Optimised ExpTime Tableaux for 𝒮ℋℐ𝒩 over Finite Residuated Lattices

Abstract

This study proposes to adopt a novel tableau reasoning algorithm for the description logic 𝒮ℋℐ𝒩 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 completion-forest with a series of sound optimization technique for checking satisfiability with respect to a TBox in the logic, a new optimized ExpTime (complexity-optimal) 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 of𝐿as 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;𝐿is 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 {0/𝑛,1/𝑛,…,𝑛/𝑛} (for an integer𝑛>1, in increasing order [1])).

Several attempts have been made at using 𝐿-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 𝒜ℒ𝒞ℐ with semantics based on a finite residuated lattice [3] and a complete De Morgan lattice equipped with a t-norm operator [4]. Further, Borgwardt and Peñaloza analysed the consistency and satisfiability problems in the description logic 𝒮ℋℐ with semantics based on a complete residuated De Morgan lattice [5] and showed that concept satisfiability in 𝒜ℒ𝒞 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 𝐿-𝒮ℋℐ𝒩, where𝐿is a complete De Morgan lattice equipped with a t-norm operator. The fuzzy description logic 𝐿-𝒮ℋℐ𝒩 is a generalization of the crisp description logic 𝒮ℋℐ𝒩 that uses the elements of 𝐿as 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 tableaux-based algorithm satisfiability becomes decidable and the ExpTime complete if 𝐿 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 𝒮ℋℐ𝒩 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 𝐿-𝒮ℋℐ𝒩, 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 lattices𝐿-𝒮ℋℐ𝒩 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 a𝐿-𝒮ℋℐ𝒩 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 logic𝐿-𝒮ℋℐ𝒩. 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 𝐿-𝒮ℋℐ𝒩

In this section we introduce a residuated De Morgan lattice extension of the𝐿-𝒮ℋℐ𝒩 DL, creating the𝐿-𝒮ℋℐ𝒩 language. The fuzzy description logic𝐿-𝒮ℋℐ𝒩, which is the extension of 𝒜ℒ𝒞_𝐿[4] and𝐿-𝒮ℋℐ [5], with the number restrictions constructor. And𝐿-𝒮ℋℐ𝒩 is a generalization of the fuzzy description logic𝑓-𝒮ℋℐ𝒩.

Definition 1 (syntax). Let𝑁_𝐶and𝑁_𝑅be pairwise disjoint sets of concepts and role, and let𝑁⁺_𝑅⊆𝑁_𝑅, 𝑁_𝑅 be a set of transitive role names. The complex roles are𝑁_𝑅∪{𝑅⁻∣𝑟∈𝑁_𝑅}.𝐿-𝒮ℋℐ𝒩complex concepts are defined inductively by the following production rule, where𝐴∈𝑁_𝐶and𝑠is a complex role: 𝐶∶=𝐴∣𝐶₁⊓𝐶₂∣𝐶₁⊔𝐶₂∣𝐶₁⟶𝐶₂∣¬𝐶₁∣∃𝑠⋅𝐶∣∀𝑠⋅𝐶∣⊤∣⟂.(1)

A complex role𝑠is transitive if either𝑠or inverse of𝑠belongs to𝑁⁺_𝑅.

Definition 2 (semantics). For a residuated De Morgan lattice𝐿, an 𝐿 interpretation is now a pair ℐ=(Δ^ℐ,⋅^ℐ), whereΔ^ℐis the domain for the classical case and⋅^ℐis an interpretation function mapping: every concept name 𝐶 a function𝐴^ℐ∶Δ^ℐ→𝐿 and every role name 𝑟 a function𝑟^ℐ∶Δ^ℐ×Δ^ℐ→𝐿. An𝐿-assertion is an expression ⟨𝑎∶𝐶⊳ℓ⟩, a role assertion of the form⟨(𝑎,𝑏)∶𝑠⊳ℓ⟩, where𝐶is a concept,𝑠is a complex role,𝑎,𝑏are individual names, ℓ∈𝐿, and⊳∈{≻,⪰}.

A finite set of assertions and terminological axioms is called a Knowledge Base (KB). A fuzzy knowledge baseΣconsists of a finite set of axioms organized in three parts: a fuzzy ABox 𝒜 (axioms about individuals), a fuzzy TBox 𝒯 (axioms about concepts), and a fuzzy RBox ℛ (axioms about roles); that is, Σ=(𝒜,ℛ,𝒯). An interpretation ℐ satisfies (is a model of) a KB Σ if and only if ℐ satisfies each element in Σ.

An interpretation ℐ satisfies the concept definition⟨𝐴≐𝐶,ℓ⟩ if and only if for all 𝑥∈Δ^ℐ,(𝐴^ℐ(𝑥)⇒𝐶^ℐ(𝑥))⊗(𝐶^ℐ(𝑥)⇒𝐴^ℐ(𝑥))≥ℓ. A general TBox is a finite set of GCIs.Σsatisfies the GCI⟨𝐶⊑𝐷,ℓ⟩ if and only if, for all 𝑥∈Δ^ℐ,𝐶^ℐ(𝑥)⇒𝐷^ℐ(𝑥)≥ℓ. 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.

Table 1 

Semantics of 𝐿-𝒮ℋℐ𝒩 concepts and 𝐿-𝒮ℋℐ𝒩 roles.

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 (ℓ-satisfiable). Let𝐶be𝐿-𝒮ℋℐ𝒩 concept descriptions, 𝒯 a TBox, and ℓ∈𝐿. 𝐶 is ℓ-satisfiable with respect to 𝒯 if there is a model ℐ of 𝒯such that∨_𝑥∈Δ^ℐ𝐶^ℐ(𝑥)≥ℓ. The best satisfiability degree for𝐶with respect to 𝒯 is the largest ℓ such that𝐶 is ℓ-satisfiable with respect to 𝒯.

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). Let𝑛∈ℕ. A model I of an KBΣis 𝑛-witnessed if for every𝑥∈Δ^ℐ, every role 𝑠, and every concept 𝐶 there are (∀𝑠⋅𝐶)^ℐ(𝑥)_𝑀=inf_𝑏∈𝑦i𝑠^ℐ𝑥,𝑦_𝑖⊗𝐶^ℐ𝑦_𝑖=𝑠^ℐ(𝑥,𝑏)⊗𝐶^ℐ(𝑏),(∃𝑠⋅𝐶)^ℐ(𝑥)_𝑀=sup_{𝑏∈𝑦𝑖}𝑠^ℐ𝑥,𝑦_𝑖⟹𝐶^ℐ𝑦_𝑖=𝑠^ℐ(𝑥,𝑏)⟹𝐶^ℐ(𝑏).(2)
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]. Since𝐿is finite, we always have the 𝑛-witnessed model property for some𝑛∈ℕ. On the basis of 𝒜ℒ𝒞ℐ_𝐿 of [6], we have deduced Lemma 5.

Lemma 5. If the cardinality of the largest antichain of L is n, then𝐿-𝒮ℋℐ𝒩has the n-witnessed model property.

For simplification of the algorithm description, we consider𝑛=1. The algorithm and the proofs of correctness can easily be adapted for any other𝑛∈ℕ.

3. Tableau Algorithm for 𝐿-𝒮ℋℐ𝒩 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 𝒯 is in NNF if all the concepts occurring in 𝒯 are in NNF. Moreover, we prove the decidability of the𝐿-𝒮ℋℐ𝒩 DL by providing a tableaux algorithm for deciding the standard DL inference problems.

3.1. 𝐿−𝒮ℋℐ𝒩 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 well-known tableau algorithm for𝑓-𝒮ℋℐ𝒩 [10], which is the basis for several highly successful implementations.

It is assumed that𝐿is finite, and we can accordingly restrict reasoning to𝑛-witnessed models. In the lattice 𝐿=⟨𝜁,⊲,⊳⟩,we employ the symbols⊲and⊳as a placeholder for the inequalities≺,⪯and≻,⪰ and the symbol ⋈ as a placeholder for all types of inequations. Furthermore we employ the symbols⊲⁻,⊳⁻and ⋈⁻to denote their reflections.

Definition 6. For a fuzzy concept𝐶, we will denote bysub(𝐶)the set that contains𝐶and it is closed under subconcepts of𝐶. The set of all subconcepts of concepts that appear within an ABox is denoted by sub(𝐶). Let𝑅_𝐴be the set of roles occurring in 𝒜 and ℛ together with their inverses, and𝐼_𝒜is the set of individuals in 𝒜. A fuzzy tableau𝐓forΣis a quadruple in(𝐒,𝒢,Υ,𝒱), such that (i)𝐒 is a nonempty set of individuals (nodes),(ii)𝒢:𝐒×sub(𝐴)→𝜁maps each element and concept, that is, a member ofsub(𝐴), to the degree of certainty of that element to the concept,(iii)Υ:𝑅_Σ×𝐒×𝐒→𝜁maps each role of𝑅_𝐴and pair of elements to the degree of certainty of the pair to the role,(iv)𝒱: maps individuals occurring in𝐴to elements in𝑆.
For all𝑠,𝑡∈𝐒,𝐶,𝐷∈sub(𝐴), ℓ∈𝜁, and𝑅∈𝑅_𝐴,𝐓satisfies the following. (1)Σ(𝑠,⟂)=0,Σ(𝑠,⊤)=1for∀𝑠∈𝐒.(2)If⟨¬𝐶,⋈,ℓ⟩∈𝒢(𝑠), then⟨𝐶,⋈⁻,∼ℓ⟩∈𝒢(𝑠).(3)If⟨𝐶⊓𝐷,⊳,ℓ⟩∈𝒢(𝑠), then⟨𝐶,⊳,ℓ⟩∈𝒢(𝑠)and⟨𝐷,⊳,ℓ⟩∈𝒢(𝑠).(4)If⟨𝐶⊔𝐷,⊲,ℓ⟩∈𝒢(𝑠), then⟨𝐶,⊲,ℓ⟩∈𝒢(𝑠)and⟨𝐷,⊲,ℓ⟩∈𝒢(𝑠).(5)If⟨𝐶⊔𝐷,⊳,ℓ⟩∈𝒢(𝑠), then⟨𝐶,⊳,ℓ⟩∈𝒢(𝑠)and⟨𝐷,⊳,ℓ⟩∈𝒢(𝑠).(6)If⟨𝐶⊓𝐷,⊲,ℓ⟩∈𝒢(𝑠), then⟨𝐶,⊲,ℓ⟩∈𝒢(𝑠)and⟨𝐷,⊲,ℓ⟩∈𝒢(𝑠).(7)If⟨∀𝑅⋅𝐶,⊳,ℓ⟩∈𝒢(𝑠)and⟨⟨𝑠,𝑡⟩,⊳^′,ℓ₁⟩∈Υ(𝑅)is conjugated with⟨⟨𝑠,𝑡⟩,⊳⁻,∼ℓ⟩, then⟨∀𝐶,⊳,ℓ⟩∈𝒢(𝑡).(8)If⟨∀𝑅⋅𝐶,⊲,ℓ⟩∈𝒢(𝑠)and⟨⟨𝑠,𝑡⟩,⊳,ℓ₁⟩ ∈ Υ(𝑅)is conjugated with⟨⟨𝑠,𝑡⟩,⊲,∼ℓ⟩, then⟨∀𝐶,⊲,ℓ⟩ ∈ 𝒢(𝑡).(9)If⟨∃𝑅⋅𝐶,⊳,ℓ⟩∈𝒢(𝑠), then there exists𝑡 ∈ 𝐒such that⟨⟨𝑠,𝑡⟩,⊳,ℓ⟩ ∈ Υ(𝑅)and⟨⟨𝑠,𝑡⟩,⊳,∼ℓ⟩, then⟨∀𝐶,⊳,ℓ⟩∈𝒢(𝑡).(10)If⟨∀𝑅⋅𝐶,⊲,ℓ⟩ ∈ 𝒢(𝑠), then there exists𝑡∈𝐒such that⟨⟨𝑠,𝑡⟩,⊲⁻,∼ℓ⟩ ∈ Υ(𝑅)and⟨⟨𝑠,𝑡⟩,⊲,∼ℓ⟩, then⟨∀𝐶,⊳,ℓ⟩ ∈ 𝒢(𝑡).(11)If⟨∃𝑠⋅𝐶,⊲,ℓ⟩∈𝒢(𝑠), and⟨⟨𝑠,𝑡⟩,⊳,ℓ₁⟩∈Υ(𝑅)is conjugated with⟨𝑠⋅𝐶,⊲,ℓ⟩, for some𝑅⊑^∗with   Trans(𝑅), then⟨∃𝑠⋅𝐶,⊲,ℓ⟩∈𝒢(𝑠).(12)If ⟨∀𝑠⋅𝐶,⊳,ℓ⟩∈𝒢(𝑠), and⟨⟨𝑠,𝑡⟩,⊳^′,ℓ₁⟩∈Υ(𝑅)is conjugated with⟨𝑠⋅𝐶,⊳⁻,∼ℓ⟩, for some𝑅⊑^∗with   Trans(𝑅), then⟨∀𝑠⋅𝐶,⊳,ℓ⟩∈𝒢(𝑠).(13)⟨⟨𝑠,𝑡⟩,⋈,ℓ⟩∈Υ(𝑅)if⟨⟨𝑡,𝑠⟩,⋈,ℓ⟩∈Υ(𝑅)(Inv(𝑅)).(14)If⟨⟨𝑠,𝑡⟩,⊳,ℓ⟩∈Υ(𝑅)and𝑅⊑^∗𝑆then,⟨⟨𝑠,𝑡⟩,⊳,ℓ⟩∈Υ(𝑆).(15)⟨≥𝑝𝑅,⊳,ℓ⟩∈Υ(𝑅), then{𝑡∈𝐒∣⟨⟨𝑠,𝑡⟩,⊳,ℓ⟩∈Υ(𝑅)}≥𝑝.(16)⟨≤𝑝𝑅,⊲,ℓ⟩∈Υ(𝑅), then{𝑡∈𝐒∣⟨⟨𝑠,𝑡⟩,⊲⁻,∼ℓ⟩∈Υ(𝑅)}≥𝑝+1.(17)⟨≥𝑝𝑅,⊲,ℓ⟩∈Υ(𝑅), then{𝑡∈𝐒∣⟨⟨𝑠,𝑡⟩,⊳,ℓ_𝑖⟩ ∈ Υ(𝑅)}≤𝑝−1, conjugated with⟨⟨𝑠,𝑡⟩, ⊲,ℓ⟩.(18)⟨≤𝑝𝑅,⊳,ℓ⟩ ∈ Υ(𝑅), then{𝑡∈𝐒∣⟨⟨𝑠,𝑡⟩,⊳^′,ℓ_𝑖⟩ ∈ Υ(𝑅)}≤𝑝, conjugated with⟨⟨𝑠,𝑡⟩,⊳⁻,∼ℓ⟩.(19)There do not exist two conjugated triples in any label of any individual𝑥∈𝐒.(20)If⟨𝑎∶𝐶⋈ℓ⟩∈𝐴, then⟨𝐶⋈ℓ⟩∈𝒢(𝒱(𝑎)).(21)If⟨(𝑎,𝑏)∶𝑅⋈ℓ⟩∈𝐴, then⟨(𝒱(𝑎),(𝒱(𝑏))⋈ℓ⟩∈Υ(𝑅).(22)If𝑎≠𝑏∈𝒜, then 𝒱(𝑎)≠𝒱(𝑏).

Lemma 7. An𝐿-𝒮ℋℐ𝒩ABox 𝒜 is consistent with respect to ℛ if and only if there exists a fuzzy tableau for 𝒜 with respect to ℛ.

Proof. Similar to that of Lemma 6.5 of [11], here gives the proof of the Lemma 7. For the direction if 𝐓 = (𝐒,𝒢,Υ,𝒱) is a fuzzy tableau for an ABox 𝒜 with respect to ℛ, we can construct a fuzzy interpretation ℐ = (Δ^ℐ,⋅^ℐ)that is a model of 𝒜. Consider the following: Δ^ℐ𝑎=𝐒,_ℐ=𝐕(𝑎),𝑎∈ℐ_𝐴,⊺^ℐ()⟂=𝒢𝑠,⊤,∀𝑠∈𝐒,_ℐ()𝐴=𝒢𝑠,⟂,∀𝑠∈𝐒,_ℐ=𝒢(𝑠,𝐴),∀𝑠∈𝐒andconceptnames𝐴.(3)
For all role𝑅∈𝐑_𝐴, 𝑅^ℐ()=𝑅𝑠,𝑡^+Υ(𝑠,𝑡),∀⟨𝑠,𝑡⟩∈𝐒×𝐒if(𝑅)𝑅𝚃𝚛𝚊𝚗𝚜_Υ(𝑠,𝑡),∀⟨𝑠,𝑡⟩∈𝐒×𝐒otherwise,(4) where𝑅_Υ(𝑠,𝑡)is a binary fuzzy relation defined as𝑅_Υ(𝑠,𝑡)=Υ(𝑅,⟨𝑠,𝑡⟩)for all∀⟨𝑠,𝑡⟩∈𝐒×𝐒, and𝑅⁺_Υ(𝑠,𝑡)represents its sup-min 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 ℐ satisfies each assertion in 𝒜 inductively.

3.2. Constructing a 𝐿-𝒮ℋℐ𝒩 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 ℱ forΣis a collection of trees whose distinguished roots are arbitrarily connected by edges. Each node𝑥is labelled with a setΣ(𝑥)={⟨𝐶,⋈,𝑙⟩}. Each edge⟨𝑥,𝑦⟩is labelled with a setΣ(⟨𝑥,𝑦⟩)={⟨𝑅,⋈,𝑙⟩}, where𝑅∈𝑅_𝐴are roles and its inverse occurring in 𝒜. A completion forest comes with an explicit inequality relation≠on nodes and an explicit equality relation≐which are implicitly assumed to be symmetric. Given a completion forest, If nodes𝑥and𝑦are connected by an edge⟨𝑣,𝑤⟩with⟨𝑅,𝑙⟩occurring inΣ(⟨𝑥,𝑦⟩), then𝑦is called an𝑅-successorof𝑥and𝑥is called an𝑅-predecessorof𝑦. A node𝑥is a positive (resp., negative) successor (resp., predecessor or neighbour) of𝑦.

A node is blocked if it is either directly or indirectly blocked. It is indirectly blocked if its𝑅-predecessor is either directly or indirectly blocked. A node𝑥is directly blocked if and only if none of its ancestors are blocked, and it has an ancestor𝑦such thatΣ(𝑥)=Σ(𝑦).

A node 𝑥 is said to contain a clash satisfying one of the conditions such that if and only if there exist two conjugated triples inΣ(⟨𝑥,𝑦⟩), or ifΣ(⟨𝑥,𝑦⟩)∪{⟨Inv(𝑅),⋈,ℓ⟩∣⟨𝑅,⋈,ℓ⟩∈Σ(⟨𝑦,𝑥⟩)} and𝑥,𝑦 are root nodes, contains two conjugated triples. Table 2 shows all the entries under which condition the row column pair of𝐿-constraints is aconjugated pair(e.g.,𝑎∶𝐴≻𝑢and𝑎∶𝐴⪰^╱𝑣is a conjugated pair as¬(∃𝑣⋅𝑣≻𝑢∧𝑣⪰^╱𝑣)).

Table 2 

Conjugated pairs.

The calculus is based on a set of constraint propagation rules transforming a set𝑆into satisfiability preserving sets𝑆_𝑖until either one of𝑆_𝑖contains an inconsistency (clash), or some𝑆_𝑖is completed and clash-free; that is, no rule can further be applied to𝑆_𝑖and𝑆_𝑖contains no clash (indicating that from𝑆_𝑖a model of𝑆can be built).

ℱ is then expanded by repeatedly applying the rules from Table 3. The notation𝑅^∗denotes either the role𝑅or the role returned byInv(𝑅), and the notation ⟨∗,⋈,ℓ⟩ denotes any role that participates in such a triple.

Table 3 

Rule description.

The completion forest is complete when for some node 𝑥∶Σ(𝑥) 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 𝐿-𝒮ℋℐ𝒩 Tableaux Algorithm

Lemma 9 (termination, soundness, completeness). For a given KBΣ: (i)the tableau algorithm terminates;(ii)if the expansion rules can be applied toΣsuch that they yield a complete completion-forest ℱ such that𝐶_ℱhas a solution, thenΣhas a model;(iii)ifΣhas a model, then the expansion rules can be applied in such a way that the tableaux algorithm yields a complete completion forest forΣsuch that𝐶_ℱhas a solution.

3.4. Computational Complexity of the Tableau Algorithm for 𝐿-𝒮ℋℐ𝒩

We have described a tableau procedure for reasoning in𝐿-𝒮ℋℐ𝒩. 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 for𝐿-𝒮ℋℐ𝒩.

Proposition 10. The concept satisfiability checking problem in𝐿-𝒮ℋℐ𝒩 with respect to witnessed models can be decided in NExpTime.

Proof (sketch). It is assumed that𝐿-𝒮ℋℐ𝒩 has the𝑛-witnessed model property for some𝑛≥1, the completion forest has to generate𝑛different 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 n individuals {y₁,…,y_n} and 2n values {ℓ¹₁,…,ℓⁿ₁}∈L, {ℓ¹₂…,ℓⁿ₂}∈L, which satisfies ⋁ⁿ_i=1ℓⁱ₁⊗ℓⁱ₂=ℓ or ⋀ⁿ_i=1ℓⁱ₁⇒ℓⁱ₂=ℓ, respectively. In the following we let n_max be the number that occurs in a number restriction, and let ℎ be the number of different lattices appearing in 𝒜.

Following [14] we set 𝑛=||𝐴|+|𝑅||=𝒪(|𝐴|+|𝑅|) = 𝒪(2𝑛); in the meantime, we let𝑚=|sub(𝐷)|=𝒪(2|𝐴||𝑅|) = 𝒪(𝑛²),𝑘=|𝑅_𝐴|=𝒪(|𝐴|+|𝑅|)=𝒪(2𝑛),𝑛_max=𝒪(2|𝐴|)=𝒪(2𝑛),ℎ=𝒪(|𝐴|)=𝒪(𝑛), and𝑞=max{𝐷_ℒ(𝑠)∣∀𝑠∈𝒯}. Then due to the proof of Lemma  6.9 in [11] and by the addition of the number of different lattices appearing in 𝒜-𝑞: a completion forest for 𝒜 becomes no longer than2^{8⋅𝑚⋅ℎ⋅𝑘⋅𝑞}and that the outdegree is bounded by2⋅ℎ⋅𝑚⋅𝑛_max. Consequently, the𝐿-𝒮ℋℐ𝒩 algorithm will construct a completion forest with no more than (2⋅ℎ⋅𝑚⋅𝑛_max)^{28⋅𝑚⋅ℎ⋅𝑘⋅𝑞} = 𝒪((2⋅𝑛⋅𝑛²⋅2^𝑛)^{228⋅𝑛⋅𝑛⋅𝑛⋅𝑞}) = 𝒪(2^{𝑛⋅248⋅𝑛⋅𝑞})=𝒪(2^{24𝑞⋅𝑛}) nodes.

For fuzzy description logics with semantics based on complete residuated De Morgan lattices (e.g.,𝐿-𝒜ℒ𝒞ℐ), 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 𝒜ℒ𝒞 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 t-norms are allowed [4]. Besides, strong concept satisfiability is in NExpTime 𝐿-𝒮ℋℐ.

When we consider finite De Morgan lattices𝐿, 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 𝐿-𝒮ℋℐ𝒩

From Proposition 10, it is obvious that the theoretical complexity of the tableau reasoning algorithm for𝐿-𝒮ℋℐ𝒩 is 2-NExptime, 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 for𝐿-𝒮ℋℐ𝒩. 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 tableau-based 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 tableau-based fuzzy reasoners and may also be useful with other kinds of fuzzy DL reasoners. Herein on the basis of𝐿-𝒮ℋℐ𝒩, 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. Let𝑎, 𝑏, 𝑐∈𝐈, ℓ∈𝐿. The connection relation between two individuals𝑎, 𝑏in an ABox 𝒜 is inductively defined through role 𝑅_𝐴: 𝑎↭_𝒜𝑏⟺𝑅_𝐴(𝑎,𝑏)⋈𝑙∈𝒜derectlyconnected𝑎↭_𝒜𝑐∪𝑏↭_𝒜𝑐inderectlyconnected.(5)

Note that𝑐may be run into an iterative process for connecting indirectly. The individuals in the ABox 𝒜 may be divided into one or more individual groups. Then individual group is partitioned into one or more independent subsets called Assertion Groups (𝐴𝐺) [16]. For instance, two assertions𝐴₁and𝐴₂are in the same 𝐴𝐺 if𝐴₁is directly or indirectly inferred from𝐴₂(through the application of completion rules), or𝐴₁and𝐴₂differ 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 assertions𝑅_𝐴.

Definition 12. We denote𝐴𝐺_[𝑎]with the partition of the ABox 𝒜 that contains only connected individuals in 𝒜𝐴𝐺_[𝑎]={𝑎}∪𝑥∣∀𝑥∈𝐈,𝑎↭_𝒜𝑥.(6)

Proposition 13. It holds that(1)∪𝐴𝐺_𝑖=𝒜,(2)𝐴𝐺_𝑖∩𝐴𝐺_𝑗=∅, for each pair𝐴𝐺_𝑖,𝐴𝐺_𝑗∈𝐴such that𝐴𝐺_𝑖=𝐴𝐺_𝑗,(3)𝒜 is consistent with respect to a TBox 𝒯 if and only if each 𝐴𝐺_𝑖 is consistent with respect to a TBox 𝒯.

Partition based on connectivity optimization technique has several advantages. First, it is of polynomial complexity and it can be applied to any fuzzy DL. In addition, If any 𝐴𝐺 is found to be inconsistent, this implies that the whole 𝒜 is inconsistent. A related benefit is to more precisely identify the assertions that cause the inconsistency. Finally, the speed of solving a few small constraint sets would be faster than solving one large constraint set.

4.1.2. Dealt with TBox Axioms

If dealt with axioms of TBox 𝒯 naively, which maybe lead to a serious degradation in reasoning performance, as each such axiom would cause a disjunction to be added to every node of the completion forest, leading to potentially enormous amounts on nondeterministic expansion [18].

Given𝜍∈𝒯, with𝜍=((𝐶₁∨𝐷₁)∧⋯∧(𝐶_𝑛∨𝐷_𝑛)), testing the satisfiability of𝜍leads to the construction of a completion forest containing𝑘nodes, then there are2^𝑘𝑛different ways to apply the ⊓- and ⊔-rules to the resulting𝑘copies of𝐴, which leads to the explosion in the size of the search space. Fortunately, some notable works of optimisations known as lazy unfolding and absorption had been used to identifying these problems [19]. Herein we choose lazy unfolding optimisation to address this issue. Due to the limitation of the paper, detailed definition of lazy unfolding is omitted here, and we refer the reader to the above literatures.

4.2. Core Satisfiability Optimisations

In the following, we present core reasoning of 𝐿-𝒮ℋℐ𝒩, a satisfiability checking algorithm, which implements a highly optimised version on the basis of the tableau algorithm described in Section 3. Before introducing the optimisations extended to the naive tableau algorithm, we present a brief introduction to backjumping and global caching and show how such algorithms are implemented in practice.

4.2.1. Dependency Directed Backtracking (Backjumping)

Backjumping is an optimisation that is crucial for effective tableau based reasoning. Inherent unsatisfiability concealed in subformulae can lead to large amounts of unproductive backtracking search known as thrashing, especially if the unsatisfiability caused by the modal subformulae had been slightly less trivial [20]. Consider the following modified formula𝜍: 𝜍=𝑝₁∨𝑞₁∧⋯∧𝑝_𝑛∨𝑞_𝑛∧(¬∃𝑅⋅(𝐴,𝐵))∧(∀𝑅⋅(¬𝐴)).(7)

To avoid an exponential (2^𝑛) search in the case of𝜍, a form of dependency directed backtracking called backjumping can be adapted, which has also been used, for example, in solving constraint satisfiability problems [21] and (in a slightly different form) in the HARP theorem prover [22]. The arithmetic labels each formula𝜍in a node𝑣with a dependency setdep(𝜍,v)indicating the branching points on which it depends.

In case node v is with dep(𝜍,¬𝜍)∈Σ(v), dep(𝜍,v) and dep(¬𝜍,v) are used to identify the most recent branching point b on which 𝜍 or ¬𝜍 depends. If the dependance exists, the algorithm can then jump back tobover intervening branching points without exploring any alternative branches (nondeterministic choices) and make a different nondeterministic choice which might not lead to the same closure condition being encountered. Figure 1 illustrates how the expansion below𝑣is pruned by backjumping, with the number of𝑅-𝑠𝑢𝑐𝑐𝑒𝑠𝑠𝑜𝑟𝑠explored being reduced by2^𝑛−1. We will go into particulars in Algorithm 2. In case no such𝑏exists, the clash did not depend on any nondeterministic choice, and the algorithm stops, reporting the unsatisfiability of the TBox [20].

Figure 1 

Runing the search using backjumping.

4.2.2. Global Caching

In order to avoid the same subproblem being solved again and again, it is possible to cache and reuse the results of such subproblems. The usual technique is to use a hash table to store the satisfiability status of node labels (i.e., sets of concepts treated as a conjunction). At present, there are various caching methods briefly outlined in [23]:(i)no caching DFS (NC),(ii)unsat caching DFS (UC),(iii)mixed caching DFS (MC),(iv)unrestricted global caching and propagation (GC-NonDFS),(v)global caching DFS with propagation (GC-DFS).

Global caching means that each possible set of concepts/formulas is expanded at most once. With respect to the size of the given problem, the notion of global caching is an ExpTime complexity procedure if the search space contains at most an exponential number of different nodes and as long as the extension of each node requires at most exponential time.

4.2.3. Optimized GLB

From Table 2, we can conclude that an 𝐿-𝒮ℋℐ𝒩 ABox 𝒜 can contain a number of positive or negative assertions without forming a contradiction. Therefore, it is useful to compute lower and upper bounds of finite lattice degree.

Given a fuzzy KB Σ and an assertion𝛼, it is absorbing to compute best lower and upper truth-value bounds of𝛼. To this end we define the greatest lower bound of𝛼with respect to Σ(denoted byglb(Σ,𝛼))to be⊕{𝑙∣Σ⊨⟨𝛼⪰𝑙⟩}. Similarly, we define the least upper bound of𝛼with respect to Σ (denoted bylub(Σ,𝛼)) to be⊗{𝑙∣Σ⊨⟨𝛼⪯𝑙⟩}. lub(Σ,𝛼))could be computed from theglbaslub(Σ,𝛼)) = ∼glb(Σ,∼𝛼)) because∼lub(Σ,𝛼)) = ∼⊗{𝑙∣Σ⊨⟨𝛼⪯𝑙⟩}=⊗{∼𝑙∣Σ⊨⟨𝛼⪯𝑙⟩} = ⊗{𝑙∣Σ⊨⟨𝛼⪯∼𝑙⟩} = ⊕{𝑙∣Σ⊨⟨∼𝛼⪰𝑙⟩} = ∼glb(Σ,∼𝛼)). Determining theluband theglbis called the Best Lattice Degree Bound (BLDB) problem.

Concerning the BCVB problem, we may have a similar algorithm as for fuzzy 𝒜ℒ𝒞 [9], and an optimization in the search space is to use binary search algorithm reducing in that way the satisfiability checks required. Let glb(Σ,𝛼) ∈ 𝑁^Σ and lub(Σ,𝛼) ∈ 1−𝑁^Σ, where 𝑁^Σ={0,0.5,1}∪{𝑙∶⟨𝛼≥𝑐⟩ ∈ Σ}∪{1−𝑙∶⟨𝛼≤𝑙⟩ ∈ Σ} and1−𝑁^Σ={(1−𝑙)∶𝑙 ∈ 𝑁^Σ}. We employ the binary search algorithm, assuming that0.5is the middle if we sort the elements of𝑁^Σ. We evaluate if Σ⊨(𝑥∶𝐶)≥0.5is satisfied, in case it is we move on to higher degree until Σ⊭(𝑥∶𝐶)≥𝑛, indicating that the previous degree is theglb(Σ,(𝑥∶𝐶)).

Let us assume that ℒ=⟨𝑙,⪯⟩is a linear order,𝑁^Σis ordered by ascending linear order, and‖𝑁^Σ‖is the numbers of lattice degrees. A simple iterative approximation algorithm computing glb(Σ,𝛼)is described in Algorithm 1. The similar algorithm can be easily constructed for thelub.

Data: Let 𝑡∈ℕ+, 𝑡≈𝑚/2, 𝑑,𝑑𝑡∈𝑁Σ, 𝑑=𝑑𝑡. Result:  glb(Σ,𝛼)=𝑑. begin   while  𝑁Σ>1  do    if  Σ⊨⟨𝛼⪰𝑑𝑡⟩  then      𝑑=𝑑𝑡      𝑁Σ←𝑁Σ∖ {𝑑′: 𝑑⪰′╱𝑑∪{𝑑}}      𝑑𝑡←𝑑𝑡+1    else      𝑁Σ←𝑁Σ∖ {𝑑′: 𝑑′⪰𝑑}      𝑑𝑡←𝑑𝑡−1

Algorithm 1 

Calculate glb.

Input: A TBox 𝒯 in NNF and a finite set of concepts 𝑋 in NNF  Output: an and-or forest 𝐺=⟨𝑉,𝐸⟩, with 𝜎∈𝑉 as the initial node such that 𝜎.𝑠𝑡𝑎𝑡𝑢𝑠={sat, unsat}    begin    create a new node 𝜎 with Σ(𝜎)∶=𝑋∪𝒯 and 𝜎.𝑠𝑡𝑎𝑡𝑢𝑠 = unexpanded;    let 𝑉∶={𝜎}, 𝐸∶=∅; (a)    while  𝑣.status∉{sat, unsat}  do (b)     choose an node 𝑣∈𝑉, 𝑣.𝑠𝑡𝑎𝑡𝑢𝑠  := unexpanded;      if no 𝐿-𝒮ℋℐ𝒩-tableau rule is applicable to Σ(𝑣)  then         𝑣.𝑠𝑡𝑎𝑡𝑢𝑠={sat};      else if  ⟂ is applicable to Σ(𝑣)  then         𝑣.𝑠𝑡𝑎𝑡𝑢𝑠={unsat};      else if  𝛾  (𝛾∈𝛾1) is applicable to Σ(𝑣) giving concept 𝑌  then         𝑣.𝑘𝑖𝑛𝑑:= and-node, Θ∶={𝑌};      else if  𝛾  (𝛾∈𝛾2) is applicable to Σ(𝑣) giving concepts 𝑌1 and 𝑌2  then         𝑣.𝑘𝑖𝑛𝑑:= or-node, Θ∶={𝑌1,𝑌2};      else         (i) 𝑣.𝑘𝑖𝑛𝑑:= and-node;         (ii) for every ∃𝑅⋅𝐶∈ℒ(𝑣), apply 𝛾  (𝛾∈𝛾3) to Σ(𝑣) giving concept 𝚝𝚛𝚊𝚗𝚜𝑅(Σ(𝑣),𝑅)∪{𝐶}∪𝒯 and add this         concept to Θ; (c)    for  ∀𝑌∈Θ  do         (i) if  𝑠𝑜𝑚𝑒  𝑤∈𝑉 has Σ(𝑤)=𝑌  then         then add edge (𝑣,𝑤) to 𝐸;         (ii) let 𝑤 be a new node, set Σ(𝑤)∶=𝑌, 𝑤.𝑠𝑡𝑎𝑡𝑢𝑠:= unexpanded, add 𝑤 to 𝑉, and add edge (𝑣,𝑤) to 𝐸; (d)    if (𝑣.kind = or-node and one of the successors of 𝑣 has status sat) or (𝑣.kind = and-node and all the      successors of 𝑣 have status sat) then         𝑣.𝑠𝑡𝑎𝑡𝑢𝑠:= sat, propagate(𝑣);      else if (𝑣.kind = and-node and one of the successors of 𝑣 has status unsat) or (𝑣.kind = or-node and all the      successors of 𝑣 have status unsat) then         𝑣.𝑠𝑡𝑎𝑡𝑢𝑠≔ unsat, propagate(𝑣);      else         𝑣.𝑠𝑡𝑎𝑡𝑢𝑠:= expanded; (e)    if  𝑣.𝑠𝑡𝑎𝑡𝑢𝑠 = unsat  and  𝑣 in 𝒯, {𝜅,¬𝜅}∈Σ(𝑣)  then         dep(𝜅,𝑣)∪dep(¬𝜅,𝑣);      else if  𝑣.𝑠𝑡𝑎𝑡𝑢𝑠:= unexpanded  then         if  Σ(𝑣) is of the form 𝜅1∨𝜅2  then         save(𝑇𝑠);         add 𝜅1 to Σ(𝑣) with dep(𝜅1,𝑣)={𝑏}∪dep(Σ(𝑣),𝑣); (f)       if  𝑏∈𝑆  then          add 𝜅2 to Σ(𝑣) with dep(𝜅2,𝑣)=𝑏∪𝑑;      else         break; (g)    If  𝜎.𝑠𝑡𝑎𝑡𝑢𝑠 = unsat  then      return false;      else      return true;

Algorithm 2 

A optimized decision procedure for checking satisfiability in 𝐿-𝒮ℋℐ𝒩.

The described optimizations in Algorithm 1 for glb are very effective since they reduce the search space of tableau independently ofΣused. Based on the previous partition based on connectivity optimization, we could select a partition𝒜^′where the individual of assertion𝛼is contained. Thus a new set of membership degrees only for it with𝑁^𝒜′⊆𝑁^Σis evaluated, resulting to greatly less satisfiability checks.

Proposition 14. By using a binary search on𝑁^Σ, theglb(Σ,𝛼)problem in𝐿-𝒮ℋℐ𝒩 with respect to witnessed models can be decided in 𝒪(log|𝑁^Σ|).

Proof. The complexity of ordering and determining𝑁^Σare 𝒪(|𝑁^Σ|⋅log|Σ|)and 𝒪(Σ), respectively. Hence, the value ofglb(Σ,𝛼)can be determined in at most 𝒪(log|𝑁^Σ|)fuzzy entailment tests.
Moreover, the performance of calculatingglbcan be further improved by preventing recurrent expansion of the expansion rules in 𝒯. When ℱ in which no expansion rule can apply, which is then added to the cached completion-forest.

5. The Optimized Algorithm

5.1. Algorithm Overview

Our optimized algorithm builds a completion forest ℱ consisting of a collection of trees whose distinguished roots are arbitrarily connected by edges. We firstly explain the structure of ℱ in more detail.

Definition 15. A node𝑣in the completion forest ℱ=⟨𝑉,𝐸⟩extends an and/or node which is a record with extra two attributes:𝑠𝑡𝑠_𝑣∈℘^?,kind_𝑣∈ℑ^?, where℘:={unexpanded,expanded,sat,unsat}, ℑ:={and-node,or-node}

The status of a non-leaf-node is computed from the status of its𝑅-𝑠𝑢𝑐𝑐𝑒𝑠𝑠𝑜𝑟using its kind (and-node/or-node) and treating satisfiability with respect to the TBox 𝒯 (i.e., sat) as true and unsatisfiability with respect to the 𝒯 (i.e. unsat) as false. When a node gets status   sat or  unsat, the status can be propagated to its𝑅-𝑝𝑟𝑒𝑑𝑒𝑐𝑒𝑠𝑠𝑜𝑟in a way appropriate to the forest and/or structure if desired.

An   and-node (resp., or-node) becomes   unsat (resp., sat) if one of its𝑅-𝑠𝑢𝑐𝑐𝑒𝑠𝑠𝑜𝑟𝑠 becomes   unsat (resp., sat). For the remaining case, an   and-node (resp.,   or-node) becomes   sat (resp.,    unsat) if the number of its𝑅-𝑠𝑢𝑐𝑐𝑒𝑠𝑠𝑜𝑟𝑠 with𝑠𝑡𝑎𝑡𝑢𝑠≠𝚜𝚊𝚝(resp.,≠𝚞𝚗𝚜𝚊t) reduces to0.

It is well known that different orders of expanding nondeterministic rules can result in huge (up to several orders of magnitude) differences in reasoning performance [24]. According to empirical analysis in [24], we have the definition as below.

Definition 16. In our terminology, the following ordering would be optimal for extending: the ⊔-rule has the lowest priority, the generating rules (like the⩾-rule and the∃-rule) have the second-lowest priority, all other rules except for⩽_∗have the second-highest priority, and the other rules have the highest priority.

Instead of a top-down approach to most systems used for ordering expansion rules, we use a new ordering heuristic rule by an OrderedRule list to control the application of the expansion rules. The fundamental idea is that expansion rules may become applicable whenever a concept is added to a node label, then the node/concept pair is added to the OrderedRule list.

Definition 17. Granting the demand of Algorithm 2, the tableau expansion rules in Table 3 are divided into three parts{𝛾₁,𝛾₂,𝛾₃}: 𝛾₁:= {⊓_⊳,⊓_⊲,∀_⊲,∀_⊳,∀₊}, 𝛾₂:= {≤_⊲,≤_⊲,≤_⊳,≥_⊲,≤_𝑟⊳,≥_𝑟⊲,¬_⋈,⊔_⊲,⊓_⊳}, 𝛾₃:= {∃_⊳,∃_⊲,∃₊}.

Additionally, we use a OrderedRule list to control the application of the expansion rules{𝛾₁,𝛾₂,𝛾₃}. The OrderedRule list sorts all these entries according to the descending order queued above and gives access to the first element in the list. The strategy for rule applications given above is the standard one, by applying 𝛾₁ and 𝛾₂ rules as much as possible and then applying 𝛾₃ rules to realise existential concepts. Besides, the strategy for choosing which node in the 𝛾_𝑖 to expand next is in the descending order, and that means that,for example, ⊓_⊳ in𝛾₁ will be the first applied rule. The set Θ contains the contents of the resultingΣ(𝑣). If the applied tableau rule is in𝛾₁, then 𝑣 has one formula in Θ; if the applied rule is in 𝛾₂ then 𝑣 has two formulae in Θ; otherwise, ∃_∗ supply one appropriate formula for Θ.

5.2. The Optimized Algorithm Design

In the following, assume𝑣is current expanded node,𝑏is a nonnegative integer indicating the 𝑏th-∨rule application in the run of the tableau algorithm,dep(𝜍,𝑣)is a formula𝜍in the label of a node𝑣with a dependency set if a node𝑣contains both{𝜅,¬𝜅}∈Σ(𝑣),𝑌is finite sets of concepts, and the setΛcontains the contents of the resulting𝑌.

To check whether a given finite set 𝑋 is satisfiable in 𝒯, where 𝒯 is a Box, the content of the initial node𝜎with status unexpanded is𝑋∪𝑇. Initially, the queue of nodes𝑉set covers initially only root node𝜎but can grow as the algorithm proceeds. Next, we present an Exptime decision procedure for𝐿-𝒮ℋℐ𝒩 to create a completion forest and/or structure.

We comment on Algorithm 2 given in pseudocode, which determines whether a concept 𝑋∈𝒜 is satisfiable with respect to a TBox 𝒯, both in NNF. The main while-loop (at line (a)) continues processing nodes until the status of𝜎is determined to be in{𝚜𝚊𝚝,unsat}, or until every node is expanded, whichever happens first.

Inside the main loop, we choose an unexpanded node (at line (b)) and try to apply one and only one of the tableau rules in the order𝛾₁, 𝛾₂, 𝛾₃to the current node𝑣. If the applied tableau rule is within𝛾₁, then𝑣has one𝑌; or, if the applied rule is within𝛾₂, then𝑣has𝑌₁, 𝑌₂; otherwise, apply(∃_⊳)and(∃_⊲)toΣ(𝑣)giving concept 𝚝𝚛𝚊𝚗𝚜_𝑅(Σ(𝑣),𝑅)∪{𝐶}∪𝒯 and add this concept toΘ.

After that, for every 𝑌 inΘ(at line (c)), we create the required successor in the ℱ only if it does not yet exist. Then, the procedure attempts to compute the status of such a non-leaf-node𝑣employing the kind (or-node/and-node) of𝑣and the status of the𝑅-𝑠𝑢𝑐𝑐𝑒𝑠𝑠𝑜𝑟of𝑣, regarding 𝚞𝚗𝚜𝚊𝚝 as irrevocably false and 𝚜𝚊𝚝 as irrevocably true.

On the condition that previous two steps cannot determine the status of𝑣as𝚜𝚊𝚝 or 𝚞𝚗𝚜𝚊𝚝, then its status is set to be expanded. On the other side, if so (at line (d)), this information is itself propagated to𝑅-𝑝𝑟𝑒𝑑𝑒𝑐𝑒𝑠𝑠𝑜𝑟of𝑣in the ℱ via the routine 𝑝𝑟𝑜𝑝𝑎𝑔𝑎𝑡𝑒(𝑣).

Sequentially, if𝑣.𝑠𝑡𝑎𝑡𝑢𝑠=𝚞𝚗𝚜𝚊𝚝 and𝑣 in 𝒯 (at line (e)),{𝜅,¬𝜅}∈Σ(𝑣), then the closure dependency set𝑆∶=dep(𝜅,𝑣)∪dep(¬𝜅,𝑣), and the algorithm backtracks to the𝑛th branching point 𝑏to apply 𝑏th-∗-rule application. 𝒯 would be restored to its state prior to adding𝜅₁toΣ(𝑣), and the rule would be applied again such that𝜅₂was added toΣ(𝑣).

Using backjumping, the algorithm will return without expanding the completion trees obtained by adding the various𝜅₂toΣ(𝑣)immediately (at line (f)) because𝑏∉𝑆. Satisfiable will eventually return∅, concluding that𝑋is unsatisfiable.

If𝜎.𝑠𝑡𝑎𝑡𝑢𝑠=unsat(at line (g)), the result is false, otherwise true.

Proposition 18. Let ℱ be constructed by Algorithm 2 for Σ(𝑣)andΓ, with𝜎as the initial node. Then Σ(𝑣)is satisfiable with respect toΓif and only if𝜎.status =   sat.

Lemma 19. Algorithm 2 assumes that Σ(𝑣)is consistent and constructs a model ofΓthat satisfies Σ(𝑣). Let ℱ=⟨𝑉,𝐸⟩be the completion forest constructed by the algorithm for Σ(𝑣)∪Γ. Suppose the set of concepts carried by the node𝑣∈𝑉isℏ(𝑣), for every𝑣∈𝑉, if𝑣.status =   unsat, thenℏ(𝑣)is inconsistent with respect toΓ.

Proof (sketch).. Since𝑣only depends on its successors and by copying nodes to ensure that the resulting structure is a completion-tree tableau, we can construct a closed tableau with respect toΓfor the set of concepts carried by the node𝑣by induction.

5.3. Computational Complexity of the Optimized Algorithm

We proceed to analyse the runtime of the global caching algorithm, under suitable sanity assumptions on the set of modal rules. Specifically, in order to ensure that executions of the algorithm run in exponential time, we need to assume𝑛is the size of input, that is, the sum of the size of Σ(𝑣)∪Γ. This construction uses both caching and backjumping techniques.

Proposition 20. Algorithm 2 runs in ExpTime.

Proof (sketch).. Sinceℏ(𝑣)⊆𝑆(Σ(𝑣)∪Γ), so the set contains2^𝒪(𝑛)concepts. Omitting the execution time of procedure propagate, every𝑣∈𝑉is expanded only once and every expansion takes2^𝒪(𝑛)time units.
When𝑣.𝑠𝑡𝑎𝑡𝑢𝑠becomes   sat or   unsat, the propagate procedure executes2^𝒪(𝑛)basic steps directly involved with𝑣, so the total time of the executions of propagate is of rank2^{2⋅𝒪(𝑛)}.
Using backjumping, the algorithm returns from𝑏th branching point immediately because𝑏∉𝑆. This is obviously true for all of the preceding branching points, so all calls to𝑏-th branching point will return without expanding the completion trees toΣ(𝑣). So that the size of Σ(𝑣)∪Γcan be further reduced.

6. Evaluation

In this section, we provide test results about the usefulness of the optimizations of our algorithm on the test from revised LogicsWork Bench (LWB) benchmarks [25] and the DL98 benchmarks (http://dL.kr.org/dl98/comparison/). All the experiments were conducted under Ubuntu 12.04 on a PC with 2.40 GHz Intel Core 2 CPU Q8300 and 2 GB RAM, using the 1s time limit for each test problem. We consider only the number of solved problems instead of the used CPU time.

6.1. Programming Language

The problem we are dealing with in this work is whether the optimized tableau algorithm proposed by us can be implemented to give an efficient prover for𝐿-𝒮ℋℐ𝒩. This problem can be raised as what optimization techniques can be implemented to obtain a prover for𝐿-𝒮ℋℐ𝒩 that is both optimal from the theoretical point of view and efficient in practice. To the best of our knowledge, there is no DL/uncertainty prover/reasoner for𝐿-𝒮ℋℐ𝒩. Thus we have developed in C++ a tableau prover for𝐿-𝒮ℋℐ𝒩, which uses serial of optimization methods described above.

C++ is chosen to implement tableau prover because of its increase efficiency, portability, and extendibility. Pointers are intensively used for the data structures of the tableau prover to reduce memory use and to allow caching objects like formulas, sets of formulas, and nodes of a completion forest. Advanced data structures of C++ with low complexity like the templates set, map, multimap, and priority queue are also intensively used. And it is more convenient to extending it for other description logics in a modular way [26].

6.2. Experimental Results on Usefulness of the Optimizations

Because neither do good benchmarks for𝐿-𝒮ℋℐ𝒩 seem to exist, so we found it hard to compile a good set of test problems. Although the Tableaux’98 and random test suites could show how optimisations perform on propositional modal logics, neither is very good for our purposes.

We therefore recast the K, KT, and S4 problem sets from the LogicsWork Bench (LWB) benchmarks and the   T98-sat problem set from the DL98 benchmarks (there are 4 sets of DL98 test suite, and only concept satisfiability tests are employed) to be compatible with the fuzzy logic𝐿-𝒮ℋℐ𝒩 over finite lattices. Due to the space limit, we omit the detailed process of transformation. The three anterior benchmarks measure the performance of the system when testing the satisfiability of large concept expressions without reference to a Tbox. Moreover, DL98 TBox problem set textttT98-sat test is added to contain TBox axioms. The correctness of the system is also tested by checking that the answer is as expected.

Our algorithm is fairly detailed, so a naive implementation should be straightforward. The preprocessing optimisations (“partition based on connectivity" and “dealting with TBox axioms”) described in Section 4.1 take at most polynomial time to run and reduce the search space more significantly than the extended framework of Donini and Massacci [27]. So it is sensible to use them in almost every case in our tableau prover. After preprocessing, it is crucial to employ core satisfiability optimisations presented in Section 4.2 that try to improve the performance of the core satisfiability tester.

We intend to show which of the optimisations are most effective, so we have had compile-time configuration options included that can be used to turn on and off or vary the previous optimisations. We chose to test various combinations with all the other optimisations enabled and then to test the same optimisations options, with each of the optimisations turned off one by one. The optimisations that were removed are as follows.(i)ALL: turn on all the optimizations.(ii)No partition based on connectivity (NP): turn off ABox partition.(iii)No dealing with TBox axioms (ND): turn off dealing with TBox axioms.(iv)No backjumping (NB): turn off backjumping.(v)No global caching (NC): turn off global caching.

We end up with 5 configurations and ran each of these configurations over the nine test suites from satisfiability checking problems. For the corresponding subset of tests and the corresponding set of options, each table cell includes the number of solved problems, where the diamond symbol denotes 21 (the number of test problems in the subset). For each configuration, there are five entries corresponding to timeouts of 1 second, 2 seconds, 4 seconds, 8 seconds, and 16 seconds.

6.2.1. K(ℒ) Tests

Table 4 obviously reveals that, amongst the tested configurations, turning on all the optimizations provides the most effective method. These two other optimisations are the most effective, whose absence in the experiments has made them unacceptably slow. For k_lin_n(ℒ) and k_lin_p(ℒ), the methods of global caching significantly outperform backjumping. There is a nice distinction between the two preprocessing optimisations methods. Partition based on connectivity and dealing with TBox axioms is less important at least within our set of test cases. For k_lin_n(ℒ), k_path_n(ℒ), and k_path_p(ℒ), “dealing with TBox axioms" significantly outperforms “partition based on connectivity," and for the other problems, “partition based on connectivity" outperforms “dealing with TBox axioms."

Table 4 

Results for K(ℒ) tests.

6.2.2. KT(ℒ) Tests

Similar testing was performed on KT(ℒ) tests, and results are showed in Table 5. Again, there is too much data to present it that global caching significantly outperforms other optimization methods in both of its incarnations. Particularly, we have undergone an increase of memory usage by a worst-case factor of two in case of backjumping in comparison to no backjumping. This time, for the most problems, the “partition based on connectivity” method is on par with “dealing with TBox axioms.”

Table 5 

Results for KT(ℒ) tests.

6.2.3. S4(ℒ) Tests

The results of S4(ℒ) tests, given in Table 6, show that backjumping performs very well compared to other methods. This result is surprising; global caching is less effective at almost all in this test suite. The probable reason for the ineffectiveness of the method is that, with such a small number of literals in the successor nodes, the purely propositional problems can always be worked out deterministically, while the performance is contingent on the efficiency of propositional reasoning at the root node.

Table 6 

Results for S4(ℒ) tests.

There appears to be no clear winner between the two methods “partition based on connectivity” and “dealing with TBox axioms.” “Dealing with TBox axioms" method outperforms the “partition based on connectivity” method for most problems, and for the problems s4_ipc_p(ℒ) and s4_path_p(ℒ), “dealing with TBox axioms” method beats the “partition based on connectivity" method. However, the differences are not great.

6.2.4. K(ℒ) TBox Tests

Due to latticed LWB benchmarks not covering TBox problem, so test on the test set DL’98 T98 kb is employed. As shown in Table 7, in this test, “dealing with TBox axioms" is by far the most important optimisation. The next most important optimisations are global caching and backjumping. That is because of explosion in the large size of the search space (2^𝑘𝑛) in the problems, which results in a destructive degradation in performance, even when optimisations such as global caching and backjumping are used. “Partition based on connectivity" method is almost totally ineffective.

Table 7 

Results for K(ℒ) TBox tests.

The results reveal that, amongst the tested configurations, global caching is more efficient over the other methods in most but not all cases. Since global caching severely prunes the search space by avoiding multiple expansions of the same node, by enabling greater propagation of   sat   and   unsat  via subset checking and hence allowing more cutoffs to be done [28]. Backjumping approach can be applied together with global caching well, which reduces the number of tested nondeterministic branches largely especially for S4(ℒ) Tests, while in particular, the method may increase memory usage. When considering TBox problem, “dealing with TBox axioms” is the most important optimisation. Sometimes “partition based on connectivity" method is very effective but not all the time.

As shown in the test results, the difference between using preprocessing optimizations or not is less significant than the core satisfiability optimization at least within our set of test cases. Furthermore, preprocessing optimizations cooperates very well with core satisfiability optimization techniques in most of the time.

7. Conclusion and Future Work

Many papers offer technical contributions but without practical motivations. Our method for𝐿-𝒮ℋℐ𝒩 is definitely promising and should be extended to more expressive logics to test whether it remains feasible. In the further way, the tableau algorithm can be used on the large scale data set such as linked data to solve the practical problem. As far as we know 𝐿-𝒮ℋℐ𝒩 is the first presentation of a reasoning algorithm for such complex fuzzy DL languages over finite residuated lattices. Besides, this is the first time that optimized techniques were used to tableau reasoning for completion forest in DLs augmented by De Morgan lattices. This enabled us to treat the case of general concept inclusions instead of the simpler acyclic TBoxes.

Of course, there remains plenty of work to do with the tableau algorithm of𝐿-𝒮ℋℐ𝒩, such as to directly and efficiently handle general TBoxes and to deal with concrete domains. We intend to extend𝐿-𝒮ℋℐ𝒩 also for more description logics under more semantics. Besides, the presented dependency management allows for more informed backjumping, while also supporting the creation of precise cache unsatisfiability entries [29]. Among others, better heuristics are needed to guide the choices needed during tableau reasoning, and better use of cached information is expected to provide significant benefits. Additionally, like other related research areas such as Semantic Web, most DL reasoners are implemented in Java for usability by many applications, so the Java implementation of the optimized algorithm will be carried out in the future work.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.