Induction (logique)

L'induction est historiquement le nom utilisé pour signifier un genre de raisonnement qui se propose de chercher des lois générales à partir de l'observation de faits particuliers, sur une base probabiliste. Remarque : Bien qu'associée dans le titre de cet article à la logique, la présentation qui suit correspond surtout à la notion bayésienne, utilisée consciemment ou non, de l'induction. Dans les ouvrages anglo-saxons de mathématiques, logique et informatique, l’induction complète, désignée sous le nom d’induction (faux-ami) désigne la récurrence, aussi bien dans le raisonnement par récurrence que dans les définitions par récurrence. Le terme est souvent employé pour les généralisations de la récurrence aux bons ordres et aux relations bien fondées. L'induction ne doit pas être confondue avec l'abduction qui, en raisonnement automatisé, est un mode de raisonnement qui vise à émettre une hypothèse pour expliquer un fait. Poussée à son paroxysme, l'induction peut conduire à l'indu

A Decision Procedure for (Co)datatypes in SMT Solvers

Andrew Joseph Reynolds

We present a decision procedure that combines reasoning about datatypes and codatatypes. The dual of the acyclicity rule for datatypes is a uniqueness rule that identifies observationally equal codatatype values, including cyclic values. The procedure decides universal problems and is composable via the Nelson-Oppen method. It has been implemented in CVC4, a state-of-the-art SMT solver. An evaluation based on problems generated from theories developed with Isabelle demonstrates the potential of the procedure.

Springer-Verlag Berlin2015

Constraint consistency techniques for continuous domains

Jamila Sam

Constraint Satisfaction Problems (CSPs) are ubiquitous in computer science. Many problems, ranging from resource allocation and scheduling to fault diagnosis and design, involve constraint satisfaction as an essential component. A CSP is given by a set of variables and constraints on small subsets of these variables. It is solved by finding assignments of values to the variables such that all constraints are satisfied. In its most general form, a CSP is combinatorial and complex. In this thesis, we consider constraint satisfaction problems with variables in continuous, numerical domains. Contrary to most existing techniques, which focus on computing a single optimal solution, we address the problem of computing a compact representation of the space of all solutions that satisfy the constraints. This has the advantage that no optimization criterion has to be formulated beforehand, and that the space of possibilities can be explored systematically. In certain applications, such as diagnosis and design, these advantages are crucial. In consistency techniques, the solution space is represented by labels assigned to individual variables or combinations of variables. When the labeling is globally consistent, each label contains only those values or combinations of values which appear in at least one solution. This kind of labeling is a compact, sound and complete representation of the solution space, and can be combined with other reasoning methods. In practice, computing a globally consistent labeling is too complex. This is usually tackled in two ways. One way is to enforce consistencies locally, using propagation algorithms. This prunes the search space and hence reduces the subsequent search effort. The other way is to identify simplifying properties which guarantee that global consistency can be enforced tractably using local propagation algorithms. When constraints are represented by mathematical expressions, implementing local consistency algorithms is difficult because it requires tools for solving arbitrary systems of equations. In this thesis, we propose to approximate feasible solution regions by 2k-trees, thus providing a means of combining constraints logically rather than numerically. This representation, commonly used in computer vision and image processing, avoids using complex mathematical tools. We propose simple and stable algorithms for computing labels of arbitrary degrees of consistency using this representation. For binary constraints, it is known that simplifying convexity properties reduces the complexity of solving a CSP. These properties guarantee that local degrees of consistency are sufficient to ensure global consistency. We show how, in continuous domains, these results can be generalized to ternary and in fact arbitrary n-ary constraints. This leads to polynomial-time algorithms for computing globally consistent labels for a large class of constraint satisfaction problems with continuous variables. We describe and justify our representation of constraints and our consistency algorithms. We also give a complete analysis of the theoretical results we present. Finally, the developed techniques are illustrated using practical examples.

EPFL1995

Constraint consistency techniques for continuous domains

Jamila Sam

EPFL1995

Popper's criticism of the inductive method

A Decision Procedure for (Co)datatypes in SMT Solvers

Constraint consistency techniques for continuous domains

Constraint consistency techniques for continuous domains

A Decision Procedure for (Co)datatypes in SMT Solvers

Popper's criticism of the inductive method