Abstract not found

Abstract not found

. In a previous paper we have introduced a method that allows one to combine decision procedures for unifiability in disjoint equational theories. Lately, it has turned out that the prerequisite for this method to apply---namely that unification with so-called linear constant restrictions is decidable in the single theories---is equivalent to requiring decidability of the positive fragment of the first order theory of the equational theories. Thus, the combination method can also be seen as a tool for combining decision procedures for positive theories of free algebras defined by equational theories. Complementing this logical point of view, the present paper isolates an abstract algebraic property of free algebras--- called combinability---that clarifies why our combination method applies to such algebras. We use this algebraic point of view to introduce a new proof method that depends on abstract notions and results from universal algebra, as opposed to technical manipul...

We show the decidability of the existential theory of term algebras with any Knuth-Bendix ordering by giving a procedure for solving Knuth-Bendix ordering constraints.

. A usual technique in symbolic constraint solving is to apply transformation rules until a solved form is reached for which the problem becomes simple. Ordering constraints are well-known to be reducible to (a disjunction of) solved forms, but unfortunately no polynomial algorithm deciding the satisfiability of these solved forms is known. Here we deal with a different notion of solved form, where fundamental properties of orderings like transitivity and monotonicity are taken into account. This leads to a new family of constraint solving algorithms for the full recursive path ordering with status (RPOS), and hence as well for other path orderings like LPO, MPO, KNS and RDO, and for all possible total precedences and signatures. Apart from simplicity and elegance from the theoretical point of view, the main contribution of these algorithms is on efficiency in practice. Since guessing is minimized, and, in particular, no linear orderings between the subterms are guessed, ...

We show that the problem of solving Knuth-Bendix ordering constraints is NP-complete, as a corollary we show that the existential first-order theory of any term algebra with the Knuth-Bendix ordering is NP-complete.

We introduce a propositional encoding of the recursive path order with status (RPO). RPO is a combination of a multiset path order and a lexicographic path order which considers permutations of the arguments in the lexicographic comparison. Our encoding allows us to apply SAT solvers in order to determine whether a given term rewrite system is RPO-terminating. Furthermore, to apply RPO within the dependency pair framework, we combined our novel encoding for RPO with an existing encoding for argument filters. We implemented our contributions in the termination prover AProVE. Our experiments show that due to our encoding, combining termination provers with SAT solvers improves the performance of RPO-implementations by orders of magnitude.

Abstract. Algorithms and datastructures form the kernel of any efficient theorem prover. In this abstract we discuss research on algorithms and datastructures for efficient theorem proving based on our experience with the theorem prover Vampire. We also briefly overview other works related to algorithms and datastructures, and to efficient theorem proving in general. 1

This paper is a collection of results on the decidability and the computational complexity of a syntactic unification problem under constrained substitutions. A number of decidable, undecidable, tractable and intractable results of the problem are presented. Since a unification problem under constrained substitutions can be regarded as an order-sorted unification problem with term declarations such that the number of sorts is only one, the results presented in this paper also indicate how the intractability of order-sorted unification problems is reduced by restricting the number of sorts to one.

2). We show that both the orientability problem is P-complete. Also we show that if a system is orientable using a real-valued instance of KBO, then it is also orientable using an integer-valued instance of KBO. Therefore, all our results hold both for the integer-valued and the real-valued KBO. 1 Introduction In this section we give an informal overview of the results proved in this paper. The formal definitions will be given in the next section.