Vincent Danos

Disjunctive Tautologies as Synchronisation Schemes

In the ambient logic of classical second order propositional calculus, we solve the specification problem for a family of excluded middle like tautologies. These are shown to be realized by sequential simulations of specific communication schemes for which they provide a safe typing mechanism.

Formal molecular biology

A language of formal proteins, the K-calculus, is introduced. Interactions are modeled at the domain level, bonds are represented by means of shared names, and reactions are required to satisfy a causality requirement of monotonicity.An example of a simplified signalling pathway is introduced to illustrate how standard biological events can be expressed in our protein language. A more comprehensive example, the lactose operon, is also developed, bringing some confidence in the formalism considered as a modeling language.Then a finer-grained concurrent model, the mK-calculus, is considered, where interactions have to be at most binary. We show how to embed the coarser-grained language in the latter, a properly which we call self-assembly.Finally we show how the finer-grained language can itself be encoded in π-calculus, a standard foundational language for concurrency theory.

Reversible communicating systems

One obtains in this paper a process algebra RCCS, in the style of CCS, where processes can backtrack. Backtrack, just as plain forward computation, is seen as a synchronization and incurs no additional cost on the communication structure. It is shown that, given a past, a computation step can be taken back if and only if it leads to a causally equivalent past.

Transactions in RCCS

We propose a formalisation of the notion of transaction, using a variant of CCS, RCCS, that distinguishes reversible and irreversible actions, and incorporates a distributed backtrack mechanism. Any weakly correct implementation of a transaction in CCS, once embedded in RCCS, automatically obtains a correct one. We show examples where this method allows for a more concise implementation and a simpler proof of correctness.

The structure of multiplicatives

Investigating Girards new propositionnal calculus which aims at a large scale study of computation, we stumble quickly on that question: What is a multiplicative connective? We give here a detailed answer together with our motivations and expectations.

Rule-based modelling of cellular signalling

Modelling is becoming a necessity in studying biological signalling pathways, because the combinatorial complexity of such systems rapidly overwhelms intuitive and qualitative forms of reasoning. Yet, this same combinatorial explosion makes the traditional modelling paradigm based on systems of differential equations impractical. In contrast, agentbased or concurrent languages, such as ? [1,2,3] or the closely related BioNetGen language [4,5,6,7,8,9,10], describe biological interactions in terms of rules, thereby avoiding the combinatorial explosion besetting differential equations. Rules are expressed in an intuitive graphical form that transparently represents biological knowledge. In this way, rules become a natural unit of model building, modification, and discussion. We illustrate this with a sizeable example obtained from refactoring two models of EGF receptor signalling that are based on differential equations [11,12]. An exciting aspect of the agent-based approach is that it naturally lends itself to the identification and analysis of the causal structures that deeply shape the dynamical, and perhaps even evolutionary, characteristics of complex distributed biological systems. In particular, one can adapt the notions of causality and conflict, familiar from concurrency theory, to ?, our representation language of choice. Using the EGF receptor model as an example, we show how causality enables the formalization of the colloquial concept of pathway and, perhaps more surprisingly, how conflict can be used to dissect the signalling dynamics to obtain a qualitative handle on the range of system behaviours. By taming the combinatorial explosion, and exposing the causal structures and key kinetic junctures in a model, agent- and rule-based representations hold promise for making modelling more powerful, more perspicuous, and of appeal to a wider audience.

Modeling and querying biomolecular interaction networks

We introduce a formalism to represent and analyze protein-protein and protein-DNA interaction networks. We illustrate the expressivity of this language, by proposing a formal counterpart of Kohns compilation on the mammalian cell-cycle control. This effectively turns an otherwise static knowledge into a discrete transition system incorporating a qualitative description of the dynamics. We then propose to use the computation tree logic (CTL) as a query language for querying the possible behaviors of the system. We provide examples of biologically relevant queries expressed in CTL about the mammalian cell-cycle control and show the effectiveness of symbolic model checking tools to evaluate CTL queries in this context.

Internal coarse-graining of molecular systems

Modelers of molecular signaling networks must cope with the combinatorial explosion of protein states generated by posttranslational modifications and complex formation. Rule-based models provide a powerful alternative to approaches that require explicit enumeration of all possible molecular species of a system. Such models consist of formal rules stipulating the (partial) contexts wherein specific protein–protein interactions occur. These contexts specify molecular patterns that are usually less detailed than molecular species. Yet, the execution of rule-based dynamics requires stochastic simulation, which can be very costly. It thus appears desirable to convert a rule-based model into a reduced system of differential equations by exploiting the granularity at which rules specify interactions. We present a formal (and automated) method for constructing a coarse-grained and self-consistent dynamical system aimed at molecular patterns that are distinguishable by the dynamics of the original system as posited by the rules. The method is formally sound and never requires the execution of the rule-based model. The coarse-grained variables do not depend on the values of the rate constants appearing in the rules, and typically form a system of greatly reduced dimension that can be amenable to numerical integration and further model reduction techniques.

The measurement calculus

Measurement-based quantum computation has emerged from the physics community as a new approach to quantum computation where the notion of measurement is the main driving force of computation. This is in contrast with the more traditional circuit model that is based on unitary operations. Among measurement-based quantum computation methods, the recently introduced one-way quantum computer [Raussendorf and Briegel 2001] stands out as fundamental. We develop a rigorous mathematical model underlying the one-way quantum computer and present a concrete syntax and operational semantics for programs, which we call patterns, and an algebra of these patterns derived from a denotational semantics. More importantly, we present a calculus for reasoning locally and compositionally about these patterns. We present a rewrite theory and prove a general standardization theorem which allows all patterns to be put in a semantically equivalent standard form. Standardization has far-reaching consequences: a new physical architecture based on performing all the entanglement in the beginning, parallelization by exposing the dependency structure of measurements and expressiveness theorems. Furthermore we formalize several other measurement-based models, for example, Teleportation, Phase and Pauli models and present compositional embeddings of them into and from the one-way model. This allows us to transfer all the theory we develop for the one-way model to these models. This shows that the framework we have developed has a general impact on measurement-based computation and is not just particular to the one-way quantum computer.

The Structure of Exponentials: Uncovering the Dynamics of Linear Logic Proofs

We construct the exponential graph of a proof π in (second order) linear logic, an artefact displaying the interdependencies of exponentials in π. Within this graph superfluous exponentials are defined, the removal of which is shown to yield a correct proof π▹ with essentially the same set of reductions.