Treffer: Expressiveness of full first-order constraints in the algebra of finite or infinite trees
Laboratoire d'Informatique Fondamentale d'Orléans, Université d'Orléans, France
CC BY 4.0
Sauf mention contraire ci-dessus, le contenu de cette notice bibliographique peut être utilisé dans le cadre d’une licence CC BY 4.0 Inist-CNRS / Unless otherwise stated above, the content of this bibliographic record may be used under a CC BY 4.0 licence by Inist-CNRS / A menos que se haya señalado antes, el contenido de este registro bibliográfico puede ser utilizado al amparo de una licencia CC BY 4.0 Inist-CNRS
Operational research. Management
Weitere Informationen
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 modes, 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 h 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 2's. 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 Vorohyov: 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.