Correctness of Linear Logic Proof Structures is NL-Complete

23 pagesInternational audienceWe provide new correctness criteria for all fragments (multiplicative, exponential, additive) of linear logic. We use these criteria for proving that deciding the correctness of a linear logic proof structure is NL-complete

A formal study of Bernstein coefficients and polynomials

International audienceBernstein coefficients provide a discrete approximation of the behavior of a polynomial inside an interval. This can be used for example to isolate real roots of polynomials. We prove a criterion for the existence of a single root in an interval and the correctness of the de Casteljau algorithm to compute efficiently Bernstein coefficients

Handsome Proof-nets: R&B-Graphs, Perfect Matchings and Series-parallel Graphs

The main interest of this paper is to provide proof-nets, a proof syntax which identify proofs with the same meaning, with a standard graph-theoretical description. More precisely, we give two such descriptions which both view proof-nets as graphs endowed with a perfect matching, and in both cases the graphs corresponding to proofs are recognized by a simple correctnes- s criterion (with and without the so-called mix rule). The first description may be viewed as a graph-theoretical reformulation of usual proof-nets; nevertheless it allows us to recover various results on proof-nets as the corollaries of a single graph theoretical result. The second description, inspired from the first one, is more innovative: a proof-structure simply consists in the set of its axioms - the perfect matching - plus one single series-parallel graph (a.k.a cograph) which encodes the whole syntactical forest of the sequent. Unlike other approaches, every such graph, with out any further specification, corresponds to a proof structure, and this description identify proof structures which only differ up to the commutativity or associativity of the connectives, or because final disjunctions have been performed or not. Thus these proof-nets are even closer to the proofs themselves than usual proof-nets

A polyhedral approach to computing border bases

Border bases can be considered to be the natural extension of Gr\"obner bases that have several advantages. Unfortunately, to date the classical border basis algorithm relies on (degree-compatible) term orderings and implicitly on reduced Gr\"obner bases. We adapt the classical border basis algorithm to allow for calculating border bases for arbitrary degree-compatible order ideals, which is \emph{independent} from term orderings. Moreover, the algorithm also supports calculating degree-compatible order ideals with \emph{preference} on contained elements, even though finding a preferred order ideal is NP-hard. Effectively we retain degree-compatibility only to successively extend our computation degree-by-degree. The adaptation is based on our polyhedral characterization: order ideals that support a border basis correspond one-to-one to integral points of the order ideal polytope. This establishes a crucial connection between the ideal and the combinatorial structure of the associated factor spaces

Chu's construction: a proof-theoretic approach

The essential interaction between classical and intuitionistic features in the system of linear logic is best described in the language of category theory. Given a symmetric monoidal closed category C with products, the category C x C^op can be given the structure of a *-autonomous category by a special case of the Chu construction. The main result of the paper is to show that the intuitionistic translations induced by Girard trips on a proof net determine the functor from the free *-autonomous category on a set of atoms {P, P',...} to C x C^op, where C is the free monoidal closed category with products and coproducts on the set of atoms {P_O, P_I, P'_O, P'_I, ...} (a pair P_O, P_I in C for each atom P of A)