Handbook of logic in computer science.

The finite π-calculus has an explicit set-theoretic functor-category model that is known to be fully abstract for strong late bisimulation congruence. We characterize this as the initial free algebra for an appropriate set of operations and equations in the enriched Lawvere theories of Plotkin and Power. Thus we obtain a novel algebraic description for models of the π-calculus, and validate an existing construction as the universal such model.

The algebraic operations are intuitive, covering name creation, communication of names over channels, and nondeterminism; the equations then combine these features in a modular fashion. We work in an enriched setting, over a “possible worlds” category of sets indexed by available names. This expands significantly on the classical notion of algebraic theories, and in particular allows us to use nonstandard arities that vary as processes evolve.

Based on our algebraic theory we describe a category of models for the π-calculus, and show that they all preserve bisimulation congruence. We develop a direct construction of free models in this category; and generalise previous results to prove that all free-algebra models are fully abstract.

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Free-algebra models for the π-calculus

We use the mathematical language of sheaf theory to give a unified treatment of non-locality and contextuality, in a setting that generalizes the familiar probability tables used in non-locality theory to arbitrary measurement covers; this includes Kochen–Specker configurations and more. We show that contextuality, and non-locality as a special case, correspond exactly to obstructions to the existence of global sections. We describe a linear algebraic approach to computing these obstructions, which allows a systematic treatment of arguments for non-locality and contextuality. We distinguish a proper hierarchy of strengths of no-go theorems, and show that three leading examples—due to Bell, Hardy and Greenberger, Horne and Zeilinger, respectively—occupy successively higher levels of this hierarchy. A general correspondence is shown between the existence of local hidden-variable realizations using negative probabilities, and no-signalling; this is based on a result showing that the linear subspaces generated by the non-contextual and no-signalling models, over an arbitrary measurement cover, coincide. Maximal non-locality is generalized to maximal contextuality, and characterized in purely qualitative terms, as the non-existence of global sections in the support. A general setting is developed for the Kochen–Specker-type results, as generic, model-independent proofs of maximal contextuality, and a new combinatorial condition is given, which generalizes the ‘parity proofs’ commonly found in the literature. We also show how our abstract setting can be represented in quantum mechanics. This leads to a strengthening of the usual no-signalling theorem, which shows that quantum mechanics obeys no-signalling for arbitrary families of commuting observables, not just those represented on different factors of a tensor product.

http://iopscience.iop.org/article/10.1088/1367-2630/13/11/113036/pdf

The sheaf-theoretic structure of non-locality and contextuality

A notion of bigraph is introduced as a model of mobile interaction. A bigraph consists of two independent structures: a topograph representing locality and an edge net representing connectivity. Bigraphs are equipped with reaction rules to form bigraphical reactive systems (BRSs), which include versions of the π-calculus and the ambient calculus. A behavioural theory is established, using the categorical notion of relative pushout; it allows labelled transition systems to be derived uniformly for a wide variety of BRSs, in such a way that familiar behavioural preorders and equivalences, in particular bisimilarity, are congruential. An example of the derivation is discussed.

Bigraphical Reactive Systems

We present the first efficient statistical zero-knowledge protocols to prove statements such as: - A committed number is a prime. - A committed (or revealed) number is the product of two safe primes, i.e., primes p and q such that (p - 1)=2 and (q - 1)=2 are prime. - A given integer has large multiplicative order modulo a composite number that consists of two safe prime factors. The main building blocks of our protocols are statistical zero-knowledge proofs of knowledge that are of independent interest. We show how to prove the correct computation of a modular addition, a modular multiplication, and a modular exponentiation, where all values including the modulus are committed to but not publicly known. Apart from the validity of the equations, no other information about the modulus (e.g., a generator whose order equals the modulus) or any other operand is exposed. Our techniques can be generalized to prove that any multivariate modular polynomial equation is satisfied, where only commitments to the variables of the polynomial and to the modulus need to be known. This improves previous results, where the modulus is publicly known. We show how these building blocks allow to prove statements such as those listed earlier.

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Proving in zero-knowledge that a number is the product of two safe primes

This paper studies presheaf models for concurrent computation An aim is to harness the general machinery around presheaves for the purposes of process calculi Traditional models like synchronisation trees and event structures have been shown to embed fully and faithfully in particular presheaf models in such a way that bisimulation, expressed through the presence of a span of open maps, is conserved As is shown in the work of Joyal and Moerdijk, presheaves are rich in constructions which preserve open maps, and so bisimulation, by arguments of a very general nature This paper contributes similar results but biased towards questions of bisimulation in process calculi It is concerned with modelling process constructions on presheaves, showing these preserve open maps, and with transferring such results to traditional models for processes One new result here is that a wide range of left Kan extensions, between categories of presheaves, preserve open maps As a corollary, this also implies that any colimit-preserving functor between presheaf categories preserves open maps A particular left Kan extension is shown to coincide with a refinement operation on event structures A broad class of presheaf models is proposed for a general process calculus General arguments are given for why the operations of a presheaf model preserve open maps and why for specific presheaf models the operations coincide with those of traditional models

Presheaf Models for Concurrency

This paper studies fundamental connections between profunctors (that is, 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 of 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.

Profunctors, open maps and bisimulation

We introduce the notion of higher dimensional transition systems as a model of concurrency providing an elementary, set-theoretic formalisation of the idea of higher dimensional transition. We show an embedding of the category of higher dimensional transition systems into that of higher dimensional automata which cuts down to an equivalence when we restrict to non-degenerate automata. Moreover, we prove that the natural notion of bisimulation for such structures is a generalisation of the strong history preserving bisimulation, and provide an abstract categorical account of it via open maps. Finally, we define a notion of unfolding for higher dimensional transition systems and characterise the structures so obtained as a generalisation of event structures.

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Higher dimensional transition systems

A systematic treatment of weak bisimulation and observational congruence on presheaf models is presented. The theory is developed with respect to a "hiding" functor from a category of paths to observable paths. Via a view of processes as bundles, we are able to account for weak morphisms (roughly only required to preserve observable paths) and to derive a saturation monad (on the category of presheaves over the category of paths). Weak morphisms may be encoded as strong ones via the Kleisli construction associated to the saturation monad. A general notion of weak open-map bisimulation is introduced, and results relating various notions of strong and weak bisimulation are provided. The abstract theory is accompanied by fine concrete study of two key models for concurrency, the interleaving model of synchronisation trees and the independence model of labelled event structures.

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Weak bisimulation and open maps

Recent work has shown that presheaf categories provide a general model of concurrency, with an inbuilt notion of bisimulation based on open maps. Here it is shown how this approach can also handle systems where the language of actions may change dynamically as a process evolves. The example is the pi-calculus, a calculus for `mobile processes' whose communication topology varies as channels are created and discarded. A denotational semantics is described for the pi-calculus within an indexed category of profunctors; the model is fully abstract for bisimilarity, in the sense that bisimulation in the model, obtained from open maps, coincides with the usual bisimulation obtained from the operational semantics of the pi-calculus. While attention is concentrated on the `late' semantics of the pi-calculus, it is indicated how the `early' and other variants can also be captured. A version of this paper appears in Category Theory and Computer Science: Proceedings of the 7th International Conference CTCS '97, Lecture Notes in Computer Science 1290. Springer-Verlag, September 1997.

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Gian Luca Cattani

Papers

Presheaf Models for Concurrency

Profunctors, open maps and bisimulation

Higher dimensional transition systems

Weak bisimulation and open maps

Presheaf Models for the pi-Calculus