There are two ways to prove an equality: reflexivity, and propositional extensionality - logically equivalent propositions are equal. This system allows for some definitional equalities that are not present in cubical type theory, namely that transport along the trivial path is identity. We present a call-by-name reduction relation for this system, and prove that the system satisfies canonicity: every closed typable term head-reduces to a canonical form. This work has been formalised in Agda

Adventures in Formalisation: Financial Contracts, Modules, and Two-Level Type Theory

We present three projects concerned with applications of proof assistants in the area of programming language theory and mathematics. The first project is about a certified compilation technique for a domain-specific programming language for financial contracts (the CL language). The code in CL is translated into a simple expression language well-suited for integration with software components implementing Monte Carlo simulation techniques (pricing engines). The compilation procedure is accompanied with formal proofs of correctness carried out in Coq. The second project presents techniques that allow for formal reasoning with nested and mutually inductive structures built up from finite maps and sets. The techniques, which build on the theory of nominal sets combined with the ability to work with isomorphic representations of finite maps, make it possible to give a formal treatment, in Coq, of a higher-order module system, including the ability to eliminate at compile time abstraction barriers introduced by the module system. The development is based on earlier work on static interpretation of modules and provides the foundation for a higher-order module language for Futhark, an optimising compiler targeting data-parallel architectures. The third project presents an implementation of two-level type theory, a version of Martin-Lof type theory with two equality types: the first acts as the usual equality of homotopy type theory, while the second allows us to reason about strict equality. In this system, we can formalise results of partially meta-theoretic nature. We develop and explore in details how two-level type theory can be implemented in a proof assistant, providing a prototype implementation in the proof assistant Lean. We demonstrate an application of two-level type theory by developing some results on the theory of inverse diagrams using our Lean implementation.Comment: PhD thesis defended in January 2018 at University of Copenhagen, Department of Computer Scienc

Nominal Abstraction

Recursive relational specifications are commonly used to describe the computational structure of formal systems. Recent research in proof theory has identified two features that facilitate direct, logic-based reasoning about such descriptions: the interpretation of atomic judgments through recursive definitions and an encoding of binding constructs via generic judgments. However, logics encompassing these two features do not currently allow for the definition of relations that embody dynamic aspects related to binding, a capability needed in many reasoning tasks. We propose a new relation between terms called nominal abstraction as a means for overcoming this deficiency. We incorporate nominal abstraction into a rich logic also including definitions, generic quantification, induction, and co-induction that we then prove to be consistent. We present examples to show that this logic can provide elegant treatments of binding contexts that appear in many proofs, such as those establishing properties of typing calculi and of arbitrarily cascading substitutions that play a role in reducibility arguments.Comment: To appear in the Journal of Information and Computatio