Quotient types in type theory

Martin-Lof's intuitionistic type theory (Type Theory) is a formal system that serves not only as a foundation of constructive mathematics but also as a dependently typed programming language. Dependent types are types that depend on values of other types. Type Theory is based on the Curry-Howard isomorphism which relates computer programs with mathematical proofs so that we can do computer-aided formal reasoning and write certified programs in programming languages like Agda, Epigram etc. Martin Lof proposed two variants of Type Theory which are differentiated by the treatment of equality. In Intensional Type Theory, propositional equality defined by identity types does not imply definitional equality, and type checking is decidable. In Extensional Type Theory, propositional equality is identified with definitional equality which makes type checking undecidable. Because of the good computational properties, Intensional Type Theory is more popular, however it lacks some important extensional concepts such as functional extensionality and quotient types. This thesis is about quotient types. A quotient type is a new type whose equality is redefined by a given equivalence relation. However, in the usual formulation of Intensional Type Theory, there is no type former to create a quotient. We also lose canonicity if we add quotient types into Intensional Type Theory as axioms. In this thesis, we first investigate the expected syntax of quotient types and explain it with categorical notions. For quotients which can be represented as a setoid as well as defined as a set without a quotient type former, we propose to define an algebraic structure of quotients called definable quotients. It relates the setoid interpretation and the set definition via a normalisation function which returns a normal form (canonical choice) for each equivalence class. It can be seen as a simulation of quotient types and it helps theorem proving because we can benefit from both representations. However this approach cannot be used for all quotients. It seems that we cannot define a normalisation function for some quotients in Type Theory, e.g. Cauchy reals and finite multisets. Quotient types are indeed essential for formalisation of mathematics and reasoning of programs. Then we consider some models of Type Theory where types are interpreted as structured objects such as setoids, groupoids or weak omega-groupoids. In these models equalities are internalised into types which means that it is possible to redefine equalities. We present an implementation of Altenkirch's setoid model and show that quotient types can be defined within this model. We also describe a new extension of Martin-Lof type theory called Homotopy Type Theory where types are interpreted as weak omega-groupoids. It can be seen as a generalisation of the groupoid model which makes extensional concepts including quotient types available. We also introduce a syntactic encoding of weak omega-groupoids which can be seen as a first step towards building a weak omega-groupoids model in Intensional Type Theory. All of these implementations were performed in the dependently typed programming language Agda which is based on intensional Martin-Lof type theory

Quotient types in type theory

Martin-Lof's intuitionistic type theory (Type Theory) is a formal system that serves not only as a foundation of constructive mathematics but also as a dependently typed programming language. Dependent types are types that depend on values of other types. Type Theory is based on the Curry-Howard isomorphism which relates computer programs with mathematical proofs so that we can do computer-aided formal reasoning and write certified programs in programming languages like Agda, Epigram etc. Martin Lof proposed two variants of Type Theory which are differentiated by the treatment of equality. In Intensional Type Theory, propositional equality defined by identity types does not imply definitional equality, and type checking is decidable. In Extensional Type Theory, propositional equality is identified with definitional equality which makes type checking undecidable. Because of the good computational properties, Intensional Type Theory is more popular, however it lacks some important extensional concepts such as functional extensionality and quotient types. This thesis is about quotient types. A quotient type is a new type whose equality is redefined by a given equivalence relation. However, in the usual formulation of Intensional Type Theory, there is no type former to create a quotient. We also lose canonicity if we add quotient types into Intensional Type Theory as axioms. In this thesis, we first investigate the expected syntax of quotient types and explain it with categorical notions. For quotients which can be represented as a setoid as well as defined as a set without a quotient type former, we propose to define an algebraic structure of quotients called definable quotients. It relates the setoid interpretation and the set definition via a normalisation function which returns a normal form (canonical choice) for each equivalence class. It can be seen as a simulation of quotient types and it helps theorem proving because we can benefit from both representations. However this approach cannot be used for all quotients. It seems that we cannot define a normalisation function for some quotients in Type Theory, e.g. Cauchy reals and finite multisets. Quotient types are indeed essential for formalisation of mathematics and reasoning of programs. Then we consider some models of Type Theory where types are interpreted as structured objects such as setoids, groupoids or weak omega-groupoids. In these models equalities are internalised into types which means that it is possible to redefine equalities. We present an implementation of Altenkirch's setoid model and show that quotient types can be defined within this model. We also describe a new extension of Martin-Lof type theory called Homotopy Type Theory where types are interpreted as weak omega-groupoids. It can be seen as a generalisation of the groupoid model which makes extensional concepts including quotient types available. We also introduce a syntactic encoding of weak omega-groupoids which can be seen as a first step towards building a weak omega-groupoids model in Intensional Type Theory. All of these implementations were performed in the dependently typed programming language Agda which is based on intensional Martin-Lof type theory

Optimal Bayes Classifiers for Functional Data and Density Ratios

Bayes classifiers for functional data pose a challenge. This is because probability density functions do not exist for functional data. As a consequence, the classical Bayes classifier using density quotients needs to be modified. We propose to use density ratios of projections on a sequence of eigenfunctions that are common to the groups to be classified. The density ratios can then be factored into density ratios of individual functional principal components whence the classification problem is reduced to a sequence of nonparametric one-dimensional density estimates. This is an extension to functional data of some of the very earliest nonparametric Bayes classifiers that were based on simple density ratios in the one-dimensional case. By means of the factorization of the density quotients the curse of dimensionality that would otherwise severely affect Bayes classifiers for functional data can be avoided. We demonstrate that in the case of Gaussian functional data, the proposed functional Bayes classifier reduces to a functional version of the classical quadratic discriminant. A study of the asymptotic behavior of the proposed classifiers in the large sample limit shows that under certain conditions the misclassification rate converges to zero, a phenomenon that has been referred to as "perfect classification". The proposed classifiers also perform favorably in finite sample applications, as we demonstrate in comparisons with other functional classifiers in simulations and various data applications, including wine spectral data, functional magnetic resonance imaging (fMRI) data for attention deficit hyperactivity disorder (ADHD) patients, and yeast gene expression data

Towards Automated Boundary Value Testing with Program Derivatives and Search

A natural and often used strategy when testing software is to use input values at boundaries, i.e. where behavior is expected to change the most, an approach often called boundary value testing or analysis (BVA). Even though this has been a key testing idea for long it has been hard to clearly define and formalize. Consequently, it has also been hard to automate. In this research note we propose one such formalization of BVA by, in a similar way as to how the derivative of a function is defined in mathematics, considering (software) program derivatives. Critical to our definition is the notion of distance between inputs and outputs which we can formalize and then quantify based on ideas from Information theory. However, for our (black-box) approach to be practical one must search for test inputs with specific properties. Coupling it with search-based software engineering is thus required and we discuss how program derivatives can be used as and within fitness functions. This brief note does not allow a deeper, empirical investigation but we use a simple illustrative example throughout to introduce the main ideas. By combining program derivatives with search, we thus propose a practical as well as theoretically interesting technique for automated boundary value (analysis and) testing

Investigating self-similar groups using their finite LL-presentation

Self-similar groups provide a rich source of groups with interesting properties; e.g., infinite torsion groups (Burnside groups) and groups with an intermediate word growth. Various self-similar groups can be described by a recursive (possibly infinite) presentation, a so-called finite $L$-presentation. Finite $L$-presentations allow numerous algorithms for finitely presented groups to be generalized to this special class of recursive presentations. We give an overview of the algorithms for finitely $L$-presented groups. As applications, we demonstrate how their implementation in a computer algebra system allows us to study explicit examples of self-similar groups including the Fabrykowski-Gupta groups. Our experiments yield detailed insight into the structure of these groups