Term Collections in λ and ρ-calculi

The ρ-calculus generalises term rewriting and the λ-calculus by defining abstractions on arbitrary patterns and by using a pattern-matching algorithm which is a parameter of the calculus. In particular, equational theories that do not have unique principal solutions may be used. In the latter case, all the principal solutions of a matching problem are stored in a “structure” that can also be seen as a collection of terms.Motivated by the fact that there are various approaches to the definition of structures in the ρ-calculus, we study in this paper a version of the λ-calculus with term collections.The contributions of this work include a new syntax and operational semantics for a λ-calculus with term collections, which is related to the λ-calculi with strict parallel functions studied by Boudol and Dezani et al. and a proof of the confluence of the β-reduction relation defined for the calculus (which is a suitable extension of the standard rule of β-reduction in the λ-calculus).     

A Proof Search Specification of the π-Calculus

We present a meta-logic that contains a new quantifier (for encoding “generic judgments”) and inference rules for reasoning within fixed points of a given specification. We then specify the operational semantics and bisimulation relations for the finite π-calculus within this meta-logic. Since we restrict to the finite case, the ability of the meta-logic to reason within fixed points becomes a powerful and complete tool since simple proof search can compute this one fixed point. The quantifier helps with the delicate issues surrounding the scope of variables within π-calculus expressions and their executions (proofs). We shall illustrate several merits of the logical specifications we write: they are natural and declarative; they contain no side conditions concerning names of variables while maintaining a completely formal treatment of such variables; differences between late and open bisimulation relations are easy to see declaratively; and proof search involving the application of inference rules, unification, and backtracking can provide complete proof systems for both one-step transitions and for bisimulation.     

A Coalgebraic Semantics for Epistemic Programs

This paper gives a fresh look at my previous work on “epistemic actions” and information updates in distributed systems, from a coalgebraic perspective. I show that the “relational” semantics of epistemic programs, given in [BMS2] in terms of epistemic updates, can be understood in terms of functors on the category of coalgebras and natural transformations associated to them. Then, I introduce a new, alternative, more refined semantics for epistemic programs: programs as “epistemic coalgebras”. I argue for the advantages of this second semantics, from a semantic, heuristic, syntactical and proof-theoretic point of view. Finally, as a step towards a generalization, I show these concepts make sense for other functors, and that apparently unrelated concepts, such as Bayesian belief updates and process transformations, can be seen to arise in the same way as our “epistemic actions”.     

Biological Transactions for Quantitative Models

In this work an extension of stochastic π-calculus with biological transactions is presented. This permits to model multi-reactant multi-product reactions as atomic actions when quantitative information are given. First, the syntax and the semantics are defined, then some transaction properties are discussed. Finally, some examples are described.     

Linearity and bisimulation

Exploiting linear type structure, we introduce a new theory of weak bisimilarity for the π-calculus in which we abstract away not only τ-actions but also non-τ actions which do not affect well-typed observers. This gives a congruence far larger than the standard bisimilarity while retaining semantic soundness. The framework is smoothly extendible to other settings involving nondeterminism and state. As an application we develop a behavioural theory of secrecy in the π-calculus which ensures secure information flow for a strictly greater set of processes than the type-based approach, while still offering compositional verification techniques.     

Parametric synchronizations in mobile nominal calculi

We present and compare P-PRISMA and F-PRISMA, two parametric calculi that can be instantiated with different interaction policies, defined as synchronization algebras with mobility of names (SAMs). In particular, P-PRISMA is based on name transmission (P-SAM), like π-calculus, and thus exploits directional (input–output) communication only, while F-PRISMA is based on name fusion (F-SAM), like Fusion calculus, and thus exploits a more symmetric form of communication. However, P-PRISMA and F-PRISMA can easily accommodate many other high-level synchronization mechanisms than the basic ones available in π-calculus and Fusion, hence allowing for the development of a general meta-theory of mobile calculi. We define for both the labeled operational semantics and a form of strong bisimilarity, showing that the latter is compositional for any SAM. We also discuss reduction semantics and weak bisimilarity. We give several examples based on heterogeneous SAMs, we investigate the case studies of π-calculus and Fusion calculus giving correspondence theorems, and we show how P-PRISMA can be encoded in F-PRISMA. Finally, we show that basic categorical tools can help to relate and to compose SAMs and PRISMA processes in an elegant way.     

Structured coalgebras and minimal HD-automata for the -calculus

The coalgebraic framework developed for the classical process algebras, and in particular its advantages concerning minimal realizations, does not fully apply to the π-calculus, due to the constraints on the freshly generated names that appear in the bisimulation.In this paper we propose to model the transition system of the π-calculus as a coalgebra on a category of name permutation algebras and to define its abstract semantics as the final coalgebra of such a category. We show that permutations are sufficient to represent in an explicit way fresh name generation, thus allowing for the definition of minimal realizations.We also link the coalgebraic semantics with a slightly improved version of history dependent (HD) automata, a model developed for verification purposes, where states have local names and transitions are decorated with names and name relations. HD-automata associated with agents with a bounded number of threads in their derivatives are finite and can be actually minimized. We show that the bisimulation relation in the coalgebraic context corresponds to the minimal HD-automaton.     

A Novel Stochastic Game Via the Quantitative μ-calculus

The quantitative μ-calculus qMμ extends the applicability of Kozen's standard μ-calculus [D. Kozen, Results on the propositional μ-calculus, Theoretical Computer Science 27 (1983) 333–354] to probabilistic systems. Subsequent to its introduction [C. Morgan, and A. McIver, A probabilistic temporal calculus based on expectations, in: L. Groves and S. Reeves, editors, Proc. Formal Methods Pacific '97 (1997), available at [PSG, Probabilistic Systems Group: Collected reports, http://web.comlab.ox.ac.uk/oucl/research/areas/probs/bibliography.html]; also appears at [A. McIver, and C. Morgan, “Abstraction, Refinement and Proof for Probabilistic Systems,” Technical Monographs in Computer Science, Springer, New York, 2005, Chap. 9], M. Huth, and M. Kwiatkowska, Quantitative analysis and model checking, in: Proceedings of 12th annual IEEE Symposium on Logic in Computer Science, 1997] it has been developed by us [A. McIver, and C. Morgan, Games, probability and the quantitative μ-calculus qMu, in: Proc. LPAR, LNAI 2514 (2002), pp. 292–310, revised and expanded at [A. McIver, and C. Morgan, Results on the quantitative μ-calculus qMμ (2005), to appear in ACM TOCL]; also appears at [A. McIver, and C. Morgan, “Abstraction, Refinement and Proof for Probabilistic Systems,” Technical Monographs in Computer Science, Springer, New York, 2005, Chap. 11], A. McIver, and C. Morgan, “Abstraction, Refinement and Proof for Probabilistic Systems,” Technical Monographs in Computer Science, Springer, New York, 2005, A. McIver, and C. Morgan, Results on the quantitative μ-calculus qMμ (2005), to appear in ACM TOCL] and by others [L. de Alfaro, and R. Majumdar, Quantitative solution of omega-regular games, Journal of Computer and System Sciences 68 (2004) 374–397]. Beyond its natural application to define probabilistic temporal logic [C. Morgan, and A. McIver, An expectation-based model for probabilistic temporal logic, Logic Journal of the IGPL 7 (1999), pp. 779–804, also appears at [A. McIver, and C. Morgan, “Abstraction, Refinement and Proof for Probabilistic Systems,” Technical Monographs in Computer Science, Springer, New York, 2005, Chap.10]], there are a number of other areas that benefit from its use.One application is stochastic two-player games, and the contribution of this paper is to depart from the usual notion of “absolute winning conditions” and to introduce a novel game in which players can “draw”.The extension is motivated by examples based on economic games: we propose an extension to qMμ so that they can be specified; we show that the extension can be expressed via a reduction to the original logic; and, via that reduction, we prove that the players can play optimally in the extended game using memoryless strategies.     

Separation of Synchronous and Asynchronous Communication Via Testing

One of the early results about the asynchronous π-calculus which significantly contributed to its popularity is the capability of encoding the output prefix of the (choiceless) π-calculus in a natural and elegant way. Encodings of this kind were proposed by Honda and Tokoro, by Nestmann and (independently) by Boudol. We investigate whether the above encodings preserve De Nicola and Hennessy's testing semantics. In this sense, it turns out that, under some general conditions, no encoding of output prefix is able to preserve the must testing. This negative result is due to (a) the non atomicity of the sequences of steps which are necessary in the asynchronous π-calculus to mimic synchronous communication, and (b) testing semantics's sensitivity to divergence.     

On the representation of McCarthy's in the -calculus

We study the encoding of , the call-by-name λ-calculus enriched with McCarthy's amb operator, into the π-calculus. Semantically, amb is a challenging operator, for the fairness constraints that it expresses. We prove that, under a certain interpretation of divergence in the λ-calculus (weak divergence), a faithful encoding is impossible. However, with a different interpretation of divergence (strong divergence), the encoding is possible, and for this case we derive results and coinductive proof methods to reason about that are similar to those for the encoding of pure λ-calculi. We then use these methods to derive the most important laws concerning amb. We take bisimilarity as behavioural equivalence on the π-calculus, which sheds some light on the relationship between fairness and bisimilarity.     

Expressing Priorities and External Probabilities in Process Algebra via Mixed Open/Closed Systems

Defining operational semantics for a process algebra is often based either on labeled transition systems that account for interaction with a context or on the so-called reduction semantics: we assume to have a representation of the whole system and we compute unlabeled reduction transitions (leading to a distribution over states in the probabilistic case). In this paper we consider mixed models with states where the system is still open (towards interaction with a context) and states where the system is already closed. The idea is that (open) parts of a system “P” can be closed via an operator “P↑G” that turns already synchronized actions whose “handle” is specified inside “G” into prioritized reduction transitions (and, therefore, states performing them into closed states). We show that we can use the operator “P↑G” to express multi-level priorities and external probabilistic choices (by assigning weights to handles inside G), and that, by considering reduction transitions as the only unobservable τ transitions, the proposed technique is compatible, for process algebra with general recursion, with both standard (probabilistic) observational congruence and a notion of equivalence which aggregates reduction transitions in a (much more aggregating) trace based manner. We also observe that the trace-based aggregated transition system can be obtained directly in operational semantics and we present the “aggregating” semantics. Finally, we discuss how the open/closed approach can be used to also express discrete and continuous (exponential probabilistic) time and we show that, in such timed contexts, the trace-based equivalence can aggregate more with respect to traditional lumping based equivalences over Markov Chains.     

Interaction Nets vs. the ρ-calculus: Introducing Bigraphical Nets

The ρ-calculus generalises both term rewriting and the λ-calculus in a uniform framework. Interaction nets are a form of graph rewriting which proved most successful in understanding the dynamics of the λ-calculus, the prime example being the implementation of optimal β-reduction. It is thus natural to study interaction net encodings of the ρ-calculus as a first step towards the definition of efficient reduction strategies. We give two interaction net encodings which bring a new understanding to the operational semantics of the ρ-calculus; however, these encodings have some drawbacks and to overcome them we introduce bigraphical nets—a new paradigm of computation inspired by Lafont's interactions nets and Milner's bigraphs.     

Completeness of Continuation Models for λμ-Calculus

We show that a certain simple call-by-name continuation semantics of Parigot's λ_μ-calculus is complete. More precisely, for every λ_μ-theory we construct a cartesian closed category such that the ensuing continuation-style interpretation of λ_μ, which maps terms to functions sending abstract continuations to responses, is full and faithful. Thus, any λ_μ-category in the sense of L. Ong (1996, in “Proceedings of LICS '96,” IEEE Press, New York) is isomorphic to a continuation model (Y. Lafont, B. Reus, and T. Streicher, “Continuous Semantics or Expressing Implication by Negation,” Technical Report 93-21, University of Munich) derived from a cartesian-closed category of continuations. We also extend this result to a later call-by-value version of λ_μ developed by C.-H. L. Ong and C. A. Stewart (1997, in “Proceedings of ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, Paris, January 1997,” Assoc. Comput. Mach. Press, New York).     

Interaction in Time and Space

Distributed π-calculus and ambient calculus are extended with timers which may trigger timeout recovery processes. Timers provide a useful notion of relative time with respect to the interaction in a distributed system. The rather flat notion of space in timed distributed π-calculus is improved by considering a hierarchical representation of space in timed mobile ambients. Some basic results are proven, making sound both formal approaches. An easily understood example is used for both extensions, showing how it is possible to describe a non-monotonic behaviour and use a decentralized control to coordinate the interacting components in time and space.     

On the Expressiveness of Pure Mobile Ambients

We consider the Pure Ambient Calculus, which is Cardelli and Gordon's Ambient Calculus (or more precisely its safe version by Levi and Sangiorgi) restricted to its mobility primitives, and we focus on its expressive power. Since it has no form of communication or substitution, we show how these notions can be simulated by mobility and modifications in the hierarchical structure of ambients. As an example, we give an encoding of the synchronous π-calculus into pure ambients and we state an operational correspondence result. In order to simplify the proof and give an intuitive understanding of the encoding, we design an intermediate language: the π-Calculus with Explicit Substitutions and Channels, which is a syntactic extension of the π-calculus with a specific operational semantics.     

A Model in κ for DNA Addition

DNA computing is a hot research topic in recent years. Formalization and verification using theories(π-calculus, bioambients, κ-calculus and etc.) in Computer Science attract attention because it can help prove and predict to a certian degree various kinds of biological processes. Combining these two aspects, formal methods can be used to verify algorithms in DNA computing, including basic arithmetic operations if they are to be included in a DNA chip. In this paper, we first introduce a newly-designed algorithm for solving binary addition with DNA, which contributes to a unit in DNA computer processor, and then formalize the algorithm in κ-calculus(a formal method well suited for describing protein interactions) to show the correctness of it in a sense, and a sensible example is provided. Finally, some discussion on the described model is made, in addition to a few possible future improvement directions.     

Timers for Distributed Systems

We deal with temporal aspects of distributed systems, introducing and studying a new model called timed distributed π-calculus. This model extends distributed π-calculus with timers, transforming the communication channels into temporary resources. Distributed π-calculus describes located interactions between processes with restricted access to resources. We introduce time constraints by considering timeout timers for channels. Combining these timers with types and locations, we provide a formal framework able to describe complex systems with constraints on time and on resource access. Its typing system and operational semantics are presented. It is proved that the passage of time does not interfere with the typing system. The new model is proved to be sound by using a method based on subject reduction.     

Probabilistic and nondeterministic aspects of anonymity

The concept of anonymity comes into play in a wide range of situations, varying from voting and anonymous donations to postings on bulletin boards and sending emails. The protocols for ensuring anonymity often use random mechanisms which can be described probabilistically, while the agents’ behavior may be totally unpredictable, irregular, and hence expressible only nondeterministically. Formal definitions of the concept of anonymity have been investigated in the past either in a totally nondeterministic framework, or in a purely probabilistic one. In this paper, we investigate a notion of anonymity which combines both probability and nondeterminism, and which is suitable for describing the most general situation in which the protocol and the users can have both probabilistic and nondeterministic behavior. We also investigate the properties of the definition for the particular cases of purely nondeterministic users and purely probabilistic users. We formulate the notions of anonymity in terms of probabilistic automata, and we describe protocols and users as processes in the probabilistic π-calculus, whose semantics is again based on probabilistic automata. Throughout the paper, we illustrate our ideas by using the example of the dining cryptographers.     

π-calculus with noisy channels

It is assumed in the π-calculus that communication channels are always noiseless. But it is usually not the case in the mobile systems that developers are faced with in the real life. In this paper, we introduce an extension of π, called π_N, in which noisy channels may be present. A probabilistic transitional semantics of π_N is given. The notions of approximate (strong) bisimilarity and equivalence between agents in π_N are proposed, and various algebraic laws for them are established. In particular, we introduce the notion of stratified bisimulation which is suited to describe behavior equivalence between infinite probabilistic processes. Some useful techniques for reasoning about approximate bisimilarity and equivalence are developed. We also introduce a notion of reliability in order to compare different behaviors of an agent in π and π_N. It is shown that reliability is preserved by the basic combinators in π. A link between reliability and bisimulation is given. This provides us with a uniform framework in which we can reason about both correctness properties and reliability of mobile systems. Also, a potential way of combing value-passing process algebras and Shannon’s information theory is pointed out.