Simone Martini

An extension of system F with subtyping

System F is a well-known typed λ-calculus with polymorphic types, which provides a basis for polymorphic programming languages. We study an extension of F, called F<:, that combines parametric polymorphism with subtyping.

Interacting with Networks: How Does Structure Relate to Controllability in Single-Leader, Consensus Networks?

As networked dynamical systems appear around us at an increasing rate, questions concerning how to manage and control such systems are becoming more important. Examples include multiagent robotics, distributed sensor networks, interconnected manufacturing chains, and data networks. In response to this growth, a significant body of work has emerged focusing on how to organize such networks to facilitate their control and make them amenable to human interactions. In this article, we summarize these activities by connecting the network topology, that is, the layout of the interconnections in the network, to the classic notion of controllability.

Controllability analysis of multi-agent systems using relaxed equitable partitions

This paper investigates how to make decentralised networks, amenable to external control, i.e., how to ensure that they are appropriately organised so that they can be effectively reprogrammed. In particular, we study networked systems whose interaction dynamics are given by a nearest-neighbour averaging rule, with one leader node providing the control input to the entire system. The main result is a necessary and sufficient condition for the controllability of such systems in terms of the graph topology. In particular, we give a graph theoretic interpretation of the controllability properties through the so-called relaxed equitable partition.

A Computational Interpretation of Modal Proofs

Proof theory of modal logics, though largely studied since the fifties, has always been a delicate subject, the main reason being the apparent impossibility to obtain elegant, natural systems for intensional operators (with the excellent exception of intuitionistic logic). For example Segerberg, not earlier than 1984 [5], observed that the Gentzen format, which works so well for truth functional and intuitionistic operators, cannot be a priori expected to remain valid for modal logics; carrying to the limit this observation one could even assert that ‘logics with no good proof theory are unnatural.’ In such a way we should mark as ‘unnatural’ all modal logics (with great delight of a large number of logicians!).

On Constructor Rewrite Systems and the Lambda-Calculus

We prove that orthogonal constructor term rewrite systems and lambda-calculus with weak (i.e., no reduction is allowed under the scope of a lambda-abstraction) call-by-value reduction can simulate each other with a linear overhead. In particular, weak call-by-value beta-reduction can be simulated by an orthogonal constructor term rewrite system in the same number of reduction steps. Conversely, each reduction in an orthogonal term rewrite system can be simulated by a constant number of weak call-by-value beta-reduction steps. This is relevant to implicit computational complexity, because the number of beta steps to normal form is polynomially related to the actual cost (that is, as performed on a Turing machine) of normalization, under weak call-by-value reduction. Orthogonal constructor term rewrite systems and lambda-calculus are thus both polynomially related to Turing machines, taking as notion of cost their natural parameters.

Controllability decompositions of networked systems through quotient graphs

In this paper we study decentralized, networked systems whose interaction dynamics are given by a nearest-neighbor averaging rule. By letting one node in the network take on the role of a leader in the sense that this node provides the control input to the entire system, we can ask questions concerning the controllability. In particular, we show that the controllable subspaces associated with such systems have a direct, graph theoretic interpretation in terms of so-called quotient graphs, providing us with a smaller, approximate bisimulation of the original network.

Derivational complexity is an invariant cost model

We show that in the context of orthogonal term rewriting systems, derivational complexity is an invariant cost model, both in innermost and in outermost reduction. This has some interesting consequences for (asymptotic) complexity analysis, since many existing methodologies only guarantee bounded derivational complexity.

Typing lambda terms in elementary logic with linear constraints

We present a type inference algorithm for λ-terms in Elementary Affine Logic using linear constraints. We prove that the algorithm is correct and complete.

The weak lambda calculus as a reasonable machine

We define a new cost model for the call-by-value lambda-calculus satisfying the invariance thesis. That is, under the proposed cost model, Turing machines and the call-by-value lambda-calculus can simulate each other within a polynomial time overhead. The model only relies on combinatorial properties of the usual beta-reduction, without any reference to a specific machine or evaluator. In particular, the cost of a single beta reduction is proportional to the difference between the size of the redex and the size of the reduct. In this way, the total cost of normalizing a lambda term will take into account the size of all intermediate results (as well as the number of steps to normal form).

On the fine structure of the exponential rule

We present natural deduction systems for fragments of intuitionistic linear logic obtained by dropping weakening and contractions also on !-preexed formulas. The systems are based on a two-dimensional generalization of the notion of sequent, which accounts for a clean formulation of the introduction/elimination rules of the modality. Moreover, the diierent subsystems are obtained in a modular way, by simple conditions on the elimination rule for !. For the proposed systems we introduce a notion of reduction and we prove a normalization theorem.