Proof Theoretic Concepts for the Semantics of Types and Concurrency

We present a method for providing semantic interpretations for languages with a type system featuring inheritance polymorphism. Our approach is illustrated on an extension of the language Fun of Cardelli and Wegner, which we interpret via a translation into an extended polymorphic lambda calculus. Our goal is to interpret inheritances in Fun via coercion functions which are definable in the target of the translation. Existing techniques in the theory of semantic domains can be then used to interpret the extended polymorphic lambda calculus, thus providing many models for the original language. This technique makes it possible to model a rich type discipline which includes parametric polymorphism and recursive types as well as inheritance. A central difficulty in providing interpretations for explicit type disciplines featuring inheritance in the sense discussed in this paper arises from the fact that programs can type-check in more than one way. Since interpretations follow the type-checking derivations, coherence theorems are required: that is, one must prove that the meaning of a program does not depend on the way it was type-checked. The proof of such theorems for our proposed interpretation are the basic technical results of this paper. Interestingly, proving coherence in the presence of recursive types, variants, and abstract types forced us to reexamine fundamental equational properties that arise in proof theory (in the form of commutative reductions) and domain theory (in the form of strict vs. non-strict functions)

Inheritance as Implicit Coercion

We present a method for providing semantic interpretations for languages with a type system featuring inheritance polymorphism. Our approach is illustrated on an extension of the language Fun of Cardelli and Wegner, which we interpret via a translation into an extended polymorphic lambda calculus. Our goal is to interpret inheritances in Fun via coercion functions which are definable in the target of the translation. Existing techniques in the theory of semantic domains can be then used to interpret the extended polymorphic lambda calculus, thus providing many models for the original language. This technique makes it possible to model a rich type discipline which includes parametric polymorphism and recursive types as well as inheritance. A central difficulty in providing interpretations for explicit type disciplines featuring inheritance in the sense discussed in this paper arises from the fact that programs can type-check in more than one way. Since interpretations follow the type-checking derivations, coherence theorems are required: that is, one must prove that the meaning of a program does not depend on the way it was type-checked. The proof of such theorems for our proposed interpretation are the basic technical results of this paper. Interestingly, proving coherence in the presence of recursive types, variants, and abstract types forced us to reexamine fundamental equational properties that arise in proof theory (in the form of commutative reductions) and domain theory (in the form of strict vs. non-strict functions)

Inductive and Coinductive Components of Corecursive Functions in Coq

In Constructive Type Theory, recursive and corecursive definitions are subject to syntactic restrictions which guarantee termination for recursive functions and productivity for corecursive functions. However, many terminating and productive functions do not pass the syntactic tests. Bove proposed in her thesis an elegant reformulation of the method of accessibility predicates that widens the range of terminative recursive functions formalisable in Constructive Type Theory. In this paper, we pursue the same goal for productive corecursive functions. Notably, our method of formalisation of coinductive definitions of productive functions in Coq requires not only the use of ad-hoc predicates, but also a systematic algorithm that separates the inductive and coinductive parts of functions.Comment: Dans Coalgebraic Methods in Computer Science (2008

CoLoR: a Coq library on well-founded rewrite relations and its application to the automated verification of termination certificates

Termination is an important property of programs; notably required for programs formulated in proof assistants. It is a very active subject of research in the Turing-complete formalism of term rewriting systems, where many methods and tools have been developed over the years to address this problem. Ensuring reliability of those tools is therefore an important issue. In this paper we present a library formalizing important results of the theory of well-founded (rewrite) relations in the proof assistant Coq. We also present its application to the automated verification of termination certificates, as produced by termination tools

Extending the Calculus of Constructions with Tarski's fix-point theorem

We propose to use Tarski's least fixpoint theorem as a basis to define recursive functions in the calculus of inductive constructions. This widens the class of functions that can be modeled in type-theory based theorem proving tool to potentially non-terminating functions. This is only possible if we extend the logical framework by adding the axioms that correspond to classical logic. We claim that the extended framework makes it possible to reason about terminating and non-terminating computations and we show that common facilities of the calculus of inductive construction, like program extraction can be extended to also handle the new functions

Practical Datatype Specializations with Phantom Types and Recursion Schemes

Datatype specialization is a form of subtyping that captures program invariants on data structures that are expressed using the convenient and intuitive datatype notation. Of particular interest are structural invariants such as well-formedness. We investigate the use of phantom types for describing datatype specializations. We show that it is possible to express statically-checked specializations within the type system of Standard ML. We also show that this can be done in a way that does not lose useful programming facilities such as pattern matching in case expressions.Comment: 25 pages. Appeared in the Proc. of the 2005 ACM SIGPLAN Workshop on M

Bricklayer: An Authentic Introduction to the Functional Programming Language SML

Functional programming languages are seen by many as instrumental to effectively utilizing the computational power of multi-core platforms. As a result, there is growing interest to introduce functional programming and functional thinking as early as possible within the computer science curriculum. Bricklayer is an API, written in SML, that provides a set of abstractions for creating LEGO artifacts which can be viewed using LEGO Digital Designer. The goal of Bricklayer is to create a problem space (i.e., a set of LEGO artifacts) that is accessible and engaging to programmers (especially novice programmers) while providing an authentic introduction to the functional programming language SML.Comment: In Proceedings TFPIE 2014, arXiv:1412.473

An Editor for Helping Novices to Learn Standard ML

This paper describes a novel editor intended as an aid in the learning of the functional programming language Standard ML. A common technique used by novices is programming by analogy whereby students refer to similar programs that they have written before or have seen in the course literature and use these programs as a basis to write a new program. We present a novel editor for ML which supports programming by analogy by providing a collection of editing commands that transform old programs into new ones. Each command makes changes to an isolated part of the program. These changes are propagated to the rest of the program using analogical techniques. We observed a group of novice ML students to determine the most common programming errors in learning ML and restrict our editor such that it is impossible to commit these errors. In this way, students encounter fewer bugs and so their rate of learning increases. Our editor, C Y NTHIA, has been implemented and is due to be tested on st..