Alexandre Boudet

Diophantine Equations, Presburger Arithmetic and Finite Automata

We investigate the use of Buchis techniques for Presburger arithmetic. More precisely, we show how to efficiently compute an automaton which accepts the set of solutions of a linear Diophantine equation (suitably encoded). Following Buchi, this gives a decision technique for the whole Presburger arithmetic. We show however how to compute more efficiently the automaton in the case of disequalities, inequalities and systems of linear Diophantine problems. We also show that such an “automaton algorithm” has a nearly optimal worst case complexity, both for the existential fragment and for the whole first-order theory.

Combining unification algorithms

Abstract An algorithm is presented for solving equations in a combination of arbitrary theories over disjoint sets of function symbols. It is an extension of (Boudet el al . 1989) in which the problem was treated for the combination of an arbitrary and a simple theory. The algorithm consists of a set of transformation rules that simplify a unification problem until a solved form is obtained. Each rule is shown to preserve solutions, and solved problems are unification problems in normal form. The rules terminate for any control that delays replacement until the end. The algorithm is more efficient than Schmidt-Schaus (1990) because nondeterministic branching is performed only when necessary, that is when theory clashes or compound cycles are encountered. The results presented here come from the authors thesis (Boudet (1990a)).

Unification in Boolean Rings and Abelian Groups

Abstract A complete unification algorithm is presented for the combination of two theories E in T(F,X) and E ’ in T(F’,X) where F and F ’ denote two disjoint sets of function symbols, E and E ’ are arbitrary equational theories for which are given, for E : a complete unification algorithm for terms in T(F ∪ C,X ), where C is a set of free constants and a complete constant elimination algorithm for eliminating a constant c from a term s; for E ’: a complete unification algorithm. E ’ is supposed to be cycle free, i.e., equations x=t where x is a variable occurring in t have no E ’-solution. The method adapts to unification of infinite trees. It is applied to two well-known open problems, when E is the theory of Boolean Rings or the theory of Abelian Groups, and E ’ is the free theory. Our interest to Boolean Rings originates in VLSI verification.

A new AC unification algorithm with an algorithm for solving systems of diophantine equations

A novel AC-unification algorithm is presented. A combination technique for regular collapse-free theories is provided along the line developed by A. Boudet et al. (1989). The number of calls to the diophantine equations solver is bounded by the number of AC symbols times the number of shared variables. The rest of the algorithm being linear, this gives a much better idea of how the complexity of AC unification is related to the complexity of solving linear diophantine equations. The termination proof is surprisingly easy. Finally, systems of constraint linear diophantine equations can be solved, rather than one equation at a time, using an algorithm which extends Fortenbachers algorithm to an arbitrary dimension. This allows a much more efficient use of the constraints than in the standard case.<<ETX>>

Unification in a Combination of Equational Theories: an Efficient Algorithm

An algorithm is presented for solving equations in a combination of arbitrary theories with disjoint sets of function symbols. It is an extension of [3] in which the problem was treated for the combination of an arbitrary and a simple theory. The algorithm consists in a set of transformation rules that simplify a unification problem until a solved form is obtained. Each rule is shown to preserve solutions, and solved problems are unification problems in normal form. The rules terminate for any control that delays replacement until the end. The algorithm is more efficient than [13] because nondeterministic branching is performed only when necessary, that is when theory clashes or compound cycles are encountered.

About the Theory of Tree Embedding

We show that the positive existential fragment of the theory of tree embedding is decidable.

Unification in Order-Sorted Algebras with Overloading

We present an algorithm for unification in the combination of a theory Th1 and one of its overloaded extensions Th2 in the order-sorted framework. This problem is a particular combination problem where the signatures are not disjoint. A major consequence is that an equality proof between two pure terms in Th1 may need the use of an axiom of Th2. This makes the usual combination techniques incomplete, in particular the solving of pure equations in the theory to which they belong. To solve the problem, we need a separated normal form as well as a complete set of normalizing substitutions.

Competing for the AC -unification race

We describe our implementation of the unification algorithm for terms involving some associative-commutative operators plus free function symbols described by Boudetet al. The first goal of this implementation is efficiency, more precisely competing for theAC Unification Race. Although our implementation has been designed for good performance when applied to non-elementaryAC-unification problems, it is also very efficient on elementary problems. Our implementation, written in C and running on Sun workstations, is to be compared with the implementations in LISP, on Symbolics LIPS machines.

AC-Complete Unification and its Application to Theorem Proving

The inefficiency of AC-completion is mainly due to the doubly exponential number of AC-unifiers and thereby of critical pairs generated. We present AC-complete E-unification, a new technique whose goal is to reduce the number of AC-critical pairs inferred by performing unification in a extension E of AC (e.g. ACU, Abelian groups, Boolean rings, ...) in the process of normalized completion [24, 25]. The idea is to represent complete sets of AC-unifiers by (smaller) sets of E-unifiers. Not only do the theories E used for unification have exponentially fewer most general unifiers than AC, but one can remove from a complete set of E-unifiers those solutions which have no E- instance which is an AC-unifier.

On n-Syntactic Equational Theories

We define the n-syntactic theories as a natural extension of the syntactic theories. A n-syntactic theory is an equational theory which admits a finite presentation in which every proof can be performed with at most n applications of an axiom at the root, but no finite presentation in which every proof can be performed with at most n − 1 applications of an axiom at the root. The n-syntactic theories inherit the good properties of the syntactic theories for solving the word problem, or matching or unification problems. We show that for any integer n ≥ 1, there exists a n-syntactic theory.