Christophe Tollu

Generating plans in linear logic I: actions as proofs

Abstract There is an increasing interest in the relation between logic and the changes involved in reasoning and, specifically, in plan generation. Up to now, several attempts in this direction have been made, either by embedding actions into a classical framework or by using nonstandard formalisms. We think that these attempts, though promising, miss their objectives, for a lack of a suitable logic, and that the effort must be pursued. In this paper, we show how to obtain a strong and clean correspondence between proofs and sequences of actions by using only Girards linear logic, eliminating from the classical logic the structural rules which are not adapted to our purpose. A theorem is presented which expresses the new adequacy between proofs and actions.

Linear Constraint Query Languages: Expressive Power and Complexity

We give an AC0 upper bound on the complexity of first-oder queries over (infinite) databases defined by restricted linear constraints. This result enables us to deduce the non-expressibility of various usual queries, such as the parity of the cardinality of a set or the connectivity of a graph in first-order logic with linear constraints.

Generating plans in linear logic

We dealt with a well-delineated domain of planning and, thanks to linear logic, we gave it an adequate logical characterization, so that we managed to solve the completeness problem. Although this domain may appear restricting at first glance, it seems that it includes most of the examples in the literature on plan analysis and it gives rise to important questions which are quite satisfactorily handled by the formalism we proposed. On the other hand, we hope that our work modestly contributed to a deeper intuitive understanding of linear logic. Besides, we think that integrating other features of linear logic, such as the ones attached to linear implication (--o) or the bounded modalities ?n and !n will enable to cope with more realistic situations.

Asymptotic probabilities of languages with generalized quantifiers

The impact of adding certain families of generalized quantifiers to first-order logic (FO) on the asymptotic behavior of sentences is studied. All the results are stated and proved for languages disallowing free variables in the scope of generalized quantifiers. For a class K of finite structures closed under isomorphism, the quantifier Q/sub K/ is said to be strongly monotonic, sm, if membership in the class is preserved under a loose form of extensions. The first theorem (O/1 law for FO with any set of sm quantifiers) subsumes a previous criterion for proving that almost no graphs satisfy a given property. A O/1 law for FO with Hartig quantifiers (equicardinality quantifiers) and a limit law for a fragment of FO with Rescher quantifiers (expressing inequalities of cardinalities) are also established. It is also proved that the O/1 law fails for the extension of FO with Hartig quantifiers if the above syntactic restriction is relaxed, giving the best upper bound for the existence of a O/1 law for FO with Hartig quantifiers.<<ETX>>

Query Languages with Counters

We investigate the expressive power of query languages with counting ability. We define a LOGSPACE extension of first order logic and a PTIME extension of fixpoint logic with counters. We develop specific techniques, such as games, for dealing with languages with counters and therefore integers. We prove in particular that the arity of the tuples which are counted induces a strict expressivity hierarchy. We also establish results about the asymptotic probabilities of sentences with counters. In particular we show that first order logic with comparison of the cardinalities of relations has a. 0/1 law.

Factorisation of Littlewood--Richardson coefficients

The hive model is used to show that the saturation of any essential Horn inequality leads to the factorisation of Littlewood-Richardson coefficients. The proof is based on the use of combinatorial objects known as puzzles. These are shown not only to account for the origin of Horn inequalities, but also to determine the constraints on hives that lead to factorisation. Defining a primitive Littlewood-Richardson coefficient to be one for which all essential Horn inequalities are strict, it is shown that every Littlewood-Richardson coefficient can be expressed as a product of primitive coefficients. Precisely the same result is shown to apply to the polynomials defined by stretched Littlewood-Richardson coefficients.

HOPF ALGEBRAS OF DIAGRAMS

We investigate several generalizations of the Hopf algebra MQSym whose constructions come from labelings of special diagrams in bijection with packed matrices. Their products come either from the Hopf algebras WSym or WQSym, respectively built on integer set partitions and set compositions. Realizations on bi-word are exhibited, and it is shown how these algebras fit into a commutative diagram. Hopf deformations and dendriform structures are also considered for some algebras in the picture.

Hyperdeterminantal computation for the Laughlin wavefunction

The decomposition of the Laughlin wavefunction in the Slater orthogonal basis appears in the discussion on the second-quantized form of the Laughlin states and is straightforwardly equivalent to the decomposition of the even powers of the Vandermonde determinants in the Schur basis. Such a computation is notoriously difficult and the coefficients of the expansion have not yet been interpreted. In our paper, we give an expression of these coefficients in terms of hyperdeterminants of sparse tensors. We use this result to construct an algorithm allowing us to compute one coefficient of the development without computing the others. Thanks to a program in C, we performed the calculation for the square of the Vandermonde up to an alphabet of 11 letters.