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

Abstract

We are interested in the expressiveness of constraints represented by general first order formulae, with equality as unique relational symbol and functional symbols taken from an infinite set F. The chosen domain is the set of trees whose nodes, in possibly infinite number, are labeled by elements of F. The operation linked to each element f of F is the mapping (a ₁ ,...,a _n ) → b, where b is the tree whose initial node is labeled f and whose sequence of daughters is a ₁ ,...,a _n .

We first consider 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 constraints, inspired by a constructive proof of a complexity result by Pawel Mielniczuk. This family involves the huge number α(k), obtained by evaluating top down a power tower of 2’s, of height k. With elements of this family, of sizes at most proportional to k, we define a finite tree having α(k) nodes, and we express the result of a Prolog machine executing at most α(k) instructions.

By replacing the Prolog machine by a Turing machine we rediscover the following result of Sergei Vorobyov: the complexity of an algorithm, deciding whether a constraint without free variables is true, cannot be bounded above by a function obtained by finite composition of elementary functions including exponentiation.

Finally, taking advantage of the fact that we have at our disposal an algorithm for solving such constraints in all their generality, we produce a set of benchmarks for separating feasible examples from purely speculative ones. Among others we solve constraints involving alternated sequences of more than 160 quantifiers.