Unnesting of Copatterns

Inductive data such as finite lists and trees can elegantly be defined by constructors which allow programmers to analyze and manipulate finite data via pattern matching. Dually, coinductive data such as streams can be defined by observations such as head and tail and programmers can synthesize infinite data via copattern matching. This leads to a symmetric language where finite and infinite data can be nested. In this paper, we compile nested pattern and copattern matching into a core language which only supports simple non-nested (co)pattern matching. This core language may serve as an intermediate language of a compiler. We show that this translation is conservative, i.e. the multi-step reduction relation in both languages coincides for terms of the original language. Furthermore, we show that the translation preserves strong and weak normalisation: a term of the original language is strongly/weakly normalising in one language if and only if it is so in the other. In the proof we develop more general criteria which guarantee that extensions of abstract reduction systems are conservative and preserve strong or weak normalisation. \ua9 2014 Springer International Publishing Switzerland

Copattern matching and first-class observations in OCaml, with a macro

International audienceInfinite data structures are elegantly defined by means of copatternmatching, a dual construction to pattern matching that expressesthe outcomes of the observations of an infinite structure. We extendthe OCaml programming language with copatterns, exploiting theduality between pattern matching and copattern matching. Pro-vided that a functional programming language has GADTs, everycopattern matching can be transformed into a pattern matching viaa purely local syntactic transformation, a macro. The developmentof this extension leads us to a generalization of previous calculus ofcopatterns: the introduction of first-class observation queries. Westudy this extension both from a formal and practical point of view

Programming with monadic CSP-style processes in dependent type theory

We introduce a library called CSP-Agda for representing processes in the dependently typed theorem prover and interactive programming language Agda. We will enhance processes by a monad structure. The monad structure facilitates combining processes in a modular way, and allows to define recursion as a direct operation on processes. Processes are defined coinductively as non-well-founded trees. The nodes of the tree are formed by a an atomic one step relation, which determines for a process the external, internal choices, and termination events it can choose, and whether the process has terminated. The data type of processes is inspired by Setzer and Hancock's notion of interactive programs in dependent type theory. The operators of CSP will be defined rather than atomic operations, and compute new elements of the data type of processes from existing ones. The approach will make use of advanced type theoretic features: the use of inductive-recursively defined universes; the definition of coinductive types by their observations, which has similarities to the notion of an object in object-oriented programming; the use of sized types for coinductive types, which allow coinductive definitions in a modular way; the handling of finitary information (names of processes) in a coinductive settings; the use of named types for automatic inference of arguments similar to its use in template Meta-programming in C++; and the use of interactive programs in dependent type theory.We introduce a simulator as an interactive program in Agda. The simulator allows to observe the evolving of processes following external or internal choices. Our aim is to use this in order to simulate railway interlocking system and write programs in Agda which directly use CSP processes

How to Reason Coinductively Informally

We start by giving an overview of the theory of indexed inductively and coinductively defined sets. We consider the theory of strictly positive indexed inductive definitions in a set theoretic setting. We show the equiv-alence between the definition as an indexed initial algebra, the definition via an induction principle, and the set theoretic definition of indexed in-ductive definitions. We review as well the equivalence of unique iteration, unique primitive recursion, and induction. Then we review the theory of indexed coinductively defined sets or final coalgebras. We construct indexed coinductively defined sets set theoretically, and show the equiv-alence between the category theoretic definition, the principle of unique coiteration, of unique corecursion, and of iteration together with bisimula-tion as equality. Bisimulation will be defined as an indexed coinductively defined set. Therefore proofs of bisimulation can be carried out corecur-sively. This fact can be considered together with bisimulation implying equality as the coinduction principle for the underlying coinductively de-fined set. Finally we introduce various schemata for reasoning about coin-ductively defined sets in an informal way: the schemata of corecursion, of indexed corecursion, of coinduction, and of corecursion for coinductively defined relations. This allows to reason about coinductively defined sets similarly as one does when reasoning about inductively defined sets using schemata of induction. We obtain the notion of a coinduction hypothesis, which is the dual of an induction hypothesis.

Defining Trace Semantics for CSP-Agda

This article is based on the library CSP-Agda, which represents the process algebra CSP coinductively in the interactive theorem prover Agda. The intended application area of CSP-Agda is the proof of properties of safety critical systems (especially the railway domain). In CSP-Agda, CSP processes have been extended to monadic form, allowing the design of processes in a more modular way. In this article we extend the trace semantics of CSP to the monadic setting. We implement this semantics, together with the corresponding refinement and equality relation, formally in CSP-Agda. In order to demonstrate the proof capabilities of CSP-Agda, we prove in CSP-Agda selected algebraic laws of CSP based on the trace semantics. Because of the monadic settings, some adjustments need to be made to these laws. The examples covered in this article are the laws of refinement, commutativity of interleaving and parallel, and the monad laws for the monadic extension of CSP. All proofs and definitions have been type checked in Agda. Further proofs of algebraic laws will be available in the repository of CSP-Agda

Programming Languages and Systems

This open access book constitutes the proceedings of the 28th European Symposium on Programming, ESOP 2019, which took place in Prague, Czech Republic, in April 2019, held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2019

Integration of the Process Algebra CSP in Dependent Type Theory - Formalisation and Verification

We introduce a library called CSP-Agda for representing processes in the dependently typed theorem prover and interactive programming language Agda. We will enhance processes by a monad structure. The monad struc-ture facilitates combining processes in a modular way, and allows to define recursion as a direct operation on processes. Processes are defined coinduc-tively as non-well-founded trees. The nodes of the tree are formed by a an atomic one step relation, which determines for a process the external, internal choices, and termination events it can choose, and whether the process has terminated. The data type of processes is inspired by Setzer and Hancock’s notion of interactive programs in dependent type theory. The operators of CSP will be defined rather than atomic operations, and compute new ele-ments of the data type of processes from existing ones.The approach will make use of advanced type theoretic features: the use of inductive-recursively defined universes; the definition of coinductive types by their observations, which has similarities to the notion of an object in object-oriented programming; the use of sized types for coinductive types, which allow coinductive definitions in a modular way; the handling of fini-tary information (names of processes) in a coinductive settings; the use of named types for automatic inference of arguments similar to its use in tem-plate Meta-programming in C++; and the use of interactive programs in dependent type theory.We introduce a simulator as an interactive program in Agda. The simula-tor allows to observe the evolving of processes following external or internal choices. Our aim is to use this in order to simulate railway interlocking system and write programs in Agda which directly use CSP processes.Then we extend the trace semantics of CSP to the monadic setting. We implement this semantics, together with the corresponding refinement and equality relation, formally in CSP-Agda. In order to demonstrate the proof capabilities of CSP-Agda, we prove in CSP-Agda selected algebraic laws of CSP based on the trace semantics. Because of the monadic settings, some adjustments need to be made to these laws.Next we implement the more advanced semantics of CSP, the stable fail-ures semantics and the failures divergences infinite traces semantics (FDI), in CSP-Agda, and define the corresponding refinement and equality relations. Direct proofs in these semantics are cumbersome, and we develop a tech-nique of showing algebraic laws in those semantics in an indirect way, which is much easier. We introduce divergence-respecting weak bisimilarity and strong bisimilarity in CSP-Agda, and show that both imply equivalence with respect to stable failures and FDI semantics. Now we show certain algebraic laws with respect to one of these two bisimilarity relations. As a case study, we model and verify a possible scenario for railways in CSP-Agda and in standard CSP tools

Programming Languages and Systems

This open access book constitutes the proceedings of the 30th European Symposium on Programming, ESOP 2021, which was held during March 27 until April 1, 2021, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2021. The conference was planned to take place in Luxembourg and changed to an online format due to the COVID-19 pandemic. The 24 papers included in this volume were carefully reviewed and selected from 79 submissions. They deal with fundamental issues in the specification, design, analysis, and implementation of programming languages and systems