Sylvain Salvati

Vector addition tree automata

We introduce a new class of automata, which we call vector addition tree automata. These automata are a natural generalization of vector addition systems with states, which are themselves equivalent to Petri-nets. Then, we prove that the decidability of provability in multiplicative exponential linear logic (which is an open problem) is equivalent to the decidability of the reachability relation for vector addition tree automata. This result generalizes the well-known connection existing between Petri nets and the !-horn fragment of multiplicative exponential linear logic.

Krivine machines and higher-order schemes

We propose a new approach to analysing higher-order recursive schemes. Many results in the literature use automata models generalising pushdown automata, most notably higher-order pushdown automata with collapse (CPDA). Instead, we propose to use the Krivine machine model. Compared to CPDA, this model is closer to lambda-calculus, and incorporates nicely many invariants of computations, as for example the typing information. The usefulness of the proposed approach is demonstrated with new proofs of two central results in the field: the decidability of the local and global model checking problems for higher-order schemes with respect to the mu-calculus.

Recognizability in the Simply Typed Lambda-Calculus

We define a notion of recognizable sets of simply typed *** -terms that extends the notion of recognizable sets of strings or trees. This definition is based on finite models. Using intersection types, we generalize the notions of automata for strings and trees so as to grasp recognizability for *** -terms. We then expose the closure properties of this notion and present some of its applications.

The T-calculus: towards a structured programing of (musical) time and space

In the field of music system programming, the T-calculus is a proposal for combining space modeling and time programming into a single programming feature: spatiotemporal tiled programming. Based on a solid algebraic model, it aims at decomposing every operation on musical objects into the sequence of a synchronization operation that describes how objects are positioned one with respect the other, and a fusion operation that describes how their values are then combined. A first simple version of such a tiled calculus is presented and studied in this paper.

Minimalist grammars in the light of logic

In this paper, we aim at understanding the derivations of minimalist grammars without the shortest move constraint. This leads us to study the relationship of those derivations with logic. In particular we show that the membership problem of minimalist grammars without the shortest move constraint is as difficult as provability in Multiplicative Exponential Linear Logic. As a byproduct, this result gives us a new representation of those derivations with linear λ-terms. We show how to interpret those terms in a homomorphic way so as to recover the sentence they analyse. As the homorphisms we describe are rather evolved, we turn to a proof-net representation and explain how Monadic Second Order Logic and related techniques allow us both to define those proof-nets and to retrieve the sentence they analyse.

Recursive schemes, krivine machines, and collapsible pushdown automata

Higher-order recursive schemes are an interesting method of approximating program semantics. The semantics of a scheme is an infinite tree labeled with built-in constants. This tree represents the meaning of the program up to the meaning of built-in constants. It is much easier to reason about properties of such trees than properties of interpreted programs. Moreover some interesting properties of programs are already expressible on the level of these trees. Collapsible pushdown automata (CPDA) give another way of generating the same class of trees. We present a relatively simple translation from recursive schemes to CPDA using Krivine machines as an intermediate step. The later are general machines for describing computation of the weak head normal form in the lambda-calculus. They provide the notions of closure and environment that facilitate reasoning about computation.

Evaluation is MSOL compatible

We consider simply-typed lambda calculus with fixpoint operators. Evaluation of a term gives as a result the Bohm tree of the term. We show that evaluation is compatible with monadic second-order logic (MSOL). This means that for a fixed finite vocabulary of terms, the MSOL properties of Bohm trees of terms are effectively MSOL properties of terms themselves. Theorems of this kind have been known for some graph operations: unfolding, and Muchnik iteration. Similarly to those results, our main theorem has diverse applications. It can be used to show decidability results, to construct classes of graphs with decidable MSOL theory, or to obtain MSOL formulas expressing behavioral properties of terms. Another application is decidability of a control-flow synthesis problem.

Loader and urzyczyn are logically related

In simply typed λ-calculus with one ground type the following theorem due to Loader holds. (i) Given the full model

Typing Weak MSOL Properties

\mathcal F

A model for behavioural properties of higher-order programs

over a finite set, the question whether some element