Computing Minimal EL-Unifiers is Hard

Unification has been investigated both in modal logics and in description logics, albeit with different motivations. In description logics, unification can be used to detect redundancies in ontologies. In this context, it is not sufficient to decide unifiability, one must also compute appropriate unifiers and present them to the user. For the description logic EL, which is used to define several large biomedical ontologies, deciding unifiability is an NP-complete problem. It is known that every solvable EL-unification problem has a minimal unifier, and that every minimal unifier is a local unifier. Existing unification algorithms for EL compute all minimal unifiers, but additionally (all or some) non-minimal local unifiers. Computing only the minimal unifiers would be better since there are considerably less minimal unifiers than local ones, and their size is usually also quite small. In this paper we investigate the question whether the known algorithms for EL-unification can be modified such that they compute exactly the minimal unifiers without changing the complexity and the basic nature of the algorithms. Basically, the answer we give to this question is negative

Dismatching and Local Disunification in EL

Unification in Description Logics has been introduced as a means to detect redundancies in ontologies. We try to extend the known decidability results for unification in the Description Logic EL to disunification since negative constraints on unifiers can be used to avoid unwanted unifiers. While decidability of the solvability of general EL-disunification problems remains an open problem, we obtain NP-completeness results for two interesting special cases: dismatching problems, where one side of each negative constraint must be ground, and local solvability of disunification problems, where we restrict the attention to solutions that are built from so-called atoms occurring in the input problem. More precisely, we first show that dismatching can be reduced to local disunification, and then provide two complementary NP-algorithms for finding local solutions of (general) disunification problems

Restricted Unification in the DL FL₀: Extended Version

Unification in the Description Logic (DL) FL₀ is known to be ExpTimecomplete, and of unification type zero. We investigate in this paper whether a lower complexity of the unification problem can be achieved by either syntactically restricting the role depth of concepts or semantically restricting the length of role paths in interpretations. We show that the answer to this question depends on whether the number formulating such a restriction is encoded in unary or binary: for unary coding, the complexity drops from ExpTime to PSpace. As an auxiliary result, which is however also of interest in its own right, we prove a PSpace-completeness result for a depth-restricted version of the intersection emptiness problem for deterministic root-to-frontier tree automata. Finally, we show that the unification type of FL₀ improves from type zero to unitary (finitary) for unification without (with) constants in the restricted setting

Theorem Proving Using Equational Matings and Rigid E-Unifications

In this paper, it is shown that the method of matings due to Andrews and Bibel can be extended to (first-order) languages with equality. A decidable version of E-unification called rigid E-unification is introduced, and it is shown that the method of equational matings remains complete when used in conjunction with rigid E-unification. Checking that a family of mated sets is an equational mating is equivalent to the following restricted kind of E-unification. Problem: Given →/E = {Ei | 1 ≤ i ≤ n} a family of n finite sets of equations and S = {〈ui, vi〉 | 1 ≤ i ≤ n} a set of n pairs of terms, is there a substitution θ such that, treating each set θ(Ei) as a set of ground equations (i.e. holding the variables in θ(Ei) rigid ), θ(ui) and θ(vi) are provably equal from θ(Ei) for i = 1, ... ,n? Equivalently, is there a substitution θ such that θ(ui) and θ(vi) can be shown congruent from θ(Ei) by the congruence closure method for i 1, ... , n? A substitution θ solving the above problem is called a rigid →/E-unifier of S, and a pair (→/E, S) such that S has some rigid →/E-unifier is called an equational premating. It is shown that deciding whether a pair 〈→/E, S〉 is an equational premating is an NP-complete problem

Dismatching and Local Disunification in EL

Unification in Description Logics has been introduced as a means to detect redundancies in ontologies. We try to extend the known decidability results for unification in the Description Logic EL to disunification since negative constraints on unifiers can be used to avoid unwanted unifiers. While decidability of the solvability of general EL-disunification problems remains an open problem, we obtain NP-completeness results for two interesting special cases: dismatching problems, where one side of each negative constraint must be ground, and local solvability of disunification problems, where we restrict the attention to solutions that are built from so-called atoms occurring in the input problem. More precisely, we first show that dismatching can be reduced to local disunification, and then provide two complementary NP-algorithms for finding local solutions of (general) disunification problems

ORDER-SORTED RIGID E-UNIFICATION

Rigid E-Unification is a special type of unification which arises naturally when extending Andrew's method of matings to logic with equality. We study the rigid E-Unification problem in the presence of subsorts. We present an order sorted method for the computation of order sorted rigid-E-unifiers. The method is based on an unsorted one which we refine and extend to handle sort information. Our approach is to incorporate the sort information within the method so as to leverage it. We show via examples how the order sorted method is able to detect failures due to sort conflicts at an early stage in the construction of potential rigid E Unifiers. The algorithm presented here is NP-complete, as is the unsorted one. This is significant, specially due to the complications presented by the sort information.Information Systems Working Papers Serie

Inheritance hierarchies: Semantics and unification

Inheritance hierarchies are introduced as a means of representing taxonomicallyorganized data. The hierarchies are built up from so-called feature types that are ordered by subtyping and whose elements are records. Every feature type comes with a set of features prescribing fields of its record elements. So-called feature terms are available to denote subsets of feature types. Feature unification is introduced as an operation that decides whether two feature terms have a nonempty intersection and computes a feature term denoting the intersection.We model our inheritance hierarchies as algebraic specifications in ordersortedequational logic using initial algebra semantics. Our framework integrates feature types whose elements are obtained as records with constructor types whose elements are obtained by constructor application. Unification in these hierarchies combines record unification with order-sorted term unification and is presented as constraint solving. We specify a unitary unification algorithm by a set of simplification rules and prove its soundness and completeness with respect to the model-theoretic semantics

Constrained completion: Theory, implementation, and results

The Knuth-Bendix completion procedure produces complete sets of reductions but can not handle certain rewrite rules such as commutativity. In order to handle such theories, completion procedure were created to find complete sets of reductions modulo an equational theory. The major problem with this method is that it requires a specialized unification algorithm for the equational theory. Although this method works well when such an algorithm exists, these algorithms are not always available and thus alternative methods are needed to attack problems. A way of doing this is to use a completion procedure which finds complete sets of constrained reductions. This type of completion procedure neither requires specialized unification algorithms nor will it fail due to unorientable identities. We present a look at complete sets of reductions with constraints, developed by Gerald Peterson, and the implementation of such a completion procedure for use with HIPER - a fast completion system. The completion procedure code is given and shown correct along with the various support procedures which are needed by the constrained system. These support procedures include a procedure to find constraints using the lexicographic path ordering and a normal form procedure for constraints. The procedure has been implemented for use under the fast HIPER system, developed by Jim Christian, and thus is quick. We apply this new system, HIPER- extension, to attack a variety of word problems. Implementation alternatives are discussed, developed, and compared with each other as well as with the HIPER system. Finally, we look at the problem of finding a complete set of reductions for a ternary boolean algebra. Given are alternatives to attacking this problem and the already known solution along with its run in the HIPER-extension system --Abstract, page iii

Solving Equality Reasoning Problems with a Connection Graph Theorem Prover

The integration of a Knuth-Bendix completion algorithm into a paramodulation theorem prover on the basis of a connection graph resolution procedure is presented. The Knuth-Bendix completion idea is compared to a decomposition approach, and some ideas to handle conditional equations are discussed. The contents of this paper is not intended to present new material on term rewriting, instead it is more a pleading for the usage of completion ideas in automated deduction. It records our experience with an actual implementation of a hybrid system, where a completion procedure was imbedded into a connection graph theorem prover, the MKRP-system, with satisfactory positive results