A Parameterized Complexity View on Description Logic Reasoning

A Parameterized Complexity View on Description Logic Reasoning

Ronald de Haan
Institute for Logic, Language and Computation
University of Amsterdam
me@ronalddehaan.eu
Abstract

Description logics are knowledge representation languages that have been designed to strike a balance between expressivity and computational tractability. Many different description logics have been developed, and numerous computational problems for these logics have been studied for their computational complexity. However, essentially all complexity analyses of reasoning problems for description logics use the one-dimensional framework of classical complexity theory. The multi-dimensional framework of parameterized complexity theory is able to provide a much more detailed image of the complexity of reasoning problems.

In this paper we argue that the framework of parameterized complexity has a lot to offer for the complexity analysis of description logic reasoning problems—when one takes a progressive and forward-looking view on parameterized complexity tools. We substantiate our argument by means of three case studies. The first case study is about the problem of concept satisfiability for the logic with respect to nearly acyclic TBoxes. The second case study concerns concept satisfiability for concepts parameterized by the number of occurrences of union operators and the number of occurrences of full existential quantification. The third case study offers a critical look at data complexity results from a parameterized complexity point of view. These three case studies are representative for the wide range of uses for parameterized complexity methods for description logic problems.

A Parameterized Complexity View on Description Logic Reasoning


Ronald de Haan Institute for Logic, Language and Computation University of Amsterdam me@ronalddehaan.eu

Copyright © 2018, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved.

Introduction

Description logics have been designed as knowledge representation formalisms that have good computational properties (?). Correspondingly, there has been a lot of research into the computational complexity of reasoning problems for different description logics. This research has, however, focused entirely on the framework of classical complexity theory to study the computational complexity (see, e.g., ? ?? ?).

The more fine-grained and multi-dimensional framework of parameterized complexity theory has hardly been applied to study the complexity of reasoning problems for description logics. Only a few works used the framework of parameterized complexity to study description logic problems (???????). Moreover, these works all use the framework in a traditional way, focusing purely on one commonly used notion of tractability (namely that of fixed-parameter tractability).

Parameterized complexity is designed to address the downside of classical complexity theory that it is largely ignorant of structural properties of problem inputs that can potentially be exploited algorithmically. It does so by distuinguishing a problem parameter , in addition to the input size , and measuring running times in terms of both of these. The parameter  can be used to measure various types of structure that are present in the problem input. Parameterized complexity theory has grown into a large and thriving research community over the last few decades (see, e.g., ? ?; ? ?; ? ?). Most results and techniques in parameterized complexity theory revolve around the notion of fixed-parameter tractability—a relaxation of polynomial-time solvability based on running times of the form , for some computable function  (possibly exponential or worse).

Due to the fact that reasoning problems related to description logics are typically of high complexity (e.g., complete for classes like PSPACE and EXPTIME), it is unsurprising that one would need very restrictive parameters to obtain fixed-parameter tractability results for such problems. It has been proposed recently that the investigation of problems that are of higher complexity can also benefit from the parameterized complexity point of view (?????)—using tools and methods that overstep the traditional focus on fixed-parameter tractability as positive results.

In this paper, we show how the complexity study of description logic problems can benefit from using the framework of parameterized complexity and all the tools and methods that it offers. We do so using three case studies: (1) parameterized results for concept satisfiability for with respect to nearly acyclic TBoxes, (2) parameterized results for concept satisfiability for fragments of that are close to , and , respectively, and (3) parameterized results addressing the notion of data complexity for instance checking and conjunctive query entailment for . The complexity results that we obtain are summarized in Tables 13 and 4—at the end of the sections where we present the case studies.

Outline.

We begin by giving an overview of the theory of parameterized complexity—including commonly used (and more traditional) concepts and tools, as well as more progressive notions. Then we present our three case studies in three separate sections, before sketching directions for future research and concluding.

Parameterized Complexity Theory

We begin by introducing relevant concepts from the theory of parameterized complexity. For more details, we refer to textbooks on the topic (??). We introduce both concepts that are used commonly in parameterized complexity analyses in the literature and less commonly used concepts, that play a role in this paper.

FPT and XP.

The core notion in parameterized complexity is that of fixed-parameter tractability, which is a relaxation of the traditional notion of polynomial-time solvability. Fixed-parameter tractability is a property of parameterized problems. A parameterized problem  is a subset of , for some finite alphabet . An instance of a parameterized problem is a pair  where  is the main part of the instance, and  is the parameter. Intuitively, the parameter captures some type of structure of the instance that could potentially be exploited algorithmically—the smaller the value of the parameter , the more structure there is in the instance. (When considering multiple parameters, we take their sum as a single parameter.) A parameterized problem is fixed-parameter tractable if instances  of the problem can be solved by a deterministic algorithm that runs in time , where  is a computable function of . Algorithms running within such time bounds are called fpt-algorithms. FPT denotes the class of all parameterized problems that are fixed-parameter tractable.

Intuitively, the idea behind fixed-parameter tractability is that whenever the parameter value  is small, the overall running time is reasonably small—assuming that the constant hidden behind  is small. In fact, for every fixed parameter value , the running time of an fpt-algorithm is polynomial (where the order of the polynomial is constant).

A related parameterized complexity class is XP, which consists of all parameterized problems for which instances  can be solved in time , for some computable function . Algorithms running within such time bounds are called xp-algorithms. That is, a parameterized problem  is in XP if there is an algorithm that solves  in polynomial time for each fixed value  of the parameter—where the order of the polynomial may grow with . It holds that . Intuitively, if a parameterized problem is in , it is not likely to be efficiently solvable in practice. Suppose, for example, that a problem is solvable in time  in the worst case. Then already for  and , it could take ages to solve this problem (see, e.g., ? ?).

Completeness Theory.

Parameterized complexity also offers a completeness theory, similar to the theory of NP-completeness, that provides a way to obtain evidence that a parameterized problem is not fixed-parameter tractable. Hardness for parameterized complexity classes is based on fpt-reductions, which are many-one reductions where the parameter of one problem maps into the parameter for the other. More specifically, a parameterized problem  is fpt-reducible to another parameterized problem  if there is a mapping  that maps instances of  to instances of  such that (i)  if and only if , (ii)  for a computable function , and (iii)  can be computed in time  for a computable function  and a constant . A problem  is hard for a parameterized complexity class K if every problem  can be fpt-reduced to . A problem  is complete for a parameterized complexity class K if  and  is K-hard.

Central to the completeness theory are the classes  of the Weft hierarchy. We will not define the classes W[] in detail (for details, see, e.g., ? ?). It suffices to note that it is widely believed that .111In fact, it holds that , assuming that -variable 3SAT cannot be solved in subexponential time, that is, in time  (???). Thus, showing that a problem  is W[1]-hard gives evidence that  is not fpt-time solvable.

An example of a W[1]-complete parameterized problem is Clique (??). Instances for this problem consist of , where  is an undirected graph, and . The parameter is , and the question is to decide whether  contains a clique of size .

Para-K.

For each classical complexity class K, we can construct a parameterized analogue para-K (?). Let K be a classical complexity class, e.g., NP. The parameterized complexity class para-K is then defined as the class of all parameterized problems  for which there exist a computable function  and a problem  in , such that for all instances  it holds that  if and only if . Intuitively, the class  consists of all problems that are in K after a precomputation that only involves the parameter. A common example of such parameterized analogues of classical complexity classes is the parameterized complexity class para-NP. Another example is .

If (the unparameterized variant of) a parameterized problem  is in the class K, then . Also, if  is already K-hard for a finite set of parameter values, then  is para-K-hard (?).

Using the classes para-K and the notion of fpt-reductions, one can also provide evidence that certain parameterized problems are not fixed-parameter tractable. If a para-K-hard parameterized problem is fixed-parameter tractable, then . For example, a para-NP-hard parameterized problem is not fixed-parameter tractable, unless .

Para-NP and para-co-NP.

The classes para-NP and para-co-NP are parameterized analogues of the classes NP and co-NP. The class para-NP can alternatively be defined as the class of parameterized problems that are solvable in fpt-time by a non-deterministic algorithm (?). Similarly, para-co-NP can be defined using fpt-algorithms using universal nondeterminism—i.e., nondeterministic fpt-algorithms that reject the input if at least one sequence of nondeterministic choices leads the algorithm to reject. It holds that .

Another alternative definition of the class para-NP—that can be motivated by the amazing practical performance of SAT solving algorithms (see, e.g., ? ?)—is using the following parameterized variant of the propositional satisfiability problem (????). Let  be the problem SAT with a constant parameter . The class para-NP consists of all problems that can be fpt-reduced to . In other words, para-NP can be seen as the class of all parameterized problems that can be solved by (1) a fixed-parameter tractable encoding into SAT, and (2) using a SAT solving algorithm to then decide the problem. The class para-co-NP can be characterized in a similar way, using UNSAT instead of SAT. Consequently, problems in para-co-NP can also be solved using the combination of an fpt-time encoding and a SAT solving algorithm.

Para-PSPACE.

The class para-PSPACE can alternatively be defined as the class of all parameterized problems  for which there exists a (deterministic or nondeterministic) algorithm deciding whether  using space , for some computable function . It holds that .

Another alternative characterization of para-PSPACE is using a parameterized variant of TQBF—the problem of deciding whether a given quantified Boolean formula is true. Let  be the problem TQBF with a constant parameter . The class para-PSPACE consists of all problems that can be fpt-reduced to . In other words, para-PSPACE can be seen as the class of all parameterized problems that can be solved by (1) an fpt-time encoding into TQBF, and (2) using a TQBF solver to then decide the problem (see, e.g., ? ?).

Yet another characterization of para-PSPACE uses alternating Turing machines (ATMs). An ATM is a nondeterministic Turing machine where the states are partitioned into existential and universal states (see, e.g., ? ?, Appendix A.1). A configuration of the ATM with an existential state is accepting if at least one successor configuration is accepting, and a configuration with a universal state is accepting if all successor configurations are accepting. Intuitively, an ATM can alternate between existential and universal nondeterminism. The class para-PSPACE consists of all parameterized problems that can be decided by an ATM in fixed-parameter tractable time.

Para-EXPTIME.

The class para-EXPTIME can be defined as the class of all parameterized problems  for which there exists a deterministic algorithm deciding whether  in time , for some computable function . It holds that  and that .

For an overview of all parameterized complexity classes that feature in this paper—and their relation—see Figure 1.

FPT

W[1]

co-W[1]

para-NP

para-co-NP

pXP

para-PSPACE

para-EXPTIME

Figure 1: An overview of the landscape of parameterized complexity classes that play a role in this paper.

Case Study 1: Concept Satisfiability for with respect to Nearly Acyclic TBoxes

In this section, we provide our first case study to illustrate how parameterized complexity can be used to obtain a more detailed image of the computational complexity of description logic reasoning. In particular, we consider the problem of concept satisfiability for the description logic with respect to general TBoxes. This problem is EXPTIME-complete in general. We consider two parameters for this problem. One of these parameters does not help to reduce the complexity of the problem—that is, for this parameter the problem is para-EXPTIME-complete. The other of the two parameters does help to reduce the complexity of the problem—that is, for this parameter the problem is para-PSPACE-complete.

We begin by revisiting the description logic , the problem of concept satisfiability with respect to acyclic and general TBoxes, and classical complexity results for this problem. We then discuss our parameterized complexity results, and how to interpret these results.

The Description Logic

Let  and  be sets of atomic concepts, roles, and individuals, respectively. The triple  is called the signature. (We will often omit the signature if this is clear from the context.)

Concepts  are defined by the following grammar in Backus-Naur form, for  and :

An interpretation  over a signature  consists of a non-empty set  called the domain, and an interpretation function  that maps (1) every individual  to an element , (2) every concept  to a subset of , and (3) every role  to a subset of , such that:
; ; ; ; ;

  • there exists some  such that ; and

  • for each  such that it holds that  .

A general concept inclusion (GCI) is a statement of the form , where  are concepts. We write  (and say that  satisfies ) if . A (general) TBox  is a finite set of GCIs. A concept definition is a statement of the form , where  is an atomic concept, and  is a concept. We write  (and say that  satisfies ) if . An acyclic TBox  is a finite set of concept definitions such that (1)  does not contain two different concept definitions  and  for any , and (2)  contains no (direct or indirect) cyclic definitions—that is, the graph  with vertex set  that contains an edge  if and only if  contains a concept definition  where  occurs in  is acyclic. An interpretation  satisfies a (general or acyclic) TBox  if  satisfies all GCIs or concept definitions in .

A concept assertion is a statement of the form , where  and  is a concept. A role assertion is a statement of the form , where  and . We write  (and say that  satisfies ) if . Moreover, we write  (and say that  satisfies ) if . An ABox  is a finite set of concept and role assertions.

Classical Complexity Results

An important reasoning problem for description logics is the problem of concept satisfiability. In this decision problem, the input consists of a concept  and a TBox , and the question is whether  is satisfiable with respect to —that is, whether there exists an interpretation  such that  and . The problem of concept satisfiability is PSPACE-complete, both for the case where  is empty and for the case where  is an acyclic TBox. For the case where  is a general TBox, the problem is EXPTIME-complete.

Proposition 1 (? ?? ?).

Concept satisfiability for the logic with respect to general TBoxes is EXPTIME-complete.

Proposition 2 (? ?? ?).

Concept satisfiability for the logic with respect to acyclic TBoxes is PSPACE-complete.

Parameterized Complexity Results

We consider a parameterized variant of the problem of concept satisfiability for where the parameter captures the distance towards acyclicity for the given TBox. That is, for this parameterized problem, the input consists of a concept , an acyclic TBox , and a general TBox . The parameter is , and the question is whether  is satisfiable with respect to —that is, whether there exists an interpretation  such that , and .

Parameterizing by the size of  does not offer an improvement in the complexity of the problem—that is, this parameter leads to para-EXPTIME-completeness.

Theorem 3.

Concept satisfiability for with respect to both an acyclic TBox  and a general TBox  is para-EXPTIME-complete when parameterized by .

Proof.

Membership in para-EXPTIME follows from the fact that the unparameterized version of the problem is in EXPTIME. To show para-EXPTIME-hardness, it suffices to show that the problem is already EXPTIME-hard for a constant value of the parameter (?). We do so by giving a reduction from the problem of concept satisfiability for with respect to general TBoxes.

Let  be a concept and let  be a general TBox. Moreover, let . We construct an acyclic TBox  and a general TBox  such that  is satisfiable with respect to  if and only if it is satisfiable with respect to . Let  be a fresh atomic concept. We let , and we let . It is straightforward to verify that  is satisfiable with respect to  if and only if it is satisfiable with respect to . Moreover,  is constant. From this, we can conclude that the problem is para-EXPTIME-hard. ∎

Intuitively, restricting only the number (and size) of the general TBox  does not restrict the problem, as we can encode a general TBox of arbitrary size in the acyclic TBox (together with a small general TBox). If we restrict the number of concepts impacted by the general TBox, however, we do get an improvement in the complexity of the problem.

Let  be an acyclic TBox and let  be a general TBox. We define the set of concepts impacted by  (w.r.t. ) as the smallest set  of concepts that is closed under (syntactic) subconcepts and that satisfies that (A) whenever , then , and (B) whenever  and , then . If we parameterize the problem of concept satisfiability with respect to both an acyclic TBox  and a general TBox  by the number of concepts impacted by , the complexity of the problem jumps down to para-PSPACE.

Theorem 4.

Concept satisfiability for with respect to both an acyclic TBox  and a general TBox  is para-PSPACE-complete when parameterized by the number  of concepts that are impacted by  (w.r.t. ).

Proof.

Hardness for para-PSPACE follows directly from the fact that the problem is already PSPACE-hard when  is empty (Proposition 2)—and thus the number of concepts impacted by  is . We show membership in para-PSPACE by exhibiting a nondeterministic algorithm to solve the problem that runs in space , for some computable function . Let  be a concept, let  be an acyclic TBox, and let  be a general TBox. We may assume without loss of generality that all concepts occurring in  are in negation normal form—that is, negations occur only directly in front of atomic concepts. If this were not the case, we could straightforwardly transform  to a TBox that does have this property in polynomial time, by introducing new atomic concepts  for any negated concept .

The algorithm that we use is the usual tableau algorithm (with static blocking) for —see, e.g., (?). That is, it aims to construct a tree that can be used to construct an interpretation satisfying  and . For each node in the tree, it first exhaustively applies the rules for the  and  operators, the rules for the concept definitions in , and the rules for the GCIs in  (for each  adding the concept  to a node), before applying the rules for the  and  operators. Moreover, it applies all rules exhaustively to one node of the tree before moving to another node. Additionally, the algorithm uses the following usual blocking condition (subset blocking): the rule for the  operator cannot be applied to a node  that has a predecessor  in the tree that is labelled with all concepts that  is labelled with (and possibly more). It is straightforward to verify that this tableau algorithm correctly decides the problem.

We argue that this algorithm requires space , where  is the number of concepts impacted by  and  denotes the input size. It is straightforward to verify that there is a polynomial  such that each node in the tree constructed by the tableau algorithm that is more than  steps away from the root of the tree is only labelled with concepts that are impacted by . Since there are only  concepts that are impacted by , we know that in each branch of the tree, the blocking condition applies at depth at most , and thus that each branch is of length at most . From this, it follows that this algorithm requires space , and thus that the problem is in para-PSPACE. ∎

Interpretation of the Results

The results in this section are summarized in Table 1. The parameterized results of Theorems 3 and 4 show that parameterized complexity theory can make a distinction between the complexity of the two variants of the problem that classical complexity theory is blind to. Classically, both variants are EXPTIME-complete, but one parameter can be used to get a polynomial-space algorithm (when an additional  factor for the parameter ), whereas the other parameter requires exponential space, no matter what additional  factor is allowed. The para-PSPACE result of Theorem 4 also yields an algorithm solving the problem using (1) an fpt-time encoding into the problem TQBF, and then (2) using a TQBF solver to decide the problem (see, e.g., ? ?).

  parameter complexity of concept
satisfiability w.r.t.  and 
   – EXPTIME-c (Proposition 1)
   para-EXPTIME-c (Theorem 3)
  # of concepts impacted
  by  (w.r.t. ) para-PSPACE-c (Theorem 4)
Table 1: The parameterized complexity of concept satisfiability w.r.t. both an acyclic TBox  and a general TBox , for different parameters.

Case Study 2: Concept Satisfiability for , , and

In this section, we provide our second case study to illustrate how parameterized complexity can be used to obtain a more detailed image of the computational complexity of description logic reasoning. In particular, we consider the problem of concept satisfiability for the description logic . This problem is PSPACE-complete in general. We consider several parameters that measure the distance to the logics and . The logics and are obtained from by disallowing concept union and full existential qualification, respectively. The parameters that we consider both help to reduce the complexity of the problem. One parameter renders the problem para-co-NP-complete. The other parameter renders the problem para-NP-complete. The combination of both parameters renders the problem fixed-parameter tractable.

We begin by revisiting the description logics and (and their intersection ), and classical complexity results for the problem of concept satisfiability for these logics. We then discuss our parameterized complexity results, and how to interpret these results.

The Description Logics , and

In order to obtain the description logics , and , we consider a (syntactic) variant of the logic where all concepts are in negation normal form. That is, negations only occur immediately followed by atomic concepts. Put differently, we consider concepts  that are defined as follows, for  and :

One can transform any concept into negation normal form in linear time (see, e.g., ? ?). The semantics of this variant of is defined exactly as described in the previous section. Throughout this section, we will only consider this variant of .

The description logic is obtained from the logic by forbidding any occurrence of the operator . The description logic is obtained from the logic by requiring that for every occurrence  of the existential quantifier it holds that ; that is, only limited existential quantification  is allowed. The description logic contains those concepts that are concepts in both and —that is, is the intersection of and .

Thus, the logics and are obtained from by means of two orthogonal restrictions: disallowing concept union and replacing full existential qualification by limited existential quantification, respectively. The logic is obtained from by using both of these restrictions.

Classical Complexity Results

In this section, we consider the problem of concept satisfiability with respect to empty TBoxes. In this decision problem, the input consists of a concept , and the question is whether  is satisfiable—that is, whether there exists an interpretation  such that . This problem is PSPACE-complete for , co-NP-complete for , NP-complete for , and polynomial-time solvable for .

Proposition 5 (? ?).

Concept satisfiability for the logic is PSPACE-complete.

Proposition 6 (? ?).

Concept satisfiability for the logic is co-NP-complete.

Proposition 7 (? ?).

Concept satisfiability for the logic is NP-complete.

Proposition 8 (? ?).

Concept satisfiability for the logic is polynomial-time solvable.

The -rule
Condition contains , but it does not contain both  and .
Action .
The -rule
Condition contains , but neither  nor .
Action , .
The -rule
Condition contains , but there is no individual  such that  and  are in .
Action ,
where  is an arbitrary individual not occurring in .
The -rule
Condition contains  and , but it does not contain .
Action .
The -rule
Condition contains  and , but it does not contain .
Action .
Table 2: Transformation rules of the tableau algorithm for concept satisfiability.

Parameterized Complexity Results

In order to conveniently describe the parameterized complexity results that we will establish in this section, we firstly describe an algorithm for deciding concept satisfiability for in polynomial space (see, e.g., ? ?, Chapter 2). To use this algorithm to prove the parameterized complexity results in this section, we describe a variant of the algorithm that can be implemented by a polynomial-time alternating Turing machine—i.e., a nondeterministic Turing machine that can alternate between existential and universal nondeterminism.

The algorithm uses ABoxes  as data structures, and works by extending these ABoxes by means of several transformation rules. These rules are described Table 2—however, not all rules are applied in the same fashion. The -rule, the -rule and the -rule are used as deterministic rules, and are applied greedily whenever they apply. The -rule and the -rule are nondeterministic rules, but are used in a different fashion. The -rule transforms an ABox  into one of two different ABoxes  or  nondeterministically. The -rule is implemented using existential nondeterminism—i.e., the algorithm succeeds if at least one choice of  and  ultimately leads to the algorithm accepting. (For more details on existential and universal nondeterminism and alternating Turing machines, see, e.g., ? ?, Appendix A.1.) The -rule, on the other hand, transforms an ABox  into a unique next ABox , but it is a nonmonotonic rule that can be applied in several ways—the condition can be instantiated in different ways, and these instantiations are not all possible anymore after having applied the rule. The -rule is implemented using universal nondeterminism—i.e., the algorithm succeeds if all ways of instantiating the condition of the -rule (and applying the rule accordingly) ultimately lead to the algorithm accepting.

The tableau algorithm works as follows. Let  be an concept for which we want to decide satisfiability. We construct an initial ABox , where  is an arbitrary individual. We proceed in two alternating phases: (I) and (II)—starting with phase (I).

In phase (I), we apply the deterministic rules (the -rule, the -rule and the -rule) and the nondeterministic -rule exhaustively, until none of these rules is applicable anymore. For the -rule we use existential nondeterminism to choose which of  and  to use. When none of these rules is applicable anymore, we proceed to phase (II). In phase (II), we apply the -rule once, using universal nondeterminism to choose how to instantiate the condition (and we apply the rule accordingly). Then, we go back to phase (I).

Throughout the execution of the algorithm, there is always a single current ABox . Whenever it holds that , the algorithm rejects. If at some point no rule is applicable anymore—that is, if at some point we are in phase (II) and the -rule is not applicable—the algorithm accepts.

This algorithm essentially works the same way as known tableau algorithms for concept satisfiability (see, e.g., ? ?, Chapter 2). The only difference is that in the algorithm described above the implementation of the -rule using existential nondeterminism and the implementation of the -rule using universal nondeterminism is built in. In the literature, typically descriptions of tableau algorithms leave freedom for different implementations of the way in which the search tree is traversed. One can think of the algorithm described above as traversing a search tree that is generated by the different (existential and universal) nondeterministic choices that are made in the execution of the algorithm. This search tree is equivalent to the search tree of the usual tableau algorithm for concept satisfiability. Thus, we get that the algorithm is correct. In fact, this algorithm is a reformulation of the standard algorithm known from the literature (?).

Proposition 9 (? ?).

The tableau algorithm described above for an alternating polynomial-time Turing machine correctly decides concept satisfiability for .

We will now consider several parameterized variants of the problem of concept satisfiability for . These parameters, in a sense, measure the distance of an concept to the logics , and , respectively. We will make use of the tableau algorithm described above to establish upper bounds on the complexity of these problems. Lower bounds follow directly from Propositions 68.

We begin with the parameterized variant of concept satisfiability where the parameter measures the distance to .

Theorem 10.

Concept satisfiability for the logic , parameterized by the number of occurrences of the union operator  in , is para-co-NP-complete.

Proof.

Hardness for para-co-NP follows from the fact that concept satisfiability is co-NP-complete (Proposition 6). Any concept is an concept with zero occurrences of the union operator . Therefore, the problem of concept satisfiability parameterized by the number  of occurrences of the union operator  in  is already co-NP-hard for the parameter value . From this, it follows that the parameterized problem is para-co-NP-hard (?).

To show that the parameterized problem is also contained in para-co-NP, we describe an algorithm that can be implemented by an alternating Turing machine that only makes use of universal nondeterminism and that runs in fixed-parameter tractable time. This algorithm is similar to the tableau algorithm for described above, with the only difference that the -rule is now not implemented using existential nondeterminism. Instead, we deterministically iterate over all possible choices that can be made in executions of the -rule. That is, whenever the -rule is applied, resulting in two possible next ABoxes  and , we firstly continue the algorithm with , and if the continuation of the algorithm with  failed, we then continue the algorithm with  instead.

Let  be the number of occurrences of the union operator  in . For each occurrence, the -rule is applied at most once. Therefore, the total number of possible choices resulting from executions of the -rule is at most . Therefore, this modification of the algorithm can be implemented by an alternating Turing machine that only uses universal nondeterminism and that runs in time . In other words, the problem is in para-co-NP, and thus is para-co-NP-complete. ∎

Theorem 11.

Concept satisfiability for the logic , parameterized by the number of occurrences of full existential qualification  in , is para-NP-complete.

Proof.

Hardness for para-NP follows from the fact that concept satisfiability is NP-complete (Proposition 7). Any concept is an concept with zero occurrences of full existential qualification . Therefore, the problem of concept satisfiability parameterized by the number of occurrences of full existential qualification  in  is already NP-hard for the parameter value . From this, it follows that the parameterized problem is para-NP-hard (?).

To show membership in para-NP, we modify the tableau algorithm for , similarly to the way we did in the proof of Theorem 10. In particular, we describe an algorithm that can be implemented by an alternating Turing machine that only makes use of existential nondeterminism and that runs in fixed-parameter tractable time. We do so by executing the -rule deterministically, instead of using universal nondeterminism. That is, instead of using universal nondeterminism to choose which instantiation of the condition of the -rule to use, we iterate over all possibilities deterministically.

Let  be the number of occurrences of full existential quantification  in . At each point, there are at most  different ways of instantiating the -rule. Moreover, after having applied the -rule for at most  times, the -rule is not applicable anymore. Therefore, the total number of possible choices to iterate over is at most . Therefore, this modification of the algorithm can be implemented by an alternating Turing machine that only uses existential nondeterminism and that runs in time . In other words, the problem is in para-NP, and thus is para-NP-complete. ∎

Theorem 12.

Concept satisfiability for the logic , parameterized by both (i) the number of occurrences of the union operator  in  and (ii) the number of occurrences of full existential qualification  in , is fixed-parameter tractable.

Proof (sketch).

We can modify the alternating polynomial-time tableau algorithm for concept satisfiability to work in deterministic fpt-time by implementing both the -rule and the -rule deterministically, iterating sequentially over all possible choices that can be made for these rules. That is, we combine the ideas behind the proofs of Theorems 10 and 11. We omit the details of this fpt-time algorithm. ∎

Interpretation of the Results

The results in this section are summarized in Table 3. Similarly as for the first case study, the results for the second case study show that parameterized complexity theory can make distinctions that classical complexity theory does not see. The problems studied in Theorems 1011 and 12 are all PSPACE-complete classically, yet from a parameterized point of view their complexity goes down to para-NP, para-co-NP and FPT. The para-NP- and para-co-NP-completeness results of Theorems 10 and 11 also yield algorithms that (1) firstly use an fpt-encoding to an instance of SAT and (2) then use a SAT solver to decide the problem (see, e.g., ? ?).

  parameter complexity of
concept satisfiability
   – PSPACE-c (Proposition 5)
  # of occurrences of  para-co-NP-c (Theorem 10)
  # of occurrences of  para-NP-c (Theorem 11)
  # of occurrences of  and  FPT (Theorem 12)
Table 3: The parameterized complexity of concept satisfiability (with no TBoxes) for different parameters.

Case Study 3: A Parameterized Complexity View on Data Complexity

In this section, we provide our third case study illustrating the use of parameterized complexity for the analysis of description logic reasoning. This third case study is about refining the complexity analysis for cases where one part of the input is much smaller than another part. Typically, these cases occur where there is a small TBox and a small query, but where there is a large database of facts (in the form of an ABox). What is often done is that the size of the TBox and the query are seen as fixed constants—and the complexity results are grouped under the name of “data complexity.”

In this section, we will look at two concrete polynomial-time data complexity results for the description logic . Even though the data complexity view gives the same outlook on the complexity of these problems, we will use the viewpoint of parameterized complexity theory to argue that these two problems in fact have a different complexity. One of these problems is more efficiently solvable than the other.

We chose the example of to illustrate our point because it is technically straightforward. More intricate fixed-parameter tractability results for conjunctive query answering in description logics have been obtained in the literature (???).

We begin by reviewing the description logic , and the two reasoning problems for this logic that we will look at (instance checking and conjunctive query entailment). We will review the classical complexity results for these two problems, including the data complexity results. We will then use results from the literature to give a parameterized complexity analysis for these two problems, and argue why the parameterized complexity perspective gives a more accurate view on the complexity of these problems.

The Description Logic

To define the logic , we first consider the logic . The description logic is obtained from the logic by forbidding any use of the negation operator (), the empty concept (), the union operator (), and universal quantification (). The description logic is obtained from the logic by introducing inverse roles. That is, concepts are defined by the following grammar in Backus-Naur form, for  and :

Interpretations  for are defined as interpretations for with the following addition:

  • there exists some  such that .

Classical Complexity Results

We consider two reasoning problems for the logic . The first problem that we consider is the problem of instance checking. In this problem, the input consists of an ABox , a (general) TBox , an individual name  and a concept , and the question is whether —that is, whether for each interpretation  such that  and  it holds that .

The second problem that we consider is the problem of conjunctive query entailment (which can be seen as a generalization of the problem of instance checking). A conjunctive query is a set  of atoms of the form  and , where  is a concept, where , and where  are variables. Let Var denote the set of variables occurring in . Let  be an interpretation and let  be a mapping from Var to . We write  if , we write  if , we write  if  for all , and we write  if  for some . For any ABox  and TBox , we write  if  for each interpretation  such that  and . In the problem of conjunctive query entailment, the input consists of a (general) TBox , an ABox , and a conjunctive query , and the question is to decide whether .

Both the problem of instance checking and the problem of conjunctive query entailment for are EXPTIME-complete in general.

Proposition 13 (? ??).

Instance checking for is EXPTIME-complete.

Corollary 14 (? ??).

Conjunctive query entailment for is EXPTIME-complete.

The results of Propositions 13 and Corollary 14 are typically called “combined complexity” results—meaning that all elements of the problem statement are given as inputs for the problem. To study how the complexity of these problems increases when the size of the ABox  grows—and when the size of the TBox  and the size of the query  remain the same—often different variants of the problems are studied. In these variants, the TBox  and the query  are fixed (and thus not part of the problem input), and only the ABox  is given as problem input. That is, there is a variant of the problem for each choice of  and . The computational complexity of these problem variants are typically called the “data complexity” of the problem.

From a data complexity perspective, the problems of instance checking and conjunctive query entailment for the logic are both polynomial-time solvable. In other words, from a data complexity point of view these problems are of the same complexity.

Claim 15 (? ?).

Instance checking for is polynomial-time solvable regarding data complexity.

Claim 16 (? ?).

Conjunctive query entailment for is polynomial-time solvable regarding data complexity.

Parameterized Complexity Results

We will argue that the computational complexity of the problems of instance checking and conjunctive query entailment for —when only the ABox  grows in size—is of a vastly different nature. We will do so by using the parameterized complexity methodology. Concretely, we will take (the size of) the TBox  and (for the case of conjunctive query entailment) the query  as parameters, and observe that the parameterized complexity of these two problems is different.

We begin by observing that the algorithm witnessing polynomial-time data complexity for the problem of instance checking for corresponds to an fpt-algorithm for the problem when parameterized by the size of the TBox .

Observation 17.

Instance checking for is fixed-parameter tractable when parameterized by .

Proof.

The algorithm to solve the problem of instance checking for described by Krisnadhi (?, Proposition 4.3) runs in time . ∎

The polynomial-time data complexity algorithm for the problem of conjunctive query entailment, on the other hand, does not translate to an fpt-algorithm, but to an xp-algorithm instead—when the parameter is (the sum of) the size of the TBox  and the size of the query .

Observation 18.

Conjunctive query entailment for is in XP when parameterized by  and .

Proof.

The algorithm to solve the problem of conjunctive query entailment for described by Krisnadhi and Lutz (?, Theorem 4) runs in time . ∎

For this parameter, the problem of conjunctive query entailment for is in fact W[1]-hard—and thus not fixed-parameter tractable, assuming the widely believed conjecture that . This follows immediately from the W[1]-hardness of conjunctive query answering over databases when parameterized by the size of the query (?, Theorem 1).

Corollary 19 (? ?).

Conjunctive query entailment for is W[1]-hard when parameterized by  and .

instance
checking conjunctive
query entailm.
combined
complexity EXPTIME-c
a (Prop 13) EXPTIME-c
a (Cor 14)
data complexity in P (Claim 15) in P (Claim 16)
combined
complexity with
parameter  in FPT
a (Obs 17) in XP (Obs 18)
W[1]-h (Cor 19)
Table 4: (Parameterized) complexity results for instance checking and conjunctive query entailment for .

Interpretation of the Results

The results in this section are summarized in Table 4. Observation 17 and Corollary 19 show that parameterized complexity can give a more accurate view on data complexity results than classical complexity theory. From a classical complexity perspective, the data complexity variants of both problems are polynomial-time solvable, whereas the parameterized data complexity variants of the problems differ in complexity. Both problems are solvable in polynomial time when only the ABox  grows in size. However, for instance checking the order of the polynomial is constant (Observation 17), and for conjunctive query entailment the order of the polynomial grows with the size of the query  (Corollary 19). This is a difference with enormous effects on the practicality of algorithms solving these problems (see, e.g., ? ?).

Directions for Future Research

The results in this paper are merely an illustrative exposition of the type of parameterized complexity results that are possible for description logic reasoning problems when using less commonly studied concepts (e.g., the classes para-NP, para-co-NP and para-PSPACE). We hope that this paper sparks a structured investigation of the parameterized complexity of different reasoning problems for the wide range of description logics that have been studied. For this, it would be interesting to consider a large assortment of different parameters that could reasonably be expected to have small values in applications. It would also be interesting to investigate to what extent, say, para-NP-membership results can be used to develop practical algorithms based on the combination of fpt-time encodings into SAT and SAT solving algorithms.

Conclusion

We showed how the complexity study of description logic problems can benefit from using the framework of parameterized complexity and all the tools and methods that it offers. We did so using three case studies. The first addressed the problem of concept satisfiability for with respect to nearly acyclic TBoxes. The second was about the problem of concept satisfiability for fragments of that are close to , and , respectively. The third case study concerned a parameterized complexity view on the notion of data complexity for instance checking and conjunctive query entailment for . Moreover, we sketched some directions for future research, applying (progressive notions from) parameterized complexity theory to the study of description logic reasoning problems.

Acknowledgments.

This work was supported by the Austrian Science Fund (FWF), project J4047.

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