An Intensional Concurrent Faithful Encoding of Turing Machines

The benchmark for computation is typically given as Turing computability; the ability for a computation to be performed by a Turing Machine. Many languages exploit (indirect) encodings of Turing Machines to demonstrate their ability to support arbitrary computation. However, these encodings are usually by simulating the entire Turing Machine within the language, or by encoding a language that does an encoding or simulation itself. This second category is typical for process calculi that show an encoding of lambda-calculus (often with restrictions) that in turn simulates a Turing Machine. Such approaches lead to indirect encodings of Turing Machines that are complex, unclear, and only weakly equivalent after computation. This paper presents an approach to encoding Turing Machines into intensional process calculi that is faithful, reduction preserving, and structurally equivalent. The encoding is demonstrated in a simple asymmetric concurrent pattern calculus before generalised to simplify infinite terms, and to show encodings into Concurrent Pattern Calculus and Psi Calculi.Comment: In Proceedings ICE 2014, arXiv:1410.701

On the Semantics of Intensionality and Intensional Recursion

Intensionality is a phenomenon that occurs in logic and computation. In the most general sense, a function is intensional if it operates at a level finer than (extensional) equality. This is a familiar setting for computer scientists, who often study different programs or processes that are interchangeable, i.e. extensionally equal, even though they are not implemented in the same way, so intensionally distinct. Concomitant with intensionality is the phenomenon of intensional recursion, which refers to the ability of a program to have access to its own code. In computability theory, intensional recursion is enabled by Kleene's Second Recursion Theorem. This thesis is concerned with the crafting of a logical toolkit through which these phenomena can be studied. Our main contribution is a framework in which mathematical and computational constructions can be considered either extensionally, i.e. as abstract values, or intensionally, i.e. as fine-grained descriptions of their construction. Once this is achieved, it may be used to analyse intensional recursion.Comment: DPhil thesis, Department of Computer Science & St John's College, University of Oxfor

In Search of Effectful Dependent Types

Real world programming languages crucially depend on the availability of computational effects to achieve programming convenience and expressive power as well as program efficiency. Logical frameworks rely on predicates, or dependent types, to express detailed logical properties about entities. According to the Curry-Howard correspondence, programming languages and logical frameworks should be very closely related. However, a language that has both good support for real programming and serious proving is still missing from the programming languages zoo. We believe this is due to a fundamental lack of understanding of how dependent types should interact with computational effects. In this thesis, we make a contribution towards such an understanding, with a focus on semantic methods.Comment: PhD thesis, Version submitted to Exam School