Research Article  Open Access
Optimised ExpTime Tableaux for over Finite Residuated Lattices
Abstract
This study proposes to adopt a novel tableau reasoning algorithm for the description logic with semantics based on a finite residuated De Morgan lattice. The syntax, semantics, and logical properties of this logic are given, and a sound, complete, and terminating tableaux algorithm for deciding fuzzy ABox consistency and concept satisfiability problem with respect to TBox is presented. Moreover, based on extended and/or completionforest with a series of sound optimization technique for checking satisfiability with respect to a TBox in the logic, a new optimized ExpTime (complexityoptimal) tableau decision procedure is presented here. The experimental evaluation indicates that the optimization techniques we considered result in improved efficiency significantly.
1. Introduction
The fuzzy DL (Description Language) over lattice is a generalization of the crisp DL that uses the elements ofas truth values, instead of just the Boolean true and false. Different to fuzzy DLs, elements of the rational unit interval provided a membership degree semantics for their concepts;is further generalized to address qualitative uncertainty reasoning (by relying, e.g., on {false, likelyfalse, unknown, likelytrue, true}) and quantitative uncertainty reasoning (by relying, e.g., on (for an integer, in increasing order [1])).
Several attempts have been made at using fuzzy set semantics [2], but only a limited kind of semantics over lattices is considered, where conjunction and disjunction are interpreted through the lattice operators meet and join, respectively. Borgwardt and Peñaloza considered the fuzzy logic with semantics based on a finite residuated lattice [3] and a complete De Morgan lattice equipped with a tnorm operator [4]. Further, Borgwardt and Peñaloza analysed the consistency and satisfiability problems in the description logic with semantics based on a complete residuated De Morgan lattice [5] and showed that concept satisfiability in under this semantics is undecidable, in general, even if a very simple class of infinite lattices is restricted.
In this paper, we extend the more general description logic , whereis a complete De Morgan lattice equipped with a tnorm operator. The fuzzy description logic  is a generalization of the crisp description logic that uses the elements of as truth values instead of just the Boolean true and false. We study fuzzy variants of the standard reasoning problems like concept satisfiability and consistency with general concept inclusions (GCI) in this setting and proved that with the help of a tableauxbased algorithm satisfiability becomes decidable and the ExpTime complete if is required to be finite.
The main contributions of this work can be highlighted twofold as follows.(i)It presented a novel tableau reasoning algorithm for the description logic with semantics based on a finite residuated De Morgan lattice.(ii)By combining a series of optimization techniques that can be applied to fuzzy DL , our framework reduces the search space of the tableau algorithm more significantly to ExpTime complete.The paper is organized as follows. Section 2 introduces the syntax and semantics of the fuzzy description logic over lattices and discusses some logical properties of the logic. In Section 3, we give a detailed presentation of the reasoning algorithm for deciding the consistency and satisfiability of a ABox, provide the proofs for the termination, soundness, and completeness of the procedure, and address the computational aspect of reasoning in it. In Section 4 we extend the previous results by adding a wide variety of optimisations to achieving a high level of performance on tableau reasoning in expressive description logic. Our empirical evaluation shows that the proposed optimisations result in significant performance improvements in Section 6. Finally, future research issues are outlined in Section 7.
2. The Logic 
In this section we introduce a residuated De Morgan lattice extension of the DL, creating the language. The fuzzy description logic, which is the extension of [4] and [5], with the number restrictions constructor. And is a generalization of the fuzzy description logic.
Definition 1 (syntax). Letandbe pairwise disjoint sets of concepts and role, and let, be a set of transitive role names. The complex roles are.complex concepts are defined inductively by the following production rule, whereandis a complex role:
A complex roleis transitive if eitheror inverse ofbelongs to.
Definition 2 (semantics). For a residuated De Morgan lattice, an interpretation is now a pair , whereis the domain for the classical case andis an interpretation function mapping: every concept name a function and every role name a function. Anassertion is an expression , a role assertion of the form, whereis a concept,is a complex role,are individual names, , and.
A finite set of assertions and terminological axioms is called a Knowledge Base (KB). A fuzzy knowledge baseconsists of a finite set of axioms organized in three parts: a fuzzy ABox (axioms about individuals), a fuzzy TBox (axioms about concepts), and a fuzzy RBox (axioms about roles); that is, . An interpretation satisfies (is a model of) a KB if and only if satisfies each element in .
An interpretation satisfies the concept definition if and only if for all ,. A general TBox is a finite set of GCIs.satisfies the GCI if and only if, for all ,. An acyclic TBox is a finite set of concept definitions such that every concept name occurs at most once in the lefthand side of an axiom, and there is no cyclic dependency between definitions [6].
The complete set of semantics is depicted in Table 1. Particularly notice that existential and universal quantifiers are not dual to each other, and the implication constructor cannot be expressed in terms of the negation and conjunction.

Next, we will pay more attention to the problem of deciding satisfiability of a concept, especially computing the highest degree with which an individual may belong to a concept.
Definition 3 (satisfiable). Letbe concept descriptions, a TBox, and . is satisfiable with respect to if there is a model of such that. The best satisfiability degree forwith respect to is the largest such that is satisfiable with respect to .
The notion of witnessed model for fuzzy DLs was first introduced by Hájek [7]. Witnessed models of fuzzy predicate logic are models in which each quantified formula is witnessed; that is, the truth value of a universally quantified formula is the minimum of the values of its instances and similarly for existential quantification (maximum) [8].
Definition 4 (witnessed model). Let. A model I of an KBis witnessed if for every, every role , and every concept there are
The semantics of the quantifiers require the computation of a supremum or infimum of the membership degrees of a possibly infinite set of elements of the domain. Reasoning is usually restricted to witnessed models in order to obtain effective decision procedures [7]. Sinceis finite, we always have the witnessed model property for some. On the basis of of [6], we have deduced Lemma 5.
Lemma 5. If the cardinality of the largest antichain of L is n, thenhas the nwitnessed model property.
For simplification of the algorithm description, we consider. The algorithm and the proofs of correctness can easily be adapted for any other.
3. Tableau Algorithm for  DL
We assume that all concept descriptions are in negation normal form (NNF); that is, negation appears only in front of concept names. We will also abuse our notation by saying that a TBox is in NNF if all the concepts occurring in are in NNF. Moreover, we prove the decidability of the DL by providing a tableaux algorithm for deciding the standard DL inference problems.
3.1. Tableau
Since most of the inference services of fuzzy DLs can be reduced to the problem of consistency checking for ABoxes [9], here we discuss the approaches to deciding consistency of fuzzy DLs over finite residuated De Morgan lattices in the presence of GCIs. The algorithm we present can be seen as an extension of the tableau presented in [5] and is inspired by the wellknown tableau algorithm for [10], which is the basis for several highly successful implementations.
It is assumed thatis finite, and we can accordingly restrict reasoning towitnessed models. In the lattice ,we employ the symbolsandas a placeholder for the inequalitiesand and the symbol as a placeholder for all types of inequations. Furthermore we employ the symbolsand to denote their reflections.
Definition 6. For a fuzzy concept, we will denote bythe set that containsand it is closed under subconcepts of. The set of all subconcepts of concepts that appear within an ABox is denoted by sub(). Letbe the set of roles occurring in and together with their inverses, andis the set of individuals in . A fuzzy tableauforis a quadruple in, such that (i) is a nonempty set of individuals (nodes),(ii):maps each element and concept, that is, a member of, to the degree of certainty of that element to the concept,(iii):maps each role ofand pair of elements to the degree of certainty of the pair to the role,(iv): maps individuals occurring into elements in.
For all,, , and,satisfies the following. (1),for.(2)If, then.(3)If, thenand.(4)If, thenand.(5)If, thenand.(6)If, thenand.(7)Ifandis conjugated with, then.(8)Ifand is conjugated with, then .(9)If, then there exists such that and, then.(10)If , then there existssuch that and, then .(11)If, andis conjugated with, for somewith Trans(), then.(12)If , andis conjugated with, for somewith Trans(), then.(13)if.(14)Ifandthen,.(15), then.(16), then.(17), then , conjugated with, .(18) , then , conjugated with,.(19)There do not exist two conjugated triples in any label of any individual.(20)If, then.(21)If, then.(22)If, then .
Lemma 7. AnABox is consistent with respect to if and only if there exists a fuzzy tableau for with respect to .
Proof. Similar to that of Lemma 6.5 of [11], here gives the proof of the Lemma 7. For the direction if = is a fuzzy tableau for an ABox with respect to , we can construct a fuzzy interpretation = that is a model of . Consider the following:
For all role,
whereis a binary fuzzy relation defined asfor all, andrepresents its supmin transitive closure [12]. The proof of this property is quite technical and omitted here. From all above properties, the interpretation of individuals and roles implies that satisfies each assertion in inductively.
3.2. Constructing a  Tableau
Like most of the tableaux algorithms, our algorithm works on completion forests rather than on completion trees which is applied in work [5]. As pointed out in [13], that is because an ABox might contain several individuals with arbitrary roles connecting them.
Definition 8 (completion forest). A completion forest foris a collection of trees whose distinguished roots are arbitrarily connected by edges. Each nodeis labelled with a set. Each edgeis labelled with a set, whereare roles and its inverse occurring in . A completion forest comes with an explicit inequality relationon nodes and an explicit equality relationwhich are implicitly assumed to be symmetric. Given a completion forest, If nodesandare connected by an edgewithoccurring in, thenis called anofandis called anof. A nodeis a positive (resp., negative) successor (resp., predecessor or neighbour) of.
A node is blocked if it is either directly or indirectly blocked. It is indirectly blocked if itspredecessor is either directly or indirectly blocked. A nodeis directly blocked if and only if none of its ancestors are blocked, and it has an ancestorsuch that.
A node is said to contain a clash satisfying one of the conditions such that if and only if there exist two conjugated triples in, or if and, are root nodes, contains two conjugated triples. Table 2 shows all the entries under which condition the row column pair ofconstraints is aconjugated pair(e.g.andis a conjugated pair as).
The calculus is based on a set of constraint propagation rules transforming a setinto satisfiability preserving setsuntil either one ofcontains an inconsistency (clash), or someis completed and clashfree; that is, no rule can further be applied toandcontains no clash (indicating that froma model ofcan be built).
is then expanded by repeatedly applying the rules from Table 3. The notationdenotes either the roleor the role returned by, and the notation denotes any role that participates in such a triple.

The completion forest is complete when for some node contains a clash, or none of the completion rules in Table 3 are applicable. The algorithm is correct in the sense that it produces a clash if and only if Σ is inconsistent. The expansion rules are based on the properties of the semantics presented in Definition 6.
3.3. Soundness and Completeness of the  Tableaux Algorithm
Lemma 9 (termination, soundness, completeness). For a given KB: (i)the tableau algorithm terminates;(ii)if the expansion rules can be applied tosuch that they yield a complete completionforest such thathas a solution, thenhas a model;(iii)ifhas a model, then the expansion rules can be applied in such a way that the tableaux algorithm yields a complete completion forest forsuch thathas a solution.
3.4. Computational Complexity of the Tableau Algorithm for 
We have described a tableau procedure for reasoning in. Due to the tableau rules are nondeterministic, every application of expansion rules to a termination after at most exponentially many rule applications. So according to the theorem in [5] that the tableaux algorithm is NExpTime complete, we get the following proposition deciding the computational complexity of the tableau algorithm for.
Proposition 10. The concept satisfiability checking problem in with respect to witnessed models can be decided in NExpTime.
Proof (sketch). It is assumed that has thewitnessed model property for some, the completion forest has to generatedifferent successors for every existential and universal restriction to ensure that the degrees guessed for these complex concepts are indeed witnessed by the model. As stated by Table 1, we have to introduce individuals and values , , which satisfies or , respectively. In the following we let be the number that occurs in a number restriction, and let be the number of different lattices appearing in .
Following [14] we set ; in the meantime, we let ,,,, and. Then due to the proof of Lemma 6.9 in [11] and by the addition of the number of different lattices appearing in : a completion forest for becomes no longer thanand that the outdegree is bounded by. Consequently, the algorithm will construct a completion forest with no more than nodes.
For fuzzy description logics with semantics based on complete residuated De Morgan lattices (e.g.,), strong satisfiability with respect to general TBoxes is undecidable for some infinite lattices while, for finite lattices, decidability is regained [5]. If one considers the without terminological axioms, concept satisfiability is ExpTime complete as in the crisp case [15]. The problem with respect to general TBoxes becomes ExpTime complete matching the complexity of the crisp case, though arbitrary (finite) lattices and tnorms are allowed [4]. Besides, strong concept satisfiability is in NExpTime .
When we consider finite De Morgan lattices, then satisfiability problem can be effectively decided. We guess that the computational complexity of it can be improved to ExpTime either by an ABox partitioning [16] or with the help of global caching and other techniques [17].
4. Optimization Techniques Employed in Tableau Algorithm for 
From Proposition 10, it is obvious that the theoretical complexity of the tableau reasoning algorithm for is 2NExptime, which is too expensive for a reasoner to deal with. So we plan to investigate some optimizations [18] developed for tableaux algorithms for crisp DLs, which can be transferred to our setting to reduce the search space created by the choice of lattice values. By using these techniques, a theoretically expensive computation could be converted to an equivalent of practically lower complexity.
In this section we describe in detail the optimisation techniques employed in tableau algorithm for. Not only focusing mainly on novel techniques and significant refinements and extensions of previously known techniques such as global caching, we also apply some simple but often very effective optimisations to our implement. First of all, we perform some preprocessing optimisations directly on the syntax of the input. Then in virtue of the ideas of both backjumping and global caching, we investigate an optimized tableaubased reasoning algorithm and prove its ExpTime complete.
4.1. Preprocessing Optimisations
In case of obvious clash detection or cycle via concept names, preprocessing optimisations (e.g., absorption or told cycle elimination) can lead to important speedup of the subsequent reasoning process. These optimisations serve to preprocess and simplify the input into a form more amenable to later processing [18]. Furthermore, these optimisations are not specific to tableaubased fuzzy reasoners and may also be useful with other kinds of fuzzy DL reasoners. Herein on the basis of, we apply two striking optimizations techniques to our design and implement.
4.1.1. Partition Based on Connectivity
Partition based on connectivity is a quite effective optimization technique to boost up the performance of reasoning, which can be applied to any fuzzy DL without nominals independently of the fuzzy operators used to provide the interpretations [17]. Termination of the expansion of the completion forest is a result of the properties of the expansion rules in Table 3, and a tableau expansion rule can either add (i) a new neighbour node to the node of examination, (ii) new membership triples in this node, or (iii) new membership triples to neighbouring nodes. The expansion rule will examine the different ways that the tableau expansion rules affect this completion forest, with the respect of different connectivity circumstances. Inspired by the work in [17], we have the following definitions.
Definition 11. Let, , , . The connection relation between two individuals, in an ABox is inductively defined through role :
Note thatmay be run into an iterative process for connecting indirectly. The individuals in the ABox may be divided into one or more individual groups. Then individual group is partitioned into one or more independent subsets called Assertion Groups () [16]. For instance, two assertionsandare in the same ifis directly or indirectly inferred from(through the application of completion rules), oranddiffer only in terms of their certainty values and/or conjunction and disjunction functions. Each group is composed of individuals that are connected to each other through role assertions.
Definition 12. We denotewith the partition of the ABox that contains only connected individuals in