Set Of Ground Terms Research Articles

Termination and normalisation under strategy Proofs in ELAN

In this paper, we review several methods for proving termination of rewrite programs or answer questions about normal forms. We concentrate on the methods that have been designed in ELAN and which are useful for studying termination and normal forms of ELAN programs. We address several problems: general termination of a rewrite program, innermost termination on sets of ground terms, computation or approximation of the set of normal forms, modular termination of the union of rewrite systems, estimation of the number of normal forms for a non-deterministic rewrite program with strategies. For each point we briefly present one method, explain its scope and give references to other methods concerning the same problem, before describing the implementation in ELAN. We conclude with some open questions.

Set Constraints in Some Equational Theories

Set constraints are relations between sets of ground terms over a given alphabet. They give a natural formalism for many problems in program analysis, type inference, order-sorted unification, and constraint logic programming. In this paper we start studies of set constraints in the environment given by equational specifications. We show that in the case of associativity (i.e., in free monoids) as well as in the case of associativity and commutativity (i.e., in commutative monoids) the problem of consistency of systems of set constraints is undecidable; in linear nonerasing shallow theories the consistency of systems of positive set constraints is NEXPTIME-complete and in linear shallow theories the problem for positive and negative set constraints is decidable.

Minimal equational representations of recognizable tree languages

A tree language is congruential if it is the union of finitely many classes of a finitely generated congruence on the term algebra. It is well known that congruential tree languages are the same as recognizable tree languages. An equational representation is an ordered pair (E, P) , where E is either a ground term equation system or a ground term rewriting system, and P is a finite set of ground terms. We say that (E, P) represents the congruential tree language L which is the union of those ?* E -classes containing an element of P, i.e., for which L=⋃{[p]? * E ∣p∈P}. We define two sorts of minimality for equational representations. We introduce the cardinality vector (∣E∣, ∣P∣) of an equational representation (E, P). Let ? l and ? a denote the lexicographic and antilexicographic orders on the set of ordered pairs of nonnegative integers, respectively. Let L be a congruential tree language. An equational representation (E, P) of L with ? l -minimal (? a -minimal) cardinality vector is called ? l -minimal (? a -minimal). We compute, for an L given by a deterministic bottom-up tree automaton, both a ? l -minimal and a ? a -minimal equational representation of L.

Partial Automata and Finitely Generated Congruences: An Extension of Nerode's Theorem

Let T_Sigma be the set of ground terms over a finite ranked alphabet Sigma. We define <em> partial autornata on</em> T_Sigma and prove that the finitely generated congruences on T_Sigma are in one-to one correspondence (up to isomorphism) with the finite partial automata on Sigma with no inaccessible and no inessential states. We give an application in term rewriting: every ground term rewrite system has a canonical equivalent system that can be constructed in polynomial time.