On the Parameterized Complexity of Default Logic and Autoepistemic Logic

10/04/2011
by   Arne Meier, et al.
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We investigate the application of Courcelle's Theorem and the logspace version of Elberfeld etal. in the context of the implication problem for propositional sets of formulae, the extension existence problem for default logic, as well as the expansion existence problem for autoepistemic logic and obtain fixed-parameter time and space efficient algorithms for these problems. On the other hand, we exhibit, for each of the above problems, families of instances of a very simple structure that, for a wide range of different parameterizations, do not have efficient fixed-parameter algorithms (even in the sense of the large class XPnu), unless P=NP.

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1 Introduction

Non-monotonic reasoning formalisms were introduced in the 1970s as a formal model for human reasoning and have developed into one of the most important topics in computational logic and artificial intelligence. However, as it turns out, most interesting reasoning tasks are computationally intractable already for propositional versions of non-monotonic logics

[7], in fact presumably much harder than for classical propositional logic. Because of this, a lot of effort has been spent to identify fragments of the logical language for which at least some of the algorithmic problems allow efficient algorithms; a survey of this line of research can be found in [13].

In this paper a different approach is chosen to deal with hard problems, namely the framework of parameterized complexity. Gottlob et al. [8]

made it clear that the tree width of a (suitable graph theoretic encoding of a) given knowledge base is a useful parameter in this context: making use of Courcelle’s Theorem it was shown that many reasoning tasks for logical formalisms such as circumscription, abduction and logic programming become tractable in the case of bounded tree width. Here we examine a family of non-monotonic logics where the semantics of a given set of formulae (axioms, knowledge base) is defined in terms of a fixed-point equation. In particular we turn to default logic

[11] and autoepistemic logic [9]. In the first, human reasoning is mimicked using so called “default rules” (in the absence of contrary information, assume this and that); in the second, a modal operator is introduced to model the beliefs of a perfect rational agent. For both logics the algorithmic tasks of satisfiability and reasoning have been shown to be complete in the second level of the polynomial hierarchy [7].

Much in the vein of [8] we here examine the parameterized complexity of these problems and, making again use of Courcelle’s Theorem and a recent improvement by Elberfeld et al., we obtain time and space efficient algorithms if the tree width of the given knowledge base is bounded. This proves once again how important this parameter is.

A second contribution of our paper concerns lower bounds: Under the assumption we show that, even for certain families of very simple knowledge bases and for any parameterization taken from a broad variety, no efficient fixed-parameter algorithms exist, not even in the sense of the quite large parameterized class . These simple families of knowledge bases are defined in terms of severe syntactic restrictions, e.g., using default rules with literals or propositions only. Restricting the input structure even further we obtain that no fixed-parameter algorithm in the sense of the space-bounded class (the logarithmic space analogue of ) exists, unless .

Unfortunately, tree width is not among the parameters for which our lower bound can be proven—otherwise we would have proven . In a third part of our paper, we show that those structurally very simple families of knowledge bases, for which we gave our lower bounds, already have unbounded tree width. For this result, we introduce the notion of pseudo-cliques and show how to embed these into our graph-theoretic encodings of knowledge bases.

2 Preliminaries

Complexity Theory.

In this paper we will make use of several standard notions of complexity theory such as the complexity classes , , , , , , and and their completeness notions under logspace-many-one reductions.

Given a problem and a parameterization , belongs to the class iff there is a deterministic algorithm solving in time ; is said to be fixed parameter tractable then. If is a parameterized problem, then is the -th slice of . Define to be a member of (in words, nonuniform) iff for all . For background on parameterized complexity we recommend [6].

Furthermore, we require space parameterized complexity classes which have been defined in [12] recently. Given a parameterized problem , we say

  • iff there exists a deterministic algorithm deciding in space ,

  • iff there exists a deterministic algorithm deciding in space for some recursive funktion , and

  • iff for all .

It holds that as well as .

Tree width.

A tree decomposition of a graph is a pair , where is a family of subsets of (the set of bags), and is a tree whose nodes are the bags , satisfying the following conditions: (i) , i.e., every node appears in at least one bag, (ii) : , i.e., every edge is ’contained’ in a bag, and (iii) : is connected in , i.e., for every node the set of bags containing is connected in .

The width of a decomposition , , is the number , , the size of the largest bag minus 1. The tree width of a graph , , is the minimum of the widths of the tree decompositions of .

Default Logic.

Following Reiter [11], a default rule is a triple ; is called the prerequisite, is called the justification, and is called the conclusion. If is a set of Boolean functions, then is a -default rule if are -formulae, i.e., formulae that use only connectors for functions in . A -default theory consists of a set of propositional -formulae and a set of -default rules .

Let be a default theory and be a set of formulae. Now define as the smallest set of formulae such that (i) , (ii) is closed under deduction, , , and (iii) for all defaults with and , it holds that . Then a stable extension of is a fix-point of , , a set such that .

A definition for stable extensions beyond fix-point semantics which was introduced by Reiter [11] uses the principle of a stage construction: for a given default theory and a set of formulae, define and . Then is a stable extension of if and only if , and the set is called the set of generating defaults.

The so to speak satisfiability problem for default logic, here called extension existence problem, , is the problem, given a theory , to decide if it has a stable extension. Gottlob [7] proved that this problem is complete for the class .

Autoepistemic Logic.

Moore in 1985 introduced a new modal operator stating that its argument is ”believed” as an extension of propositional logic [9]. Further the expression is treated as an atomic formula with respect to the consequence relation . Given a set of Boolean functions , we define with the set of all autoepistemic -formulae through for being a Boolean functions in and a proposition . If , then a set is a stable expansion of if it satisfies the condition , where and , and .

Let () define the set of ( preceded) subformulae for an autoepistemic formula , and let us use the shorthand for a set of (autoepistemic) formulae . Given a set of autoepistemic -formulae , we say a set is -full if for each it holds iff , and iff .

The connection of -full sets and stable expansions of has been observed by Niemelä [10]: if is a set of autoepistemic formulae and is a -full set, then for every either or . The stable expansions of and -full sets are in one-to-one correspondence.

The expansion existence problem, , is the problem, given a set of autoepistemic formulae , to decide if it has a stable expansion. Again, Gottlob proved that this problem is complete for the class .

3 MSO-Encodings

It will be the aim of this paper to apply Courcelle’s theorem to obtain fixed-parameter algorithms in the context of default and autoepistemic logic. For this, we will have to describe the relevant decision problems by monadic second-order formulae. In this section we will explain how to do this and obtain a preliminary result for the implication problem. Our approach is similar to the one of Gottlob, Pichler, and Wei [8] where MSO encodings for algorithmic problems from logic programming, abduction, and circumscription where developed.

Now fix a finite set of Boolean functions. Denote by the vocabulary . With respect to a set of propositional -formulae we associate a -structure where such that the universe of is the set of subformulae of the formulae in , and (i) holds iff represents a variable, (ii) holds iff represents a formula from , (iii) holds iff represents the constant , and (iv) holds iff represents the th argument of the function at the root of the formula tree represented by .

Lemma 1

Let be a finite set of Boolean functions. Then there exists an MSO-formula over such that for any it holds that

Proof

We define the MSO-formulae , , and over the vocabulary as follows:

The formula states that if an individual is not representing a formula , then there must be at least one subformula in which it occurs. If an individual is not a variable, then it represents either a constant or a Boolean function and needs to have corresponding individuals.

Let denote the maximal arity of , i.e., .

where is iff holds and otherwise. Now define

It is easy to verify that satisfies the lemma.∎

Let be a finite set of Boolean functions and be sets of -formulae. Answering the implication problem of sets of propositional formulae, i.e., the question whether , requires to extend our vocabulary to as well as our structure which we will denote by : is true iff represents a formula from , and is true iff represents a formula from . Now it is straightforward to formalize implication.

Lemma 2

Let be a finite set of Boolean functions. Then there exists an MSO-formula over such that for any and any it holds that

Proof

Define the MSO-formulae , , and as follows:

Then, we can define the formula as , where and are defined as above in Lemma 1.∎

The application of Courcelle’s Theorem [3] and the logspace version of Elberfeld  [5] directly leads to the following theorem.

Theorem 3.1

Let be a finite set of Boolean functions, let be fixed, and let be sets of -formulae such that has tree width bounded by . Then the implication problem for sets of -formulae is solvable in time and space .

In other words, the implication problem of sets of formulae parameterized by the tree width of is fixed-parameter tractable, and even in . In the following sections we will extend this result to default logic and autoepistemic logic.

4 Default Logic

Let be a finite set of Boolean functions. Write as a shorthand for the set of formulae . To any -default theory , we associate a -structure such that the universe of is the union of the set of subformulae of together with a set corresponding to the defaults in , the relations from are interpreted as in Section 3, and

  • holds iff represents a formula from the knowledge base ,

  • holds iff represents a default ,

  • (resp. , ) holds iff represents the premise (resp. justification , conclusion ) and represents the default rule .

Lemma 3

Let be a finite set of Boolean functions and let be a -default theory. There exists an MSO-formula such that possesses a stable extension iff .

Proof

First the formula expresses the fact that one formula is the negation of another formula: Observe that and are not formulae but placeholders for individuals. The following two formulae define the applicability of defaults, , whether a premise is entailed or a justification is violated which uses the shortcut :

Now we can define the MSO-formulae (a default is applicable), (a set of defaults is stable), (a set of defaults is generating) as follows.

Then is true under iff has a stable extension.∎

As a consequence of Lemma 3, we obtain from Courcelle’s Theorem [3] and the logspace version of Elberfeld  [5] that, given the tree width of as a parameter, the extension existence problem for default logic is fixed-parameter tractable, and in fact, in .

Theorem 4.1

Let be a finite set of Boolean functions, let be fixed, and let be a -default theory such that has tree width bounded by . Then the extension existence problem for -default logic is solvable in time and space .

So again and maybe with no big surprise, similar to the study by Gottlob et al. [8] for different nonmonotonic formalisms, we see here that bounding the tree width of a default theory yields time and space efficient algorithms for satisfiability. In the following we want to contrast this with a strong lower bound. We consider knowledge bases with very simple defaults rules, namely consisting only of literals (and in a second step even only propositions). Then we consider any parameterization of the extension existence problem that is bounded for all knowledge bases that obey this restriction. It follows that even for these very restricted knowledge bases, the parameterized extension existence problem is not even in the class , unless .

We want to point out that this theorem comprises for example the usual parameterizations for (in terms of, e.g., backdoor sets or formula tree width): For all these, we have -algorithms for propositional satisfiability, but still the extension existence problem is not in .

Theorem 4.2

Let be a finite set of Boolean functions such that and let be the set of sets of default rules such that each default is composed of literals only. Further let be a parameterization function for which there exists a such that for all . If , then the extension existence problem for -default logic, parameterized by , is not contained in .

Proof

The reduction from to default logic restricted to default theories with and default rules composed of literals only, shown in Lemma 5.6 of [2], proves that the extension existence problem of default logic restricted to theories of this kind (which will be denoted by ) is -hard. Now let be such a parameterization and suppose . For contradiction assume . Hence, by definition of , it holds for every . As also holds we can compose a deterministic polynomial time algorithm which solves . This contradicts and concludes the proof.∎

Theorem 4.3

Let be a finite set of Boolean functions such that and let be the set of sets of default rules such that each default is composed of propositions or the constant only. Further let be a parameterization function for which there exists a such that for all and all that consists of at most one proposition. If , then the extension existence problem for -default logic, parameterized by , is not contained in .

Proof

The reduction from to default logic restricted to default theories with and default rules composed of propositions or the constant only, shown in Lemma 5.8 of [2], proves that the extension existence problem of default logic restricted to theories of this kind (which will be denoted by ) is -hard. Following the argumentation in the proof of Theorem 4.2, we conclude for and that holds for every . This eventually leads to the desired contradiction proving the theorem.∎

5 Autoepistemic Logic

Let be a finite set of Boolean functions. To any set of autoepistemic -formulae, we associate a -structure such that the universe of is the union of the set of subformulae of , the relations from are interpreted as in Section 3, and holds iff the subformulae represented by is prefixed by an , and holds iff represents a formula in .

Lemma 4

Let be a finite set of Boolean functions and let be a set of autoepistemic -formulae. There exists an MSO-formula such that possesses a stable expansion iff .

Proof

For a set of formulae and a formula , similar to in the proof of Lemma 3, define be the MSO-formula

to test for . Now define the MSO-formula as

Then is true under iff has a -full set , which is the case iff has a stable expansion.∎

As above we obtain from Lemma 4 that, given the tree width of as a parameter, the expansion existence problem for autoepistemic logic is fixed-parameter tractable, and in fact in .

Theorem 5.1

Let be a finite set of Boolean functions, let be fixed, and let be a set of autoepistemic -formulae such that has tree width bounded by . Then the expansion problem is solvable in time and space .

On the other hand, analogues of Theorems 4.2 and 4.3 are easily obtained:

Theorem 5.2

Let be a finite set of Boolean functions such that and let be the set of sets of autoepistemic -formulae such that all are disjunctions of propositions or -prefixed propositions. Further let be a parameterization function for which there exists a such that for all . If , then the expansion existence problem for sets of autoepistemic -formulae, parameterized by , is not contained in .

Proof

Observe that there exists a reduction from to autoepistemic logic restricted to -formulae shown in Lemma 4.5 of [4]. This implies our claim, as membership in implies a polynomial-time algorithm for any fixed .∎

Theorem 5.3

Let be a finite set of Boolean functions such that Further let be a parameterization function for which there exists a such that for all . If , then the expansion existence problem for sets of autoepistemic -formulae, parameterized by , is not contained in .

Proof

Observe that there exists a reduction from the implication problem restricted to -formulae shown in Lemma 4.8 of [1]. This implies our claim, as membership in implies a logspace algorithm for any fixed .∎

We remark that similar lower bounds as given for default logic in the previous section and for autoepistemic logic here hold for the implication problem as well, see Appendix 0.A.

6 Pseudo-Cliques

Looking at Theorems 4.2 and 4.3 one might hope that the syntactic restriction imposed there, namely allowing only defaults that involve literals or propositions, is so severe that it will bound the tree width of every such input structure. Combining this with Theorem 4.1 would then yield (or , resp.). Stated the other way round, if then the tree width of is a non-trivial parameterization, , a parameterization for which there exists no such that holds for all consisting of defaults rules involving only literals.

In the following we directly prove the non-triviality of the parameterization by tree width (, without any complexity hypothesizes). As a tool we utilize the subsequent definition of pseudo-cliques.

Definition 1

Let be an undirected graph. A pseudo-clique is a set of vertices that can be partitioned into the set of main-nodes and sets of edge-nodes for each such that the following holds: for the nodes in form a simple path from to , , it holds that and no other edges are present.

The size of a pseudo-clique is , , the number of main-nodes. The cardinality of a pseudo-clique is , , the length of the longest simple path between edge-nodes. A pseudo-clique is said to have exact cardinality if : .

The first five pseudo-cliques of exact cardinality , and one of cardinality 3 are visualized in Figure 1. The thick vertices correspond to the main-nodes whereas the small dots correspond to the edge-nodes.

Figure 1: Pseudo-cliques of exact card. and size , one of card. and size , and a tree decomposition of a pseudo-clique of exact card. and size .

The important fact for us is the observation that pseudo-cliques of size have the same tree width as the clique of size .

Theorem 6.1

Let be a pseudo-clique of size and cardinality . Then the tree width of is .

Proof

Is proven in Appendix 0.B.

Whenever one wants to show that a parametrization by tree width is non-trivial, the most obvious method is to show that the family of graphs has (sub-) cliques of arbitrary size. Now Theorem 6.1 provides an alternative when this method is prohibited: it suffices to construct pseudo-cliques. Corollary 1 (1.) shows that, for families used for the lower bounds in the previous sections, it is not possible to use cliques in order to prove unbounded tree width and therefore additionally motivates the definition and purpose of pseudo-cliques.

Corollary 1

Let be a finite set of Boolean functions such that and let be a -default theory in the sense of Theorem 4.2, i.e., each default in is composed of literals only. Then there exists an MSO formula fulfilling the property iff , and

  1. is neither -connected nor contains a clique of size for any .

  2. There exists a family of default theories such that the tree width of is not constant.

Proof

For (1.) we construct the MSO formula according to Lemma 3. At first observe that the universe of comprises only literals and defaults. Further, there are no edges between literals, and no edges between defaults. Every default can be connected to at most three different literals. Obviously the graph does not contain a clique of size . Furthermore, the graph is not -connected for any by the following observation. Let be some individual representing the default . Then there are individuals to represent the respective parts of which are all connected to . If now and are removed from the graph, then there is no other individual to which is connected yielding a contradiction to the connectivity.

Turning to (2.) observe that (1.) prohibits using -cliques or -connectivity for any to measure the tree width of . Now define a default theory complying with Theorem 5.2, where and are variables for . Application of Theorem 6.1 concludes the proof. ∎

An analogous result holds for autoepistemic logic.

Corollary 2

Let be a finite set of Boolean functions. There exists a family of autoepistemic -formulae and all are disjunctions of propositions or -prefixed propositions such that there exists an MSO formula fulfilling the property iff and the tree width of is not constant.

Proof

Define as Then the structure consist of cliques of size , in fact.∎

Corollary 3

Let be a finite set of Boolean functions such that . Let be the set of sets of formulae in monotone -CNF and let be the set of sets of formulae in DNF. There exists a family of sets of -formulae with such that there exists an MSO formula fulfilling the property iff and the tree width of is not constant.

7 Conclusion

In this paper we applied Courcelle’s Theorem [3] and the logspace version of Elberfeld [5] to the most prominent decision problems in the nonmontonic default logic and autoepistemic logic. Thereby we showed that the extension existence problem for a given default theory is solvable in time and space if the tree width of the corresponding MSO structure is bounded by ; similarly for the expansion existence problem for a set of autoepistemic formulae, and as well for the implication problem for sets of formulae .

We mention that furthermore one can achieve similar results for the credulous (resp. brave) and skeptical (resp. cautious) reasoning problems of the nonmontone logics from above by slight extensions of the constructed MSO-formulae.

Furthermore we introduced with pseudo-cliques a weaker notion of cliques: basically we have a clique where each edge is divided into two edges by a fresh node (or even a longer path). There we showed that the tree width of a graph is bounded from below by the size of its largest sub-pseudo-clique. If we investigate default theories which contain an empty knowledge base and only defaults which are composed of propositions or the constant only, then for constant parameterizations we show collapses of and (resp. and ) if the corresponding parameterized problem is in (resp. ). Thus through the concept of pseudo-cliques we construct a family of default theories whose tree width of its MSO-structures is unbounded. Therefore this kind of parameterization cannot be used to prove such complexity class collapses. Analogue claims can be made for the expansion existence problem in autoepistemic logic and the implication problem for sets of formulae.

For further research it would be very interesting to find a parameterization that is non-trivial in the sense of Theorem 4.2 but uses many different values. Also insights on new types of parameterizations, in particular in the context of the new space parameterized complexity classes, would be very engaging.

Acknowledgement.

For helpful hints and discussions we are grateful to Nadia Creignou (Marseille) and Thomas Schneider (Bremen).

References

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Appendix 0.A Lower Bounds for the Implication Problem

Analogues of Theorems 4.2 and 4.3 for autoepistemic logic were given in Section 5. Here, we want to point out that also for the implication problems, similar results hold.

Theorem 0.A.1

Let be a finite set of Boolean functions such that and let be the set of sets of formulae such that each formula is composed of functions only. Further let be a parameterization function for which there exists a such that for all . If , then the implication problem for sets of -formulae, parameterized by , is not contained in .

Proof

From Lemma 4.4 in [1] it follows that the implication problem for these sets of formulae is -hard. Suppose and let be a parameterization such that for every . Now following the argumentation of Theorem 4.2 yields a contradiction to .

Theorem 0.A.2

Let be a finite set of Boolean functions such that . Let be the set of sets of formulae in monotone -CNF and let be the set of sets of formulae in DNF. Further let be a parameterization function for which there exists a such that for all . If , then the implication problem for sets of -formulae, parameterized by , is not contained in .

Proof

From Lemma 4.2 in [1] it follows that the implication problem for these sets of formulae is -hard. Following an analogue argumentation as in the proof of Theorem 0.A.1 implies this theorem.∎

Appendix 0.B Tree Width of Pseudo-Cliques

Theorem 0.B.1

Let be a pseudo-clique of size and cardinality . Then the tree width of is .

Proof

Let be the following undirected graph, where

  • contains main-nodes, labeled for and a number of edge-nodes labeled for and , , and

  • .

Claim

Let . Any tree decomposition of can be transformed into a tree decomposition of such that

  1. ,

  2. in every edge-node appears exactly once, in particular in a bag of size .

Proof of Claim

We execute consecutively for every pair of main-nodes the following procedure on the (valid) tree decomposition .

Consider , , all bags containing edge-nodes between and . Observe that all bags in are connected in . Pick up one bag such that . Such a does exist, since is a valid tree decomposition and . Now replace in every bag of every occurrence of any edge-node by . Add to the hierarchical chain of children .

One verifies that after each round of this procedure the so obtained structure is a valid tree decomposition of satisfying condition 1. Finally, it is obvious that after the whole procedure satisfies even condition 2.

Further, the following claim shows that the tree width of depends on .

Claim

The graph has tree width .

Proof of Claim

It suffices to show that the transformed tree decomposition of the preceding claim has width . We observe that all bags in of size greater than are bags containing solely main-nodes. Thus we have .

For a lower bound observe that for each pair there is at least one bag such that . Therefore, restricted to bags containing only main-nodes (, any is removed from its bag) is a valid tree decomposition of the edge-complete graph with node set . Thus, .

This concludes the proof.∎