Recovering purity with comonads and capabilities

© 2020 Owner/Author. In this paper, we take a pervasively effectful (in the style of ML) typed lambda calculus, and show how to extend it to permit capturing pure expressions with types. Our key observation is that, just as the pure simply-typed lambda calculus can be extended to support effects with a monadic type discipline, an impure typed lambda calculus can be extended to support purity with a comonadic type discipline. We establish the correctness of our type system via a simple denotational model, which we call the capability space model. Our model formalises the intuition common to systems programmers that the ability to perform effects should be controlled via access to a permission or capability, and that a program is capability-safe if it performs no effects that it does not have a runtime capability for. We then identify the axiomatic categorical structure that the capability space model validates, and use these axioms to give a categorical semantics for our comonadic type system. We then give an equational theory (substitution and the call-by-value ß and • laws) for the imperative lambda calculus, and show its soundness relative to this semantics. Finally, we give a translation of the pure simply-typed lambda calculus into our comonadic imperative calculus, and show that any two terms which are ß•-equal in the STLC are equal in the equational theory of the comonadic calculus, establishing that pure programs can be mapped in an equation-preserving way into our imperative calculus

Recovering purity with comonads and capabilities

© 2020 Owner/Author. In this paper, we take a pervasively effectful (in the style of ML) typed lambda calculus, and show how to extend it to permit capturing pure expressions with types. Our key observation is that, just as the pure simply-typed lambda calculus can be extended to support effects with a monadic type discipline, an impure typed lambda calculus can be extended to support purity with a comonadic type discipline. We establish the correctness of our type system via a simple denotational model, which we call the capability space model. Our model formalises the intuition common to systems programmers that the ability to perform effects should be controlled via access to a permission or capability, and that a program is capability-safe if it performs no effects that it does not have a runtime capability for. We then identify the axiomatic categorical structure that the capability space model validates, and use these axioms to give a categorical semantics for our comonadic type system. We then give an equational theory (substitution and the call-by-value ß and • laws) for the imperative lambda calculus, and show its soundness relative to this semantics. Finally, we give a translation of the pure simply-typed lambda calculus into our comonadic imperative calculus, and show that any two terms which are ß•-equal in the STLC are equal in the equational theory of the comonadic calculus, establishing that pure programs can be mapped in an equation-preserving way into our imperative calculus

Categorical structure of continuation passing style

Laboratory for Foundations of Computer ScienceThis thesis attempts to make precise the structure inherent in Continuation Passing Style (CPS). We emphasize that CPS translates lambda-calculus into a very basic calculus that does not have functions as primitive. We give an abstract categorical presentation of continuation semantics by taking the continuation type constructor (cont in Standard ML of New Jersey) as primitive. This constructor on types extends to a contravariant functor on terms which is adjoint to itself on the left; restricted to the subcategory of those programs that do not manipulate the current continuation, it is adjoint to itself on the right. The motivating example of such a category is built from (equivalence classes of typing judgements for) continuation passing style (CPS) terms. The categorical approach suggests a notion of effect-free term as well as some operators for manipulating continuations. We use these for writing programs that illustrate our categorical approach and refute some conjectures about control effects. A call-by-value lambda-calculus with the control operator callcc can be interpreted. Arrow types are broken down into continuation types for argument/result-continuations pairs, reflecting the fact that CPS compiles functions into a special case of continuations. Variant translations are possible, among them lazy call-by-name, which can be derived by way of argument thunking, and a genuinely call-by-name transform. Specialising the semantics to the CPS term model allows a rational reconstruction of various CPS transforms

Applying quantitative semantics to higher-order quantum computing

Finding a denotational semantics for higher order quantum computation is a long-standing problem in the semantics of quantum programming languages. Most past approaches to this problem fell short in one way or another, either limiting the language to an unusably small finitary fragment, or giving up important features of quantum physics such as entanglement. In this paper, we propose a denotational semantics for a quantum lambda calculus with recursion and an infinite data type, using constructions from quantitative semantics of linear logic