Limitations of Applicative Bisimulation (Preliminary Report)

We present a series of examples that illuminate an important aspect of the semantics of higher-order functions with local state. Namely that certain behaviour of such functions can only be observed by pro- viding them with arguments that contain the functions themselves. This provides evidence for the necessity of complex conditions for functions in modern semantics for state, such as logical relations and Kripke-like bisimulations, where related functions are applied to related arguments (that may contain the functions). It also suggests that simpler semantics, such as those based on applicative bisimulations where functions are ap- plied to identical arguments, would not scale to higher-order languages with local state

Calculi for higher order communicating systems

This thesis develops two Calculi for Higher Order Communicating Systems. Both calculi consider sending and receiving processes to be as fundamental as nondeterminism and parallel composition. The first calculus called CHOCS is an extension of Milner's CCS in the sense that all the constructions of CCS are included or may be derived from more fundamental constructs. Most of the mathematical framework of CCS carries over almost unchanged. The operational semantics of CHOCS is given as a labelled transition system and it is a direct extension of the semantics of CCS with value passing. A set of algebraic laws satisfied by the calculus is presented. These are similar to the CCS laws only introducing obvious extra laws for sending and receiving processes. The power of process passing is underlined by a result showing that the recursion operator is unnecessary in the sense that recursion can be simulated by means of process passing and communication. The CHOCS language is also studied by means of a denotational semantics. A major result is the full abstractness of this semantics with respect to the operational semantics. The denotational semantics is used to provide an easy proof of the simulation of recursion. Introducing processes as first class objects yields a powerful metalanguage. It is shown that it is possible to simulate various reduction strategies of the untyped λ-Calculus in CHOCS. As pointed out by Milner, CCS has its limitations when one wants to describe unboundedly expanding systems, e.g. an unbounded number of procedure invocations in an imperative concurrent programming language P with recursive procedures. CHOCS may neatly describe both call-by-value and call-by-reference parameter mechanisms for P. We also consider call-by-name and lazy parameter mechanisms for P. The second calculus is called Plain CHOCS. Essential to the new calculus is the treatment of restriction as a static binding operator on port names. This calculus is given an operational semantics using labelled transition systems which combines ideas from the applicative transition systems described by Abramsky and the transition systems used for CHOCS. This calculus enjoys algebraic properties which are similar to those of CHOCS only needing obvious extra laws for the static nature of the restriction operator. Processes as first class objects enable description of networks with changing interconnection structure and there is a close connection between the Plain CHOCS calculus and the π-Calculus described by Milner, Parrow and Walker: the two calculi can simulate one another. Recently object oriented programming has grown into a major discipline in computational practice as well as in computer science. From a theoretical point of view object oriented programming presents a challenge to any metalanguage since most object oriented languages have no formal semantics. We show how Plain CHOCS may be used to give a semantics to a prototype object oriented language called 0.Open Acess

Environmental Bisimulations for Probabilistic Higher-Order Languages

International audienceEnvironmental bisimulations for probabilistic higher-order languages are studied. In contrast with applicative bisimulations, environmental bisimulations are known to be more robust and do not require sophisticated techniques such as Howe's in the proofs of congruence. As representative calculi, call-by-name and call-by-value λ-calculus, and a (call-by-value) λ-calculus extended with references (i.e., a store) are considered. In each case full abstraction results are derived for probabilistic environmental similarity and bisimilarity with respect to contextual preorder and contextual equivalence, respectively. Some possible enhancements of the (bi)simulations, as 'up-to techniques', are also presented. Probabilities force a number of modifications to the definition of environmental bisimulations in non-probabilistic languages. Some of these modifications are specific to probabilities, others may be seen as general refinements of environmental bisimulations, applicable also to non-probabilistic languages. Several examples are presented, to illustrate the modifications and the differences

A supposedly fun thing i may have to do again : a HOAS encoding of Howe's method

We formally verify in Abella that similarity in the call-by-name lambda calculus is a pre-congruence, using Howe\u2019s method. This turns out to be a very challenging task for HOAS-based systems, as it entails a demanding combination of inductive and coinductive reasoning on open terms, for which no other existing HOAS-based system is equipped for. We also o er a proof using a version of Abella supplemented with predicate quantification; this results in a more structured presentation that is largely independent of the operational semantics as well of the chosen notion of (bi)similarity. While the end result is significantly more succinct and elegant than previous attempts, the exercise highlights some limitations of the two-level approach in general and of Abella in particular

Profunctors, Open Maps and Bisimulation

This paper studies fundamental connections between profunctors (i.e., distributors, or bimodules), open maps and bisimulation. In particular, it proves that a colimit preserving functor between presheaf categories (corresponding to a profunctor) preserves open maps and open map bisimulation. Consequently, the composition of profunctors preserves open maps as 2-cells. A guiding idea is the view that profunctors, and colimit preserving functors, are linear maps in a model of classical linear logic. But profunctors, and colimit preserving functors, as linear maps, are too restrictive for many applications. This leads to a study of a range of pseudo-comonads and how non-linear maps in their co-Kleisli bicategories preserve open maps and bisimulation. The pseudo-comonads considered are based on finite colimit completion, ``lifting'', and indexed families. The paper includes an appendix summarising the key results on coends, left Kan extensions and the preservation of colimits. One motivation for this work is that it provides a mathematical framework for extending domain theory and denotational semantics of programming languages to the more intricate models, languages and equivalences found in concurrent computation. But the results are likely to have more general applicability because of the ubiquitous nature of profunctors

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

Names and higher-order functions.

Many functional programming languages rely on the elimination of 'impure' features: assignment to variables, exceptions and even input/output. But some of these are genuinely useful, and it is of real interest to establish how they can be reintroducted in a controlled way. This dissertation looks in detail at one example of this: the addition to a functional language of dynamically generated names. Names are created fresh, they can be compared with each other and passed around, but that is all. As a very basic example of state, they capture the graduation between private and public, local and global, by their interaction with higher-order functions. The vehicle for this study is the nu-calculus, an extension of the simply-typed lambdacalculus. The nu-calculus is equivalent to a certain fragment of Standard ML, omitting side-effects, exceptions, datatypes and recursion. Even without all these features, the interaction of name creation with higher-order functions can be complex and subtle. Various operational and denotational methods for reasoning about the nu-calculus are developed. These include a computational metalanguage in the style of Moggi, which distinguishes in the type system between values and computations. This leads to categorical models that use a strong monad, and examples are devised based on functor categories. The idea of logical relations is used to derive powerful reasoning methods that capture some of the distinction between private and public names. These techniques are shown to be complete for establishing contextual equivalence between first-order expressions; they are also used to construct a correspondingly abstract categorical model. All the work with the nu-calculus extends cleanly to Reduced ML, a larger language that introduces integer references: mutable storage cells that are dynamically allocated. It turns out that the step up is quite simple, and both the computational metalanguage and the sample categorical models can be reused