Guarded recursion in the sense of Nakano allows general recursive types and terms to be added to type theory without breaking consistency. Recent work has demonstrated applications of guarded recursion such as programming with codata, reasoning about coinductive types, as well as constructing and reasoning about denotational models of general recursive types. Guarded recursion can also be used as an abstract form of step-indexing for reasoning about programming languages with advanced features.Ultimately, we would like to have an implementation of a type theory with guarded recursion in which all these applications can be carried out and machine-checked. In this paper, we take a step towards this goal by devising a suitable reduction semantics.We present Clocked Type Theory, a new type theory for guarded recursion that is more suitable for reduction semantics than the existing ones. We prove confluence, strong normalisation and canonicity for its reduction semantics, constructing the theoretical basis for a future implementation. We show how coinductive types as exemplified by streams can be encoded, and derive that the type system ensures productivity of stream definitions.

Functional reactive programming (FRP) is a paradigm for programming with signals and events, allowing the user to describe reactive programs on a high level of abstraction. For this to make sense, an FRP language must ensure that all programs are causal, and can be implemented without introducing space leaks and time leaks. To this end, some FRP languages do not give direct access to signals, but just to signal functions.Recently, modal types have been suggested as an alternative approach to ensuring causality in FRP languages in the synchronous case, giving direct access to the signal and event abstractions. This paper presents Simply RaTT, a new modal calculus for reactive programming. Unlike prior calculi, Simply RaTT uses a Fitch-style approach to modal types, which simplifies the type system and makes programs more concise. Echoing a previous result by Krishnaswami for a different language, we devise an operational semantics that safely executes Simply RaTT programs without space leaks.We also identify a source of time leaks present in other modal FRP languages: The unfolding of fixed points in delayed computations. The Fitch-style presentation allows an easy way to rules out these leaks, which appears not to be possible in the more traditional dual context approach.

In this article we present a new approach to the problem of calculating compilers. In particular, we develop a simple but general technique that allows us to derive correct compilers from highlevel semantics by systematic calculation, with all details of the implementation of the compilers falling naturally out of the calculation process. Our approach is based upon the use of standard equational reasoning techniques, and has been applied to calculate compilers for a wide range of language features and their combination, including arithmetic expressions, exceptions, state, various forms of lambda calculi, bounded and unbounded loops, non-determinism, and interrupts. All the calculations in the article have been formalised using the Coq proof assistant, which serves as a convenient interactive tool for developing and verifying the calculations.

Domain-specific languages (DSLs) for complex financial contracts are in practical use in many banks and financial institutions today. Given the level of automation and pervasiveness of software in the sector, the financial domain is immensely sensitive to software bugs. At the same time, there is an increasing need to analyse (and report on) the interaction between multiple parties. In this paper, we present a multi-party contract language that rigorously relegates any artefacts of simulation and computation from its core, which leads to favourable algebraic properties, and therefore allows for formalising domain-specific analyses and transformations using a proof assistant. At the centre of our formalisation is a simple denotational semantics independent of any stochastic aspects. Based on this semantics, we devise certified contract analyses and transformations. In particular, we give a type system, with an accompanying type inference procedure, that statically ensures that contracts follow the principle of causality. Moreover, we devise a reduction semantics that allows us to evolve contracts over time, in accordance with the denotational semantics. From the verified Coq definitions, we automatically extract a Haskell implementation of an embedded contract DSL along with the formally verified contract management functionality. This approach opens a road map towards more reliable contract management software, including the possibility of analysing contracts based on symbolic instead of numeric methods.

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