Thi-Bich-Hanh Dao

Expressiveness of Full First-Order Constraints in the Algebra of Finite or Infinite Trees

We are interested in the expressiveness of constraints represented by general first order formulae, with equality as unique relation symbol and function symbols taken from an infinite set F. The chosen domain is the set of trees whose nodes, in possibly infinite number, are labelled by elements of F. The operation linked to each element f of F is the mapping (a1,..., an) ↦ b, where b is the tree whose initial node is labelled f and whose sequence of daughters is a1,..., an.We first consider tree constraints involving long alternated sequences of quantifiers ∃∀∃∀.... We show how to express winning positions of two-person games with such constraints and apply our results to two examples.We then construct a family of strongly expressive tree constraints, inspired by a constructive proof of a complexity result by Pawel Mielniczuk. This family involves the huge number α(k), obtained by top down evaluating a power tower of 2s, of height k. By a tree constraint of size proportional to k, it is then possible to define a tree having exactly α(k) nodes or to express the multiplication table computed by a Prolog machine executing up to α(k) instructions.By replacing the Prolog machine with a Turing machine we show the quasi-universality of tree constraints, that is to say, the ability to concisely describe trees which the most powerful machine will never have time to compute. We also rediscover the following result of Sergei Vorobyov: the complexity of an algorithm, deciding whether a tree constraint without free variables is true, cannot be bounded above by a function obtained from finite composition of simple functions including exponentiation.Finally, taking advantage of the fact that we have at our disposal an algorithm for solving such constraints in all their generalities, we produce a set of benchmarks for separating feasible examples from purely speculative ones. Among others we notice that it is possible to solve a constraint of 5000 symbols involving 160 alternating quantifiers.

A declarative framework for constrained clustering

In recent years, clustering has been extended to constrained clustering, so as to integrate knowledge on objects or on clusters, but adding such constraints generally requires to develop new algorithms. We propose a declarative and generic framework, based on Constraint Programming, which enables to design clustering tasks by specifying an optimization criterion and some constraints either on the clusters or on pairs of objects. In our framework, several classical optimization criteria are considered and they can be coupled with different kinds of constraints. Relying on Constraint Programming has two main advantages: the declarativity, which enables to easily add new constraints and the ability to find an optimal solution satisfying all the constraints (when there exists one). On the other hand, computation time depends on the constraints and on their ability to reduce the domain of variables, thus avoiding an exhaustive search.

Constrained clustering by constraint programming

Cluster analysis is an important task in Data Mining with hundreds of different approaches in the literature. Since the last decade, the cluster analysis has been extended to constrained clustering, also called semi-supervised clustering, so as to integrate previous knowledge on data to clustering algorithms. In this dissertation, we explore Constraint Programming (CP) for solving the task of constrained clustering. The main principles in CP are: (1) users specify declaratively the problem in a Constraint Satisfaction Problem; (2) solvers search for solutions by constraint propagation and search. Relying on CP has two main advantages: the declarativity, which enables to easily add new constraints and the ability to find an optimal solution satisfying all the constraints (when there exists one). We propose two models based on CP to address constrained clustering tasks. The models are flexible and general and supports instance-level constraints and different cluster-level constraints. It also allows the users to choose among different optimization criteria. In order to improve the efficiency, different aspects have been studied in the dissertation. Experiments on various classical datasets show that our models are competitive with other exact approaches. We show that our models can easily be embedded in a more general process and we illustrate this on the problem of finding the Pareto front of a bi-criterion optimization process.

A model-theoretic framework for grammaticality judgements

Although the observation of grammaticality judgements is well acknowledged, their formal representation faces problems of different kinds: linguistic, psycholinguistic, logical, computational. In this paper we focus on addressing some of the logical and computational aspects, relegating the linguistic and psycholinguistic ones in the parameter space. We introduce a model-theoretic interpretation of Property Grammars, which lets us formulate numerical accounts of grammaticality judgements. Such a representation allows for both clear-cut binary judgements, and graded judgements. We discriminate between problems of Intersective Gradience (i.e., concerned with choosing the syntactic category of a model among a set of candidates) and problems of Subsective Gradience (i.e., concerned with estimating the degree of grammatical acceptability of a model). Intersective Gradience is addressed as an optimisation problem, while Subsective Gradience is addressed as an approximation problem.

Theory of finite or infinite trees revisited

We present in this paper a first-order axiomatization of an extended theory T of finite or infinite trees, built on a signature containing an infinite set of function symbols and a relation finite(t), which enables to distinguish between finite and infinite trees. We show that T has at least one model and prove its completeness by giving not only a decision procedure, but a full first-order constraint solver that gives clear and explicit solutions for any first-order constraint satisfaction problem in T. The solver is given in the form of 16 rewriting rules that transform any first-order constraint ϕ into an equivalent disjunction φ of simple formulas such that φ is either the formula true or the formula false or a formula having at least one free variable, being equivalent neither to true nor to false and where the solutions of the free variables are expressed in a clear and explicit way. The correctness of our rules implies the completeness of T. We also describe an implementation of our algorithm in CHR (Constraint Handling Rules) and compare the performance with an implementation in C++ and that of a recent decision procedure for decomposable theories.

Constrained Minimum Sum of Squares Clustering by Constraint Programming

The Within-Cluster Sum of Squares (WCSS) is the most used criterion in cluster analysis. Optimizing this criterion is proved to be NP-Hard and has been studied by different communities. On the other hand, Constrained Clustering allowing to integrate previous user knowledge in the clustering process has received much attention this last decade. As far as we know, there is a single approach that aims at finding the optimal solution for the WCSS criterion and that integrates different kinds of user constraints. This method is based on integer linear programming and column generation. In this paper, we propose a global optimization constraint for this criterion and develop a filtering algorithm. It is integrated in our Constraint Programming general and declarative framework for Constrained Clustering. Experiments on classic datasets show that our approach outperforms the exact approach based on integer linear programming and column generation.

Property grammar parsing seen as a constraint optimization problem

Blache [1] introduced Property Grammar as a formalism where linguistic information is represented in terms of non hierarchical constraints. This feature gives it an adequate expressive power to handle complex linguistic phenomena, such as long distance dependencies, and also agrammatical sentences [2]. Recently, Duchier et al. [3] proposed a model-theoretic semantics for property grammar. The present paper follows up on that work and explains how to turn such a formalization into a constraint optimization problem, solvable using constraint programming techniques. This naturally leads to an implementation of a fully constraint-based parser for property grammars.

Extension of first-order theories into trees

We present in this paper an automatic way to combine any first-order theory T with the theory of finite or infinite trees. First of all, we present a new class of theories that we call zero-infinite-decomposable and show that every decomposable theory T accepts a decision procedure in the form of six rewriting which for every first order proposition give either true or false in T. We present then the axiomatization T* of the extension of T into trees and show that if T is flexible then its extension into trees T* is zero-infinite-decomposable and thus complete. The flexible theories are theories having elegant properties which enable us to eliminate quantifiers in particular cases.

Intermediate (learned) consistencies

What makes a good consistency? Depending on the constraint, it may be a good pruning power or a low computational cost. By using machine learning techniques (search in an hypothesis space and clustering), we propose to define new automatically generated solvers which form a sequence of consistencies intermediate between bound- and arc-consistency.

Repetitive branch-and-bound using constraint programming for constrained minimum sum-of-squares clustering

Minimum sum-of-squares clustering (MSSC) is a widely studied task and numerous approximate as well as a number of exact algorithms have been developed for it. Recently the interest of integrating prior knowledge in data mining has been shown, and much attention has gone into incorporating user constraints into clustering algorithms in a generic way. Exact methods for MSSC using integer linear programming or constraint programming have been shown to be able to incorporate a wide range of constraints. However, a better performing method for unconstrained exact clustering is the Repetitive Branch-and-Bound Algorithm (RBBA) algorithm. In this paper we show that both approaches can be combined. The key idea is to replace the internal branch-and-bound of RBBA by a constraint programming solver, and use it to compute tight lower and upper bounds. To achieve this, we integrate the computed bounds into the solver using a novel constraint. Our method combines the best of both worlds, and is generic as well as performing better than other exact constrained methods. Furthermore, we show that our method can be used for multi-objective MSSC clustering, including constrained multi-objective clustering.