On Bijections vs. Unary Functions - CiteSeerX
On Bijections vs. Unary Functions
Thomas Schwentick Universitat Mainz Institut fur Informatik, Johannes{Gutenberg Universitat Mainz, Germany
Abstract. A set of nite structures is in Binary NP if it can be characterized by existential second order formulas in which second order quanti cation is over relations of arity 2. In [DLS95] subclasses of Binary NP were considered, in which the second order quanti ers range only over certain classes of relations. It was shown that many of these subclasses coincide and that all of them can be ordered in a three-level linear hierarchy, the levels of which are represented by bijections, successor relations and unary functions respectively. In this paper it is shown that { Graph Connectivity is expressible by bijections, thereby showing that the two lower levels of the hierarchy coincide; { the set of graphs with exactly as many vertices as arcs is expressible by unary functions but not by bijections. This shows that level 3 is strictly stronger than the other two levels.
1 Introduction Fagin [Fag74] showed that NP coincides with the class of sets of nite structures that are characterized by existential second order (SO) formulas. If follows that NP 6= coNP, if and only if the expressive power of existential and universal SO formulas on nite structures di er. As proving this is not an easy task it seems natural to look at restrictions of existential SO logic. One way is to restrict the arity of the relations that are quanti ed. We call the class of sets that can be characterized by existential SO formulas, in which only relations of arity (at most) k are quanti ed, k-ary NP. For k = 1 and k = 2 the corresponding classes are called Monadic NP [FSV93] and Binary NP, respectively. Ajtai [Ajt83] already showed that k-ary NP is, for every k, not closed under complementation. In fact, he showed that Monadic coNP is not included in kary NP for any xed k. He also proved that the parity of the number of arcs of a graph is not in Monadic NP, even not with arbitrary built-in relations (see Theorem 8 below - we will make use of a slightly strengthened version of this theorem in the proof of our main result). In the case of Monadic NP there are a lot of nonexpressibility results for more natural graph problems. Fagin [Fag75] showed that connectivity of undirected graphs - a Monadic coNP property - is not in Monadic NP. This result was extended by a variety of papers [dR87, FSV93, Sch94, Sch95, Nur95] to cases where some kinds of built-in relations or generalized quanti ers are allowed. Ajtai
and Fagin [AF90] proved that directed reachability, in contrast to undirected reachability, is not in Monadic NP. By means of reductions some of these results can be transferred to other problems [Cos93]. The proofs of the results about connectivity and directed reachability make use of Ehrenfeucht games (see Section 2 below). In many of them sucient conditions for the existence of a winning strategy for Duplicator, one of the two players in an Ehrenfeucht game, play an important role ([FSV93, AF94, Sch94, Sch95]). One such sucient condition, the Weak Extension Theorem from [Sch95], will be used in the proof of the main result of this paper. As there has been some success in proving inexpressibility results for Monadic NP, it seems reasonable to turn to the next stage, Binary NP. But for full Binary NP inexpressibility results by means of Ehrenfeucht games seem already rather dicult. No proof has yet been given for the unexpressibility of a feasible, say PSPACE, graph problem in Binary NP (of course one can encode e.g. a NEXPcomplete set of strings into graphs and get a graph problem that is not in NP hence not in Binary NP). Therefore it makes sense to investigate subclasses of Binary NP that are still stronger than Monadic NP. One such subclass is obtained by restricting the SO quanti cation to unary functions. Grandjean has shown that this class contains all sets of strings that can be accepted in linear time by a nondeterministic successor RAM [Gra84, Gra85, Gra90]1 . In [DLS95] other, semantical restrictions of the SO quanti cation were examined, e.g. linear orders, successor relations, equivalence relations, bijections. It turned out that all the resulting classes stay in a 3-level hierarchy with the respective representatives bijections, successor relations and unary functions. It was left open whether this hierarchy is strict. In this paper we prove that
{ Graph Connectivity is expressible by bijections, thereby showing that the two lower levels of the mentioned hierarchy coincide;
{ the set of graphs with exactly as many vertices as arcs is expressible by unary
functions but not by bijections. This shows that level 3 is strictly stronger than the other two levels.
Combining this with the results of [DLS95] we get the picture that is shown in Figure 1. The paper is organized as follows. Section 2 gives some basic de nitions, reviews several variants of Ehrenfeucht games and states the Weak Extension Theorem. In Section 3 it is shown that Graph Connectivity is expressible by bijections. In Section 4 it is shown that unary functions are stronger than bijections. I want to thank Arnaud Durand for many very helpful discussions and Clemens Lautemann for giving the inspiration for the proof of Theorem 5. 1 for a similar connection between BinNP and quadratic time on NTMs see [Lyn82].
{ partial order relations { arbitrary binary relations j =6 { linear order relations { unary functions { equivalence relations { graphs with bounded outdegree j =6 { successor relations { connected graphs with bounded degree { graphs with bounded degree { bijective unary functions j =6 { sets Fig. 1. The structure of Binary NP
2 Preliminaries In this paper a nite structure G consists of a nite set U (the universe) and some relations on U . Most of the time our structures are graphs, possibly with additional relations. E.g., we write hG; R1; R2 i for the graph G with the additional relations R1 and R2 . Given a class C of binary relations, we call a set S of nite structures expressible by C -relations, i there is a rst order formula such that:
A 2 S () there are R1 ; : : : ; Rk 2 C such that (A; R1 ; : : : ; Rk ) j= : It is important to note that we view unary functions and bijections always as binary relations. (Hence our formulas contain no function symbols.) Two vertices of a graph are adjacent if there is an edge between them. In the presence of other relations two vertices are also adjacent if they are in some tuple of these relations. The distance d(x; y) between vertices x and y is the minimal k such that there exist x0 = x; x1 ; : : : ; xk = y and every xi is adjacent to xi+1 for i < k. For a subset H of the vertices of G we set d(x; H ) := miny2H d(x; y). One of the main tools for proving inexpressibility results are Ehrenfeucht games [Ehr61]. The rules of a rst-order (FO) Ehrenfeucht game are as follows. There are two players, Spoiler and Duplicator. They play on two (not necessarily nite) structures G1; G2 . Spoiler's aim is to prove a di erence between G1 and G2, whereas Duplicator tries to let them look alike.
They play a xed number, k, of rounds. In every round, Spoiler chooses an element of one of the two structures. Then Duplicator chooses an element of the other structure. We write xi for the element of G1 , chosen in round i, and x0i for the element of G2, chosen in round i. Finally, Duplicator wins if the structures induced by the chosen elements are isomorphic under an isomorphism which maps xi to x0i for every i. The importance of Ehrenfeucht games results from the following theorem. Theorem 1. [Ehr61, Fra54] A set M of structures is rst order de nable, if and only if there is a xed k, such that, whenever G1 2 M and G2 62 M , then Spoiler has a winning strategy in the k-round FO Ehrenfeucht game on G1 and G2 . Ehrenfeucht games can be extended to characterize second order expressibility [Fag75, Ten75, Loe91]. For Monadic NP Ajtai and Fagin [AF90] invented such a game which can be easily transferred to other existential SO logics. We give here a version for expressibility by bijections. The Bijection game for a set M of graphs consists of the following steps. (1) Spoiler chooses numbers k and l. (2) Duplicator selects a graph G1 2 M . (3) Spoiler chooses a tuple f = (f1; : : : ; fl ) of bijections on G1 . (4) Duplicator selects a graph G2 62 M and a tuple f 0 = (f10 ; : : : ; fl0 ) of bijections on G2. (5) Spoiler and Duplicator play a k-round FO Ehrenfeucht game on the structures hG1; f i and hG2; f 0 i. Analogously to the result of Ajtai and Fagin we get the following. Theorem 2. A set M of graphs is expressible by bijections, if and only if Spoiler has a winning strategy in the bijection game over M . The unary function game (UF game) is de ned analgously in an obvious way. For the proof of our main result we need a modi ed version of this game. The modi ed UF game has the additional feature that Duplicator has to choose a graph G2 with as many vertices as G1 and is not allowed to choose any functions. The rst order game is then played on the structures hG1; f i and hG2; f i. This makes the game more dicult for Duplicator. In fact, a winning strategy of Spoiler in the modi ed UF game on a set S doesn't imply the expressibility of S by unary functions. But we will show in Section 4, by using techniques of Ajtai [Ajt83] that Duplicator still has a winning strategy in the modi ed UF game on the set of graphs with an even number of arcs. For the proof of this result we will need the following de nition. We say that two structures G1 and G2 are k-equivalent, if Duplicator has a winning strategy in the k-round Ehrenfeucht game on G1 and G2. The k-type, k (G), of a structure G is its equivalence class with respect to k-equivalence. We will make use of the fact that for xed k and a xed signature (i.e., the number and arities of the of the relations and functions of the structure are xed) the number of di erent k-types is nite.
Finally we state a version of the Weak Extension Theorem from [Sch95]. It says that under certain circumstances a winning strategy of Duplicator on substructures H1 of G1 and H2 of G2 can be extended to a winning strategy on G1 and G2. Let the e-neighbourhood of H1 in G1 be the set of all vertices that have a distance of at most e from one of the vertices of H1 . We say that Duplicator has a distance respecting winning strategy on neighbourhhoods of H1 and H2 , if he can play in a way such that d(xi; H1 ) = d(x0i; H2 ) for every i. Theorem 3. Let k > 0. Let G1; G2 be two structures and let H1 and H2 be (induced) substructures of G1 and G2 , respectively. Let N (H1 ) denote the 2k -neighbourhood of H1 in G1. Analogously N (H2 ). Duplicator has a winning strategy in the k-round FO Ehrenfeucht game on G1 and G2, if the following conditions are ful lled. (i) Duplicator has a distance respecting winning strategy in the k-round Ehrenfeucht game on N (H1 ) and N (H2 ). (ii) There is a distance respecting isomorphism from G1 ? H1 to G2 ? H2 (i.e., d(x; H1 ) = d( (x); H2) for every x 2 G1).
3 Expressing Connectivity by Bijective Functions The proof of the result of this section makes use of the following characterization of Graph Connectivity which was shown by Sekanina. Proposition 4. [Sek60] A graph G is connected if and only if its cube2 contains a hamiltonian cycle.
Theorem 5. Connectivity of undirected graphs can be expressed by bijective unary functions. Proof. We show that the following two statements are equivalent for an undirected graph G.
(1) G is connected. (2) There are bijections s and t and a subset A of the vertices of G such that (a) For every x it holds that s(x) 6= x and d(x; s(x)) 3. (b) For every y, except at most one, either y 2 t(A) or s(y) 2 t(A). (c) There is exactly one x0 2 A with s(x0) 62 A. (d) For all other x 2 A it holds t(s(x)) = s(s(t(x))). (e) For the unique x1 62 A that ful ls s(x1) 2 A it holds that t(s(x1)) = s(s(x1)). (f) For every x 2 A it holds t(x) 6= x. 2 In the cube of G two vertices are adjacent i their distance in G is at most 3.
Here t(A) denotes the set ft(x) j x 2 Ag. The role of s and t will become clear from the following construction. (1) =) (2) Let G be an undirected connected graph. From Lemma 4 it follows that there is a successor relation s such that in G it holds d(x; s(x)) 3 for every x (we extend s to a bijection by setting s(max) = min). Let N be the numbering induced by s. I.e., N (min) = 1 and N (s(x)) = N (x) + 1 for every x 6= max. Let A be the set of vertices x ful lling N (x) N (max) 2 . For every x 2 A let t(x) be the vertex y with N (y) = 2N (x). For x 62 A let t be de ned in an arbitrary way such that t becomes a bijection. It is easy to verify that s and t ful l conditions (a) - (f) above. Figure 2 illustrates the de nition of s; t and A for a cycle of 9 vertices. Informally, s plays the part of the successor relation that veri es the connectivity of G by means of Lemma 4 and t veri es that s itself is connected, i.e. that it consists only of one cycle instead of a union of several cycles. In a sense s reduces the problem of testing connectivity from arbitrary graphs to unions of cycles and t decides it for such graphs.
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Fig. 2. s; t and A for a cycle with 9 vertices. The upper four vertices constitute the set A. s ist indicated by solid arcs, t by dotted arcs (only for vertices in A). (2) =) (1) Let G be an undirected graph and let s be a bijection that ful ls (a). We identify s in a natural way with the graph that it induces on the vertices of G. It follows immediately from (a) that G is connected, if s is connected. We show in the following that s is connected if there exists a bijection t such that (b) - (f) are ful lled. We prove this by contradiction. Assume s and t ful l (a)-(f) but s is not connected. Hence s consists of at least two cycles. Because of (a) all cycles have length at least two. By (c) it follows that there is exactly one cycle C0 which contains vertices from A as well as vertices that are not in A. All other cycles consist either entirely of vertices in A (A-cycles) or entirely of vertices not in A (A-cycles).
We write C ! C 0 if in C there is a vertex x such that x 2 A and t(x) 2 C 0. Now we prove some properties of the graph H the vertices of which are the cycles of G and the arcs of which are given by !. Claim 1 H has indegree at least 1. This follows directly from (b). Claim 2 C0 ! C0. By de nition of C0 we have s(x1) 2 C0 and by (e) it follows that t(s(x1)) = s(s(x1)) 2 C0 as well. Claim 3 H has no other loops than that in C0 . Assume C ! C . C cannot be an A-cycle, by the de nition of !. We show that C cannot be an A-cycle either. For a contradiction assume x; y 2 C , t(x) = y and C is an A-cycle. Hence there is a k such that sk (y) = x. Let z := sk (x). By assumption z 2 A and si (z) 2 A for every i and all these vertices are di erent from x0 . By applying (d) it follows t(z) = t(sk(x)) = s2k (t(x)) = s2k (y) = sk (x) = z; contradicting (f). Claim 4 H has maximum outdegree at most one. Let C1 ! C2 and C1 ! C3. By de nition C1 is not an A-cycle. If C1 is an A-cycle then C2 = C3 follows from (d). So C1 = C0. From (c) it follows that there are x; y 2 C0 and a k such that x; s(x); : : : ; sk (x) = y are exactly those vertices of C0 that lie in A. Again it follows by (d) that t(x); t(s(x)); : : :; t(sk (x)) are on the same cycle. Claim 5 If C1 ! C2 and C1 is an A-cycle then jC1 j jC22j+1 . Because of (b) at most jC22j+1 vertices of C2 are in t(C1). From Claims 1 to 4 it follows that H consists of a single loop (C0) and a (directed) cycle of length at least two consisting of A-cycles. But because of Claim 5 a cycle of A-cycles cannot exist, so we get the desired contradiction. Hence we have shown that (1) and (2) are equivalent. It is easy to see that the conditions in (2) can be expressed by rst order formulas3. In [DLS95] it was shown how the quanti cation of a set can be simulated by the quanti cation of some bijections. ut
Corollary 6. The classes of sets of nite structures expressible by bijections and successor relations, respectively, coincide.
4 Unary Functions are Stronger than Bijections First we are going to show that Duplicator has a winning strategy in the modi ed UF game (cf. Section 2) on the set of graphs with an even number of arcs. 3 By quantifying some more bijections it can be even expressed by a rst order formula
with only one, universally quanti ed, variable. That such a characterization (with unary functions) of Connectivity is possible was already shown in [Gra90].
We make use of Ajtai's result [Ajt83] that rst order logic fails dramatically in distinguishing between sets of even and odd size. If A is a nite structure, T an additional unary relation symbol and ' a rst order formula, we write S'even (A) for the number of relations T with an even number of elements such that ' holds in hA; T i (analogously S'odd (A)). Ajtai showed the following theorem. Theorem 7. [Ajt83] Let T be a unary relation symbol, ' a rst order formula and > 0. Then for all but nitely many n and every structure A of size n it holds that 1? 2?njS'even (A) ? S'odd (A)j 2?n : Using this theorem Ajtai showed Theorem 8. [Ajt83] The set of graphs with an even number of arcs is not expressible in Monadic NP even in the presence of arbitrary built-in relations. Let EvenArc be the set of all nite graphs with an even number of arcs and EvenArcn be the set of graphs in EvenArc with vertices f1; : : : ; ng. Correspondingly OddArcn . From Theorem 8 it follows immediately that Duplicator has a winning strategy in the (Monadic NP) Ajtai-Fagin game over EvenArc (see also [LS95]). The following result strengthens this a bit. The proof is similar to the original one of Ajtai [Ajt83]. Theorem 9. Duplicator has a winning strategy in the modi ed UF game on EvenArc. Proof. The proof is by contradiction. Let us assume that Spoiler has a winning strategy in the game on EvenArc. This means that there are k and l such that for every n and for every graph G1 2 EvenArcn there exist unary functions f = f1 ; : : : ; fl such that for all graphs G2 2 OddArcn it holds that Spoiler has a winning strategy in the k-round game on hG1; f i and hG2; f i), (i.e., k (hG1; f i) 6= k (hG2; f i)). Let b be a constant such that nn 2bn log n , for every n. We will make use of the fact that the number of graphs in EvenArcn (2n2?1 ) is, for large n, much greater than the number of l-tuples of unary functions on f1; : : : ; ng (at most 2lbn log n ). We conclude that for every n there are some f and a set A(n) of k-types such that { for at least 2n2?1?lbn log n graphs G 2 EvenArcn it holds k (hG; f i) 2 A(n), { for all graphs G 2 OddArcn it holds k (hG; f i) 62 A(n). As there is only a nite number of k-types some set A0 must occur in nitely often within the A(n). Let be a rst order formula such that k (hG; f i) 2 A0 if and only if hG; f i j= (see [EFT92]). Hence for in nitely many n there is some f such that { for at least 2n2?1?lbn log n graphs G 2 EvenArcn it holds hG; f i j= , but { for all graphs G 2 OddArcn it holds hG; f i 6j= .
Let, for the moment, one of these in nitely many n and f be xed. We de ne a structure S consisting of { the universe f1; : : : ; n2 g, { a unary relation R = f1; : : : ; ng; { unary functions g1 ; : : : ; gl with gi (x) = fi (x) for every x n and gi (x) = x for every other x; { unary functions h1 ; h2 , such that for every x it holds that h1 (x) n, h2 (x) n and every x is uniquely determined by h1 (x) and h2 (x). Let T be an additional unary relation symbol. It is easy to see that every choice of arcs on f1; : : : ; ng corresponds to a choice of a unary relation T on f1; : : : ; n2 g. Furthermore there is a formula ' (only depending on , not on n) such that hG; f i j= if and only if for the corresponding T it holds hS; T i j= '. Hence for in nitely many 2n there is a structure S of size n2 such that odd S' (S ) = 0 and S'even (S ) 2n ?1?lbn log n . For these structures it follows 2?n jS'even (S ) ? S'odd (S )j 2?1?lbn log n : But from Theorem 7 we know that for every and suciently large n it holds 2
2?n2 jS'even (S ) ? S'odd (S )j 2?n2? ;
ut
the desired contradiction.
Let HalfArc be the set of graphs in which the number of arcs equals b n22 c, where n is the number of vertices. Corollary 10. Duplicator has a winning strategy in the modi ed UF game on HalfArc. The proof is almost word for word the same as that of Theorem 9. The di erence is 2that instead of 2n2 ?1 graphs with an even number of arcs there are at least 2n 2?1 graphs with b n2 c arcs. The result remains true if Spoiler is allowed to n 2 choose also unary relations (which again have to be copied by Duplicator). This is the exact version of the result that we are going to use below. Now we are ready to prove the separation between bijections and unary functions. Let nArc be the set of all (directed) graphs in which the number of arcs equals the number of vertices. We show that nArc is expressible by unary functions but not by bijections. Theorem 11. nArc can be expressed by unary functions. Proof. Let G 2 nArc. We represent the arcs of G by unary functions f1 ; f2 by assigning to every arc e = (x; y) of G a unique vertex a and de ning
f1 (a) := x;
f2 (a) := y:
We say that a represents e. That G is in nArc is mirrored by the fact that every vertex represents exactly one arc and every arc is represented by exactly one vertex. Of course, functions f1 ; f2 with this property do not exist if G 62 nArc. Hence G is in nArc if and only if G ful ls 9f1 ; f2 8x; y 9a (x; y) 2 E ! (f1 (a) = x ^ f2 (a) = y)^ 8x; y; a (f1(a) = x ^ f2(a) = y) ! (x; y) 2 E:
ut
It is important that the functions in the proof of Theorem 11 are allowed to map many vertices to the same vertex. We will show that, in general, this behaviour cannot be simulated by bijections. On the other hand it is easy to show by a similar proof that the restriction of nArc to graphs with a xed degree bound k is expressible by bijections. Now we complete the separation of the power of bijections and unary functions. Theorem 12. nArc cannot be expressed by bijections. Proof. We rst give the idea of the proof. Duplicator chooses a graph G1 2 nArc in which most vertices are isolated and all the arcs appear in a very small subgraph H1 of G1. Let f be the bijections that are chosen by Spoiler. Duplicator chooses a graph G2 of the same kind, but with a di erent number of arcs in the subgraph H2 . As the functions fi are bijections the neighbourhoods N (H1 ) and N (H2 ) contain only a few vertices. By making use of more unary functions one can encode the structure of N (H1 ) and N (H2 ) into H1 and H2 respectively. We show that it follows from Theorem 9, that Duplicator has a winning strategy in the rst order game on structures that consist of H1 and H2 and these encoding functions. It follows that he has also a winning strategy on N (H1 ) and N (H2 ). Then, because G1 and G2 are isomorphic outside of H1 and H2 , we can extend, by using the Weak Extension Theorem, this winning strategy to G1 and G2 . Let a number, k, of rounds, and a number, l, of bijections be given and let p := (2l)2k+1 . Let H1 be a graph with vertices 1; : : : ; m that can be chosen by Duplicator within his winning strategy in the modi ed UF game on HalfArc with k rounds, p2 l unary functions and (2k + 1)p unary relations. In particular, 2 2 m m H1 has b 2 c arcs. Let n := b 2 c. In the bijection game Duplicator chooses the graph G1 on the vertices 1; : : : ; n which equals H1 on f1; : : : ; mg and has no other arcs. By de nition G1 2 nArc. Let the bijections f = f1 ; : : : ; fl be chosen by Spoiler. Let N (H1 ) be the set of vertices x of G1 ful lling d(x; H1) 2k in the structure hG1; f i. Analogously we de ne N (H2 ) Claim Duplicator can de ne a graph G2 on f1; : : : ; ng such that (1) G2 only has graph arcs on f1; : : : ; mg (we call this subgraph H2 ); 2 (2) H2 (and therefore G2) has a number of arcs di erent from n = b m2 c. (3) Duplicator has a distance respecting winning strategy in the k-round Ehrenfeucht game on the substructures of hG1; f i and hG2; f i that are induced by N (H1 ) and N (H2 ) respectively.
It is most important for the following that in (3) both structures are equipped with the same bijections. Now we are going to encode the function values of f on N (H1 ) into p2l additional unary functions and (2k +1)p additional unary relations on H1 . Then the claim follows because Duplicator chose H1 according to his winning strategy in the modi ed UF game with these parameters. Because the number of vertices in N (H1 ) is at most pm, there exists a function h which maps the vertices of N (H1) one to one to pairs (y; i) where y 2 H1 and i p is a natural number. Let the relations Aij , for every i p and every j 2k , be de ned by
y 2 Aij () h(x) = (y; i) for some x 2 N (H1 ) and d(x; H1) = j: Finally let the unary functions gji1 i2 for every j l and i1 ; i2 p be de ned as follows. gji1 i2 (y1 ) = y2 ; if for some x1; x2 2 N (H1 ) it holds that fj (x1) = x2; h(x1 ) = (y1 ; i1 ); and h(x2) = (y2 ; i2 ): All other values of gji1 i2 are de ned arbitrarily (Notice that for the following argument it doesn't hurt if these functions encode more information than f . Notice also that these functions don't need to be bijections.) By the choice of H1 Duplicator can de ne H2 such that { H2 has a number of arcs which is di erent from n, and { he has a k-round winning strategy on the structures hH1; A; g i and hH2 ; A; g i. It is easy to see that this winning strategy induces a distance respecting kround winning strategy on the substructures of hG1; f i and hG2; f i that are induced by N (H1 ) and N (H2 ) respectively. On the other hand4 hG1 ? H1 ; f i and hG2 ? H2; f i are of course isomorphic via an isomorphism which respects the distance from H1 (resp. H2 ). By Theorem 3 it follows that Duplicator has a k-round winning strategy on G1 and G2. ut
References [AF90] M. Ajtai and R. Fagin. Reachability is harder for directed than for undirected nite graphs. Journal of Symbolic Logic, 55(1):113{150, 1990. [AF94] S. Arora and R. Fagin. On winning strategies in Ehrenfeucht-Frasse games. Unpublished manuscript, 1994. [Ajt83] M. Ajtai. 11 formulae on nite structures. Ann. of Pure and Applied Logic, 24:1{48, 1983. [Cos93] S. Cosmadakis. Logical reducibility and monadic NP. In Proc. 34th IEEE Symp. on Foundations of Computer Science, pages 52{61, 1993. 4 Here only those "function arcs" are considered, which have both vertices outside Hi
[DLS95] A. Durand, C. Lautemann, and T. Schwentick. Fragments of binary NP. In Annual Conference of the EACSL, 1995. [dR87] M. de Rougemont. Second-order and inductive de nability on nite structures. Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik, 33:47{63, 1987. [EFT92] H.-D. Ebbinghaus, J. Flum, and W. Thomas. Einfuhrung in die mathematische Logik. BI, Mannheim, 3rd edition, 1992. [Ehr61] A. Ehrenfeucht. An application of games to the completeness problem for formalized theories. Fund. Math., 49:129{141, 1961. [Fag74] R. Fagin. Generalized rst{order spectra and polynomial{time recognizable sets. In R. M. Karp, editor, Complexity of Computation, SIAM-AMS Proceedings, Vol. 7, pages 43{73, 1974. [Fag75] R. Fagin. Monadic generalized spectra. Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik, 21:89{96, 1975. [Fra54] R. Frasse. Sur quelques classi cations des systemes de relations. Publ. Sci. Univ. Alger. Ser. A, 1:35{182, 1954. [FSV93] R. Fagin, L. Stockmeyer, and M. Vardi. On monadic NP vs. monadic coNP. In The Proceedings of the 8th Annual IEEE Conference on Structure in Complexity Theory, pages 19{30, 1993. [Gra84] E. Grandjean. The spectra of rst-order sentences and computational complexity. SIAM Journal on Computing, 13:356{373, 1984. [Gra85] E. Grandjean. Universal quanti ers and time complexity of random access machines. Mathematical System Theory, 13:171{187, 1985. [Gra90] E. Grandjean. First-order spectra with one variable. Journal of Computer and System Sciences, 40:136{153, 1990. [Loe91] B. Loescher. Begrundung, Verallgemeinerung und Anwendung der Ehrenfeucht-Spiele in der Relationentheorie von Roland Frasse. Informatikbericht 2/91, Institut fur Informatik, Universitat Mainz, 1991. [LS95] B. Loescher and A. Sharell. Functions vs. relations on nite structures - a ner hierarchy in existential second order logic. presented at ASL-Logic Colloquium, FMT-23, Haifa, 1995. [Lyn82] J. F. Lynch. Complexity classes and theories of nite models. Mathematical System Theory, 15:127{144, 1982. [Nur95] J. Nurmonen. On winning strategies with unary quanti ers. Preprint 77, Department of mathematics, University of Helsinki, 1995. [Sch94] T. Schwentick. Graph connectivity and monadic NP. In Proc. 35th IEEE Symp. on Foundations of Computer Science, pages 614{622, 1994. [Sch95] T. Schwentick. Graph connectivity, monadic NP and built-in relations of moderate degree. In Proc. 22nd International Colloq. on Automata, Languages, and Programming, pages 405{416, 1995. [Sek60] M. Sekanina. On an ordering of the set of vertices of a connected graph. Spisy Prrod. Fak. Univ. Brno, pages 137{141, 1960. [Ten75] R. Tenney. Second-order Ehrenfeucht games and the decidability of the second-order theory of an equivalence relation. Journal of the Australian Mathematical Society, 20:323{331, 1975.
This article was processed using the LATEX macro package with LLNCS style
Thomas Schwentick Universitat Mainz Institut fur Informatik, Johannes{Gutenberg Universitat Mainz, Germany
Abstract. A set of nite structures is in Binary NP if it can be characterized by existential second order formulas in which second order quanti cation is over relations of arity 2. In [DLS95] subclasses of Binary NP were considered, in which the second order quanti ers range only over certain classes of relations. It was shown that many of these subclasses coincide and that all of them can be ordered in a three-level linear hierarchy, the levels of which are represented by bijections, successor relations and unary functions respectively. In this paper it is shown that { Graph Connectivity is expressible by bijections, thereby showing that the two lower levels of the hierarchy coincide; { the set of graphs with exactly as many vertices as arcs is expressible by unary functions but not by bijections. This shows that level 3 is strictly stronger than the other two levels.
1 Introduction Fagin [Fag74] showed that NP coincides with the class of sets of nite structures that are characterized by existential second order (SO) formulas. If follows that NP 6= coNP, if and only if the expressive power of existential and universal SO formulas on nite structures di er. As proving this is not an easy task it seems natural to look at restrictions of existential SO logic. One way is to restrict the arity of the relations that are quanti ed. We call the class of sets that can be characterized by existential SO formulas, in which only relations of arity (at most) k are quanti ed, k-ary NP. For k = 1 and k = 2 the corresponding classes are called Monadic NP [FSV93] and Binary NP, respectively. Ajtai [Ajt83] already showed that k-ary NP is, for every k, not closed under complementation. In fact, he showed that Monadic coNP is not included in kary NP for any xed k. He also proved that the parity of the number of arcs of a graph is not in Monadic NP, even not with arbitrary built-in relations (see Theorem 8 below - we will make use of a slightly strengthened version of this theorem in the proof of our main result). In the case of Monadic NP there are a lot of nonexpressibility results for more natural graph problems. Fagin [Fag75] showed that connectivity of undirected graphs - a Monadic coNP property - is not in Monadic NP. This result was extended by a variety of papers [dR87, FSV93, Sch94, Sch95, Nur95] to cases where some kinds of built-in relations or generalized quanti ers are allowed. Ajtai
and Fagin [AF90] proved that directed reachability, in contrast to undirected reachability, is not in Monadic NP. By means of reductions some of these results can be transferred to other problems [Cos93]. The proofs of the results about connectivity and directed reachability make use of Ehrenfeucht games (see Section 2 below). In many of them sucient conditions for the existence of a winning strategy for Duplicator, one of the two players in an Ehrenfeucht game, play an important role ([FSV93, AF94, Sch94, Sch95]). One such sucient condition, the Weak Extension Theorem from [Sch95], will be used in the proof of the main result of this paper. As there has been some success in proving inexpressibility results for Monadic NP, it seems reasonable to turn to the next stage, Binary NP. But for full Binary NP inexpressibility results by means of Ehrenfeucht games seem already rather dicult. No proof has yet been given for the unexpressibility of a feasible, say PSPACE, graph problem in Binary NP (of course one can encode e.g. a NEXPcomplete set of strings into graphs and get a graph problem that is not in NP hence not in Binary NP). Therefore it makes sense to investigate subclasses of Binary NP that are still stronger than Monadic NP. One such subclass is obtained by restricting the SO quanti cation to unary functions. Grandjean has shown that this class contains all sets of strings that can be accepted in linear time by a nondeterministic successor RAM [Gra84, Gra85, Gra90]1 . In [DLS95] other, semantical restrictions of the SO quanti cation were examined, e.g. linear orders, successor relations, equivalence relations, bijections. It turned out that all the resulting classes stay in a 3-level hierarchy with the respective representatives bijections, successor relations and unary functions. It was left open whether this hierarchy is strict. In this paper we prove that
{ Graph Connectivity is expressible by bijections, thereby showing that the two lower levels of the mentioned hierarchy coincide;
{ the set of graphs with exactly as many vertices as arcs is expressible by unary
functions but not by bijections. This shows that level 3 is strictly stronger than the other two levels.
Combining this with the results of [DLS95] we get the picture that is shown in Figure 1. The paper is organized as follows. Section 2 gives some basic de nitions, reviews several variants of Ehrenfeucht games and states the Weak Extension Theorem. In Section 3 it is shown that Graph Connectivity is expressible by bijections. In Section 4 it is shown that unary functions are stronger than bijections. I want to thank Arnaud Durand for many very helpful discussions and Clemens Lautemann for giving the inspiration for the proof of Theorem 5. 1 for a similar connection between BinNP and quadratic time on NTMs see [Lyn82].
{ partial order relations { arbitrary binary relations j =6 { linear order relations { unary functions { equivalence relations { graphs with bounded outdegree j =6 { successor relations { connected graphs with bounded degree { graphs with bounded degree { bijective unary functions j =6 { sets Fig. 1. The structure of Binary NP
2 Preliminaries In this paper a nite structure G consists of a nite set U (the universe) and some relations on U . Most of the time our structures are graphs, possibly with additional relations. E.g., we write hG; R1; R2 i for the graph G with the additional relations R1 and R2 . Given a class C of binary relations, we call a set S of nite structures expressible by C -relations, i there is a rst order formula such that:
A 2 S () there are R1 ; : : : ; Rk 2 C such that (A; R1 ; : : : ; Rk ) j= : It is important to note that we view unary functions and bijections always as binary relations. (Hence our formulas contain no function symbols.) Two vertices of a graph are adjacent if there is an edge between them. In the presence of other relations two vertices are also adjacent if they are in some tuple of these relations. The distance d(x; y) between vertices x and y is the minimal k such that there exist x0 = x; x1 ; : : : ; xk = y and every xi is adjacent to xi+1 for i < k. For a subset H of the vertices of G we set d(x; H ) := miny2H d(x; y). One of the main tools for proving inexpressibility results are Ehrenfeucht games [Ehr61]. The rules of a rst-order (FO) Ehrenfeucht game are as follows. There are two players, Spoiler and Duplicator. They play on two (not necessarily nite) structures G1; G2 . Spoiler's aim is to prove a di erence between G1 and G2, whereas Duplicator tries to let them look alike.
They play a xed number, k, of rounds. In every round, Spoiler chooses an element of one of the two structures. Then Duplicator chooses an element of the other structure. We write xi for the element of G1 , chosen in round i, and x0i for the element of G2, chosen in round i. Finally, Duplicator wins if the structures induced by the chosen elements are isomorphic under an isomorphism which maps xi to x0i for every i. The importance of Ehrenfeucht games results from the following theorem. Theorem 1. [Ehr61, Fra54] A set M of structures is rst order de nable, if and only if there is a xed k, such that, whenever G1 2 M and G2 62 M , then Spoiler has a winning strategy in the k-round FO Ehrenfeucht game on G1 and G2 . Ehrenfeucht games can be extended to characterize second order expressibility [Fag75, Ten75, Loe91]. For Monadic NP Ajtai and Fagin [AF90] invented such a game which can be easily transferred to other existential SO logics. We give here a version for expressibility by bijections. The Bijection game for a set M of graphs consists of the following steps. (1) Spoiler chooses numbers k and l. (2) Duplicator selects a graph G1 2 M . (3) Spoiler chooses a tuple f = (f1; : : : ; fl ) of bijections on G1 . (4) Duplicator selects a graph G2 62 M and a tuple f 0 = (f10 ; : : : ; fl0 ) of bijections on G2. (5) Spoiler and Duplicator play a k-round FO Ehrenfeucht game on the structures hG1; f i and hG2; f 0 i. Analogously to the result of Ajtai and Fagin we get the following. Theorem 2. A set M of graphs is expressible by bijections, if and only if Spoiler has a winning strategy in the bijection game over M . The unary function game (UF game) is de ned analgously in an obvious way. For the proof of our main result we need a modi ed version of this game. The modi ed UF game has the additional feature that Duplicator has to choose a graph G2 with as many vertices as G1 and is not allowed to choose any functions. The rst order game is then played on the structures hG1; f i and hG2; f i. This makes the game more dicult for Duplicator. In fact, a winning strategy of Spoiler in the modi ed UF game on a set S doesn't imply the expressibility of S by unary functions. But we will show in Section 4, by using techniques of Ajtai [Ajt83] that Duplicator still has a winning strategy in the modi ed UF game on the set of graphs with an even number of arcs. For the proof of this result we will need the following de nition. We say that two structures G1 and G2 are k-equivalent, if Duplicator has a winning strategy in the k-round Ehrenfeucht game on G1 and G2. The k-type, k (G), of a structure G is its equivalence class with respect to k-equivalence. We will make use of the fact that for xed k and a xed signature (i.e., the number and arities of the of the relations and functions of the structure are xed) the number of di erent k-types is nite.
Finally we state a version of the Weak Extension Theorem from [Sch95]. It says that under certain circumstances a winning strategy of Duplicator on substructures H1 of G1 and H2 of G2 can be extended to a winning strategy on G1 and G2. Let the e-neighbourhood of H1 in G1 be the set of all vertices that have a distance of at most e from one of the vertices of H1 . We say that Duplicator has a distance respecting winning strategy on neighbourhhoods of H1 and H2 , if he can play in a way such that d(xi; H1 ) = d(x0i; H2 ) for every i. Theorem 3. Let k > 0. Let G1; G2 be two structures and let H1 and H2 be (induced) substructures of G1 and G2 , respectively. Let N (H1 ) denote the 2k -neighbourhood of H1 in G1. Analogously N (H2 ). Duplicator has a winning strategy in the k-round FO Ehrenfeucht game on G1 and G2, if the following conditions are ful lled. (i) Duplicator has a distance respecting winning strategy in the k-round Ehrenfeucht game on N (H1 ) and N (H2 ). (ii) There is a distance respecting isomorphism from G1 ? H1 to G2 ? H2 (i.e., d(x; H1 ) = d( (x); H2) for every x 2 G1).
3 Expressing Connectivity by Bijective Functions The proof of the result of this section makes use of the following characterization of Graph Connectivity which was shown by Sekanina. Proposition 4. [Sek60] A graph G is connected if and only if its cube2 contains a hamiltonian cycle.
Theorem 5. Connectivity of undirected graphs can be expressed by bijective unary functions. Proof. We show that the following two statements are equivalent for an undirected graph G.
(1) G is connected. (2) There are bijections s and t and a subset A of the vertices of G such that (a) For every x it holds that s(x) 6= x and d(x; s(x)) 3. (b) For every y, except at most one, either y 2 t(A) or s(y) 2 t(A). (c) There is exactly one x0 2 A with s(x0) 62 A. (d) For all other x 2 A it holds t(s(x)) = s(s(t(x))). (e) For the unique x1 62 A that ful ls s(x1) 2 A it holds that t(s(x1)) = s(s(x1)). (f) For every x 2 A it holds t(x) 6= x. 2 In the cube of G two vertices are adjacent i their distance in G is at most 3.
Here t(A) denotes the set ft(x) j x 2 Ag. The role of s and t will become clear from the following construction. (1) =) (2) Let G be an undirected connected graph. From Lemma 4 it follows that there is a successor relation s such that in G it holds d(x; s(x)) 3 for every x (we extend s to a bijection by setting s(max) = min). Let N be the numbering induced by s. I.e., N (min) = 1 and N (s(x)) = N (x) + 1 for every x 6= max. Let A be the set of vertices x ful lling N (x) N (max) 2 . For every x 2 A let t(x) be the vertex y with N (y) = 2N (x). For x 62 A let t be de ned in an arbitrary way such that t becomes a bijection. It is easy to verify that s and t ful l conditions (a) - (f) above. Figure 2 illustrates the de nition of s; t and A for a cycle of 9 vertices. Informally, s plays the part of the successor relation that veri es the connectivity of G by means of Lemma 4 and t veri es that s itself is connected, i.e. that it consists only of one cycle instead of a union of several cycles. In a sense s reduces the problem of testing connectivity from arbitrary graphs to unions of cycles and t decides it for such graphs.
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Fig. 2. s; t and A for a cycle with 9 vertices. The upper four vertices constitute the set A. s ist indicated by solid arcs, t by dotted arcs (only for vertices in A). (2) =) (1) Let G be an undirected graph and let s be a bijection that ful ls (a). We identify s in a natural way with the graph that it induces on the vertices of G. It follows immediately from (a) that G is connected, if s is connected. We show in the following that s is connected if there exists a bijection t such that (b) - (f) are ful lled. We prove this by contradiction. Assume s and t ful l (a)-(f) but s is not connected. Hence s consists of at least two cycles. Because of (a) all cycles have length at least two. By (c) it follows that there is exactly one cycle C0 which contains vertices from A as well as vertices that are not in A. All other cycles consist either entirely of vertices in A (A-cycles) or entirely of vertices not in A (A-cycles).
We write C ! C 0 if in C there is a vertex x such that x 2 A and t(x) 2 C 0. Now we prove some properties of the graph H the vertices of which are the cycles of G and the arcs of which are given by !. Claim 1 H has indegree at least 1. This follows directly from (b). Claim 2 C0 ! C0. By de nition of C0 we have s(x1) 2 C0 and by (e) it follows that t(s(x1)) = s(s(x1)) 2 C0 as well. Claim 3 H has no other loops than that in C0 . Assume C ! C . C cannot be an A-cycle, by the de nition of !. We show that C cannot be an A-cycle either. For a contradiction assume x; y 2 C , t(x) = y and C is an A-cycle. Hence there is a k such that sk (y) = x. Let z := sk (x). By assumption z 2 A and si (z) 2 A for every i and all these vertices are di erent from x0 . By applying (d) it follows t(z) = t(sk(x)) = s2k (t(x)) = s2k (y) = sk (x) = z; contradicting (f). Claim 4 H has maximum outdegree at most one. Let C1 ! C2 and C1 ! C3. By de nition C1 is not an A-cycle. If C1 is an A-cycle then C2 = C3 follows from (d). So C1 = C0. From (c) it follows that there are x; y 2 C0 and a k such that x; s(x); : : : ; sk (x) = y are exactly those vertices of C0 that lie in A. Again it follows by (d) that t(x); t(s(x)); : : :; t(sk (x)) are on the same cycle. Claim 5 If C1 ! C2 and C1 is an A-cycle then jC1 j jC22j+1 . Because of (b) at most jC22j+1 vertices of C2 are in t(C1). From Claims 1 to 4 it follows that H consists of a single loop (C0) and a (directed) cycle of length at least two consisting of A-cycles. But because of Claim 5 a cycle of A-cycles cannot exist, so we get the desired contradiction. Hence we have shown that (1) and (2) are equivalent. It is easy to see that the conditions in (2) can be expressed by rst order formulas3. In [DLS95] it was shown how the quanti cation of a set can be simulated by the quanti cation of some bijections. ut
Corollary 6. The classes of sets of nite structures expressible by bijections and successor relations, respectively, coincide.
4 Unary Functions are Stronger than Bijections First we are going to show that Duplicator has a winning strategy in the modi ed UF game (cf. Section 2) on the set of graphs with an even number of arcs. 3 By quantifying some more bijections it can be even expressed by a rst order formula
with only one, universally quanti ed, variable. That such a characterization (with unary functions) of Connectivity is possible was already shown in [Gra90].
We make use of Ajtai's result [Ajt83] that rst order logic fails dramatically in distinguishing between sets of even and odd size. If A is a nite structure, T an additional unary relation symbol and ' a rst order formula, we write S'even (A) for the number of relations T with an even number of elements such that ' holds in hA; T i (analogously S'odd (A)). Ajtai showed the following theorem. Theorem 7. [Ajt83] Let T be a unary relation symbol, ' a rst order formula and > 0. Then for all but nitely many n and every structure A of size n it holds that 1? 2?njS'even (A) ? S'odd (A)j 2?n : Using this theorem Ajtai showed Theorem 8. [Ajt83] The set of graphs with an even number of arcs is not expressible in Monadic NP even in the presence of arbitrary built-in relations. Let EvenArc be the set of all nite graphs with an even number of arcs and EvenArcn be the set of graphs in EvenArc with vertices f1; : : : ; ng. Correspondingly OddArcn . From Theorem 8 it follows immediately that Duplicator has a winning strategy in the (Monadic NP) Ajtai-Fagin game over EvenArc (see also [LS95]). The following result strengthens this a bit. The proof is similar to the original one of Ajtai [Ajt83]. Theorem 9. Duplicator has a winning strategy in the modi ed UF game on EvenArc. Proof. The proof is by contradiction. Let us assume that Spoiler has a winning strategy in the game on EvenArc. This means that there are k and l such that for every n and for every graph G1 2 EvenArcn there exist unary functions f = f1 ; : : : ; fl such that for all graphs G2 2 OddArcn it holds that Spoiler has a winning strategy in the k-round game on hG1; f i and hG2; f i), (i.e., k (hG1; f i) 6= k (hG2; f i)). Let b be a constant such that nn 2bn log n , for every n. We will make use of the fact that the number of graphs in EvenArcn (2n2?1 ) is, for large n, much greater than the number of l-tuples of unary functions on f1; : : : ; ng (at most 2lbn log n ). We conclude that for every n there are some f and a set A(n) of k-types such that { for at least 2n2?1?lbn log n graphs G 2 EvenArcn it holds k (hG; f i) 2 A(n), { for all graphs G 2 OddArcn it holds k (hG; f i) 62 A(n). As there is only a nite number of k-types some set A0 must occur in nitely often within the A(n). Let be a rst order formula such that k (hG; f i) 2 A0 if and only if hG; f i j= (see [EFT92]). Hence for in nitely many n there is some f such that { for at least 2n2?1?lbn log n graphs G 2 EvenArcn it holds hG; f i j= , but { for all graphs G 2 OddArcn it holds hG; f i 6j= .
Let, for the moment, one of these in nitely many n and f be xed. We de ne a structure S consisting of { the universe f1; : : : ; n2 g, { a unary relation R = f1; : : : ; ng; { unary functions g1 ; : : : ; gl with gi (x) = fi (x) for every x n and gi (x) = x for every other x; { unary functions h1 ; h2 , such that for every x it holds that h1 (x) n, h2 (x) n and every x is uniquely determined by h1 (x) and h2 (x). Let T be an additional unary relation symbol. It is easy to see that every choice of arcs on f1; : : : ; ng corresponds to a choice of a unary relation T on f1; : : : ; n2 g. Furthermore there is a formula ' (only depending on , not on n) such that hG; f i j= if and only if for the corresponding T it holds hS; T i j= '. Hence for in nitely many 2n there is a structure S of size n2 such that odd S' (S ) = 0 and S'even (S ) 2n ?1?lbn log n . For these structures it follows 2?n jS'even (S ) ? S'odd (S )j 2?1?lbn log n : But from Theorem 7 we know that for every and suciently large n it holds 2
2?n2 jS'even (S ) ? S'odd (S )j 2?n2? ;
ut
the desired contradiction.
Let HalfArc be the set of graphs in which the number of arcs equals b n22 c, where n is the number of vertices. Corollary 10. Duplicator has a winning strategy in the modi ed UF game on HalfArc. The proof is almost word for word the same as that of Theorem 9. The di erence is 2that instead of 2n2 ?1 graphs with an even number of arcs there are at least 2n 2?1 graphs with b n2 c arcs. The result remains true if Spoiler is allowed to n 2 choose also unary relations (which again have to be copied by Duplicator). This is the exact version of the result that we are going to use below. Now we are ready to prove the separation between bijections and unary functions. Let nArc be the set of all (directed) graphs in which the number of arcs equals the number of vertices. We show that nArc is expressible by unary functions but not by bijections. Theorem 11. nArc can be expressed by unary functions. Proof. Let G 2 nArc. We represent the arcs of G by unary functions f1 ; f2 by assigning to every arc e = (x; y) of G a unique vertex a and de ning
f1 (a) := x;
f2 (a) := y:
We say that a represents e. That G is in nArc is mirrored by the fact that every vertex represents exactly one arc and every arc is represented by exactly one vertex. Of course, functions f1 ; f2 with this property do not exist if G 62 nArc. Hence G is in nArc if and only if G ful ls 9f1 ; f2 8x; y 9a (x; y) 2 E ! (f1 (a) = x ^ f2 (a) = y)^ 8x; y; a (f1(a) = x ^ f2(a) = y) ! (x; y) 2 E:
ut
It is important that the functions in the proof of Theorem 11 are allowed to map many vertices to the same vertex. We will show that, in general, this behaviour cannot be simulated by bijections. On the other hand it is easy to show by a similar proof that the restriction of nArc to graphs with a xed degree bound k is expressible by bijections. Now we complete the separation of the power of bijections and unary functions. Theorem 12. nArc cannot be expressed by bijections. Proof. We rst give the idea of the proof. Duplicator chooses a graph G1 2 nArc in which most vertices are isolated and all the arcs appear in a very small subgraph H1 of G1. Let f be the bijections that are chosen by Spoiler. Duplicator chooses a graph G2 of the same kind, but with a di erent number of arcs in the subgraph H2 . As the functions fi are bijections the neighbourhoods N (H1 ) and N (H2 ) contain only a few vertices. By making use of more unary functions one can encode the structure of N (H1 ) and N (H2 ) into H1 and H2 respectively. We show that it follows from Theorem 9, that Duplicator has a winning strategy in the rst order game on structures that consist of H1 and H2 and these encoding functions. It follows that he has also a winning strategy on N (H1 ) and N (H2 ). Then, because G1 and G2 are isomorphic outside of H1 and H2 , we can extend, by using the Weak Extension Theorem, this winning strategy to G1 and G2 . Let a number, k, of rounds, and a number, l, of bijections be given and let p := (2l)2k+1 . Let H1 be a graph with vertices 1; : : : ; m that can be chosen by Duplicator within his winning strategy in the modi ed UF game on HalfArc with k rounds, p2 l unary functions and (2k + 1)p unary relations. In particular, 2 2 m m H1 has b 2 c arcs. Let n := b 2 c. In the bijection game Duplicator chooses the graph G1 on the vertices 1; : : : ; n which equals H1 on f1; : : : ; mg and has no other arcs. By de nition G1 2 nArc. Let the bijections f = f1 ; : : : ; fl be chosen by Spoiler. Let N (H1 ) be the set of vertices x of G1 ful lling d(x; H1) 2k in the structure hG1; f i. Analogously we de ne N (H2 ) Claim Duplicator can de ne a graph G2 on f1; : : : ; ng such that (1) G2 only has graph arcs on f1; : : : ; mg (we call this subgraph H2 ); 2 (2) H2 (and therefore G2) has a number of arcs di erent from n = b m2 c. (3) Duplicator has a distance respecting winning strategy in the k-round Ehrenfeucht game on the substructures of hG1; f i and hG2; f i that are induced by N (H1 ) and N (H2 ) respectively.
It is most important for the following that in (3) both structures are equipped with the same bijections. Now we are going to encode the function values of f on N (H1 ) into p2l additional unary functions and (2k +1)p additional unary relations on H1 . Then the claim follows because Duplicator chose H1 according to his winning strategy in the modi ed UF game with these parameters. Because the number of vertices in N (H1 ) is at most pm, there exists a function h which maps the vertices of N (H1) one to one to pairs (y; i) where y 2 H1 and i p is a natural number. Let the relations Aij , for every i p and every j 2k , be de ned by
y 2 Aij () h(x) = (y; i) for some x 2 N (H1 ) and d(x; H1) = j: Finally let the unary functions gji1 i2 for every j l and i1 ; i2 p be de ned as follows. gji1 i2 (y1 ) = y2 ; if for some x1; x2 2 N (H1 ) it holds that fj (x1) = x2; h(x1 ) = (y1 ; i1 ); and h(x2) = (y2 ; i2 ): All other values of gji1 i2 are de ned arbitrarily (Notice that for the following argument it doesn't hurt if these functions encode more information than f . Notice also that these functions don't need to be bijections.) By the choice of H1 Duplicator can de ne H2 such that { H2 has a number of arcs which is di erent from n, and { he has a k-round winning strategy on the structures hH1; A; g i and hH2 ; A; g i. It is easy to see that this winning strategy induces a distance respecting kround winning strategy on the substructures of hG1; f i and hG2; f i that are induced by N (H1 ) and N (H2 ) respectively. On the other hand4 hG1 ? H1 ; f i and hG2 ? H2; f i are of course isomorphic via an isomorphism which respects the distance from H1 (resp. H2 ). By Theorem 3 it follows that Duplicator has a k-round winning strategy on G1 and G2. ut
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