The Isomorphism Problem for omega-Automatic Trees

The Isom orphism Pr oblem f or ω -A utomatic T re es Dietrich Kuske 1 , Jiamou Liu 2 , and Markus Lohrey 2 , ⋆ 1 Laboratoire Bordelais de Recherche en Informatique (LaBRI), CNRS and Uni versit ´ e Bordeaux I, Bordeaux, France 2 Univ ersit ¨ at Leipzig, Institut f ¨ ur Informatik, Germany kuske@labri.f r, liujiamou@gma il.com, lohrey@inform atik.uni-leipz ig.de Abstract. The main result of this pap er states that the isomorphism for ω -au- tomatic trees of finite height i s at least has hard as second-order arithmetic and therefore not analytical. This strengthens a recent r esult by Hjorth, Khoussaino v , Montalb ´ an, and Nies [HKMN08] sho wing that the isomorphism problem for ω - automatic structures is not in Σ 1 2 . Moreo ver , assuming the continuum hypothesis CH , we can show that the isomorphism problem for ω -automatic trees of fi nite height is recursively equiv alent with second-order arit hmetic. On t he way to our main results, we show lo wer and upper bounds for the isomorphism problem for ω -automatic trees of every finite height: (i) It is decidable ( Π 0 1 -complete, resp,) for height 1 (2, resp.), (ii) Π 1 1 -hard and in Π 1 2 for height 3, and (iii) Π 1 n − 3 - and Σ 1 n − 3 -hard and in Π 1 2 n − 4 (assuming CH ) for all n ≥ 4 . All proofs are elementary and do not rely on theorems from set theory . 1 Intr oduction A graph is compu table if its domain is a computab le set of natural num bers an d the edge relation is com putable as well. Hen ce, on e can compute ef fecti vely in the graph . On the other hand, pr actically all other properties are un decidable fo r computable grap hs (e.g ., reachability , connectedn ess, and e ven the existence of isolated nodes). In particular , the isomorph ism problem is highly undecida ble in th e sense that it is complete for Σ 1 1 (the first existential le vel of the analytical hierarc hy [Odi89]) ; see e .g. [ CK06, GK0 2] for further in vestigations of the isomorph ism pro blem for computable st ructures. These al- gorithmic deficiencies have moti v ated in computer science the study of mor e restricted classes of fin itely presented infin ite graphs. For instance, pu shdown g raphs, equation al graphs, and pr efix recogn izable grap hs have a decidable m onadic second- order th eory and for the for mer two the isomorphism problem is known to be decidable [Cou89 ] ( for prefix recognizab le grap hs the status of the isomorphism problem seems to be open). Automatic gr aphs [KN95 ] are in between prefix recog nizable and computable grap hs. In essence, a grap h is automatic if the elemen ts of the uni verse can be r epresented as strings from a regular language and the edge relatio n can be recogn ized by a finite state automaton with se veral heads that proceed synch ronously . Automatic grap hs (and mo re general, a utomatic structures) r eceiv ed increasing interest over the last years [BG04, ⋆ The second and third author are supported by the DFG research project GELO. IKR02, KNRS07, KR S05, Rub08]. One of the main motivations for inv estigating auto- matic graphs is th at their first-order th eories can b e decided uniformly ( i.e., the input is an au tomatic presentation and a first-or der sentence). On the o ther hand, the isomor- phism proble m for automatic graph s is Σ 1 1 -complete [KNRS07] and h ence as complex as for co mputable g raphs ( see [KL10] for the recursion theo retic com plexity of some more natural prop erties of autom atic graphs). In our recent p aper [KLL1 0], we studied th e isomor phism problem for restricted classes of automatic g raphs. Amo ng o ther results, we proved that (i) the isomorphism problem f or automatic trees of height at most n ≥ 2 is complete for the le vel Π 0 2 n − 3 of the ar ithmetical hiera rchy and (ii) tha t the isom orphism problem fo r automatic trees of finite h eight is recur si vely eq uiv alent to true arithmetic. In this paper, we extend our technique s from [KLL 10] to ω -automatic tr e es . T he class of ω -au tomatic structures was introdu ced in [Blu99 ], it gener alizes automatic structure s by re placing ordin ary finite automata by B ¨ uchi automata o n ω -w ords. In this way , unco untable grap hs can be specified. So me recent results on ω -autom atic structu res can be f ound in [KL0 8, HKMN08, KRB08, Kus10]. On the log ical sid e, many of the positive results f or au- tomatic structures carry over to ω -automatic structu res [Blu99, KRB08]. On the other hand, the isomorp hism proble m of ω - automatic structures is m ore complicated th an that of automatic structu res ( which is Σ 1 1 -complete) . Hjorth et al. [HKMN08 ] constructed two ω -automatic structures for which the existence o f an isom orphism d epends o n the axioms of set theory . Using Schoenfield’ s absoluteness theore m, they infer that isomor- phism of ω -au tomatic stru ctures d oes not belon g to Σ 1 2 . The extension o f our elemen - tary techniques from [KLL10] to ω -a utomatic trees allows us to s how directly (with out a “d etour” thr ough set th eory) that the isomo rphism pro blem f or ω -automatic tr ees o f finite height is not analytical (i.e., does not belong to any of the levels Σ 1 n ). For this, we prove that the isom orphism p roblem f or ω -au tomatic trees of h eight n ≥ 4 is har d for both levels Σ 1 n − 3 and Π 1 n − 3 of the analytical hierar chy (ou r pr oof is unifor m in n ). A mor e precise a nalysis moreover reveals at which heig ht the complexity jump f or ω -auto matic trees oc curs: For au tomatic as well as for ω -au tomatic tre es of height 2, the isomo rphism prob lem is Π 0 1 -complete and hence arithmetical. But the isomorph ism problem for ω - automatic tree s o f height 3 is hard for Π 1 1 (and therefo re outside of the arithmetical h ierarchy) while the isomorph ism problem for automa tic trees of heig ht 3 is Π 0 3 -complete [KLL10]. Our lower b ounds for ω -automatic trees even hold f or the smaller class of injectiv ely ω - automatic trees. W e prove our results by reduc tions from monadic second-or der (fra gments o f) num- ber theo ry . The first step in the pro of is a nor mal form for analytical p redicates. The basic idea o f the reductio n then is th at a sub set X ⊆ N can be encoded by an ω - word w X over { 0 , 1 } , wher e the i -th symb ol is 1 if and only if i ∈ X . The co mbina- tion of this basic observation with our techniques from [KLL10] allows us to encode monadic secon d-order formulas over ( N , + , × ) by ω - automatic trees of fin ite heigh t. This yields the lower bou nds mentioned above. W e also give an upper bound for the isomorph ism pro blem: for ω -auto matic trees of height n , the is omorph ism problem be- longs to Π 1 2 n − 4 . While the lo wer bound holds in the usua l system ZFC of set theory , we can prove the up per bound only assuming in addition the continuum hypothesis. The 2 precise r ecursion th eoretic comp lexity of th e isomo rphism pr oblem for ω -automatic trees remains open, it might depend on the underly ing axiom s for set theory . Related work Results on isomorp hism problem s f or various subclasses of automa tic structures can be fou nd in [KNRS07 , KRS05, KLL10, Rub04]. Some com pleteness r e- sults for lo w le vels of the analytical hierarchy for decision pr oblems on infinitar y ratio- nal relations were shown in [ Fin09]. 2 Pr eliminaries Let N + = { 1 , 2 , 3 , . . . } . W ith x we d enote a tup le ( x 1 , . . . , x m ) of variables, wh ose length m does not matter . 2.1 The analytical hierarchy In this paper w e fo llow the definitions of the a rithmetical an d analytical hierarchy from [Odi89 ]. In order to av oid some tec hnical complication s, it is useful to exclude 0 in the follo wing, i.e., to con sider subsets of N + . In the following, f i ranges ov er unary func tions on N + , X i over subsets o f N + , and u, x, y , z , x i , . . . over elem ents o f N + . The class Σ 0 n ⊆ 2 N + is the collection of all sets A ⊆ N + of the form A = { x ∈ N + | ( N , + , × ) | = ∃ y 1 ∀ y 2 · · · Qy n : ϕ ( x, y 1 , . . . , y n ) } , where Q = ∀ (resp. Q = ∃ ) if n is even (resp. odd ) an d ϕ is a quantifier-free formula over the signature containing + and × . Th e class Π 0 n is the class of all co mplements of Σ 0 n sets. The classes Σ 0 n , Π 0 n ( n ≥ 1 ) make up the arithmetical hier ar chy . The analytical hierarch y extends the arithmetical hierarch y and is defin ed an alo- gously u sing fu nction qu antifiers: The class Σ 1 n ⊆ 2 N + is th e collection of a ll sets A ⊆ N + of the form A = { x ∈ N + | ( N , + , × ) | = ∃ f 1 ∀ f 2 · · · Qf n : ϕ ( x, f 1 , . . . , f n ) } , where Q = ∀ (resp. Q = ∃ ) if n is even (r esp. odd ) and ϕ is a first-ord er formula over the signature contain ing + , × , and the f unctions f 1 , . . . , f n . The class Π 1 n is the class of all c omplemen ts of Σ 1 n sets. The classes Σ 1 n , Π 1 n ( n ≥ 1 ) make up the analytica l hierar c hy , see Figure 1 for an inclusion diagram . The class of an alytical sets 3 is exactly S n ≥ 1 Σ 1 n . As usual in computability theory , a G ¨ o del numbering of all finite objects of interest allows to quantify over , say , finite automata as well. W e will always assume such a number ing without men tioning it explicitly . 3 Here the notion of analytical sets is defined for sets of natural numbers and is not to be con- fused with the analytic sets studied in descriptiv e set theory [K ec95]. 3 S n ≥ 1 Σ 0 n Σ 1 1 Π 1 1 Σ 1 2 Π 1 2 Σ 1 3 Π 1 3 . . . Fig. 1. The analytical hierarchy 2.2 B ¨ uchi automata For d etails o n B ¨ uchi automata, see [GTW02, PP04 , Tho97]. Let Γ be a finite alpha- bet. W ith Γ ∗ we denote the set of all finite words over th e alphabet Γ . The set of all nonemp ty fin ite word s is Γ + . An ω -word over Γ is an infinite sequence w = a 1 a 2 a 3 · · · with a i ∈ Γ . W e set w [ i ] = a i for i ∈ N + . The set of all ω -words ov er Γ is denoted by Γ ω . A (non deterministic) B ¨ uchi autom aton is a tuple M = ( Q, Γ, ∆, I , F ) , where Q is a finite s et of states, I , F ⊆ Q are resp. t he sets of initial and final states, and ∆ ⊆ Q × Γ × Q is the transition relation. If Γ = Σ n for some alph abet Σ , then we refer to M as an n - dimensional B ¨ uchi automato n over Σ . A r un of M on an ω - word w = a 1 a 2 a 3 · · · is an ω -word r = ( q 1 , a 1 , q 2 )( q 2 , a 2 , q 3 )( q 3 , a 3 , q 4 ) · · · ∈ ∆ ω such that q 1 ∈ I . Th e run r is accep ting if there exists a fin al state from F that occurs infinitely o ften in r . The languag e L ( M ) ⊆ Γ ω defined by M is the set of all ω -words f or which there exists an accepting run. An ω -languag e L ⊆ Γ ω is r egular if there exists a B ¨ u chi automaton M with L ( M ) = L . The class o f all regular ω -langu ages is ef fecti vely closed under Boolean operation s an d projections. For ω - words w 1 , . . . , w n ∈ Γ ω , the convolution w 1 ⊗ w 2 ⊗ · · · ⊗ w n ∈ ( Γ n ) ω is defined by w 1 ⊗ w 2 ⊗ · · · ⊗ w n = ( w 1 [1] , . . . , w n [1])( w 1 [2] , . . . , w n [2])( w 1 [3] , . . . , w n [3]) · · · . For w = ( w 1 , . . . , w n ) , we write ⊗ ( w ) for w 1 ⊗ · · · ⊗ w n . An n -ary relation R ⊆ ( Γ ω ) n is called ω -a utomatic if th e ω -langu age ⊗ R = {⊗ ( w ) | w ∈ R } is r egular , i.e. , it is accepted by so me n -dimensional B ¨ uch i au - tomaton. W e denote with R ( M ) ⊆ ( Γ ω ) n the re lation defined by an n -d imensional B ¨ u chi-autom aton over the alphabet Γ . T o also define the con v olution of finite words ( and of finite words with infinite words), we identify a finite word u ∈ Γ ∗ with the ω -word u ⋄ ω , where ⋄ is a new symbol. T hen, fo r u, v ∈ Γ ∗ , w ∈ Γ ω , we write u ⊗ v for the ω -word u ⋄ ω ⊗ v ⋄ ω and u ⊗ w (resp. w ⊗ u ) for u ⋄ ω ⊗ w (resp. w ⊗ u ⋄ ω ). In the following we describe some simple operatio ns on B ¨ uchi a utomata that are used in this paper . – Gi ven two B ¨ uchi automata M 0 = ( Q 0 , Γ, I 0 , ∆ 0 , F 0 ) an d M 1 = ( Q 1 , Γ, I 1 , ∆ 1 , F 1 ) , we use M 0 ⊎ M 1 to deno te the automaton obtaine d by taking the disjoin t union of M 0 and M 1 . Note that fo r any word u ∈ Γ ω , the num ber of a ccepting runs of M 0 ⊎ M 1 on u equa ls the sum of the numbers of accepting runs of M 0 and M 1 on u . 4 – Let, ag ain, M i = ( Q i , Γ, I i , ∆ i , F i ) for i ∈ { 0 , 1 } be two B ¨ uchi automata. Th en the intersection of their language s is accep ted by the B ¨ uchi autom aton M = ( Q 0 × Q 1 × { 0 , 1 } , Γ , I 0 × I 1 × { 0 } , ∆, F 0 × Q 1 × { 0 } ) , where (( p 0 , p 1 , m ) , a, ( q 0 , q 1 , n )) ∈ ∆ if and only if • ( p 0 , a, q 0 ) ∈ ∆ 1 and ( p 1 , a, q 1 ) ∈ ∆ 1 , and • if p m 6∈ F m then n = m and if p m ∈ F m then n = 1 − m . Hence the run s of M on th e ω -word u consist of a run of M 0 and of M 1 on u . The “flag” m ∈ { 0 , 1 } in ( p 0 , p 1 , m ) signals that the a utomaton waits for an acce pting state o f M m . As so on a s such a n accep ting state is seen, the flag togg les its value. Hence accepting r uns o f M correspo nd to pair s o f accep ting runs of M 0 and of M 1 . Therefo re, the nu mber of accepting runs of M on u equ als the prod uct of the number s o f acceptin g runs o f M 0 and o f M 1 on u . This c onstruction is known as the flag or Chouek a constru ction (cf. [Cho74, Tho90 , PP04]). – Le t Σ be an alphabet and M = ( Q, Γ , I , ∆, F ) be a B ¨ uchi automa ton. W e use Σ ω ⊗ M to den ote the autom aton obtained fro m M by expan ding the alpha bet to Σ × Γ : Σ ω ⊗ M = ( Q , Σ × Γ , I , ∆ ′ , F ) , where ∆ ′ = { ( p, ( σ, a ) , q ) | ( p, a, q ) ∈ ∆, σ ∈ Σ } . Note that L ( Σ ω ⊗ M ) = Σ ω ⊗ L ( A ) . 2.3 ω -auto matic s tructures A sign atur e is a finite set τ of relatio nal symb ols to gether with an arity n S ∈ N + for ev ery relational symb ol S ∈ τ . A τ -structur e is a tuple A = ( A, ( S A ) S ∈ τ ) , where A is a set (the universe of A ) and S A ⊆ A n S . When the context is clear , we denote S A with S , an d we write a ∈ A for a ∈ A . L et E ⊆ A 2 be an equ i valence relatio n on A . Th en E is a congruen ce on A if ( u 1 , v 1 ) , . . . , ( u n S , v n S ) ∈ E an d ( u 1 , . . . , u n S ) ∈ S imply ( v 1 , . . . , v n S ) ∈ S for all S ∈ τ . Then the qu otient structur e A /E can be defined: – Th e univ erse of A /E is the set of all E -equivalence classes [ u ] for u ∈ A . – Th e interpretatio n of S ∈ τ is the relation { ([ u 1 ] , . . . , [ u n S ]) | ( u 1 , . . . , u n S ) ∈ S } . Definition 2.1. An ω -automatic presentation over the signatur e τ is a tuple P = ( Γ , M , M ≡ , ( M S ) S ∈ τ ) with the following pr operties: – Γ is a finite alph abet – M is a B ¨ uchi automaton o ver the alphab et Γ . – F or every S ∈ τ , M S is an n S -dimensiona l B ¨ uchi automaton o ver the alphab et Γ . – M ≡ is a 2 -dimensiona l B ¨ uchi automaton over the a lphabet Γ such that R ( M ≡ ) is a congruen ce r elation on ( L ( M ) , ( R ( M S )) S ∈ τ ) . The τ -stru cture defined by the ω -auto matic pr esenta tion P is the quotien t st ructur e S ( P ) = ( L ( M ) , ( R ( M S )) S ∈ τ ) /R ( M ≡ ) . 5 If R ( M ≡ ) is the identity r elation on Γ ω , then P is called injective . A structure A is (injectively) ω -auto matic if th ere is an (injecti vely) ω -au tomatic presen tation P with A ∼ = S ( P ) . In [HKMN08 ] it w as shown that there exist ω -au tomatic structures that are not inje cti vely ω -auto matic. W e simp lify our statements by saying “gi ven/compute an (injectively) ω -auto matic structu re A ” fo r “gi ven/compute an (injectively) ω -automa tic presentation P of a structu re S ( P ) ∼ = A ”. A utomatic structur e s [KN9 5] are defined analogo usly to ω - automatic struc tures, but instead of B ¨ uchi auto mata o rdinary finite automata over finite words are used. For this, on e has to pad shorter strings with the padding symbol ⋄ whe n d efining the co n volution of finite strings. More details o n ω - automatic structures can be fou nd i n [BG04, HKMN08, KRB08]. In particular, a count- able structure is ω -autom atic if and o nly if it is automatic [KRB08]. Let F O [ ∃ ℵ 0 , ∃ 2 ℵ 0 ] be first-ord er logic extended by the quan tifiers ∃ κ x . . . ( κ ∈ {ℵ 0 , 2 ℵ 0 } ) saying that there exist exactly κ many x satisfying . . . . Th e fo llowing theo - rem lays out the main motiv ation for investigating ω -au tomatic s tructures. Theorem 2.2 ( [Blu99, KRB08] ). F r om an ω -au tomatic pr esenta tion P = ( Γ , M , M ≡ , ( M S ) S ∈ τ ) and a fo rmula ϕ ( x ) ∈ FO [ ∃ ℵ 0 , ∃ 2 ℵ 0 ] in the sign atur e τ with n free variables, o ne ca n compute a B ¨ uchi automaton for the r elation { ( a 1 , . . . , a n ) ∈ L ( M ) n | S ( P ) | = ϕ ([ a 1 ] , [ a 2 ] , . . . , [ a n ]) } . In particular , the F O [ ∃ ℵ 0 , ∃ 2 ℵ 0 ] theory o f any ω -auto matic structur e A is (unifo rmly) decidab le. Definition 2.3. Let K be a class of ω -auto matic presentations. The i somorph ism pro b- lem Iso ( K ) is the set of pairs ( P 1 , P 2 ) ∈ K 2 of ω -auto matic pr esenta tions fr om K with S ( P 1 ) ∼ = S ( P 2 ) . If S 1 and S 2 are two structures over the same signature, we write S 1 ⊎ S 2 for the disjoint union of the two structu res. W e use S κ to denote the disjoint union of κ many copies of the structure S , w here κ is any card inal. The d isjoint unio n as well as the countable or u ncoun table power o f a n autom atic structure are effecti vely autom atic, again. In this pap er , we will only need this pr operty (in a more explicite form) for injecti vely ω -automa tic structure s. Lemma 2.4. Let P i = ( Γ, M i , M i ≡ , ( M i S ) S ∈ τ ) be injective ω -automatic pr esentations of structur es S i for i ∈ { 1 , 2 } . On e can effectively construct injectively ω -automa tic copies of S 1 ⊎ S 2 , S ℵ 0 1 , and S 2 ℵ 0 1 such that – The universe of the injectively ω -au tomatic copy S of S 1 ⊎ S 2 equals L ( M 1 ) ∪ L ( M 2 ) an d the r elatio ns a r e g iven by S S = R ( M 1 S ) ∪ R ( M 2 S ) pr ovided L ( M 1 ) and L ( M 2 ) ar e disjoint. – The universe of the injectively ω -a utomatic copy S of S ℵ 0 1 is $ ∗ ⊗ L ( M 1 ) wher e $ is a fr esh symbol. F or i ∈ N , the r estriction o f S to { $ i } ⊗ L ( M 1 ) forms a copy of S 1 . – The universe of the injectively ω - automatic copy S of S 2 ℵ 0 1 is { $ 1 , $ 2 } ω ⊗ L ( M 1 ) wher e $ 1 and $ 2 ar e fr esh symbols. F or w ∈ { $ 1 , $ 2 } ω , the r estriction of S to { w } ⊗ L ( M 1 ) forms a copy of S 1 . 6 2.4 T rees A fores t is a p artial order F = ( V , ≤ ) su ch that fo r every x ∈ V , the set { y | y ≤ x } of ancestors of x is finite and linearly ord ered by ≤ . The level of a node x ∈ V is |{ y | y < x }| ∈ N . Th e h eight o f F is the supremu m of the levels of all nod es in V ; it ma y be infin ite. Note th at a forest of infin ite height can be well-fo unded, i.e., all its paths are finite. In th is paper we on ly d eal w ith fore sts of fi nite he ight . For all u ∈ V , F ( u ) denotes the restriction of F to the set { v ∈ V | u ≤ v } of successors of u . W e will speak of the sub tr ee r oo ted at u . A tree is a forest that has a minimal element, called the r oo t . For a forest F and r not belong ing to the domain o f F , w e den ote with r ◦ F the tree that results fr om adding r to F as a ne w root. The edge r elation E o f the forest F is the set of p airs ( u, v ) ∈ V 2 such that u is the largest element in { x | x < v } . Note that a forest F = ( V , ≤ ) of finite height is (injectiv ely) ω -automatic if and only if the graph ( V , E ) (wher e E is the ed ge relatio n of E ) is ( injectiv ely) ω -au tomatic, since each of these structure s is first-orde r interpretable in the other structure. Th is does n ot hold for trees of infinite heig ht. For any node u ∈ V , we use E ( u ) to denote the set of children (or immediate successors) of u . W e use T n (resp. T i n ) to d enote the class of (injectively) ω -automatic p resentations of trees o f heig ht at most n . Note th at it is decidable wheth er a given ω -auto matic presentation P b elongs to T n and T i n , resp ., since the class of trees o f height at most n can be axiomatized in first-order logic. 3 ω -automatic tr ees of height 1 and 2 For ω -automatic trees of height 2 we need the following result: Theorem 3.1 ( [KRB08]). Let A be an ω - automatic structur e and let ϕ ( x 1 , . . . , x n , y ) be a fo rmula o f FO [ ∃ ℵ 0 , ∃ 2 ℵ 0 ] . Then, for all a 1 , . . . , a n ∈ A , the car dinality of the set { b ∈ A | A | = ϕ ( a 1 , . . . , a n , b ) } belongs to N ∪ { ℵ 0 , 2 ℵ 0 } . Theorem 3.2. The following holds: – The isomorp hism pr ob lem Iso ( T 1 ) for ω -au tomatic tr ees of height 1 is decida ble. – The r e exists a tr ee U su ch that { P ∈ T i 2 | S ( P ) ∼ = U } is Π 0 1 -hard. The isomor- phism pr oblems Iso ( T 2 ) and Iso ( T i 2 ) for (injec tively) ω -a utomatic tr ee s of heig ht 2 ar e Π 0 1 -complete. Pr o of. T wo trees o f height 1 are isomor phic if an d only if they have the same size. By Theorem 3.1, the number of elemen ts in an ω -au tomatic tree S ( P ) with P ∈ T 1 is either finite, ℵ 0 or 2 ℵ 0 and th e exact size can be computed using Theorem 2.2 (by check ing successiv ely v alidity of the sentences ∃ κ x : x = x f or κ ∈ N ∪ {ℵ 0 , 2 ℵ 0 } 4 ). Now , let us take two trees T 1 and T 2 of heigh t 2 and let E i be the edge re lation o f T i and r i its ro ot. For i ∈ { 1 , 2 } an d a card inal λ let κ λ,i be the cardinality of the set of all u ∈ E i ( r i ) su ch that | E i ( u ) | = λ . T hen T 1 ∼ = T 2 if and only if κ λ, 1 = κ λ, 2 for 4 Where ∃ n x : ϕ ( x ) for n ∈ N is shorthand for the obvious first-order formula expressing that there are exactly n elements satisfying ϕ . 7 any cardinal λ . Now ass ume that T 1 and T 2 are both ω -autom atic. By Th eorem 3.1, for all i ∈ { 1 , 2 } an d e very u ∈ E i ( r i ) we ha ve | E i ( u ) | ∈ N ∪ {ℵ 0 , 2 ℵ 0 } . Moreover , a gain by The orem 3.1, every cardinal κ λ, 1 ( λ ∈ N ∪ {ℵ 0 , 2 ℵ 0 } ) belo ngs to N ∪ { ℵ 0 , 2 ℵ 0 } as well. Hence, T 1 ∼ = T 2 if and only if, for all κ, λ ∈ N ∪ {ℵ 0 , 2 ℵ 0 } : T 1 | = ∃ κ x : (( r 1 , x ) ∈ E ∧ ∃ λ y : ( x, y ) ∈ E ) if and only if T 2 | = ∃ κ x : (( r 2 , x ) ∈ E ∧ ∃ λ y : ( x, y ) ∈ E ) . By T heorem 2.2, this equiv alence is decidab le for all κ, λ . Since it has to h old f or all κ, λ , the isomor phism of two ω -autom atic tr ees of heigh t 2 is expressible by a Π 0 1 - statement. Hardness f or Π 0 1 follows fr om the cor respondin g re sult on au tomatic trees of height 2. ⊓ ⊔ 4 A normal form f or analytical sets T o p rove our lo wer bou nd for the isomo rphism prob lem of ω -autom atic trees of heigh t n ≥ 3 , we will use the follo wing normal f orm of analy tical sets. A f ormula of th e form x ∈ X or x 6∈ X is called a set c onstraint . Th e c onstruction s in the proof of the following lemma are standar d. Proposition 4.1. F or e very od d ( res p. even ) n ∈ N + and e very Π 1 n ( r esp. Σ 1 n ) r elatio n A ⊆ N r + , ther e exist polynomials p i , q i ∈ N [ x, y , z ] a nd disjunctions ψ i (1 ≤ i ≤ ℓ ) of set c onstraints (on the set variables X 1 , . . . , X n and in dividual variables x, y , z ) such that x ∈ A if and only if Q 1 X 1 Q 2 X 2 · · · Q n X n ∃ y ∀ z : ℓ ^ i =1 p i ( x, y , z ) 6 = q i ( x, y , z ) ∨ ψ i ( x, y , z , X 1 , . . . , X n ) , wher e Q 1 , Q 2 , . . . , Q n ar e alternating quantifiers with Q n = ∀ . Pr o of. For notational simplicity , we present the p roof only fo r the case wh en n is odd. The o ther ca se can be proved in a similar way by ju st add ing an existential qu antification ∃ X 0 at the be ginning . W e will write Σ m ( SC , REC ) for the set of Σ m -formu las over set constraints and recur si ve pred icates, Π m ( SC , REC ) is to b e understood similarly and B Σ m ( SC , REC ) is the set of boo lean combinations of formulas from Σ m ( SC , REC ) . W ith C k : N k + → N k we will denote some computable bijection. Fix an od d number n . It is well known that every Π 1 n -relation A ⊆ N r + can be written as A = { x ∈ N r + | ∀ f 1 ∃ f 2 · · · ∀ f n ∃ y : P ( x, y , f 1 , . . . , f n ) } , (1) where P is a recursive pred icate relative to the fu nctions f 1 , . . . , f n (see [Od i89, p.3 78]). In o ther word s, there exists an oracle Turing-machine which computes th e Boolean value P ( x, y , f 1 , . . . , f n ) from inpu t ( x, y ) . The o racle T u ring-mach ine can compu te a value f i ( a ) fo r a previously co mputed numb er a ∈ N + in a sin gle step. Th erefore we can ea sily o btain an oracle Turing machine M which h alts o n inp ut x if an d on ly if ∃ y : P ( x, y , f 1 , . . . , f n ) holds. 8 Follo wing [Odi89] , we can replace th e function quantifiers in (1) by set q uantifiers as follows. A function f : N + → N + is encoded by the set { C 2 ( x, y ) | f ( x ) = y } . Let func ( X ) be the following formula, wh ere X is a set v ariable: func ( X ) = ( ∀ x, y , z , u, v : C 2 ( x, y ) = u ∧ C 2 ( x, z ) = v ∧ u, v ∈ X → y = z ) ∧ ( ∀ x ∃ y , z : C 2 ( x, y ) = z ∧ z ∈ X ) Hence, func ( X ) is a Π 2 ( SC , REC ) -formula, which expresses that X en codes a total function on N . Then, the set A in (1) can be defined by the formula ∀ X 1 : ¬ func ( X 1 ) ∨ ∃ X 2 : func ( X 2 ) ∧ · · · ∀ X n : ¬ func ( X n ) ∨ R ( x, X 1 , . . . , X n ) . ( 2) The predicate R c an be derived fro m th e oracle T uring-mach ine M as follows: Con- struct fr om M a new oracle Turing machine N with o racle sets X 1 , . . . , X n . If the machine M wants to compu te the value f i ( a ) , then the machine N starts to enumer ate all b ∈ N + until it finds b ∈ N + with C 2 ( a, b ) ∈ X i . Then it con tinues its comp utation with b for f i ( a ) . The n the predicate R ( x, X 1 , . . . , X n ) expresses that machine N halts on input x . Fix a computab le bijec tion D : N + → Fin ( N + ) , where Fin ( N + ) is the set of all finite subsets of N + . Let in ( x, y ) be an abbreviation for x ∈ D ( y ) . This is a computab le predicate. Next, con sider the pred icate R ( x, X 1 , . . . , X n ) . In every run o f the m achine N on input x , the mach ine N makes only finitely m any or acle qu eries. Hen ce, the predicate R ( x, X 1 , . . . , X n ) is equiv alent to ∃ b ∃ ( s 1 , . . . , s n ) : S ( x, b , ( s 1 , . . . , s n )) ∧ n ^ i =1 ∀ z ≤ b ( in ( z , s i ) ↔ z ∈ X i ) , where the predicate S is d eriv ed fr om th e Turing-machine N as follows: Let T be the T uring -machine that on input ( x, b, ( s 1 , . . . , s n )) behav es as N , but if N ask s the oracle whether z ∈ X i , then T first check s whether z ≤ b (if not, then T diverges) and then checks, wheth er in ( z , s i ) holds. Then S ( x, b, ( s 1 , . . . , s n )) if an d only if T halts on input ( x, b, ( s 1 , . . . , s n )) . He nce, the predica te S ( x, b, ( s 1 , . . . , s n )) is recursively enumera ble, i.e., can b e described b y a formula from Σ 1 ( REC , SC ) . Hence the p redicate R can be describ ed by a f ormula from Σ 2 ( REC , SC ) . Note that the formula from (2) is equiv alent with a for mula ∀ X 1 ∃ X 2 · · · ∀ X n : ϕ ( x, X ) , (3) where ϕ is a Boolean combinatio n of R and f ormulas of the fo rm func ( X i ) . Since all these fo rmulas belo ng to Π 2 ( REC , SC ) ∪ Σ 2 ( REC , SC ) , the for mula ϕ belong s to B Σ 2 ( REC , SC ) ⊆ Π 3 ( REC , SC ) . Hence (3) is equiv alent with ∀ X 1 ∃ X 2 · · · ∀ X n ∀ a ∃ b ∀ c : β (4) where β is a bo olean combination of recursive predica tes and set constrain ts. W e can eliminate th e quantifier block ∀ a by merging it with ∀ X n : Fir st, we c an reduce ∀ a to a single quantifier ∀ a . For th is, assume that th e leng th of the tu ple a is k . 9 Then, ∀ a · · · in (4) can be replac ed by ∀ a ∃ a : C k ( a ) = a ∧ · · · . Since C k ( a ) = a is again recu rsiv e and since we can merge ∃ a ∃ b into a sin gle block of quan tifiers ∃ b , we obtain indeed an equivalent formula of the form ∀ X 1 ∃ X 2 · · · ∀ X n ∀ a ∃ b ∀ c : β ′ (5) where β ′ is a boolean combination of recursive predica tes an d set constraints. Next, we encode the pair ( X n , a ) by the set { 2 x | x ∈ X n } ∪ { 2 a + 1 } . Let α ( X ) be the formula α ( X ) = ( ∀ x, y , x ′ , y ′ : x = 2 x ′ + 1 ∧ y = 2 y ′ + 1 ∧ x, y ∈ X → x = y ) ∧ ( ∃ x, u : x ∈ X ∧ x = 2 u + 1) Hence, α ( X ) expresses that X contains exactly one o dd number . Hence, we obtain a formu la equi v alent to (5) by – re placing ∀ X n ∀ a · · · with ∀ X n : ¬ α ( X n ) ∨ ∃ a, a ′ , a ′′ : a ′′ ∈ X n ∧ a ′′ = a ′ + 1 ∧ a ′ = 2 a ∧ · · · and – re placing every existential quantifier ∃ b i · · · (resp. un i versal q uantifier ∀ c i · · · ) in (5) with ∃ b i ∃ b ′ i : b ′ i = 2 b i ∧ · · · (resp. ∀ c i ∀ c ′ i : c ′ i 6 = 2 c i ∨ · · · ), an d – re placing e very sub-formula a ∈ X n , b i ∈ X n or c i ∈ X n with a ′ ∈ X n , b ′ i ∈ X n , and c ′ i ∈ X n , resp.. All n e w quantifiers can be merged with either th e block ∃ b or the block ∀ c in (5). W e now ha ve obtaine d an equi valent form ula of the for m ∀ X 1 ∃ X 2 · · · ∀ X n ∃ b ∀ c : β ′′ (6) where β ′′ is a Boolean combinatio n of recu rsi ve p redicates and set constraints. The block ∃ b · · · ca n be replaced by ∃ b ∀ b : C ℓ ( b ) 6 = b ∨ · · · , wher e ℓ is the length of the tuple b . Since C ℓ ( b ) 6 = b is a com putable pr edicate, this results in an eq uiv alent formu la of the fo rm ∀ X 1 ∃ X 2 · · · ∀ X n ∃ b ∀ c : β ′′′ where β ′′′ is a Boolean combinatio n of recu rsi ve p redicates and set constraints. Note th at the set of recu rsi ve predicates is closed u nder Boo lean comb inations and that the set of set con straints is closed un der negation. This allows to obtain an equiva- lent formula of the form ∀ X 1 ∃ X 2 · · · ∀ X n ∃ b ∀ c : ℓ ^ i =1 ( R i ∨ ψ i ) , where the R i are recursive predica tes an d the ψ i are disjunctions of set constraints. Since the recur si ve pr edicates R i are co-Diop hantine, th ere are poly nomials p i , q i ∈ N [ b, c, z ] such th at R i ( b, c ) is equivalent with ∀ z : p i ( b, c, z ) 6 = q i ( b, c, z ) . Replacing R i in the above f ormula by this equivalent formula and m erging the new universal quantifiers ∀ z with ∀ c re sults in a formula as required. ⊓ ⊔ It is known that the first-o rder q uantifier block ∃ y ∀ z in Pro position 4.1 cannot b e r e- placed by a block with only one type of first-order quantifiers, see e.g. [Odi89]. 10 5 ω -automatic tr ees of height at least 4 W e prove the following theorem for injectively ω -automatic trees of height at least 4 . Theorem 5.1. Let n ≥ 1 an d Θ ∈ { Σ , Π } . Th er e exists a tr ee U n,Θ of height n + 3 such that the set { P ∈ T i n +3 | S ( P ) ∼ = U n,Θ } is har d for Θ 1 n . Hence, – th e isomorphism pr oblem Is o ( T i n +3 ) for th e class o f injec tively ω -automa tic tr ees of heigh t n + 3 is hard for bo th the classes Π 1 n and Σ 1 n , – a nd the isomo rphism pr oblem Iso ( T i ) for the class of injectively ω - automatic tr ees of finite height is not analytica l. Theorem 5.1 will be deri ved from the following pro position who se pro of occup ies Sec- tions 5.1 and 5.2. Proposition 5.2. Let n ≥ 1 . Ther e ar e trees U [0] and U [1] of heigh t n + 3 such that for any set A ⊆ N + that is Π 1 n if n is od d an d Σ 1 n if n is even, one can compu te fr om x ∈ N + an in jectively ω -au tomatic tr ee T [ x ] of height n + 3 with T [ x ] ∼ = U [0] if and only if x ∈ A and T [ x ] ∼ = U [1] otherwise. Pr o of of Theor em 5.1 fr om P r op osition 5.2. Let n ≥ 1 be odd. Let A be an arbitrary set from Π 1 n and set U n,Π = U [0] and U n,Σ = U [1] . Then the mapping x 7→ T [ x ] is a red uction from A to { P ∈ T i n +3 | S ( P ) ∼ = U n,Π } and, a t the same tim e, a re duction from the Σ 1 n -set N + \ A to { P ∈ T i n +3 | S ( P ) ∼ = U n,Σ } . Since A was chosen arbitrary from Π 1 n , th e first statemen t follows for n odd. If n is ev en, we can p roceed similarly exchanging the roles of U [0 ] and U [1] . W e now derive the second statement. By the first o ne, the trees U [0] and U [1 ] are in p articular injectively ω -automatic and of heig ht n + 3 , so let P 0 and P 1 be injective ω -auto matic pre sentations of these two trees. Then P 7→ ( P, P 0 ) is a reduction from the set { P ∈ T i n +3 | S ( P ) ∼ = U n,Π } to Iso ( T i n +3 ) which is therefore hard f or Π 1 n +3 . Analogou sly , this isomo rphism problem is hard for Σ 1 n +3 . Finally , we p rove th e th ird statemen t. For any n ≥ 1 , the set T i n +3 is decidable (since the set of trees of heig ht at most 3 is first-order ax iomatizable). With P ′ , P ′′ ∈ T i n +3 arbitrary with S ( P ′ ) 6 ∼ = S ( P ′′ ) , the mapping ( P 1 , P 2 ) 7→ ( ( P 1 , P 2 ) if P 1 , P 2 ∈ T i n +3 ( P ′ , P ′′ ) otherwise is a redu ction from Iso ( T i n +3 ) to Iso ( T i ) . Hen ce Iso ( T i ) is hard for all le vels Σ 1 n and therefor e not analy tical. ⊓ ⊔ The c onstruction of the trees T [ x ] , U [0] , and U [1] is unifo rm in n and the f ormula defining A . Hence the second-o rder theory of ( N , + , × ) can be reduced to S n ≥ 1 { n } × Iso ( T i n ) and therefo re to th e isomorphism problem Iso ( S n ≥ 1 T i n ) . Corollary 5.3. The second-order theory of ( N , + , × ) can be r ed uced to th e isomor - phism pr oblem Iso ( S n ∈ N + T i n ) for the class of all injectively ω -au tomatic tr ees of finite height. 11 W e n ow start to prove Propo sition 5.2. Let A be a set that is Π 1 n if n is od d and Σ 1 n otherwise. By Proposition 4.1 it can be written in the form A = { x ∈ N + | Q 1 X 1 . . . Q n X n ∃ y ∀ z : ℓ ^ i =1 p i ( x, y , z ) 6 = q i ( x, y , z ) ∨ ψ i ( x, y , z , X ) } where – Q 1 , Q 2 , . . . , Q n are alternating quantifiers with Q n = ∀ , – p i , q i (1 ≤ i ≤ ℓ ) ar e polynomials in N [ x, y , z ] wher e z has length k , and – every ψ i is a disjun ction of set constraints on the set variables X 1 , . . . , X n and the individual v ariables x , y , z . Let ϕ − 1 ( x, y , X 1 , . . . , X n ) be the formula ∀ z : ℓ ^ i =1 p i ( x, y , z ) 6 = q i ( x, y , z ) ∨ ψ i ( x, y , z , X ) . For 0 ≤ m ≤ n , we will also consider the formu la ϕ m ( x, X 1 , . . . , X n − m ) defined by Q n +1 − m X n +1 − m . . . Q n X n ∃ y : ϕ − 1 ( x, y , X 1 , . . . , X n ) such that ϕ 0 ( x, X 1 , . . . , X n ) is a first-ord er fo rmula and ϕ n ( x ) holds if and on ly if x ∈ A . T o prove Prop osition 5.2, we construct by induction on 0 ≤ m ≤ n height- ( m + 3) trees T m [ X 1 , . . . , X n − m , x ] and U m [ i ] where X 1 , . . . , X n − m ⊆ N + , x ∈ N + , and i ∈ { 0 , 1 } such that the following ho lds: ∀ X ∈ (2 N + ) n − m ∀ x ∈ N + : T m [ X , x ] ∼ = ( U m [0] if ϕ m ( x, X ) holds U m [1] otherwise (7) Setting T [ x ] = T n [ x ] , U [0] = U n [0] , and U [1] = U n [1] and constructing from x an injectively ω -au tomatic presentation of T [ x ] then proves Proposition 5.2. 5.1 Construction of trees In the following, we will u se the injective polynom ial fun ction C : N 2 + → N + with C ( x, y ) = ( x + y ) 2 + 3 x + y . (8) For e 1 , e 2 ∈ N + , let S [ e 1 , e 2 ] den ote the height-1 tree contain ing C ( e 1 , e 2 ) leav es. For ( X , x, y , z , z k +1 ) ∈ (2 N + ) n × N k +3 + and 1 ≤ i ≤ ℓ , define the following height-1 tree, where ℓ , p i , and q i refer to the definition of the set A above: 5 T ′ [ X , x, y , z , z k +1 , i ] = ( S [1 , 2] if ψ i ( x, y , z , X ) S [ p i ( x, y , z ) + z k +1 , q i ( x, y , z ) + z k +1 ] otherwise. (9) 5 The choice of S [1 , 2] in the first case is arbitrary . Any S [ a, b ] wit h a 6 = b would be acceptab le. 12 Next, we d efine the f ollowing height- 2 trees, where κ ∈ N + ∪ { ω } (we consider the natural order on N + ∪ { ω } with n < ω fo r all n ∈ N + ): T ′′ [ X , x, y ] = r ◦ ] { S [ e 1 , e 2 ] | e 1 6 = e 2 } ⊎ ] { T ′ [ X , x, y , z , z k +1 , i ] | z ∈ N k + , z k +1 ∈ N + , 1 ≤ i ≤ ℓ } ! ℵ 0 (10) U ′′ [ κ ] = r ◦  ] { S [ e 1 , e 2 ] | e 1 6 = e 2 } ⊎ ] { S [ e, e ] | κ ≤ e < ω }  ℵ 0 . (11) Note that all the trees T ′′ [ X , x, y ] and U ′′ [ κ ] are build fr om trees of the form S [ e 1 , e 2 ] . Furthermo re, if S [ e, e ] appears as a building b lock, th en S [ e + a, e + a ] also app ears as one for all a ∈ N . In addition, any building block S [ e 1 , e 2 ] appea rs either infinitely often or not at all. In this sense, U ′′ [ κ ] encodes the set of pairs { ( e 1 , e 2 ) | e 1 6 = e 2 } ∪ { ( e, e ) | κ ≤ e < ω } and T ′′ [ X , x, y ] en codes the set of pairs { ( e 1 , e 2 ) | e 1 6 = e 2 }∪ { ( p i ( x, y , z ) + z k +1 , q i ( x, y , z ) + z k +1 ) | 1 ≤ i ≤ ℓ, x, y , z k +1 ∈ N + , z ∈ N k + } . These observations allow to prove the following: Lemma 5.4. Let X ∈ (2 N + ) n and x, y ∈ N + . Then the following hold: (a) T ′′ [ X , x, y ] ∼ = U ′′ [ κ ] for some κ ∈ N + ∪ { ω } (b) T ′′ [ X , x, y ] ∼ = U ′′ [ ω ] if and only if ϕ − 1 ( x, y , X ) ho lds Pr o of. Let us start with the second property . Supp ose ϕ − 1 ( x, y , X ) holds. Let z ∈ N k + , z k +1 ∈ N , and 1 ≤ i ≤ ℓ . Since p i ( x, y , z ) 6 = q i ( x, y , z ) , there are n atural numbers e 1 6 = e 2 with T ′ [ X , x, y , z , z k +1 , i ] = S [ e 1 , e 2 ] . Hence T ′′ [ X , x, y ] ∼ = U ′′ [ ω ] . Con versely , supp ose T ′′ [ X , x, y ] ∼ = U ω . Let z ∈ N k , z k +1 ∈ N , and 1 ≤ i ≤ ℓ . T hen T ′ [ X , x, y , z , z k +1 , i ] is a heigh t-2 subtree of T ′′ [ X , x, y ] ∼ = U ′′ [ ω ] . Hence there are natural n umbers e 1 6 = e 2 with T ′ [ X , x, y , z , z k +1 , i ] ∼ = S [ e 1 , e 2 ] . By (9), this implies p i ( x, y , z ) 6 = q i ( x, y , z ) ∨ ψ i ( x, y , z , X ) . Hence we showed th at ∀ z : V ℓ i =1 p i ( x, y , z ) 6 = q i ( x, y , z ) ∨ ψ i ( x, y , z , X ) ho lds. Now it suffices to prove th e first statement in case ϕ − 1 ( x, y , X ) do es not hold. Then there exist some z ∈ N k + and 1 ≤ i ≤ ℓ with p i ( x, y , z ) = q i ( x, y , z ) ∧ ¬ ψ i ( x, y , z , X ) . Hence th ere is some e ∈ N + such th at S [ e, e ] appears in th e d efinition of T ′′ [ X , x, y ] . Let m = min { e ∈ N + | S [ e , e ] appears in T ′′ [ X , x, y ] } . Then, for all a ∈ N , also S [ m + a, m + a ] appear s in T ′′ [ X , x, y ] . Hence T ′′ [ X , x, y ] ∼ = U ′′ [ m ] . ⊓ ⊔ In a next step, we collect the trees T ′′ [ X , x, y ] and U ′′ [ κ ] into the trees T 0 [ X , x ] , U 0 [0] , and U 0 [1] as follows: T 0 [ X , x ] = r ◦  ] { U ′′ [ m ] | m ∈ N + } ⊎ ] { T ′′ [ X , x, y ] | y ∈ N + }  ℵ 0 U 0 [0] = r ◦  ] { U ′′ [ κ ] | κ ∈ N + ∪ { ω }}  ℵ 0 U 0 [1] = r ◦  ] { U ′′ [ m ] | m ∈ N + }  ℵ 0 13 By Lemm a 5 .4(a), these trees are b uild from copies of the tree s U ′′ [ κ ] (an d are therefore of height 3), each appearin g either infin itely often or not at all. Lemma 5.5. Let X ∈ (2 N + ) n and x ∈ N + . Then T 0 [ X , x ] ∼ = ( U 0 [0] if ϕ 0 ( x, X ) holds and U 0 [1] otherwise. Pr o of. If T 0 [ X , x ] ∼ = U 0 [0] , th en there m ust be some y ∈ N + such th at T ′′ [ X , x, y ] ∼ = U ′′ [ ω ] . By Lemma 5. 4(b), this means that ϕ 0 ( x, X ) holds. On the other hand , suppo se T 0 [ X , x ] 6 ∼ = U 0 [0] . Then T ′′ [ X , x, y ] 6 ∼ = U ′′ [ ω ] for a ll y ∈ N + . From L emma 5.4 (b) again, we ob tain for all y ∈ N + : T ′′ [ X , x, y ] ∼ = U ′′ [ m y ] for some m y ∈ N + . Hence T 0 [ X , x ] ∼ = U 0 [1] in this case. ⊓ ⊔ Now , we com e to the ind uction step in th e co nstruction of our trees. Suppose that for some 0 ≤ m < n we have h eight- ( m + 3) trees T m [ X 1 , . . . , X n − m , x ] , U m [0] an d U m [1] satisfying (7). Let X stand for ( X 1 , . . . , X n − m − 1 ) an d let α = m mo d 2 . W e define the following heig ht- ( m + 4 ) trees: T m +1 [ X , x ] = r ◦  U m [ α ] ⊎ ]  T m [ X , X n − m , x ] | X n − m ⊆ N +   2 ℵ 0 U m +1 [ i ] = r ◦ ( U m [ α ] ⊎ U m [ i ]) 2 ℵ 0 for i ∈ { 0 , 1 } Note that the tr ees T m +1 [ X , x ] , U m +1 [0] , and U m +1 [1] con sist of 2 ℵ 0 many co pies of U m [ α ] and possibly 2 ℵ 0 many copies of U m [1 − α ] . Lemma 5.6. Let X 1 , . . . , X n − m − 1 ⊆ N + and x ∈ N + . Then T m +1 [ X 1 , . . . , X n − m − 1 , x ] ∼ = ( U m +1 [0] if ϕ m +1 ( x, X 1 , . . . X n − m − 1 ) holds U m +1 [1] otherwise. Pr o of. W e have to hand le the cases o f odd and e ven m sep arately an d star t assumin g m to be even (i.e., α = 0 ) such that the outerm ost qua ntifier Q n − m of the formula ϕ m +1 ( x, X 1 , . . . , X n − m − 1 ) is universal. Suppose that ϕ m +1 ( X 1 , . . . , X n − m − 1 , x ) ho lds. Then, by t he induc ti ve hypo thesis, for each X n − m ⊆ N + , T m [ X 1 , . . . , X n − m , x ] ∼ = U m [0] . Hence all h eight- ( m + 3) subtrees of T m +1 [ X 1 , . . . , X n − m − 1 , x ] are isomorphic to U m [0] and thus T m +1 [ X 1 , . . . , X n − m − 1 , x ] ∼ = r ◦ U m [0] 2 ℵ 0 = U m +1 [0] . On the other hand, suppose that ¬ ϕ m +1 ( X 1 , . . . , X n − m − 1 , x ) holds. Then there e xists some set X n − m such that ¬ ϕ m ( X 1 , . . . , X n − m , x ) is true. He nce, by th e induction hypoth esis, T m ( X 1 , . . . , X n − m , x ) ∼ = U m [1] , i.e., T m +1 ( X 1 , . . . , X n − m − 1 , x ) contains on e (and the refore 2 ℵ 0 many) height- ( m + 3) subtrees iso morphic to U m [1] . T his im plies T m +1 ( X 1 , . . . , X n − m − 1 , x ) ∼ = U m +1 [1] since m is e ven. The arguments for m o dd are very similar a nd therefore left to the reader . ⊓ ⊔ 14 The following lem ma follows from Lemma 5.6 with m = n and the fact that ϕ n ( x ) holds if and only if x ∈ A . Lemma 5.7. F or all x ∈ N + , we have T n [ x ] ∼ = U n [0] if x ∈ A and T n [ x ] ∼ = U n [1] otherwise. 5.2 Injective ω - automaticity Injectively ω -automatic presentations o f the tr ees T m [ X , x ] , U m [0] , and U m [1] will be constructed inductively . Note that the con struction of T m +1 [ X , x ] inv olves all the trees T m [ X , X n − m , x ] for X n − m ⊆ N + . Hence we n eed one single injectively ω -au tomatic pr esentation for the for est co nsisting o f all these tr ees. T herefor e, we will deal with forests. T o move from one forest to the next, we will a lw ays pr oceed as follows: add a set of new roo ts an d connec t them to some of th e old r oots which results in a dir ected acyclic graph (or d ag) and n ot necessarily in a forest. The next forest will then be the unfold ing of this dag . The height of a dag D is the len gth (numb er of edges) of a longest directed path in D . W e only consider dags of finite height. A r oot of a dag is a node without incomin g edges. A dag D = ( V , E ) can be u nfolded in to a fo rest unfold( D ) in the u sual way: Nodes o f unfold( D ) are directed paths in D th at start in a root and the order relation is the prefix relation between these paths. For a root v ∈ V of D , we define the tree unfold( D, v ) as the restriction of unfold ( D ) to those p aths that start in v . W e will make use of the following lemma whose proo f is based on the immediate ob servation that the set of conv olutions of path s in D is again a regular ω - languag e. Lemma 5.8. F r om a given k ∈ N and an injectively ω -automa tic presentation for a dag D of h eight at mo st k , one can con struct effectively an injectively ω -automatic pr esentation for unfold( D ) such that the r oo ts of unfold( D ) coincide with the r oots of D and unfold( D , r ) = (unfold( D ))( r ) for an y r oot r . Pr o of. Let D = ( V , E ) = S ( P ) , i.e., V is an ω -regular language an d the binar y relation E ⊆ V × V is ω -auto matic. The univ erse for our injectively ω - automatic co py of unfold( D ) is the set L of all conv olutions v 0 ⊗ v 1 ⊗ v 2 ⊗ · · · ⊗ v m , where v 0 is a root and ( v i , v i +1 ) ∈ E for a ll 0 ≤ i < m . Since the dag D has height at most k , we have m ≤ k . Since th e edge relation of D is ω -automa tic an d since the set of all roots in D is FO -definable an d h ence ω -regular by Theorem 2.2, L is indeed an ω -regular set. Moreover , the edge relation of unfold( D ) becom es clear ly ω -autom atic on L . ⊓ ⊔ For a symbol a and a tuple e = ( e 1 , . . . , e k ) ∈ N k + , we write a e for the ω -word a e 1 ⊗ a e 2 ⊗ · · · ⊗ a e k = ( a e 1 ⋄ ω ) ⊗ ( a e 2 ⋄ ω ) ⊗ · · · ⊗ ( a e k ⋄ ω ) . For an ω -langu age L , we write ⊗ k ( L ) for ⊗ ( L k ) . The following lemm a was shown in [KLL10] for finite words instead of ω -words. Lemma 5.9. Give n a non-zer o po lynomial p ( x ) ∈ N [ x ] in k variab les, one ca n effec- tively construct a B ¨ uchi automato n B [ p ( x )] over th e alph abet { a, ⋄ } k with L ( B [ p ( x )]) = ⊗ k ( a + ) such that for all c ∈ N k + : B [ p ( x )] has exactly p ( c ) a ccepting r uns on input a c . 15 Pr o of. B ¨ uc hi autom ata for th e polyn omials p ( x ) = 1 and p ( x ) = x i are easily b uild. Inductively , let B [ p 1 ( x ) + p 2 ( x )] be the disjoint union of B [ p 1 ( x )] and B [ p 2 ( x )] and let B [ p 1 ( x ) · p 2 ( x )] be obtained from B [ p 1 ( x )] and B [ p 2 ( x )] by the flag construction . ⊓ ⊔ For X ⊆ N + , let w X ∈ { 0 , 1 } ∗ be th e char acteristic word (i.e., w X [ i ] = 1 if and on ly if i ∈ X ) and , fo r X = ( X 1 , . . . , X n ) ∈ (2 N + ) n , write w X for the conv olution of th e words w X i . Lemma 5.10. F r om a given Boolean comb ination ψ ( x 1 , . . . , x m , X 1 , . . . , X n ) of set constraints on set variables X 1 , . . . , X n and individu al variab les x 1 , . . . , x m one can construct effectively a d eterministic B ¨ uchi automa ton A ψ over th e alph abet { 0 , 1 } n × { a, ⋄} m such that for all X 1 , . . . , X n ⊆ N + , c ∈ N m + , the following hold s: w X 1 ⊗ · · · ⊗ w X n ⊗ a c ∈ L ( A ψ ) ⇐ ⇒ ψ ( c, X 1 , . . . , X n ) holds. Pr o of. W e can assume that ψ is a positi ve Boolean comb ination, since the ω -word w N + \ X is simply ob tained from w X by e xchangin g the symbo ls 0 and 1 . Then the claim is trivial for a single set constrain t. Since ω -lang uages accepted by d eterministic B ¨ u chi automata are effecti vely closed und er intersectio n and union, the result follows . ⊓ ⊔ Lemma 5.11. F or 1 ≤ i ≤ ℓ , ther e exists a B ¨ uchi-automa ton A i with the fo llowing pr o perty: F or all X ∈ (2 N + ) n , z ∈ N k + , and x, y , z k +1 ∈ N + , the nu mber of accepting runs of A i on the wor d w X ⊗ a ( x,y ,z ,z k +1 ) equals ( C (1 , 2) if ψ i ( x, y , z , X ) holds C ( p i ( x, y , z ) + z k +1 , q i ( x, y , z ) + z k +1 ) otherwise. Pr o of. By Lemma 5.9, one can construct a B ¨ uc hi automaton B i , which has precisely C ( p i ( x, y , z ) + z k +1 , q i ( x, y , z ) + z k +1 ) many accepting ru ns on the ω -word w X ⊗ a ( x,y , z,z k +1 ) . Seco ndly , on e builds dete rministic B ¨ uch i auto mata C i and C i accepting a word w X ⊗ a ( x,y , z,z k +1 ) if an d on ly if the disjunc tion ψ i ( x, y , z , X ) of set constraints is satisfied (not satisfied, resp.) which is possible by Lemma 5.10 . Let A b e the result o f applying the flag construction to C i and B i . If X ∈ (2 N + ) n , z ∈ N k + , and x, y , z k +1 ∈ N + , then the nu mber o f accepting runs of A on the word w X ⊗ a ( x,y ,z ,z k +1 ) equals ( 0 if ψ i ( x, y , z , X ) h olds C ( p i ( x, y , z ) + z k +1 , q i ( x, y , z ) + z k +1 ) otherwise. Hence the disjoint union of A and C (1 , 2) many co pies of C i has the desired properties. ⊓ ⊔ Proposition 5.12. Ther e exists a n in jectively ω -auto matic for est H ′ = ( L ′ , E ′ ) of height 1 such that – th e set of r oo ts equals { 1 , . . . , ℓ } ⊗ ( { 0 , 1 } ω ) n ⊗ ( ⊗ k +3 ( a + )) ∪ ( b + ⊗ b + ) , 16 – fo r 1 ≤ i ≤ ℓ , X ∈ (2 N + ) n , x, y , z k +1 ∈ N + and z ∈ N k + , we have H ′ ( i ⊗ w X ⊗ a ( x,y , z,z k +1 ) ) ∼ = T ′ [ X , x, y , z , z k +1 , i ] and – fo r e 1 , e 2 ∈ N + , we have H ′ ( b ( e 1 ,e 2 ) ) ∼ = S [ e 1 , e 2 ] . Pr o of. Using L emma 5 .9 (with the polynomial p = C ( x 1 , x 2 ) ) an d Lemma 5.11 , we can construct a B ¨ uchi-au tomaton A accep ting { 1 , . . . , ℓ } ⊗ ( { 0 , 1 } ) n ⊗ ( ⊗ k +3 ( a + )) ∪ ( b + ⊗ b + ) such that the numb er of accep ting runs of A on the ω -word u equals (i) C ( e 1 , e 2 ) if u = b ( e 1 ,e 2 ) , (ii) C (1 , 2) if u = i ⊗ w X ⊗ a ( x,y ,z ,z k +1 ) such that ψ i ( x, y , z , X ) ho lds, and (iii) C ( p i ( x, y , z ) + z k +1 , q i ( x, y , z ) + z k +1 ) if u = i ⊗ w X ⊗ a ( x,y ,z,z k +1 ) such that ψ i ( x, y , z , X ) do es not hold. Let Run A denote the set o f accepting ru ns of A . No te that this is a regular ω -lang uage over the alph abet ∆ of transitions of A . No w the forest H ′ is defined as follows: – Its u niv erse equals L ( A ) ∪ Run A . – Th ere is an edge ( u, v ) if and only if v ∈ Run A is a accepting ru n of A on u ∈ L ( A ) . It is clear that H ′ is an injectively ω - automatic for est of heigh t 1 with set of roo ts L ( A ) as requ ired. Note that (i)-(iii) describe the n umber of leav es of th e height-1 tre e roo ted at u ∈ L ( A ) . B y (i), we therefore get imm ediately H ′ ( b ( e 1 ,e 2 ) ) ∼ = S [ e 1 , e 2 ] . Comp aring the nu mbers in ( ii) and (iii) with the definition of the tree T ′ [ X , x, y , z , z k +1 , i ] in (9) completes the proo f. ⊓ ⊔ From H ′ = ( L ′ , E ′ ) , we build an injecti vely ω -a utomatic dag D as follows: – Th e domain of D is the set ( ⊗ n ( { 0 , 1 } ω ) ⊗ a + ⊗ a + ) ∪ b ∗ ∪  $ ∗ ⊗ L ′ ) . – For u, v ∈ L ′ , the words $ i ⊗ u and $ j ⊗ v are c onnected if and only if i = j and ( u, v ) ∈ E ′ . In other words, the restriction of D to $ ∗ ⊗ L ′ is isomorph ic to H ′ℵ 0 . – For all X ∈ (2 N + ) n , x, y ∈ N + , the new root w X ⊗ a ( x,y ) is co nnected to all nod es in $ ∗ ⊗  ( { 1 , . . . , ℓ } ⊗ w X ⊗ a ( x,y ) ⊗ ( ⊗ k +1 ( a + ))) ∪ { b ( e 1 ,e 2 ) | e 1 6 = e 2 }  . – Th e new roo t ε is connected to all nodes in $ ∗ ⊗ { b ( e 1 ,e 2 ) | e 1 6 = e 2 } . – For all m ∈ N + , the new root b m is connected to all nodes in $ ∗ ⊗ { b ( e 1 ,e 2 ) | e 1 6 = e 2 ∨ e 1 = e 2 ≥ m } . It is easily seen that D is an injectively ω -au tomatic dag. Let H ′′ = unfold ( D ) which is also injectively ω -automatic by L emma 5.8 . Then, for all X ∈ (2 N + ) n , x, y , m ∈ N + , 17 we have H ′′ ( w X ⊗ a ( x,y ) ) ∼ = ( w X ⊗ a ( x,y ) ) ◦ U {H ′ ( i ⊗ w X ⊗ a ( x,y ,z ) ) | 1 ≤ i ≤ ℓ , z ∈ N k +1 + }⊎ U {H ′ ( b ( e 1 ,e 2 ) ) | e 1 6 = e 2 } ! ℵ 0 Prop. 5.12 ∼ = r ◦ U { T ′ [ X , x, y , z , i ] | z ∈ N k +1 + , 1 ≤ i ≤ ℓ } ⊎ U { S [ e 1 , e 2 ] | e 1 6 = e 2 } ! ℵ 0 (10) = T ′′ [ X , x, y ] H ′′ ( ε ) ∼ = ε ◦  ] {H ′ ( b ( e 1 ,e 2 ) ) | e 1 6 = e 2 }  ℵ 0 Prop. 5.12 ∼ = r ◦ ]  { S [ e 1 , e 2 ] | e 1 6 = e 2 }  ℵ 0 (11) = U ′′ [ ω ] H ′′ ( b m ) ∼ = b m ◦  ] {H ′ ( b ( e 1 ,e 2 ) ) | e 1 6 = e 2 ∨ e 1 = e 2 ≥ m }  ℵ 0 Prop. 5.12 ∼ = r ◦  ] { S [ e 1 , e 2 ] | e 1 6 = e 2 ∨ e 1 = e 2 ≥ m }  ℵ 0 (11) = U ′′ [ m ] From H ′′ = ( L ′′ , E ′′ ) , we build an injecti vely ω -au tomatic dag D 0 as follows: – Th e domain of D 0 is the set ( ⊗ n { 0 , 1 } ω ) ⊗ a + ∪ { ε , b } ∪ ($ ∗ ⊗ L ′′ ) . – For u, v ∈ L ′′ , the words $ i ⊗ u and $ j ⊗ v a re connected by an edge if and only if i = j and ( u, v ) ∈ E ′′ , i.e., the restriction of D 0 to $ ∗ ⊗ L ′′ is isom orphic to H ′′ ℵ 0 . – For X ∈ (2 N + ) n , x ∈ N + , conn ect the new root w X ⊗ a x to all nodes in $ ∗ ⊗  w X ⊗ a x ⊗ a + ∪ b +  . – Con nect the new roo t ε to all nodes in $ ∗ ⊗ b ∗ . – Con nect the new roo t b to all nodes in $ ∗ ⊗ b + . Then D 0 is an injecti vely ω -autom atic dag o f heigh t 3 and we set H 0 = unfold( D 0 ) . Then, we have the fo llowing: – Th e set of roots of H 0 is (( ⊗ n ( { 0 , 1 } ω )) ⊗ a + ) ∪ { ε, b } . – For all X ∈ (2 N + ) n , x ∈ N + we have: 18 H 0 ( w X ⊗ a x ) ∼ = r ◦ ] {H ′′ ( b m ) | m ∈ N + }⊎ ] {H ′′ ( w X ⊗ a x ⊗ a y ) | y ∈ N + } ! 2 ℵ 0 ∼ = r ◦  ] { U ′′ [ m ] | m ∈ N + } ⊎ ] { T ′′ [ X , x, y ] | y ∈ N + }  ℵ 0 ∼ = T 0 [ X , x ] H 0 ( ε ) ∼ = r ◦  ] {H ′′ ( b m ) | m ∈ N }  ℵ 0 ∼ = r ◦  ] { U ′′ [ κ ] | κ ∈ N + ∪ { ω }}  ℵ 0 ∼ = U 0 [0] H 0 ( b ) ∼ = r ◦  ] {H ′′ ( b m ) | m ∈ N + }  ℵ 0 ∼ = r ◦  ] { U ′′ [ m ] | m ∈ N + }  ℵ 0 ∼ = U 0 [1] W e now construct the forest H 1 , H 2 , H 3 , . . . , H n inductively . For 0 ≤ m < n , s uppose we hav e ob tained an injectively ω -auto matic for est H m = ( L m , E m ) as de scribed in the lemma. The forest H m +1 is constru cted as follo ws, wher e α = m mo d 2 : – Th e domain of H m +1 is ⊗ n − m − 1 ( { 0 , 1 } ω ) ⊗ a + ∪ { ε , b } ∪ ( { $ 1 , $ 2 } ω ⊗ L m ) . – For u, v ∈ L m and u ′ , v ′ ∈ { $ 1 , $ 2 } ω , the words u ′ ⊗ u and v ′ ⊗ v are conn ected by an edge if and on ly if u ′ = v ′ and ( u, v ) ∈ E m , i.e., the r estriction of D m +1 to { $ 1 , $ 2 } ω ⊗ L m is isomor phic to H 2 ℵ 0 m . – For all X ∈ (2 N + ) n − m − 1 , x ∈ N + , connect the ne w ro ot w X ⊗ a x to all nodes from { $ 1 , $ 2 } ω ⊗  w X ⊗ { 0 , 1 } ω ⊗ a x ∪ b α  . – Con nect the new roo t ε to all nodes from { $ 1 , $ 2 } ω ⊗ { ε , b α } . – Con nect the new roo t b to all nodes from { $ 1 , $ 2 } ω ⊗ { b , b α } . In this way we obtain the injecti vely ω -au tomatic forest H m +1 such that: – Th e set of roots of H m +1 is (( ⊗ n − m − 1 ( { 0 , 1 } ω )) ⊗ a + ) ∪ { ε, b } . 19 – For X ∈ (2 N + ) n − m − 1 and x ∈ N + we hav e: H m +1 ( w X ⊗ a x ) ∼ = r ◦  ] {H m ( w X ⊗ w X n − m ⊗ x ) | X n − m ⊆ N + } ⊎ H m ( b α )  2 ℵ 0 ∼ = r ◦  ] { T m [ X , X n − m , x ] | X n − m ⊆ N + } ⊎ U m [ α ]  2 ℵ 0 ∼ = T m +1 [ X , x ] H m +1 ( ε ) ∼ = r ◦ ( H m ( ε ) ⊎ H m ( b α )) 2 ℵ 0 ∼ = r ◦ ( U m [0] ⊎ U m [ α ]) 2 ℵ 0 ∼ = U m +1 [0] H m +1 ( b ) ∼ = r ◦ ( H m ( b α ) ⊎ H m ( b )) 2 ℵ 0 ∼ = r ◦ ( U m [ α ] ⊎ U m [1]) 2 ℵ 0 ∼ = U m +1 [1] Hence we proved: Lemma 5.13. F r om each 0 ≤ m ≤ n , on e can ef fectively con struct an injectively ω -au tomatic f or est H m such that – th e set of r oo ts of H m is  ⊗ n − m ( { 0 , 1 } ω ) ⊗ a +  ∪ { ε , b } , – H m ( w X ⊗ a x ) ∼ = T m [ X , x ] for a ll X ∈ (2 N + ) n − m and x ∈ N + , – H m ( ε ) ∼ = U m [0] , and – H m ( b ) ∼ = U m [1] . Note that T n [ x ] is the tr ee in H n rooted at a x . Hence T n [ x ] is (effectiv ely) a n injectively ω -auto matic tree. Now Lemm a 5.7 finishes the proof of Proposition 5.2 an d there fore of Theorem 5.1. 6 ω -automatic tr ees of height 3 Recall tha t the isomorphism pr oblem Iso ( T i 2 ) is arithmetical by Theorem 3 .2 and th at Iso ( T i 4 ) is no t by T heorem 5 .1. In this sectio n, we m odify the proof of T heorem 5.1 in order to show that a lready Iso ( T i 3 ) is not arithmetical: Theorem 6.1. Ther e exists a tr ee U su ch that { P ∈ T i 3 | S ( P ) ∼ = U } is Π 1 1 -hard. Hence the i somorphism pr ob lem Iso ( T i 3 ) for injectively ω - automatic t r ees of height 3 is Π 1 1 -hard. So let A ⊆ N + be some set from Π 1 1 . By Proposition 4.1, it can be written as A = { x ∈ N + : ∀ X ∃ y ∀ z : ℓ ^ i =1 p i ( x, y , z ) 6 = q i ( x, y , z ) ∨ ψ i ( x, y , z , X ) } , where p i and q i are poly nomials with c oefficients in N and ψ i is a disjunc tion of set constraints. As in Section 5 , let ϕ − 1 ( x, y , X ) den ote the sub formula starting with ∀ z , 20 and let ϕ 0 ( x, X ) = ∀ y : ϕ − 1 ( x, y , X ) . W e r euse the trees T ′ [ X , x, y , z , z k +1 , i ] of height 1 . Recall th at they are all of the form S [ e 1 , e 2 ] a nd therefore h av e a n e ven numb er of leaves (since the range of the po lynomial C : N 2 + → N + consists of even numbers). For e ∈ N + , let S [ e ] deno te the he ight-1 tree with 2 e + 1 leaves. Recall th at the tree T ′′ [ X , x, y ] encodes the set of pairs ( e 1 , e 2 ) ∈ N 2 + such th at e 1 6 = e 2 or there exist z , z k +1 , and i with e 1 = p i ( x, y , z ) + z k +1 and e 2 = q i ( x, y , z ) + z k +1 . W e now modify the construction o f this tre e such that, in ad dition, it also encodes th e set X ⊆ N + : b T [ X , x, y ] = r ◦ ] { S [ e ] | e ∈ X } ⊎ ] { S [ e 1 , e 2 ] | e 1 6 = e 2 }⊎ ] { T ′ [ X , x, y , z , z k +1 i ] | z ∈ N k + , z k +1 ∈ N + , 1 ≤ i ≤ ℓ } ! ℵ 0 In a similar spirit, we define b U [ κ, X ] for X ⊆ N + and κ ∈ N + ∪ { ω } : b U [ κ, X ] = r ◦ ] { S [ e ] | e ∈ X } ⊎ ] { S [ e 1 , e 2 ] | e 1 6 = e 2 }⊎ ] { S [ e, e ] | κ ≤ e < ω } ! ℵ 0 Then b T [ X , x, y ] ∼ = b U [ ω , Y ] if and only if X = Y and T ′′ [ X , x, y ] ∼ = U ′′ [ ω ] , i.e., if and only if X = Y and ϕ − 1 ( x, y , X ) holds by Lemma 5.4 (b). Finally , we set T [ x ] = r ◦  ] { b U [ κ, X ] | X ⊆ N + , κ ∈ N + } ⊎ ] { b T [ X , x, y ] | X ⊆ N + , y ∈ N + }  ℵ 0 U = r ◦  ] { b U [ κ, X ] | X ⊆ N + , κ ∈ N + ∪ { ω }}  ℵ 0 . Lemma 6.2. Let x ∈ N + . Then T [ x ] ∼ = U if and only if x ∈ A . Pr o of. Suppose x ∈ A . T o prove T [ x ] ∼ = U , it suffices to show that any heig ht-2 subtree of T [ x ] is a subtree of U and vice versa. First, let X ⊆ N + and y ∈ N + . Then, by Lemm a 5.4 , the re exists κ ∈ N + ∪ { ω } with T [ X , x, y ] ∼ = U κ and therefor e b T [ X , x, y ] ∼ = b U [ X , κ ] , i.e., b T [ X , x, y ] ap pears in U . Secondly , let X ⊆ N + . From x ∈ A , we ca n infer th at there exists some y ∈ N + with ∀ z : V ℓ i =1 p i ( x, y , x ) 6 = q i ( x, y , z ) ∨ ψ i ( x, y , z , X ) . Then Lemma 5.4 implies U ω ∼ = T [ X , x, y ] and therefore b U [ X , ω ] ∼ = b T [ X , x, y ] , i .e., b U [ X , ω ] ap pears in T [ x ] . Thus, any height-2 sub tree of T [ x ] is a subtree of U and vice versa. Con versely supp ose T [ x ] ∼ = U . Let X ⊆ N + . Th en b U [ X , ω ] appe ars in U and therefor e in T [ x ] . Since U κ 6 ∼ = U ω for κ ∈ N + , there exists some y ∈ N + with U ω ∼ = T [ X , x, y ] . From Lemma 5.4 we then get x ∈ A . ⊓ ⊔ 6.1 Injective ω - automaticity W e follow closely the procedur e for m = 0 from Section 5.2. Proposition 6.3. Ther e e xists an injectively ω -a utomatic for est H ′ = ( L ′ , E ′ ) of heig ht 1 such that 21 – th e set of r oo ts equals { 1 , . . . , ℓ } ⊗ { 0 , 1 } ω ⊗ ( ⊗ k +3 ( a + )) ∪ ( b + ⊗ b + ) ∪ c + – fo r 1 ≤ i ≤ ℓ , X ⊆ N + , x, y , z k +1 ∈ N + and z ∈ N k + , we have H ′ ( i ⊗ w X ⊗ a ( x,y , z,z k +1 ) ) ∼ = T ′ [ X , x, y , z , z k +1 , i ] – fo r e 1 , e 2 ∈ N + , we have H ′ ( b ( e 1 ,e 2 ) ) ∼ = S [ e 1 , e 2 ] – fo r e ∈ N + , we have H ′ ( c e ) ∼ = S [ e ] Pr o of. Using Lemma 5.9 twice (with th e polynom ial C ( x 1 , x 2 ) and with the poly- nomial 2 x 1 + 1 ) and Lemma 5.11, we ca n construct a B ¨ u chi-autom aton A accepting { 1 , . . . , ℓ } ⊗ { 0 , 1 } ω ⊗ ( ⊗ k +3 ( a + )) ∪ ( b + ⊗ b + ) ∪ c + such that the nu mber of accepting runs of A on the ω -word u equals (i) C ( e 1 , e 2 ) if u = b ( e 1 ,e 2 ) , (ii) 2 e + 1 if u = c e , (iii) C (1 , 2 ) if u = i ⊗ w X ⊗ a ( x,y ,z ,z k +1 ) such that ψ i ( x, y , z , X ) ho lds, and (iv) C ( p i ( x, y , z ) + z k +1 , q i ( x, y , z ) + z k +1 ) if u = i ⊗ w X ⊗ a ( x,y , z,z k +1 ) such that ψ i ( x, y , z , X ) do es not hold. The rest of the proo f is the same as that of Prop osition 5.12. ⊓ ⊔ From H ′ = ( L ′ , E ′ ) , we build an injecti vely ω -a utomatic dag D as follows: – Th e domain of D is the set ( { 0 , 1 } ω ⊗ a + ⊗ a + ) ∪ ( { 0 , 1 } ω ⊗ b ∗ ) ∪ ($ ∗ ⊗ L ′ ) . – For u, v ∈ L ′ , the words $ i ⊗ u and $ j ⊗ v are c onnected if and only if i = j and ( u, v ) ∈ E ′ . In other words, the restriction of D to $ ∗ ⊗ L ′ is isomorph ic to H ′ℵ 0 . – For all X ⊆ N + , x, y ∈ N + , the new root w X ⊗ a ( x,y ) is connected to all nodes in $ ∗ ⊗  ( { 1 , . . . , ℓ } ⊗ w X ⊗ a ( x,y ) ⊗ ( ⊗ k +1 ( a + ))) ∪ { b ( e 1 ,e 2 ) | e 1 6 = e 2 } ∪ { c e | e ∈ X }  . – For all X ⊆ N + , the new root w X ⊗ ε is co nnected to all nodes in $ ∗ ⊗ ( { b ( e 1 ,e 2 ) | e 1 6 = e 2 } ∪ { c e | e ∈ X } ) . – For all X ⊆ N + and m ∈ N + , the ne w root w X ⊗ b m is conn ected to all n odes in $ ∗ ⊗ ( { b ( e 1 ,e 2 ) | e 1 6 = e 2 ∨ e 1 = e 2 ≥ m } ∪ { c e | e ∈ X } ) . It is easily seen that D is an injectively ω -au tomatic dag. Let H ′′ = unfold ( D ) which is also injectively ω -a utomatic by Lemma 5.8. Now compu tations analog ous to those on page 12 (using Pr oposition 6.3 instead of Pro position 5.12) yield for all X ⊆ N + and x, y , m ∈ N + : H ′′ ( w X ⊗ a ( x,y ) ) ∼ = b T [ X , x, y ] H ′′ ( w X ⊗ ε ) ∼ = b U [ ω , X ] H ′′ ( w X ⊗ b m ) ∼ = b U [ m, X ] From H ′′ = ( L ′′ , E ′′ ) , we build an injecti vely ω -au tomatic dag D 0 as follows: 22 – Th e domain of D 0 equals a ∗ ∪ $ ∗ ⊗ L ′′ . – For u, v ∈ L ′′ , the words $ i ⊗ u and $ j ⊗ v are con nected b y an edge if and only if i = j and ( u, v ) ∈ E ′′ . Hen ce the restriction o f D 0 to $ ∗ ⊗ L ′′ is isom orphic to H ′′ ℵ 0 . – For x ∈ N + , conn ect the new root a x to all nodes in $ ∗ ⊗  { 0 , 1 } ω ⊗ b + ∪ { 0 , 1 } ω ⊗ a x ⊗ a +  . – Con nect the new roo t ε to all nodes in $ ∗ ⊗ { 0 , 1 } ω ⊗ b ∗ . Then D 0 is an injecti vely ω -autom atic dag o f heigh t 3 and we set H 0 = unfold( D 0 ) . The set o f roo ts of H 0 is a ∗ . Calculations similar to those on page 20 the n yield H 0 ( ε ) ∼ = U an d H 0 ( a x ) ∼ = T [ x ] fo r x ∈ N + . Hence, T [ x ] is (effecti vely) an injecti vely ω - automatic tree. Now Lemma 6.2 finishes the proof of the first statement of Theore m 6.1 , the second follows imm ediately . Remark 6.4. In ou r previous paper [KL L10], we used a n iterated application of a con- struction very similar to th e o ne in this section in order to prove tha t th e iso morphism problem for auto matic tr e es of heigh t n ≥ 2 is h ard (in fact comp lete) for le vel Π 0 2 n − 3 of th e arith metical hierarch y . This con struction allows to h andle a ∀∃ -quantifier block, while increasing the height of the trees by only 1 . Unfortu nately we ca nnot iterate the co nstruction of this section for ω -automatic tr ees of height n in or der to prove a lower bound o f the form Π 1 2 n − 5 for n ≥ 3 . On the technical level, its Lemma 3.2 from [KLL10] , which do es not hold for second-ord er formu lae. 7 Upper bounds assuming CH W e den ote with CH the continuum hypothesis: Every infinite subset of 2 N has either cardinality ℵ 0 or cardinality 2 ℵ 0 . By seminal w ork of Cohen a nd G ¨ odel, CH is ind e- penden t of the ax iom system ZFC . In the f ollowing, we will identify an ω -word w ∈ Γ ω with the function w : N + → Γ , (and hence with a second-ord er object) where w ( i ) = w [ i ] . W e need th e f ollowing lemma: Lemma 7.1. F r om a given B ¨ uchi automaton M over an alphab et Γ on e can construct an arithmetical pr edicate acc M ( u ) (where u : N + → Γ ) such that: u ∈ L ( M ) if and only if acc M ( u ) holds. Pr o of. Recall that a Muller automaton is a tuple M = ( Q, Γ , ∆, I , F ) , where Q , Γ , ∆ , and I have the same meaning as for B ¨ uchi automata but F ⊆ 2 Q . Th e languag e L ( M ) accepted by M is th e set of all ω - words u ∈ Γ ω for which the re exists a run ( q 1 , u [1] , q 2 )( q 2 , u [2] , q 3 ) · · · ( q 1 ∈ I ) such that { q ∈ Q | ∃ ℵ 0 i : q = q i } ∈ F . The Muller automaton M is deterministic and co mplete , if | I | = 1 and fo r all q ∈ Q, a ∈ Γ there exists a unique p ∈ Q such that ( q , a, p ) ∈ ∆ . It is well known th at from th e given B ¨ uch i autom aton M one can ef fecti vely con - struct a de terministic an d complete Muller automaton M ′ = ( Q, Γ , ∆, { q 0 } , F ) such that L ( M ) = L ( M ′ ) , see e.g. [PP04, Tho 97]. F or a given ω -word u : N + → Γ and 23 i ∈ N let q ( u, i ) ∈ Q be the unique state that is rea ched by M ′ after reading the length - i pre fix of u . Note that q ( u, i ) is co mputable from i ( if u is g i ven as an oracle), he nce q ( u, i ) is arith metically d efinable. No w , the form ula acc M ( u ) can be defin ed as follo ws: _ A ∈F ∃ x ∈ N + ∀ y ≥ x ^ p ∈ A  q ( u, y ) ∈ A ∧ ∃ z ≥ y : q ( u, z ) = p  ⊓ ⊔ Theorem 7.2. Assuming CH , the isomo rphism pr oblem Iso ( T n ) b elongs to Π 1 2 n − 4 for n ≥ 3 . Pr o of. Consider trees T i = S ( P i ) for P 1 , P 2 ∈ T n . De fine th e f orest F = ( V , E ) as F = T 1 ⊎ T 2 For v ∈ V let E ( v ) = { w ∈ V : ( v , w ) ∈ E } be the set of children of v . Let u s fix an ω -automa tic presen tation P = ( Σ , M , M ≡ , M E ) for F . Here, M E recogn izes the edge r elation E of F . In the follo wing, f or u ∈ L ( M ) we write F ( u ) for the subtree F ([ u ] R ( M ≡ ) ) rooted in the F -node [ u ] R ( M ≡ ) represented by the ω -word u . Similar ly , we wr ite E ( u ) for E ([ u ] R ( M ≡ ) ) . W e will define a Π 1 2 n − 2 k − 4 -predicate iso k ( u 1 , u 2 ) , where u 1 , u 2 ∈ L ( M ) are on level k in F . This predicate expresses that F ( u 1 ) ∼ = F ( u 2 ) . As inductio n b ase, le t k = n − 2 . Then the trees F ( u 1 ) an d F ( u 2 ) have heig ht at most 2 . Th en, as in the p roof of Th eorem 3.2, we h a ve F ( u 1 ) ∼ = F ( u 2 ) if an d only if the following holds for all κ, λ ∈ N ∪ {ℵ 0 , 2 ℵ 0 } : F | =  ∃ κ x ∈ V : (([ u 1 ] , x ) ∈ E ∧ ∃ λ y ∈ V : ( x, y ) ∈ E )  ↔  ∃ κ x ∈ V : (([ u 2 ] , x ) ∈ E ∧ ∃ λ y ∈ V : ( x, y ) ∈ E )  . Note that by Theorem 2.2, one can compute from κ, λ ∈ N ∪ {ℵ 0 , 2 ℵ 0 } a B ¨ u chi au- tomaton M κ,λ accepting the set o f con volutions of pairs of ω -word s ( u 1 , u 2 ) satisfying the above form ula. Hence F ( u 1 ) ∼ = F ( u 2 ) if and only if the following arithmetical predicate holds: ∀ κ, λ ∈ N ∪ {ℵ 0 , 2 ℵ 0 } : acc M κ,λ ( u 1 , u 2 ) . Now let 0 ≤ k < n − 2 . W e fir st introdu ce a few notatio ns. For a set A , let count ( A ) denote th e set of all cou ntable (possibly finite) subsets of A . For κ ∈ N ∪ {ℵ 0 } we denote with [ κ ] the set { 0 , . . . , κ − 1 } (resp. N ) in case κ ∈ N ( κ = ℵ 0 ). F or a function f : ( A × B ) → C and a ∈ A let f [ a ] : B → C denote the function with f [ a ]( b ) = f ( a, b ) . On an abstract le vel, t he form ula iso k ( u 1 , u 2 ) is  ∀ x ∈ E ( u 1 ) ∃ y ∈ E ( u 2 ) : iso k +1 ( x, y )  ∧ (12)  ∀ x ∈ E ( u 2 ) ∃ y ∈ E ( u 1 ) : iso k +1 ( x, y )  ∧ (13) ∀ X 1 ∈ co unt ( E ( u 1 )) ∀ X 2 ∈ count ( E ( u 2 )) : (14) ∃ x, y ∈ X 1 ∪ X 2 : ¬ iso k +1 ( x, y ) ∨ (15) ∃ x ∈ X 1 ∪ X 2 ∃ y ∈ ( E ( u 1 ) ∪ E ( u 2 )) \ ( X 1 ∪ X 2 ) : is o k +1 ( x, y ) ∨ ( 16) | X 1 | = | X 2 | . (17) 24 Line (12) and (13) express that the child ren of u 1 and u 2 realize the same isomo rphism types of trees of height n − k − 1 . The rest of the formu la expresses that if a certain isomorph ism type τ of h eight- ( n − k − 1) trees app ears countably many tim es below u 1 then it appears with the same m ultiplicity below u 2 and v ice versa. Assum ing CH and the corre ctness of iso k +1 , the form ula iso k ( u 1 , u 2 ) expr esses indeed that F ( u 1 ) ∼ = F ( u 2 ) . In th e above defin ition of iso k ( u 1 , u 2 ) we ac tually h av e to fill in some d etails. T he countab le s et X i ∈ count ( E ( u i )) ⊆ 2 V of childr en of [ u i ] R ( M ≡ ) (which is universally quantified in (14)) can be represented as a f unction f i : [ | X i | ] × N → Σ such that th e following holds: ∀ j ∈ [ | X i | ] : ac c M E ( u i ⊗ f i [ j ]) ∧ ∀ j, l ∈ [ | X i | ] : j = l ∨ ¬ acc M ≡ ( f i [ j ] ⊗ f i [ l ]) . Hence, ∀ X i ∈ count ( E ( u i )) · · · in (14) can be replaced by: ∀ κ i ∈ N ∪ {ℵ 0 } ∀ f i : [ κ i ] × N → Σ : ( ∃ j ∈ [ κ i ] : ¬ acc M E ( u i ⊗ f i [ j ])) ∨ ( ∃ j, l ∈ [ κ i ] : j 6 = l ∧ acc M ≡ ( f i [ j ] ⊗ f i [ l ])) ∨ · · · . Next, the formula ∃ x, y ∈ X 1 ∪ X 2 : ¬ iso k +1 ( x, y ) in (15 ) can be replaced by: _ i ∈{ 1 , 2 } ∃ j, l ∈ [ κ i ] : ¬ iso k +1 ( f i [ j ] , f i [ l ]) ∨ ∃ j ∈ [ κ 1 ] ∃ l ∈ [ κ 2 ] : ¬ iso k +1 ( f 1 [ j ] , f 2 [ l ]) . Similarly , the fo rmula ∃ x ∈ X 1 ∪ X 2 ∃ y ∈ ( E ( u 1 ) ∪ E ( u 2 )) \ ( X 1 ∪ X 2 ) : is o k +1 ( x, y ) in (16) can be replaced by _ i ∈{ 1 , 2 } ∃ j ∈ [ κ i ] ∃ v : N → Σ : iso k +1 ( f i [ j ] , v ) ∧ ( acc M E ( u 1 ⊗ v ) ∨ acc M E ( u 2 ⊗ v )) ∧ ∀ l ∈ [ κ 1 ] : ¬ acc M ≡ ( f 1 [ l ] ⊗ v ) ∧ ∀ l ∈ [ κ 2 ] : ¬ acc M ≡ ( f 2 [ l ] ⊗ v ) . Note tha t in line (12) and (13) we intro duce a new ∀∃ seco nd-ord er blo ck of quantifiers. The same hold s fo r the rest of the formu la: W e intr oduce two universal set quan ti- fiers in (14) followed by the existential quantifier ∃ v : N → Σ in the above for mula. Since by induction , iso k +1 is a Π 1 2 n − 2( k +1) − 4 -statement, it follows tha t iso k ( u 1 , u 2 ) is a Π 1 2 n − 2 k − 4 -statement. ⊓ ⊔ Corollary 5.3 and 7.2 imply: Corollary 7.3. Assuming CH , the isomorp hism pr oblem for (in jectively) ω -a utomatic tr ees of finite height is r ec ursively equivalent to the second-or der theo ry of ( N ; + , × ) . Remark 7.4. For the ca se n = 3 we can a v oid th e u se of CH in The orem 7.2 : Let us consider th e p roof o f Th eorem 7.2 for n = 3 . Th en, th e binar y relation iso 1 (which holds between tw o ω - words u, v in F if and only if [ u ] and [ v ] are on lev el 1 and 25 F ( u ) ∼ = F ( v ) ) is a Π 0 1 -predicate. It follows that this relation is Borel (see e .g. [Kec95] for bac kgroun d on Borel sets). Now let u be a n ω -word on level 1 in F . It fo llows that the set of all ω - words v on level 1 with is o 1 ( u, v ) is again Borel. Now , every uncou ntable Borel set has cardinality 2 ℵ 0 (this h olds even for an alytic sets [ Kec95]). It follows that the definition of iso 0 in the pro of of Theorem 7.2 is correct e ven without assuming CH . Hence, Iso ( T 3 ) belong s to Π 1 2 (recall that we proved Π 1 1 -hardn ess for this problem in Section 6), this can be shown in ZFC . 8 Open prob lems The m ain o pen pro blem con cerns upp er bound s in case we assume the negation of the continuu m hypo thesis. Assuming ¬ CH , is the isomorphism pro blem for (injecti vely) ω -auto matic trees of height n still analytical? In our paper [KLL10] we also proved that the isomorp hism pr oblem for auto matic linear orders is not a rithmetical. This leads to the question whether our techniqu es fo r ω - automatic trees can be also used for proving lower bound s on the isomo rphism problem for ω -au tomatic linear orders. 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