Logical De nability of NP{Optimisation Problems with ... - CiteSeerX
Logical De nability of NP{Optimisation Problems with Monadic Auxiliary Predicates Clemens Lautemann Institut fur Informatik, Johannes Gutenberg{Universitat Mainz
Abstract. Given a rst{order formula ' with predicate symbols e1 : : : el ,
s0 ; : : : ; sr , an NP{optimisation problem on <e1 ; : : : ; el >{structures can be de ned as follows: for every <e1; : : : ; el >{structure G, a sequence <S0 ; : : : ; Sr > of relations on G is a feasible solution i satis es ', and the value of such a solution is de ned to be jS0 j. In a strong sense, every polynomially bounded NP{optimisation problem has such a representation, however, it is shown here that this is no longer true if the predicates s1 ; : : : ; sr are restricted to be monadic. The result is proved by an Ehrenfeucht{Frasse game and remains true in several more general situations.
1 Introduction With his seminal paper [Fa74], Ronald Fagin introduced nite model theory into computational complexity theory. He provided a non{computational characterisation of NP by proving that, for any nite signature , a set L of nite {structures is in NP i there is an extension 0 of and a rst{order sentence ' over 0 such that a {structure G belongs to L if and only if it can be extended to a 0{structure which satis es '. This result has since been developed further, mainly in two directions. On the one hand similar characterisations have been found for many other complexity classes (for a survey see [Im89]), on the other hand the possibility of restricting the syntactical form of the formula ' has been used to di erentiate between problems which cannot be distinguished by their computational complexity (cf.[dR87]). One motivation for this line of research is the fact that in nite model theory there are means of showing separation results, something we do not seem to be able to do in computational complexity. However, so far, none of the model theoretic separations have carried over to a separation of complexity classes. Recently, Papadimitriou and Yannakakis took this approach of looking at restricted formulae one step further, and applied it to NP{optimisation problems [PY91]. Their aim was to nd an explanation for the di erent behaviour of such problems with respect to polynomial{time approximation algorithms. The main result of their paper is the approximability to within a constant factor of all those optimisation problems which are de ned in a certain way by a formula of the form 8x9y'.1 Their way of de ning optimisation problems by rst{order formulae can be rephrased as follows. 1 Here 8x is short for 8x1 ; : : : ; 8xm , for some m 0, similarly, 9y.
Let be a signature and 0 an extension of , such that 0 ? = <s0 ; s1 ; : : : ; sr >. Then every rst order sentence ' over 0 gives rise to the following optimisation problem: Given A {structure G. Wanted A 0{extension of G which maximises (or minimises) jS0 j, subject to the condition that j= '. As an example consider the problem MAXCLIQUE: Given A graph G = (U; E ). Wanted A maximum{size subset V U such that G induces a complete graph on V . If we take to consist of one binary predicate symbol e, 0 ? of one unary symbol v, we can express the fact that the elements of V form a clique by ' 8x8y (x 6= y ^ v(x) ^ v(y)) ! (e(x; y) _ e(y; x)) : Consequently the clique number !(G) can be expressed as: !(G) = maxfjV j = j= 'g: Given this de nition of optimisation problems by formulae, Papadimitriou and Yannakakis' result can be stated in the following way.
1.1 Theorem ([PY91]) Let 0 ? = <s ; : : : ; sr >, and let be a quanti er{free rst{order formula over 0 ? fs g with free variables x; y. If ' = 8x9ys (x) ! , then the maximisation 0
0
0
problem de ned by ' can be solved in polynomial time approximatively to within a constant factor. ut How strong is this result? Although the restriction for ' seems rather severe compared to full rst{order logic, the example above shows that we cannot hope for much better results along these lines. In fact the formula for the clique problem violates the restrictions of the theorem "only just"; however, there seems to be little hope of nding a polynomial{time approximation algorithm with guaranteed performance for MAXCLIQUE2. On the other hand, at rst sight, it is not clear which class of optimisation problems can be de ned at all by rst{ order formulae in the way described above. A rst answer to this question was given by Kolaitis and Thakur in [KT90], where they showed that the optimal value of every NP{optimisation problem (with polynomially bounded values) equals the optimal value of an optimisation problem which is de ned, in the way sketched above, by some rst{order formula ', in fact by a formula of the form 8x9y . Their result is, however, unsatisfactory in that the logically de ned version of a given NP{optimisation problem might only have the same optimal 2
In fact, in a recent paper [ALMSS], Arora et al show that approximation to within n" would imply P = NP.
values, not necessarily the same range of approximate values. In section 3, I will strengthen their result and show that the structure of any (polynomially bounded) NP{optimisation problem can be completely modeled by a logically de ned one, in fact by one which is de ned by a sentence of the form 8x9y . Although this is not a deep result (its proof consists mainly of a detailed analysis of Fagin's construction in [Fa74], I consider it important since it shows that we can move from any (polynomially bounded) NP{optimisation problem to a logically de ned one, without losing any information. In particular, approximate solutions of the logically de ned version represent approximate solutions of the original problem. Given that with ' = 8x9y we can de ne any NP{optimisation problem, but with ' = 8x9ys0(x) ! only those with a constant factor approximation, the question arises as to the impact of other syntactical restrictions. A number of authors have looked at restrictions of the quanti er{free part of 8x9y , e.g., concerning positive or negative occurrences of predicates, cf.[KT90], [KT91], [PR90], [Ih90]. I want to complement their work by considering restrictions on the signature extension 0 ? , rather than the syntactic structure of '. The main result here is that auxiliary non{monadic predicates are necessary, i.e., that there are NP{ optimisation problems (e.g. the vertex coloring problem) which cannot be de ned in the way described above with the predicate symbols s1 ; : : : ; sr all being unary, even if ' can make use of full rst{order. The proof is by an Ehrenfeucht{Frasse game, and it can be extended to show an even stronger non{expressibility result: the chromatic number of a graph (and other optimisation problems) cannot be expressed as minfh(jS0j; : : : ; jSr j) = j= 'g for a large class of functions h, with a monadic second{order sentence ', no matter what the arities of S0 ; : : : ; Sr .
2 Preliminaries 2.1 NP{Optimisation Problems The notion of an NP{optimisation problem is not always made precise in the literature. I will use the following de nition, which is equivalent to the one found in [PR90]. Let be a nite alphabet (w.l.o.g. can be assumed to be f0; 1g). An NP{ optimisation problem is given by a triple , where For all x; y 2 : <x; y> 2 R ) jyj p(jxj), for some polynomial p; R 2 P; f : R ! IN is computable in polynomial time; opt 2 fmin; maxg. For every NP{optimisation problem there is a mapping optR;f , de ned by optR;f (x) = optff (x; y)=<x; y> 2 Rg.
If there is a polynomial q such that f (x; y) q(jxj) for every <x; y> 2 R, the optimisation problem is called polynomially bounded. For every , the sets L = f(x; k)=9y : <x; y> 2 R ^ f (x; y) kg, and L = f(x; k)=9y : <x; y> 2 R ^ f (x; y) kg are in NP, and it is mainly in this form that optimisation problems have been treated in complexity theory (cf. [GJ79]), other approaches are surveyed in [BJY90].
2.2 Logic A signature is a nite sequence of pairwise distinct predicate symbols. Each predicate symbol s has an arity, a(s) 2 IN n f0g; if a(s) = k s is called k{ary. 1{ary predicate symbols are also called monadic, 2{ary ones binary. For the rest of this section, x a signature = <e1 ; : : : ; em >. Let U be a set, k 2 IN n f0g. A k{ary relation on U is a subset of U k . A { sequence on U is a sequence <E1; : : : ; Em > of relations on U such that Ei is a(ei){ary, for i = 1; : : : ; m. A {structure G = consists of a nonempty nite3 set UG together with a {sequence on UG . For U UG , the {structure induced by G on U , G[U ], is de ned as , where Ei(U ) = Ei \ U a(ei) , for i = 1; : : : ; m. For any other signature = <s0 ; : : : ; sr >, a {extension of G is a {structure4 , also denoted by . A rst{order {formula (also: f.o. formula over ) is a rst{order formula ' all of whose atoms are of one of the forms x = y, or ei(x1; : : : ; xa(ei ) ), for some i 2 f1; : : : ; mg. Here x; y; x1 ; : : : ; xa(ei ) 2 X , where X is the set of (f.o.) variables. ' is called a sentence if all its variables are bound by quanti ers. Let ' be a f.o. {sentence, G a {structure. G is a model of ', written G j= ', if ' holds true in G when every predicate symbol is interpreted by the corresponding relation.
3 Logically De ned NP{Optimisation Problems Let = <e1 ; : : : ; em > be a signature. A {structure G can be represented by a string over f0; 1g in some canonical way, for instance as 1n0w1 : : : wm, where UG is identi ed with f0; : : : ; n ? 1g, and jwij = na(ei ) , for i = 1; : : : ; m. For every 1 ; : : : ; a(ei ) 2 UG , if < 1 ; : : : ; a(ei ) > 2 Ei then wi has 1 in position (ei ) j := la=1 l nl?1 + 1, otherwise 0. On the other hand, every string w 2 f0; 1g+ can be viewed as the representation of a nite structure. Either w is of the form described above, or it represents the 3 4
All structures in this paper are nite. We can assume the symbols of to be distinct from those of .
structure , where U = f0; : : : ; jwj ? 1g and E U , such that i 2 E () wi+1 = 1. It is thus possible to view every set of strings as the representation of a set of {structures, for some . The starting point for the following development is Fagin's celebrated theorem. 3.1 Theorem ([Fa74]) A set L of (representations of) {structures belongs to NP if and only if there is a signature and a f.o. sentence ' over such that for every {structure G: G 2 L () there is a {extension G' of G such that G' j= '. ut Let ; be signatures, ' a f.o. sentence over . By Fagin's theorem, the pair de nes the NP{optimisation problem , where R ;' := f>= j= 'g, f ;' (G; <S0 ; : : : ; Sr >) = jS0 j. An NP{optimisation problem of this form is called de ned by ', or de ned in rst{order logic. Clearly, every NP{optimisation problem of the form is polynomially bounded; but can every polynomially bounded NP{optimisation problem be de ned in rst{order logic? Kolaitis and Thakur showed in [KT90] that for every such there are ,' and opt0 such that optR;f = opt0R ;' ;f ;' ; here ' can be chosen to be of the form 8x9y , where is quanti er{free. However, a careful analysis of (a modi ed version of) Fagin's proof shows that a stronger5 result holds. A proof can be found in [Kn92].
3.2 Theorem
Let be a polynomially bounded NP{optimisation problem. There are signatures ; and a rst{order sentence ' over such that the following hold. 1. There is a polynomial{time computable mapping which, for every pair constructs a {sequence <S0 ; : : : ; Sr > such that 2 R () j= ', and jS0 j = f (G; y), if 2 R. 2. There is a polynomial{time computable mapping which, for all constructs y such that j= ' () 2 R, and f (G; y) = jS0 j, if j= '. Moreover, ' can be chosen to be of the form 8x9y , with quanti er{free . ut Thus in a very detailed way, logically de ned NP{optimisation problems represent the structure of all (polynomially bounded) NP{optimisation problems. 5
In the context of [KT90], perhaps, no better result is possible, since there a weaker notion of NP{optimisation problem is used. In particular, R is not required to be decidable in polynomial time.
4 Ehrenfeucht{Frasse Games for First{Order De nability of NP{Optimisation Problems Fagin's theorem gives rise to a hierarchy within NP, by restricting the arities of the auxiliary predicate symbols in . It is well{known that the rst level of this hierarchy, although it does contain some NP{complete problems, does not even include all of P (or even NL): Fagin showed in [Fa75] that the class of connected graphs cannot be de ned by be a signature. Assume that there is a {structure G 2 L such that the following holds: 6 Note that here, a strong correspondence of and as 0
discussed in the last section is not required, only the coincidence of the optimal values. 7 More precisely, this should be written as fjS0 j =9S1 ; : : : ; Sr : j= 'g, but I will use the abridged notation throughout.
For every {extension G0 of G there is a {structure H 2= L, and a {extension H 0 of H such that Duplicator has a winning strategy in the n{round Ehrenfeucht{ Frasse game on G0, H 0 . Then there is no rst{order {sentence ' with at most n quanti ers such that L = fG=9S0; : : : ; Sr j= 'g. Proof: Assume to the contrary that L = fG = 9S0; : : : ; Sr j= 'g, for some f.o. sentence ' over with at most n quanti ers. Let G 2 L be as in the premiss of the lemma, and let G0 be a {extension of G such that G0 j= '. Then the premiss of the lemma asserts the existence of H 2= L, and a {extension H 0 of H , such that Duplicator has a winning strategy in the n{round Ehrenfeucht{ Frasse game on G0, H 0 . On the other hand H 0 6j= ', since H 2= L. But then, by Fact 4.1, Spoiler has a winning strategy on G0, H 0 , a contradiction. ut This lemma can easily be adapted so as to provide a tool for proving nonde nability of functions.
4.3 Lemma
Let f be a mapping on {structures with values in IN , let n 2 IN , and let = <s0 ; : : : ; sr > be a signature. Assume that there is a {structure G such that the following holds: For every {sequence <S0 ; : : : ; Sr > on UG for which jS0 j = f (G) there is a {structure H , and there is a {sequence on UH such that f (H ) > f (G) jT0 j = jS0 j Duplicator has a winning strategy in the n{round Ehrenfeucht{Frasse game on , . Then there is no f.o. {sentence ' with at most n quanti ers such that for every {structure G; f (G) = minfjS0 j = j= 'g.8 Proof: Assume to the contrary that 8G f (G) = minfjS0 j = j= 'g, for some {sentence ' with at most n quanti ers. Let G be as in the premiss of the lemma, and let S0 ; : : : ; Sr be given such that j= ', and jS0 j=f (G). By assumption, there are H; T0 ; : : : ; Tr such that f (H )>f (G), jT0 j = jS0 j, and Duplicator has a winning strategy in the n{round Ehrenfeucht{ Frasse game on ; . On the other hand, since jT0 j=jS0 j=f (G){structures onto natural numbers. 8
Of course, a similar lemma can be proved for max instead of min.
4.4 Theorem
Let = <s0 ; : : : ; sr > be a signature in which the predicate symbols s1 ; : : : ; sr are monadic. There is no f.o. <e; s0 ; : : : ; sr >{sentence ' such that for every graph G (G) = minfjS0j = j= 'g: Proof: Let k = a(s0 ), and let n be given. Let m > n 2r , and p > k m. In order to apply Lemma 4.3 chose G to be the disjoint union of p copies of Km , the complete graph on m vertices. Now, let S0 VGk ; S1; : : : ; Sr VG be given, such that jS0 j = (G) = m. Consider the set
U (S0) :=
[
2S0
fa ; : : : ; ak g: 1
Since jS0 j = m; jU (S0)j k m < p, so there must be some component, G0 , of G with UG0 \ U (S0) = ;. Then G can be written as the disjoint union of G0 with some graph G0. Now, for every 2 f0; 1gr , de ne U to be the set
U := UG0 \
\
i =1
Si \
\
i =0
(UG n Si ):
Then the sets U form a partition of UG0 into at most 2r parts, so one of them, say U , must contain more than n elements. Let W be a set with jU j + 1 elements, and let H0 be the complete graph on W ] (UG0 n U ), i.e., H0 is a copy of Km+1 . De ne the graph H to be the disjoint union of H0 and G0. Then (H ) = m+1 > (G), as required. Finally, T0 ; : : : ; Tr are de ned as follows: T0 := S0 , this is possible since S0 UGk UHk . For i = 1; : : : ; r, set Ti \ (UH n W) := Si \ (UG n U ), and if i = 1 Ti \ (W ) = W; ;; if i = 0: With this de nition, jT0 j = jS0 j, and it remains to show that Duplicator has a winning strategy in the n{round Ehrenfeucht{Frasse game on , . The strategy is the following: in round j , if Spoiler plays a vertex which was already played in some round l < j , then Duplicator plays the other vertex played in round l; if Spoiler plays a vertex v 2 UG n U (= UH n W ), then Duplicator plays the same vertex v; if Spoiler plays a new vertex v 2 U , then Duplicator plays a new vertex w 2 W; if Spoiler plays a new vertex w 2 W , then Duplicator plays a new vertex v 2 U . It is not dicult to show that with this strategy Duplicator always wins the game, thus all assumptions of Lemma 4.3 are satis ed, and the theorem follows. 0
ut
It is easy to see that the graph colouring problem can be logically de ned, using a signature = <s0 ; s1 > where s0 is monadic and s1 is binary. In fact, let ' be the <e; s0 ; s1 >{sentence (8x9ys0(y) ^ s1 (x; y)) ^ (8x8y8z(s1(x; z) ^ s1 (y; z) ! : e(x; y))) : Then, whenever j= ', the binary relation S1 includes an admissible colouring of G with colours from S0 , and (G) = minfjS0 j = j= 'g. However, Theorem 4.4 says that (G) cannot be de ned logically with only monadic auxiliary predicates even if full rst{order logic is available for '. Here, the predicates s1 ; : : : ; sr are called auxiliary since the cardinalities of S1; : : : ; Sr do not contribute to the value of a solution . 9 In particular, there is no ' for which (G) = minfjS0j = j= 'g. This proves a conjecture of Kolaitis and Thakur in [KT91]10 . In the proof of Theorem 4.4 all graphs are unions of cliques. For such graphs chromatic number and clique number coincide, and it follows that the theorem remains true if is replaced by ! in its statement.
4.4.1 Corollary
Let = <s0 ; : : : ; sr > be a signature in which the predicate symbols s1; : : : ; sr are monadic. There is no rst{order sentence ' over <e; s0 ; : : : ; sr > such that for every graph G, !(G) = minfjS0j = j= 'g. ut Why should it be interesting to have a logical de nition of the clique number as a minimum? After all, the problem MAXCLIQUE seems inherently a maximisation problem, and, in fact, can easily be de ned as such without auxiliary predicates. However, in [KT91], Kolaitis and Thakur showed that NP = CoNP if and only if the clique number has a f.o. de nition as a minimum. They also proved for certain restrictions on the form of the de ning f.o. sentence that this is indeed not possible. Corollary 4.4.1 now complements their results (leaving, however, the NP = CoNP question wide open).
5 Extensions The main ingredient in the proof of Theorem 4.4 is an Ehrenfeucht{Frasse game on two complete graphs of di erent sizes. It is well known that Duplicator can win such a game even in more complicated situations, and accordingly, Theorem 4.4 can be extended in several ways. In particular, I will present extensions to monadic second{order logic, ordered structures, and a more powerful graph logic. 9 Note that the arity of s0 is not restricted. 10 In fact, the conjecture in [KT91] was weaker, in that it involved only sentences ' of rather restricted form.
Furthermore, the restriction that f ;' (G; <S0 ; : : : ; Sr >) = jS0 j can be weakened to f ;' (G; <S0 ; : : : ; Sr >) = h(jS0 j; : : : ; jSr j), for certain functions h. Let us rst consider the extension to monadic second{order logic. ' is a mso formula over if ' is a f.o. formula over , or ' is of one of the forms 8s , or 9s , where s is a monadic predicate symbol, and is a mso formula over <s>, or ' is of one of the forms 8x , or 9x , where x is an individual variable, and is a mso formula over . Let MSOn () denote the set of mso sentences over with at most n quanti ers ( rst or second order). The following fact will be basic for the rest of this paper.
5.1 Fact Let be the equivalence relation on MSOn () de ned by ' i M j= ' , M j= , for every {structure M . Then has nite index (i.e., there are only nitely many equivalence classes). ut
The n{round mso{Ehrenfeucht{Frasse game on two structures G, H is played in the same way as the game de ned in Section 4, with the additional possibility to play sets instead of elements. More precisely, in every round, Spoiler rst decides whether this is to be a set round or an element round. In an element round, both players play according to the rules de ned in Section 4. In a set round, Spoiler chooses a set U UG , or a set V UH , and Duplicator replies by choosing a subset of the universe of the other structure. Let Ai be the subset of UG , Bi the subset of UH played in set round i, for i = 1; : : : ; l, and let aj 2 UG , bj 2 UH be the elements played in element round j , for j = 1; : : : ; m, where l + m = n. Duplicator wins, if the mapping aj 7! bj , j = 1; : : : ; m, de nes an isomorphism between the two structures [fa1 ; : : : ; am g] and [fb1 ; : : : ; bm g]. In analogy to the rst{order game we have the following fact.
5.2 Fact Assume that there is a mso sentence ' with at most n quanti ers such that G j= ', but H 6j= '. Then Spoiler has a winning strategy in the n{round mso{ Ehrenfeucht{Frasse game on G; H . ut Fact 5.1 will be used in the following form which can easily be derived from Fact 5.2.
5.3 Lemma Let M be an in nite set of {structures. For every n there are M , M 2 M 1
2
such that Duplicator has a winning strategy in the n{round mso{Ehrenfeucht{ Frasse game on M1 , M2. ut The following lemma corresponds to Lemma 4.3. It can be proved in much the same way as Lemma 4.3, using Fact 5.2 instead of Fact 4.1.
5.4 Lemmar Let h : IN ! IN , let f be a mapping on {structures with values in IN , let n 2 IN , and let = <s ; : : : ; sr > be a signature. Assume that there is a +1
0
{structure G, such that the following holds: For every {sequence <S0; : : : ; Sr > on UG for which h(jS0 j; : : : ; jSr j) = f (G) there is a {structure H , and there is a {sequence on UH such that f (H ) > f (G) h(jT0 j; : : : ; jTr j) = h(jS0j; : : : ; jSr j) Duplicator has a winning strategy in the n{round mso{Ehrenfeucht{Frasse game on ; . Then there is no mso {sentence ' with at most n quanti ers such that for every {structure G; f (G) = minfh(jS0j; : : : ; jSr j) = j= 'g. ut The existence of a linear order can make a crucial di erence in expressive power. It is easy, e.g., to nd a mso sentence which holds for an ordered structure if and only if its universe has an even number of elements, whereas without an order relation no such sentence exists. However, for the problems considered here, the addition of a linear order does not help, as will now be shown. An ordered graph is a <e; o>{structure , where is a graph and O is a linear order on V .
5.5 Theorem Let = <s ; : : : ; sr > be a signature and let h : IN r ! IN be such that for every p 2 IN the set h? (p) is nite. Then there is no mso sentence ' over 0
+1
1
<e; o> such that for every ordered graph G (G) = minfh(jS0j; : : : ; jSr j) = j= 'g. Proof: Let n be given and let, for every p 2 IN , Kp be an ordered complete graph on p vertices. By Lemma 5.3 there are p; q 2 IN , p < q such that Duplicator has a winning strategy in the n{round mso{Ehrenfeucht{Frasse game on Kp; Kq . Pr Let ki be the arity of si, for i = 0; : : : ; r, and let m be such that m > miki, i=0 whenever h(m0; : : : ; mr ) = p. Let, for i = 1; : : : ; m, Gi be a copy of Kp and let G be the disjoint union of the Gi, ordered in such a way that ui < ui+1, for all ui 2 UGi ; ui+1 2 UGi+1 ; i = 1; : : : ; m ? 1. Now, if S0 ; : : : ; Sr is a {sequence on G with h(jS0j; : : : ; jSr j) = p(= (G)), then one of G's components, say Gi0 , has no vertex in common with the set [r [ fu1; : : : ; uki g: i=0 2Si In order to apply Lemma 5.4, let H be G with Gi0 replaced by Hi0 , a copy of Kq , and let Ti := Si , for i = 0; : : : ; r. Then (H ) = q > p = (G), and jTi j = jSi j, for i = 0; : : : ; r, so h(jT0 j; : : : ; jTr j) = h(jS0 j; : : : ; jSr j). All that remains to show is that Duplicator has a winning strategy in the n{round mso{Ehrenfeucht{Frasse game on G, H . But this follows from her strategy on Kp; Kq : all Duplicator has to do is mimic Spoilers behaviour on G n Gi0 ; H n Hi0
(which is no problem because these structures are identical), and play according to her strategy for Kp; Kq on Gi0 ; Hi0 . The theorem now follows from Lemma 5.4. ut As a nal application I want to show that the problems CHROMATIC NUMBER and CHROMATIC INDEX11 are not "linear emso extremum problems" as de ned by Arnborg et al. in [ALS91]. Here emso stands for "extended monadic second{order" and with the notation used in this paper, linear emso extremum problems can be de ned as those optimisation problems consists of monadic predicate symbols only, andr f () = a + P ai jSij, for some a; ai 2 Q. i=o In the given context, graphs are represented by their incidences rather than their adjacencies. More precisely, the signature consists of one binary predicate inc, and a graph with vertex set V and edge set E is represented as the { structure , where U = V [ E , and 2 I , u1 2 V; u2 2 E , and u1 is an endpoint of u2 . This representation of graphs allows us to quantify over edge sets in mso sentences, hence we can talk about arbitrary, not only induced subgraphs, which considerably increases the expressive power of mso sentences (for a more detailed discussion, see [Co90]).
5.6 Theorem
CHROMATIC NUMBER is not a linear emso minimum problem. ut The proof has to be omitted, due to space restrictions. Again, its main part is an Ehrenfeucht{Frasse game on cliques of di erent sizes. Since for cliques with an odd number of vertices chromatic number and chromatic index are the same, we get the following corollary.
5.6.1 Corollary
CHROMATIC INDEX is not a linear emso minimum problem. Of course, Theorem 5.6 can also be extended to ordered graphs.
ut
6 Conclusion The main result of this paper, the fact that chromatic number cannot be de ned in rst{order logic with only monadic auxiliary predicates, is in accordance with known results about decision problems. However, whereas in the presence of a linear order on inputs, the proofs there are quite involved ([dR87, AF90]), here the extension of the proof to ordered structures is rather straightforward. In the light of Lynch's results in [Ly82], it would be interesting to see if it can even be extended to addition structures.
11
The chromatic index of a graph is the chromatic number of the corresponding edge graph.
Acknowledgements I would like to thank Elias Dahlhaus for his insightful comments and suggestions.
References [AF90] Miklos Ajtai, Ronald Fagin, Reachability is harder for directed than for undirected nite graphs. JSL 55, pp. 113-150. [ALMSS] Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, Mario Szegedy, Proof veri cation and intractability of approximation problems (preliminary version). Preprint, Computer Science Division, University of California, Berkeley, April 1992. [ALS91] Stefan Arnborg, Jens Lagergren, Detlev Seese, Easy problems for tree decomposable graphs. J. of Algorithms, 12, pp.308{340. [BJY90] Danilo Bruschi, Deborah Joseph, Paul Young, A structural overview of NP optimization problems. Rapporto Interno 75/90, Dipartimento di Scienze dell'Informazione, Universita degli Studi di Milano. [Co90] Bruno Courcelle, On the expression of monadic second-order graph properties without quanti cations over sets of edges. Proc. 5th Ann. IEEE Symposium on Logic in Computer Science, pp. 190{196. [dR87] Michel de Rougemont, Second-order and inductive de nability on nite structures. Zeitschrift f. math. Logik und Grundlagen d. Math. 33, pp. 47{63. [Fa74] Ronald Fagin, Generalized rst{order spectra and polynomial-time recognizable sets. In Richard Karp, (Ed.):"Complexity of Computation", SIAM-AMS Proc. 7, pp. 43{73. [Fa75] Ronald Fagin, Monadic generalized spectra. Zeitschr. f. math. Logik und Grundlagen d. Math. 21, pp. 89{96. [GJ79] Michael R. Garey, David S. Johnson, Computers and Intractability. Freeman, N.Y. [Ih90] Edmund Ihler, Approximation and existential second{order logic. Bericht 26, Institut fur Informatik, Universitat Freiburg. [Im89] Neil Immerman, Descriptive and computational complexity. In: Juris Hartmanis (Ed.):"Computational Complexity Theory", Proc. AMS Symp. Appl. Math. 38, pp. 75{91. [Kn92] Dieter Knobloch, Zur Komplexitat kombinatorischer Optimierungsprobleme. Diplomarbeit, FB Mathematik, Johannes Gutenberg{Universitat Mainz. [KT90] Phokion G. Kolaitis, Madhukar N. Thakur, Logical de nability of NP optimization problems. Technical Report UCSC-CRL-90-48, Computer Research Laboratory, University of California, Santa Cruz. [KT91] Phokion G. Kolaitis, Madhukar N. Thakur, Approximation properties of NP minimization classes. Proc. 7th Structure in Complexity Theory Conference, pp. 353{366. [Ly82] James F. Lynch, Complexity Classes and Theories of Finite Models. Math. Syst. Theory 15, pp. 127{144. [PR90] Alessandro Panconesi, Desh Ranjan, Quanti ers and approximation (extended abstract). Proc. 22nd ACM STOC, pp. 446{456. [PY91] Christos H. Papadimitriou, Mihalis Yannakakis, Optimization, approximation, and complexity classes. JCSS 43, pp. 425{440. This article was processed using the LaTEX macro package with LLNCS style
Abstract. Given a rst{order formula ' with predicate symbols e1 : : : el ,
s0 ; : : : ; sr , an NP{optimisation problem on <e1 ; : : : ; el >{structures can be de ned as follows: for every <e1; : : : ; el >{structure G, a sequence <S0 ; : : : ; Sr > of relations on G is a feasible solution i satis es ', and the value of such a solution is de ned to be jS0 j. In a strong sense, every polynomially bounded NP{optimisation problem has such a representation, however, it is shown here that this is no longer true if the predicates s1 ; : : : ; sr are restricted to be monadic. The result is proved by an Ehrenfeucht{Frasse game and remains true in several more general situations.
1 Introduction With his seminal paper [Fa74], Ronald Fagin introduced nite model theory into computational complexity theory. He provided a non{computational characterisation of NP by proving that, for any nite signature , a set L of nite {structures is in NP i there is an extension 0 of and a rst{order sentence ' over 0 such that a {structure G belongs to L if and only if it can be extended to a 0{structure which satis es '. This result has since been developed further, mainly in two directions. On the one hand similar characterisations have been found for many other complexity classes (for a survey see [Im89]), on the other hand the possibility of restricting the syntactical form of the formula ' has been used to di erentiate between problems which cannot be distinguished by their computational complexity (cf.[dR87]). One motivation for this line of research is the fact that in nite model theory there are means of showing separation results, something we do not seem to be able to do in computational complexity. However, so far, none of the model theoretic separations have carried over to a separation of complexity classes. Recently, Papadimitriou and Yannakakis took this approach of looking at restricted formulae one step further, and applied it to NP{optimisation problems [PY91]. Their aim was to nd an explanation for the di erent behaviour of such problems with respect to polynomial{time approximation algorithms. The main result of their paper is the approximability to within a constant factor of all those optimisation problems which are de ned in a certain way by a formula of the form 8x9y'.1 Their way of de ning optimisation problems by rst{order formulae can be rephrased as follows. 1 Here 8x is short for 8x1 ; : : : ; 8xm , for some m 0, similarly, 9y.
Let be a signature and 0 an extension of , such that 0 ? = <s0 ; s1 ; : : : ; sr >. Then every rst order sentence ' over 0 gives rise to the following optimisation problem: Given A {structure G. Wanted A 0{extension of G which maximises (or minimises) jS0 j, subject to the condition that j= '. As an example consider the problem MAXCLIQUE: Given A graph G = (U; E ). Wanted A maximum{size subset V U such that G induces a complete graph on V . If we take to consist of one binary predicate symbol e, 0 ? of one unary symbol v, we can express the fact that the elements of V form a clique by ' 8x8y (x 6= y ^ v(x) ^ v(y)) ! (e(x; y) _ e(y; x)) : Consequently the clique number !(G) can be expressed as: !(G) = maxfjV j = j= 'g: Given this de nition of optimisation problems by formulae, Papadimitriou and Yannakakis' result can be stated in the following way.
1.1 Theorem ([PY91]) Let 0 ? = <s ; : : : ; sr >, and let be a quanti er{free rst{order formula over 0 ? fs g with free variables x; y. If ' = 8x9ys (x) ! , then the maximisation 0
0
0
problem de ned by ' can be solved in polynomial time approximatively to within a constant factor. ut How strong is this result? Although the restriction for ' seems rather severe compared to full rst{order logic, the example above shows that we cannot hope for much better results along these lines. In fact the formula for the clique problem violates the restrictions of the theorem "only just"; however, there seems to be little hope of nding a polynomial{time approximation algorithm with guaranteed performance for MAXCLIQUE2. On the other hand, at rst sight, it is not clear which class of optimisation problems can be de ned at all by rst{ order formulae in the way described above. A rst answer to this question was given by Kolaitis and Thakur in [KT90], where they showed that the optimal value of every NP{optimisation problem (with polynomially bounded values) equals the optimal value of an optimisation problem which is de ned, in the way sketched above, by some rst{order formula ', in fact by a formula of the form 8x9y . Their result is, however, unsatisfactory in that the logically de ned version of a given NP{optimisation problem might only have the same optimal 2
In fact, in a recent paper [ALMSS], Arora et al show that approximation to within n" would imply P = NP.
values, not necessarily the same range of approximate values. In section 3, I will strengthen their result and show that the structure of any (polynomially bounded) NP{optimisation problem can be completely modeled by a logically de ned one, in fact by one which is de ned by a sentence of the form 8x9y . Although this is not a deep result (its proof consists mainly of a detailed analysis of Fagin's construction in [Fa74], I consider it important since it shows that we can move from any (polynomially bounded) NP{optimisation problem to a logically de ned one, without losing any information. In particular, approximate solutions of the logically de ned version represent approximate solutions of the original problem. Given that with ' = 8x9y we can de ne any NP{optimisation problem, but with ' = 8x9ys0(x) ! only those with a constant factor approximation, the question arises as to the impact of other syntactical restrictions. A number of authors have looked at restrictions of the quanti er{free part of 8x9y , e.g., concerning positive or negative occurrences of predicates, cf.[KT90], [KT91], [PR90], [Ih90]. I want to complement their work by considering restrictions on the signature extension 0 ? , rather than the syntactic structure of '. The main result here is that auxiliary non{monadic predicates are necessary, i.e., that there are NP{ optimisation problems (e.g. the vertex coloring problem) which cannot be de ned in the way described above with the predicate symbols s1 ; : : : ; sr all being unary, even if ' can make use of full rst{order. The proof is by an Ehrenfeucht{Frasse game, and it can be extended to show an even stronger non{expressibility result: the chromatic number of a graph (and other optimisation problems) cannot be expressed as minfh(jS0j; : : : ; jSr j) = j= 'g for a large class of functions h, with a monadic second{order sentence ', no matter what the arities of S0 ; : : : ; Sr .
2 Preliminaries 2.1 NP{Optimisation Problems The notion of an NP{optimisation problem is not always made precise in the literature. I will use the following de nition, which is equivalent to the one found in [PR90]. Let be a nite alphabet (w.l.o.g. can be assumed to be f0; 1g). An NP{ optimisation problem is given by a triple , where For all x; y 2 : <x; y> 2 R ) jyj p(jxj), for some polynomial p; R 2 P; f : R ! IN is computable in polynomial time; opt 2 fmin; maxg. For every NP{optimisation problem there is a mapping optR;f , de ned by optR;f (x) = optff (x; y)=<x; y> 2 Rg.
If there is a polynomial q such that f (x; y) q(jxj) for every <x; y> 2 R, the optimisation problem is called polynomially bounded. For every , the sets L = f(x; k)=9y : <x; y> 2 R ^ f (x; y) kg, and L = f(x; k)=9y : <x; y> 2 R ^ f (x; y) kg are in NP, and it is mainly in this form that optimisation problems have been treated in complexity theory (cf. [GJ79]), other approaches are surveyed in [BJY90].
2.2 Logic A signature is a nite sequence of pairwise distinct predicate symbols. Each predicate symbol s has an arity, a(s) 2 IN n f0g; if a(s) = k s is called k{ary. 1{ary predicate symbols are also called monadic, 2{ary ones binary. For the rest of this section, x a signature = <e1 ; : : : ; em >. Let U be a set, k 2 IN n f0g. A k{ary relation on U is a subset of U k . A { sequence on U is a sequence <E1; : : : ; Em > of relations on U such that Ei is a(ei){ary, for i = 1; : : : ; m. A {structure G = consists of a nonempty nite3 set UG together with a {sequence on UG . For U UG , the {structure induced by G on U , G[U ], is de ned as , where Ei(U ) = Ei \ U a(ei) , for i = 1; : : : ; m. For any other signature = <s0 ; : : : ; sr >, a {extension of G is a {structure4 , also denoted by . A rst{order {formula (also: f.o. formula over ) is a rst{order formula ' all of whose atoms are of one of the forms x = y, or ei(x1; : : : ; xa(ei ) ), for some i 2 f1; : : : ; mg. Here x; y; x1 ; : : : ; xa(ei ) 2 X , where X is the set of (f.o.) variables. ' is called a sentence if all its variables are bound by quanti ers. Let ' be a f.o. {sentence, G a {structure. G is a model of ', written G j= ', if ' holds true in G when every predicate symbol is interpreted by the corresponding relation.
3 Logically De ned NP{Optimisation Problems Let = <e1 ; : : : ; em > be a signature. A {structure G can be represented by a string over f0; 1g in some canonical way, for instance as 1n0w1 : : : wm, where UG is identi ed with f0; : : : ; n ? 1g, and jwij = na(ei ) , for i = 1; : : : ; m. For every 1 ; : : : ; a(ei ) 2 UG , if < 1 ; : : : ; a(ei ) > 2 Ei then wi has 1 in position (ei ) j := la=1 l nl?1 + 1, otherwise 0. On the other hand, every string w 2 f0; 1g+ can be viewed as the representation of a nite structure. Either w is of the form described above, or it represents the 3 4
All structures in this paper are nite. We can assume the symbols of to be distinct from those of .
structure , where U = f0; : : : ; jwj ? 1g and E U , such that i 2 E () wi+1 = 1. It is thus possible to view every set of strings as the representation of a set of {structures, for some . The starting point for the following development is Fagin's celebrated theorem. 3.1 Theorem ([Fa74]) A set L of (representations of) {structures belongs to NP if and only if there is a signature and a f.o. sentence ' over such that for every {structure G: G 2 L () there is a {extension G' of G such that G' j= '. ut Let ; be signatures, ' a f.o. sentence over . By Fagin's theorem, the pair de nes the NP{optimisation problem , where R ;' := f>= j= 'g, f ;' (G; <S0 ; : : : ; Sr >) = jS0 j. An NP{optimisation problem of this form is called de ned by ', or de ned in rst{order logic. Clearly, every NP{optimisation problem of the form is polynomially bounded; but can every polynomially bounded NP{optimisation problem be de ned in rst{order logic? Kolaitis and Thakur showed in [KT90] that for every such there are ,' and opt0 such that optR;f = opt0R ;' ;f ;' ; here ' can be chosen to be of the form 8x9y , where is quanti er{free. However, a careful analysis of (a modi ed version of) Fagin's proof shows that a stronger5 result holds. A proof can be found in [Kn92].
3.2 Theorem
Let be a polynomially bounded NP{optimisation problem. There are signatures ; and a rst{order sentence ' over such that the following hold. 1. There is a polynomial{time computable mapping which, for every pair constructs a {sequence <S0 ; : : : ; Sr > such that 2 R () j= ', and jS0 j = f (G; y), if 2 R. 2. There is a polynomial{time computable mapping which, for all constructs y such that j= ' () 2 R, and f (G; y) = jS0 j, if j= '. Moreover, ' can be chosen to be of the form 8x9y , with quanti er{free . ut Thus in a very detailed way, logically de ned NP{optimisation problems represent the structure of all (polynomially bounded) NP{optimisation problems. 5
In the context of [KT90], perhaps, no better result is possible, since there a weaker notion of NP{optimisation problem is used. In particular, R is not required to be decidable in polynomial time.
4 Ehrenfeucht{Frasse Games for First{Order De nability of NP{Optimisation Problems Fagin's theorem gives rise to a hierarchy within NP, by restricting the arities of the auxiliary predicate symbols in . It is well{known that the rst level of this hierarchy, although it does contain some NP{complete problems, does not even include all of P (or even NL): Fagin showed in [Fa75] that the class of connected graphs cannot be de ned by be a signature. Assume that there is a {structure G 2 L such that the following holds: 6 Note that here, a strong correspondence of and as 0
discussed in the last section is not required, only the coincidence of the optimal values. 7 More precisely, this should be written as fjS0 j =9S1 ; : : : ; Sr : j= 'g, but I will use the abridged notation throughout.
For every {extension G0 of G there is a {structure H 2= L, and a {extension H 0 of H such that Duplicator has a winning strategy in the n{round Ehrenfeucht{ Frasse game on G0, H 0 . Then there is no rst{order {sentence ' with at most n quanti ers such that L = fG=9S0; : : : ; Sr j= 'g. Proof: Assume to the contrary that L = fG = 9S0; : : : ; Sr j= 'g, for some f.o. sentence ' over with at most n quanti ers. Let G 2 L be as in the premiss of the lemma, and let G0 be a {extension of G such that G0 j= '. Then the premiss of the lemma asserts the existence of H 2= L, and a {extension H 0 of H , such that Duplicator has a winning strategy in the n{round Ehrenfeucht{ Frasse game on G0, H 0 . On the other hand H 0 6j= ', since H 2= L. But then, by Fact 4.1, Spoiler has a winning strategy on G0, H 0 , a contradiction. ut This lemma can easily be adapted so as to provide a tool for proving nonde nability of functions.
4.3 Lemma
Let f be a mapping on {structures with values in IN , let n 2 IN , and let = <s0 ; : : : ; sr > be a signature. Assume that there is a {structure G such that the following holds: For every {sequence <S0 ; : : : ; Sr > on UG for which jS0 j = f (G) there is a {structure H , and there is a {sequence on UH such that f (H ) > f (G) jT0 j = jS0 j Duplicator has a winning strategy in the n{round Ehrenfeucht{Frasse game on , . Then there is no f.o. {sentence ' with at most n quanti ers such that for every {structure G; f (G) = minfjS0 j = j= 'g.8 Proof: Assume to the contrary that 8G f (G) = minfjS0 j = j= 'g, for some {sentence ' with at most n quanti ers. Let G be as in the premiss of the lemma, and let S0 ; : : : ; Sr be given such that j= ', and jS0 j=f (G). By assumption, there are H; T0 ; : : : ; Tr such that f (H )>f (G), jT0 j = jS0 j, and Duplicator has a winning strategy in the n{round Ehrenfeucht{ Frasse game on ; . On the other hand, since jT0 j=jS0 j=f (G){structures onto natural numbers. 8
Of course, a similar lemma can be proved for max instead of min.
4.4 Theorem
Let = <s0 ; : : : ; sr > be a signature in which the predicate symbols s1 ; : : : ; sr are monadic. There is no f.o. <e; s0 ; : : : ; sr >{sentence ' such that for every graph G (G) = minfjS0j = j= 'g: Proof: Let k = a(s0 ), and let n be given. Let m > n 2r , and p > k m. In order to apply Lemma 4.3 chose G to be the disjoint union of p copies of Km , the complete graph on m vertices. Now, let S0 VGk ; S1; : : : ; Sr VG be given, such that jS0 j = (G) = m. Consider the set
U (S0) :=
[
2S0
fa ; : : : ; ak g: 1
Since jS0 j = m; jU (S0)j k m < p, so there must be some component, G0 , of G with UG0 \ U (S0) = ;. Then G can be written as the disjoint union of G0 with some graph G0. Now, for every 2 f0; 1gr , de ne U to be the set
U := UG0 \
\
i =1
Si \
\
i =0
(UG n Si ):
Then the sets U form a partition of UG0 into at most 2r parts, so one of them, say U , must contain more than n elements. Let W be a set with jU j + 1 elements, and let H0 be the complete graph on W ] (UG0 n U ), i.e., H0 is a copy of Km+1 . De ne the graph H to be the disjoint union of H0 and G0. Then (H ) = m+1 > (G), as required. Finally, T0 ; : : : ; Tr are de ned as follows: T0 := S0 , this is possible since S0 UGk UHk . For i = 1; : : : ; r, set Ti \ (UH n W) := Si \ (UG n U ), and if i = 1 Ti \ (W ) = W; ;; if i = 0: With this de nition, jT0 j = jS0 j, and it remains to show that Duplicator has a winning strategy in the n{round Ehrenfeucht{Frasse game on , . The strategy is the following: in round j , if Spoiler plays a vertex which was already played in some round l < j , then Duplicator plays the other vertex played in round l; if Spoiler plays a vertex v 2 UG n U (= UH n W ), then Duplicator plays the same vertex v; if Spoiler plays a new vertex v 2 U , then Duplicator plays a new vertex w 2 W; if Spoiler plays a new vertex w 2 W , then Duplicator plays a new vertex v 2 U . It is not dicult to show that with this strategy Duplicator always wins the game, thus all assumptions of Lemma 4.3 are satis ed, and the theorem follows. 0
ut
It is easy to see that the graph colouring problem can be logically de ned, using a signature = <s0 ; s1 > where s0 is monadic and s1 is binary. In fact, let ' be the <e; s0 ; s1 >{sentence (8x9ys0(y) ^ s1 (x; y)) ^ (8x8y8z(s1(x; z) ^ s1 (y; z) ! : e(x; y))) : Then, whenever j= ', the binary relation S1 includes an admissible colouring of G with colours from S0 , and (G) = minfjS0 j = j= 'g. However, Theorem 4.4 says that (G) cannot be de ned logically with only monadic auxiliary predicates even if full rst{order logic is available for '. Here, the predicates s1 ; : : : ; sr are called auxiliary since the cardinalities of S1; : : : ; Sr do not contribute to the value of a solution . 9 In particular, there is no ' for which (G) = minfjS0j = j= 'g. This proves a conjecture of Kolaitis and Thakur in [KT91]10 . In the proof of Theorem 4.4 all graphs are unions of cliques. For such graphs chromatic number and clique number coincide, and it follows that the theorem remains true if is replaced by ! in its statement.
4.4.1 Corollary
Let = <s0 ; : : : ; sr > be a signature in which the predicate symbols s1; : : : ; sr are monadic. There is no rst{order sentence ' over <e; s0 ; : : : ; sr > such that for every graph G, !(G) = minfjS0j = j= 'g. ut Why should it be interesting to have a logical de nition of the clique number as a minimum? After all, the problem MAXCLIQUE seems inherently a maximisation problem, and, in fact, can easily be de ned as such without auxiliary predicates. However, in [KT91], Kolaitis and Thakur showed that NP = CoNP if and only if the clique number has a f.o. de nition as a minimum. They also proved for certain restrictions on the form of the de ning f.o. sentence that this is indeed not possible. Corollary 4.4.1 now complements their results (leaving, however, the NP = CoNP question wide open).
5 Extensions The main ingredient in the proof of Theorem 4.4 is an Ehrenfeucht{Frasse game on two complete graphs of di erent sizes. It is well known that Duplicator can win such a game even in more complicated situations, and accordingly, Theorem 4.4 can be extended in several ways. In particular, I will present extensions to monadic second{order logic, ordered structures, and a more powerful graph logic. 9 Note that the arity of s0 is not restricted. 10 In fact, the conjecture in [KT91] was weaker, in that it involved only sentences ' of rather restricted form.
Furthermore, the restriction that f ;' (G; <S0 ; : : : ; Sr >) = jS0 j can be weakened to f ;' (G; <S0 ; : : : ; Sr >) = h(jS0 j; : : : ; jSr j), for certain functions h. Let us rst consider the extension to monadic second{order logic. ' is a mso formula over if ' is a f.o. formula over , or ' is of one of the forms 8s , or 9s , where s is a monadic predicate symbol, and is a mso formula over <s>, or ' is of one of the forms 8x , or 9x , where x is an individual variable, and is a mso formula over . Let MSOn () denote the set of mso sentences over with at most n quanti ers ( rst or second order). The following fact will be basic for the rest of this paper.
5.1 Fact Let be the equivalence relation on MSOn () de ned by ' i M j= ' , M j= , for every {structure M . Then has nite index (i.e., there are only nitely many equivalence classes). ut
The n{round mso{Ehrenfeucht{Frasse game on two structures G, H is played in the same way as the game de ned in Section 4, with the additional possibility to play sets instead of elements. More precisely, in every round, Spoiler rst decides whether this is to be a set round or an element round. In an element round, both players play according to the rules de ned in Section 4. In a set round, Spoiler chooses a set U UG , or a set V UH , and Duplicator replies by choosing a subset of the universe of the other structure. Let Ai be the subset of UG , Bi the subset of UH played in set round i, for i = 1; : : : ; l, and let aj 2 UG , bj 2 UH be the elements played in element round j , for j = 1; : : : ; m, where l + m = n. Duplicator wins, if the mapping aj 7! bj , j = 1; : : : ; m, de nes an isomorphism between the two structures [fa1 ; : : : ; am g] and [fb1 ; : : : ; bm g]. In analogy to the rst{order game we have the following fact.
5.2 Fact Assume that there is a mso sentence ' with at most n quanti ers such that G j= ', but H 6j= '. Then Spoiler has a winning strategy in the n{round mso{ Ehrenfeucht{Frasse game on G; H . ut Fact 5.1 will be used in the following form which can easily be derived from Fact 5.2.
5.3 Lemma Let M be an in nite set of {structures. For every n there are M , M 2 M 1
2
such that Duplicator has a winning strategy in the n{round mso{Ehrenfeucht{ Frasse game on M1 , M2. ut The following lemma corresponds to Lemma 4.3. It can be proved in much the same way as Lemma 4.3, using Fact 5.2 instead of Fact 4.1.
5.4 Lemmar Let h : IN ! IN , let f be a mapping on {structures with values in IN , let n 2 IN , and let = <s ; : : : ; sr > be a signature. Assume that there is a +1
0
{structure G, such that the following holds: For every {sequence <S0; : : : ; Sr > on UG for which h(jS0 j; : : : ; jSr j) = f (G) there is a {structure H , and there is a {sequence on UH such that f (H ) > f (G) h(jT0 j; : : : ; jTr j) = h(jS0j; : : : ; jSr j) Duplicator has a winning strategy in the n{round mso{Ehrenfeucht{Frasse game on ; . Then there is no mso {sentence ' with at most n quanti ers such that for every {structure G; f (G) = minfh(jS0j; : : : ; jSr j) = j= 'g. ut The existence of a linear order can make a crucial di erence in expressive power. It is easy, e.g., to nd a mso sentence which holds for an ordered structure if and only if its universe has an even number of elements, whereas without an order relation no such sentence exists. However, for the problems considered here, the addition of a linear order does not help, as will now be shown. An ordered graph is a <e; o>{structure , where is a graph and O is a linear order on V .
5.5 Theorem Let = <s ; : : : ; sr > be a signature and let h : IN r ! IN be such that for every p 2 IN the set h? (p) is nite. Then there is no mso sentence ' over 0
+1
1
<e; o> such that for every ordered graph G (G) = minfh(jS0j; : : : ; jSr j) = j= 'g. Proof: Let n be given and let, for every p 2 IN , Kp be an ordered complete graph on p vertices. By Lemma 5.3 there are p; q 2 IN , p < q such that Duplicator has a winning strategy in the n{round mso{Ehrenfeucht{Frasse game on Kp; Kq . Pr Let ki be the arity of si, for i = 0; : : : ; r, and let m be such that m > miki, i=0 whenever h(m0; : : : ; mr ) = p. Let, for i = 1; : : : ; m, Gi be a copy of Kp and let G be the disjoint union of the Gi, ordered in such a way that ui < ui+1, for all ui 2 UGi ; ui+1 2 UGi+1 ; i = 1; : : : ; m ? 1. Now, if S0 ; : : : ; Sr is a {sequence on G with h(jS0j; : : : ; jSr j) = p(= (G)), then one of G's components, say Gi0 , has no vertex in common with the set [r [ fu1; : : : ; uki g: i=0 2Si In order to apply Lemma 5.4, let H be G with Gi0 replaced by Hi0 , a copy of Kq , and let Ti := Si , for i = 0; : : : ; r. Then (H ) = q > p = (G), and jTi j = jSi j, for i = 0; : : : ; r, so h(jT0 j; : : : ; jTr j) = h(jS0 j; : : : ; jSr j). All that remains to show is that Duplicator has a winning strategy in the n{round mso{Ehrenfeucht{Frasse game on G, H . But this follows from her strategy on Kp; Kq : all Duplicator has to do is mimic Spoilers behaviour on G n Gi0 ; H n Hi0
(which is no problem because these structures are identical), and play according to her strategy for Kp; Kq on Gi0 ; Hi0 . The theorem now follows from Lemma 5.4. ut As a nal application I want to show that the problems CHROMATIC NUMBER and CHROMATIC INDEX11 are not "linear emso extremum problems" as de ned by Arnborg et al. in [ALS91]. Here emso stands for "extended monadic second{order" and with the notation used in this paper, linear emso extremum problems can be de ned as those optimisation problems consists of monadic predicate symbols only, andr f () = a + P ai jSij, for some a; ai 2 Q. i=o In the given context, graphs are represented by their incidences rather than their adjacencies. More precisely, the signature consists of one binary predicate inc, and a graph with vertex set V and edge set E is represented as the { structure , where U = V [ E , and 2 I , u1 2 V; u2 2 E , and u1 is an endpoint of u2 . This representation of graphs allows us to quantify over edge sets in mso sentences, hence we can talk about arbitrary, not only induced subgraphs, which considerably increases the expressive power of mso sentences (for a more detailed discussion, see [Co90]).
5.6 Theorem
CHROMATIC NUMBER is not a linear emso minimum problem. ut The proof has to be omitted, due to space restrictions. Again, its main part is an Ehrenfeucht{Frasse game on cliques of di erent sizes. Since for cliques with an odd number of vertices chromatic number and chromatic index are the same, we get the following corollary.
5.6.1 Corollary
CHROMATIC INDEX is not a linear emso minimum problem. Of course, Theorem 5.6 can also be extended to ordered graphs.
ut
6 Conclusion The main result of this paper, the fact that chromatic number cannot be de ned in rst{order logic with only monadic auxiliary predicates, is in accordance with known results about decision problems. However, whereas in the presence of a linear order on inputs, the proofs there are quite involved ([dR87, AF90]), here the extension of the proof to ordered structures is rather straightforward. In the light of Lynch's results in [Ly82], it would be interesting to see if it can even be extended to addition structures.
11
The chromatic index of a graph is the chromatic number of the corresponding edge graph.
Acknowledgements I would like to thank Elias Dahlhaus for his insightful comments and suggestions.
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