Reversibility, allowing one to execute a program not only forwards as usual, but also backwards, has emerged as a fundamental concept in computing, with applications ranging from debugging and fault tolerance to biological and quantum systems. CCSK, a reversible extension of CCS, is a paradigmatic model of reversible concurrent computation. In this paper, we investigate the encodability of CCSK into classical forward-only concurrent models. We establish a separation theorem showing that there is no basic, success-sensitive encoding of CCSK into CCS or the π-calculus, highlighting the strong impact of reversibility on expressive power. We then present an encoding of CCSK processes with only top-level parallel composition into the internal π-calculus, correct up to strong bisimilarity. We also identify a fundamental limitation: no parallel-preserving encoding of CCSK (with arbitrary parallel composition) into the π-calculus can be correct up to strong bisimilarity. Finally, we provide a parallel-preserving encoding correct under a weaker behavioural correspondence: weak mutual simulation. Our findings extend the literature of encodability results to reversible process calculi.

In addition to forward computations, a reversible system also features backward computations along which the effects of forward ones can be undone. This is accomplished by reverting executed actions starting from the last one. Since the last performed action may not be uniquely identifiable in a concurrent setting, Danos and Krivine proposed causal reversibility: an executed action can be undone provided that all of its consequences have been undone already. Phillips and Ulidowski then showed how to define nondeterministic process calculi that meet causal reversibility by construction. Lanese, Phillips, and Ulidowski subsequently classified the basic properties that ensure causal reversibility. In this paper we investigate the extent to which those techniques apply to reversible nondeterministic process calculi that include quantitative aspects. Firstly, we consider the introduction of time described via numeric delays with action execution separated from time passing like in the calculus of Moller and Tofts, where actions can be lazy or eager and time is subject to time determinism and time additivity. Secondly, we address the introduction of probabilities like in the calculus of Hansson and Jonsson, in which action execution and probabilistic choices alternate. We show that both resulting reversible calculi satisfy causal reversibility provided that suitable variants of the aforementioned techniques are developed to guarantee the proper forward and backward interplay of nondeterminism and quantitative aspects. The use of the former calculus is illustrated on a timeout mechanism, whereas the use of the latter is exemplified on quantum teleportation.

Abstract We provide two alternative characterizations of hereditary history-preserving bisimilarity: a denotational one, on stable configuration structures, and an operational one, on a reversible process calculus. The characterizing equivalence is forward-reverse bisimilarity extended with a check for backward ready multiset equality. Unlike previous approaches, the focus is thus on counting identically labeled events rather than uniquely identifying them. We also investigate the relationships between event identifier logic, characterizing the former bisimilarity, and backward ready multiset logic, characterizing the latter bisimilarity.

Causal nets (CNs) are Petri nets where causal dependencies are modelled via inhibitor arcs. They play the role of occurrence nets when representing the behaviour of a concurrent and distributed system, even when reversibility is considered. In this paper we extend CNs to account also for asymmetric conflicts and study (i) how this kind of nets, and their reversible versions, can be turned into a category; and (ii) their relation with the categories of reversible asymmetric event structures.

To react to unforeseen circumstances or amend abnormal situations in communication-centric systems, programmers are in charge of "undoing" the interactions which led to an undesired state. To assist this task, session-based languages can be endowed with reversibility mechanisms. In this paper we propose a language enriched with programming facilities to commit session interactions, to roll back the computation to a previous commit point, and to abort the session. Rollbacks in our language always bring the system to previous visited states and a rollback cannot bring the system back to a point prior to the last commit. Programmers are relieved from the burden of ensuring that a rollback never restores a checkpoint imposed by a session participant different from the rollback requester. Such undesired situations are prevented at design-time (statically) by relying on a decidable compliance check at the type level, implemented in MAUDE. We show that the language satisfies error-freedom and progress of a session.

Reversible debuggers help programmers to find the causes of misbehaviours in concurrent programs more quickly, by executing a program backwards from the point where a misbehaviour was observed, and looking for the bug(s) that caused it. Reversible debuggers can be founded on the well-studied theory of causal-consistent reversibility, which only allows one to undo an action provided that its consequences, if any, are undone beforehand. Causal-consistent reversibility yields more efficient debugging by reducing the number of states to be explored when looking backwards. Till now, causal-consistent reversibility has never considered time, which is a key aspect in real-world applications. Here, we study the interplay between reversibility and time in concurrent systems via a process algebra. The Temporal Process Language (TPL) by Hennessy and Regan is a well-understood extension of CCS with discrete-time and a timeout operator. We define revTPL, a reversible extension of TPL, and we show that it satisfies the properties expected from a causal-consistent reversible calculus. We show that, alternatively, revTPL can be interpreted as an extension of reversible CCS with time.

Reversible CCS (RCCS) is a well-established, formal model for reversible communicating systems, which has been built on top of the classical Calculus of Communicating Systems (CCS). In its original formulation, each CCS process is equipped with a memory that records its performed actions, which is then used to reverse computations. More recently, abstract models for RCCS have been proposed in the literature, basically, by directly associating RCCS processes with (reversible versions of) event structures. In this paper we propose a different abstract model: starting from one of the well-known encoding of CCS into Petri nets we apply a recently proposed approach to incorporate causally-consistent reversibility to Petri nets, obtaining as result the (reversible) net counterpart of every RCCS term.

Event structures have emerged as a foundational model for concurrent computation, explaining computational processes by outlining the events and the relationships that dictate their execution. They play a pivotal role in the study of key aspects of concurrent computation models, such as causality and independence, and have found applications across a broad range of languages and models, spanning realms like persistence, probabilities, and quantum computing. Recently, event structures have been extended to address reversibility, where computational processes can undo previous computations. In this context, reversible event structures provide abstract representations of processes capable of both forward and backward steps in a computation. Since their introduction, event structures have played a crucial role in bridging operational models, traditionally exemplified by Petri nets and process calculi, with denotational ones, i.e., algebraic domains. In this context, we revisit the standard connection between Petri nets and event structures under the lenses of reversibility. Specifically, we introduce a subset of contextual Petri nets, dubbed reversible causal nets , that precisely correspond to reversible prime event structures. The distinctive feature of reversible causal nets lies in deriving causality from inhibitor arcs, departing from the conventional dependence on the overlap between the postset and preset of transitions. In this way, we are able to operationally explain the full model of reversible prime event structures.

To react to unforeseen circumstances or amend abnormal situations in communication-centric systems, programmers are in charge of “undoing” the interactions which led to an undesired state. To assist this task, session-based languages can be endowed with reversibility mechanisms. In this paper we propose a language enriched with programming facilities to commit session interactions, to roll back the computation to a previous commit point, and to abort the session. Rollbacks in our language always bring the system to previous visited states and a rollback cannot bring the system back to a point prior to the last commit. Programmers are relieved from the burden of ensuring that a rollback never restores a checkpoint imposed by a session participant different from the rollback requester. Such undesired situations are prevented at design-time (statically) by relying on a decidable compliance check at the type level, implemented in MAUDE. We show that the language satisfies error-freedom and progress of a session.

Causal reversibility blends reversibility and causality for concurrent systems. It indicates that an action can be undone provided that all of its consequences have been undone already, thus making it possible to bring the system back to a past consistent state. Time reversibility is instead considered in the field of stochastic processes, mostly for efficient analysis purposes. A performance model based on a continuous-time Markov chain is time reversible if its stochastic behavior remains the same when the direction of time is reversed. We bridge these two theories of reversibility by showing the conditions under which causal reversibility and time reversibility are both ensured by construction. This is done in the setting of a stochastic process calculus, which is then equipped with a variant of stochastic bisimilarity accounting for both forward and backward directions.

Reversible prime event structures extend the well-known model of prime event structures to represent reversible computational processes. Essentially, they give abstract descriptions of processes capable of undoing computation steps. Since their introduction, event structures have played a pivotal role in connecting operational models (traditionally, Petri nets and process calculi) with denotational ones (algebraic domains). For this reason, there has been a lot of interest in linking different classes of operational models with different kinds of event structures. Hence, it is natural to ask which is the operational counterpart of reversible prime event structures. Such question has been previously addressed for a subclass of reversible prime event structures in which the interplay between causality and reversibility is restricted to the so-called cause-respecting reversible structures. In this paper, we present an operational characterisation of the full-fledged model and show that reversible prime event structures correspond to a subclass of contextual Petri nets, called reversible causal nets. The distinctive feature of reversible causal nets is that causality is recovered from inhibitor arcs instead of the usual overlap between post and presets of transitions. In this way, we are able to operationally explain also out-of-causal order reversibility.

In programming models with a reversible semantics, computational steps can be undone. This paper addresses the integration of reversible semantics into process languages for communication-centric systems equipped with behavioral types. In prior work, we introduced a monitors-as-memories approach to seamlessly integrate reversible semantics into a process model in which concurrency is governed by session types (a class of behavioral types), covering binary (two-party) protocols with synchronous communication. The applicability and expressiveness of the binary setting, however, is limited. Here we extend our approach, and use it to define reversible semantics for an expressive process model that accounts for multiparty (n-party) protocols, asynchronous communication, decoupled rollbacks, and abstraction passing. As main result, we prove that our reversible semantics for multiparty protocols is causally-consistent. A key technical ingredient in our developments is an alternative reversible semantics with atomic rollbacks, which is conceptually simple and is shown to characterize decoupled rollbacks.

Data availability is a key aspect of modern distributed systems. We discuss an extension of coordination languages based on tuple spaces with programming abstractions for sharing data and guaranteeing availability with different consistency guarantees. Data can be spread over the system according to user-specified replica placement strategies and user-specified consistency requirements. The framework takes care then of low-level management of the replicas, so that the programmer can just focus on the business logic of the application. We advocate that the proposed programming primitives are beneficial for data-oriented applications where different kinds of data may have different needs in terms of availability and consistency.

Summary Among the paradigms for parallel and distributed computing, the one popularized with Linda, and based on tuple spaces, is one of the least used, despite the fact of being intuitive, easy to understand, and easy to use. A tuple space is a repository, where processes can add, withdraw, or read tuples by means of atomic operations. Tuples may contain different values, and processes can inspect their content via pattern matching. The lack of a reference implementation for this paradigm has prevented its wide spreading. In this paper, first we perform an extensive analysis of a number of actual implementations of the tuple space paradigm and summarize their main features. Then, we select four such implementations and compare their performances on four different case studies that aim at stressing different aspects of computing, such as communication, data manipulation, and CPU usage. After reasoning on strengths and weaknesses of the four implementations, we conclude with some recommendations for future work towards building an effective implementation of the tuple space paradigm.

Under a reversible semantics, computation steps can be undone. This paper addresses the integration of reversible semantics into a process model of multiparty protocols (choreographies). Building upon the monitors-as-memories approach that we developed in prior work for reversible binary protocols, we present a reversible process framework for multiparty communication, which improves on prior models by seamlessly integrating asynchrony, decoupled rollbacks, and process passing. As main technical result, we prove that our multiparty, reversible semantics is causally-consistent.

We study the theory of safety and liveness in a reversible calculus where reductions are totally ordered and rollbacks lead systems to past states. Liveness and safety in this setting naturally correspond to the should-testing and inverse may-testing preorders, respectively. In reversible languages, however, the natural models of these preorders would need to be based on both forward and backward transitions, thus offering complex proof techniques for verification. Here we develop novel fully abstract models of liveness and safety which are based on forward transitions and limited rollback points, giving rise to considerably simpler proof techniques. Moreover, we show that, with respect to safety, total reversibility is a conservative extension to CCS. With respect to liveness, we prove that adding total reversibility to CCS distinguishes more systems. To our knowledge, this work provides the first testing theory for a reversible calculus, and paves the way for a testing theory for causal reversibility.

Causal-consistent reversibility is a natural way of undoing concurrent computations. We study causal-consistent reversibility in the context of μKlaim, a formal coordination language based on distributed tuple spaces. We consider both uncontrolled reversibility, suitable to study the basic properties of the reversibility mechanism, and controlled reversibility based on a rollback operator, more suitable for programming applications. The causality structure of the language, and thus the definition of its reversible semantics, differs from all the reversible languages in the literature because of its generative communication paradigm. In particular, the reversible behavior of μKlaim read primitive, reading a tuple without consuming it, cannot be matched using channel-based communication. We illustrate the reversible extensions of μKlaim on a simple, but realistic, application scenario.