Stefano Guerrini

Parsing MELL Proof Nets

We propose a new formulation for full (weakening and constants included) multiplicative and exponential (MELL) proof nets, allowing a complete set of rewriting rules to parse them. The recognizing grammar defined by such a rewriting system (confluent and strong normalizing on the new proof nets) gives a correctness criterion that we show equivalent to the Danos-Regnier one.

Strong Normalization of Proof Nets Modulo Structural Congruences

This paper proposes a notion of reduction for the proof nets of Linear Logic modulo an equivalence relation on the contraction links, that essentially amounts to consider the contraction as an associative commutative binary operator that can float freely in and out of proof net boxes. The need for such a system comes, on one side, from the desire to make proof nets an even more parallel syntax for Linear Logic, and on the other side from the application of proof nets to l-calculus with or without explicit substitutions, which needs a notion of reduction more flexible than those present in the literature. The main result of the paper is that this relaxed notion of rewriting is still strongly normalizing.

Proof nets, garbage, and computations

We study the problem of local and asynchronous computation in the context of multiplicative exponential linear logic (MELL) proof nets. The main novelty is in a complete set of rewriting rules for cut-elimination in presence of weakening (which requires garbage collection). The proposed reduction system is strongly normalizing and confluent.

Coherence for Sharing Proof Nets

Sharing graphs are a way of representing linear logic proof-nets in such a way that their reduction never duplicates a redex. In their usual presentations, they present a problem of coherence: if the proof-net N reduces by standard cut-elimination to N′, then, by reducing the sharing graph of N we do not obtain the sharing graph of N′. We solve this problem by changing the way the information is coded into sharing graphs and introducing a new reduction rule (absorption). The rewriting system is confluent and terminating.

An analysis of (linear) exponentials based on extended sequents

We apply the sequents approach to the analysis of several logics derived from linear logic In particular we present a uniform formal system for Linear Logic Elementary Linear Logic and Light Linear Logic

A general theory of sharing graphs

Abstract Sharing graphs are the structures introduced by Lamping for the implementation of optimal reductions of lambda-calculus. Gonthiers reformulation of Lampings technique inside Geometry of Interaction and Asperti and Laneves work on Interaction Systems have shown that sharing graphs implement a wide class of calculi. We give a semantical characterization of sharing graphs independent of the calculus to be implemented. By means of an algebraic interpretation of sharing graphs, we define a subclass of them, the so-called proper sharing graphs, on which the usual notion of graph unfolding gives a lower semi-lattice. The least-shared-instance of a proper sharing graph is its maximal proper unfolding, that is, the unique proper unshared graph that the unfolding partial order associates to it. Exploiting a simulation property between the reductions of a proper sharing graph and the reductions of its least-shared-instance, we prove that the read-back of a proper sharing graph can be computed via an unfolding or read-back reduction. Proper sharing graphs implement in a distributed and local way any graph calculus with a global reduction in the style of the beta-rule of lambda-calculus. In fact, correctness of the sharing implementation requires the so-called box nesting property only, or equivalently, it is proved under the only assumption that two redexes never partially overlap. Thus, sharing graphs constitute an abstract machine, say the sharing graph machine, that seems to be the most natural low-level computational model for functional languages. Moreover, Levys optimal reductions correspond to lazy reductions of that sharing machine. We stress on the proof strategy followed in the paper: it rests on an amazing interplay between standard rewriting system properties (strong normalization, confluence, and unique normal form) and algebraic properties definable via the techniques of Geometry of Interaction.

A linear algorithm for MLL proof net correctness and sequentialization

The paper presents in full detail the first linear algorithm given in the literature (Guerrini (1999) 6) implementing proof structure correctness for multiplicative linear logic without units. The algorithm is essentially a reformulation of the Danos contractibility criterion in terms of a sort of unification. As for term unification, a direct implementation of the unification criterion leads to a quasi-linear algorithm. Linearity is obtained after observing that the disjoint-set union-find at the core of the unification criterion is a special case of union-find with a real linear time solution.

Parallel depth-merge: A paradigm for hidden surface removal

Abstract Visualization is a key aspect in geometric modeling. Since the processing required for realistic display is often considerable, the design of parallel visualization algorithms is a trying taks. In this paper a new parallel approach to Hidden Surface Removal, called Parallel Depth-Merge (PDM), is proposed. PDM, intrinsically parallel, couples any already known HSR object-space algorithm with an image-space hierarchical merger. Resulting data are returned in a format well-suited both for plotting on vector devices and for use in shadow computations and shading algorithms. The proposed approach is designed for MIMD architectures, which use either shared or nonshared memory, and it does not require any special purpose hardware. The term paradigm is used to stress that the proposed approach is not based on the parallel redesigning of a known HSR algorithm. Conversely, it can be considered a general parallel solution to the problem, which allows different HSR object-space algorithms to be implemented uniformly on different MIMD architectures.

Sharing Implementations of Graph Rewriting Systems

Sharing graphs are a brilliant solution to the implementation of Levy optimal reductions of λ-calculus. Sharing graphs are interesting on their own and optimal sharing reductions are just a particular reduction strategy of a more general sharing reduction system.The paper is a gentle introduction to sharing graphs and tries to confute some of the common myths on the difficulty of sharing implementations.

Proofs, tests and continuation passing style

The concept of syntactical duality is central in logic. In particular, the duality defined by classical negation, or more syntactically by left and right in sequents, has been widely used to relate logic and computations. We study the proof/test duality proposed by Girard in his 1999 paper on the meaning of logical rules. In detail, starting from the notion of “test” proposed by Girard, we develop a notion of test for intuitionistic logic and we give a complete deductive system whose computational interpretation is the target language of the call-by-value and call-by-name continuation passing style translations.