Table Of Content

Word Automaticity of Tree Automatic Scattered Linear Orderings Is Decidable Martin Huschenbett Institut fu¨r Theoretische Informatik, Technische Universität Ilmenau, Germany 2 [email protected] 1 0 2 n Abstract A tree automatic structure is a structure whose domain can a be encoded by a regular tree language such that each relation is recog- J nisable by a finite automaton processing tuples of trees synchronously. 4 Words can be regarded as specific simple trees and a structure is word 2 automatic if it is encodable using only these trees. The question naturally arises whether a given tree automatic structure is already word ] O automatic. We prove that this problem is decidable for tree automatic scattered linear orderings. Moreover, we show that in case of a posit- L ive answer a word automatic presentation is computable from the tree . s automatic presentation. c [ 1 Introduction 1 v 0 The fundamental idea of automatic structures can be traced back to the 1960s 7 when Bu¨chi, Elgot, Rabin, and others used finite automata to provide decision 0 5 proceduresforthefirst-ordertheoryofPresburgerarithmetic (N;+)andseveral . other logical problems. Hodgson generalised this idea to the concept of auto- 1 maton decidable first-order theories. Independently of Hodgson and inspired by 0 2 thesuccessfulemploymentoffiniteautomataandtheirmethodsingrouptheory, 1 Khoussainov and Nerode [4] initiated the systematic investigation of automatic v: structures. Recalling the efforts from the 1960s, Blumensath [2] extended this i concept notion beyond finite automata to finite automaton models recognising X infinite words, finite trees, or infinite trees. r Basically, a countable relational structure is tree automatic or tree automat- a ically presentable ifitselementscanbeencodedbyfinitetreesinsuchawaythat itsdomainanditsrelationsarerecognisablebyfiniteautomataprocessingeither single trees ortuples oftrees synchronously.Astructure is word automatic if its elements can be encoded using only specific simple trees which effectively rep- resent words. In contrast to the more general concept of computable structures and based on the strong closure properties of recognisability, automatic struc- turesprovidepleasantalgorithmicfeatures.Inparticular,theypossessdecidable first-order theories. Due to this latter fact, the concept of automatic structures gained a lot at- tentionwhichledtonoticeableprogress(cf.[1,6]).Automaticpresentationswere foundformanystructures,somestructureswhereshowntobetreebutnotword 2 Martin Huschenbett automatic,for instance Skolem arithmetic (N;×), whereas other structures,like therandomgraph,wereproventobe neitherwordnortreeautomatic.Forsome classesof structures it was evenpossible to characteriseits automatic members, for example an ordinal is word automatic respectively tree automatic precisely if it is less than ωω respectively ωωω. Certainextensionsof first-orderlogic were showntopreservedecidabilityofthecorrespondingtheory.Thequestionwhether two automatic structures areisomorphic turned outto be highly undecidable in generalaswellasforsomerestrictedclassesofstructures.Atthe sametime,the isomorphism problem for word automatic ordinals was proven to be decidable. Lastbutnotleast,thedifferentclassesofautomaticstructureswascharacterised by means of interpretations in universal structures. Due to the factthatwordautomaticityisa specialcaseoftree automaticity, thequestionnaturallyariseswhetheragiventreeautomaticstructureisalready word automatic. As far as we know, this problem was neither solved in general nor for any restricted class of structures. For that reason, we investigate the respectivequestionforscatteredlinearorderingsinthispaper.Actually,weprove the corresponding problem to be decidable and our main result is as follows: Theorem 1.1. Given a tree automatic presentation P of a scattered linear ordering L, it is decidable whether L is word automatic. In case L is word automatic, one can compute a word automatic presentation of L from P. Sinceeverywell-orderingisscattered,thisresultstillholdsifLisassumedtobe anordinal.The proofofTheorem1.1splits into three parts.First,we introduce the notion of slim tree languages and prove this property to be decidable (The- orem3.2). Second, we show that a slim domainis sufficient for a tree automatic structure to be word automatic (Theorem 4.1). Last, we demonstrate that this condition is also necessary in case of scattered linear orderings (Theorem 5.1). Altogether, Theorem 1.1 follows from the three mentioned theorems.1 2 Background In this section we recall the necessary notions of logic, automatic structures (cf.[1,6]),treeautomata(cf.[3]),andlinearorderings.Weagreethatthenatural numbers N include 0 and that [m,n]={m,m+1,...,n}⊆N for all m,n∈N. Logic. A (relational) signature τ = (R,ar) is a finite set R of relation symbols together with a map ar: R → N assigning to each R ∈ R its arity ar(R) ≥ 1. A τ-structure A= A;(RA) consists ofa setA=dom(A), its domain, and R∈R (cid:0) (cid:1) an ar(R)-ary relation RA ⊆ Aar(R) for each R ∈ R. First order logic FO over τ τ is defined as usual, including an equality predicate. A sentence is a formula without free variables.Writing ϕ(x¯) means that allfree variables of the formula ϕ are among the entries of the tuple x¯ =(x ,...,x ). The set ϕA is comprised 1 n of all a¯∈An satisfying A|=ϕ(a¯), where the latter is defined as usual. 1 ProofsofallseeminglyunprovenlemmasaswellastheinterpretationsfromSection4 can befound in the appendix. Title Suppressed Dueto Excessive Length 3 AutomaticStructures. Thesetofall(finite)words overanalphabetΣ isΣ⋆,the empty word is ε, and the length of w is |w|. Subsets of Σ⋆ are called languages and L⊆Σ⋆ is regular if it can be recognised by some (non-deterministic) finite automaton. Let26∈Σ beanewsymbolandΣ2 =Σ∪{2}.Forn≥1considerann-tuple w¯ = (w ,...,w ) ∈ (Σ⋆)n of words with w = a a ...a for all i ∈ [1,n]. 1 n i i,1 i,2 i,mi Let m = max{m ,...,m } and a = 2 for j ∈ [m +1,m]. The convolution 1 n i,j i of w¯ is the word ⊗w¯ = a¯ ...a¯ ∈ (Σn)⋆ with a¯ = (a ,...,a ) ∈ Σn 1 m 2 j 1,j n,j 2 for all j ∈ [1,m]. An n-ary relation R ⊆ (Σ⋆)n is automatic if the language ⊗R⊆(Σn)⋆, which is comprised of all ⊗w¯ with w¯ ∈R, is regular. 2 Aτ-structureA withdom(A)⊆Σ⋆ is (word) automatic ifdom(A)is regular and RA is automatic for all R ∈ R. A (word) automatic presentation of A is a tuple A ;(A ) of finite automata such that A recognises dom(A) dom R R∈R dom and A(cid:0) recognises ⊗R(cid:1)A. Abusing notation, we call any structure B which is R isomorphic to some word automatic structure A also (word) automatic. Tree Automata. A tree domain is a non-empty, finite, and prefix-closed subset D ⊆ {0,1}⋆ satisfying u0∈D iff u1∈ D for all u ∈D. A tree over Σ is a map t: D →Σ wheredom(t)=D is a treedomain.The setofalltrees isdenotedby T and its subsets are called (tree) languages. For some t∈T and u∈dom(t) Σ Σ the subtree of t rooted at u is the tree t↾u∈T defined by Σ dom(t↾u)={v ∈{0,1}⋆ |uv ∈dom(t)} and (t↾u)(v)=t(uv). A (deterministic bottom-up) tree automaton A = (Q,ι,δ,F) over Σ consists of a finite set Q of states, a start state function ι: Σ → Q, a transition function δ: Σ ×Q×Q → Q, and a set F ⊆ Q of accepting states. For each t ∈ T a Σ state A(t) ∈ Q is defined recursively by A(t) = ι t(ε) if dom(t) = {ε} and (cid:0) (cid:1) A(t) = δ t(ε),A(t↾0),A(t↾1) otherwise. The language recognised by A is the (cid:0) (cid:1) set of all t ∈ T with A(t) ∈ F. A language L ⊆ T is regular if it can be Σ Σ recognised by some tree automaton. The convolution of t¯= (t ,...,t ) ∈ (T )n is the tree ⊗t¯∈ T defined 1 n Σ Σ2n by dom(⊗t¯)=dom(t )∪···∪dom(t ) and (⊗t¯)(u)= t′(u),...,t′ (u) , where 1 n (cid:0) 1 n (cid:1) t′(u) = t (u) if u ∈ dom(t ) and t′(u) = 2 otherwise. A relation R ⊆ (T )n is i i i i Σ automatic if the language ⊗R⊆T is regular. Σ2n Tree automatic structures and tree automatic presentations are defined like in the word automatic case, but based on trees and tree automata. Linear Orderings. A linear ordering is a structure A = A;<A where <A is a strict linear orderrelationonA. The orderingA is scatte(cid:0)red if ((cid:1)Q;<)cannotbe embeddedintoA.Obviously,everywell-orderingisscattered.Foranytwolinear orderings A and B we define another linear ordering A·B by dom(A·B) = dom(A)×dom(B) and (a ,b )<A·B (a ,b ) iff either a <A a or a =a and 1 1 2 2 1 2 1 2 b <B b . Finally, if A can be embedded into B and A into B , then A ·A 1 2 1 1 2 2 1 2 can be embedded into B ·B . 1 2 4 Martin Huschenbett 3 Slim and Fat Tree Languages In this section, we introduce the notion of slim tree languages and show that it is decidable whether the languagerecognisedby a given tree automaton is slim. Definition 3.1. The thickness (cid:31)(t) of a tree t∈T is the maximal number of Σ nodes on any level, i.e., (cid:31)(t)=max dom(t)∩{0,1}ℓ ℓ≥0 ∈N. (cid:8)(cid:12) (cid:12)(cid:12) (cid:9) (cid:12) (cid:12)(cid:12) For every K ≥ 1 the set of all t ∈ T with (cid:31)(t) ≤ K is denoted by T . Σ Σ,K A tree language L⊆T is slim if there exists some K ≥1 such that L⊆T , Σ Σ,K otherwise L is fat. A tree automaton A is reduced if for every state q of A there is a tree t ∈ T Σ with A(t) = q. For every tree automaton A one can compute a reduced tree automaton which recognises the same languageand has no more states than A. Theorem 3.2. Given a reduced tree automaton A, it is decidable whether the tree language L recognised by A is slim or fat. If L is slim, then L ⊆ T , Σ,2n−1 where n is the number of states of A. FortherestofthissectionwefixareducedtreeautomatonA=(Q,ι,δ,F).The proofof Theorem3.2 essentially depends onan inspection ofthe directed graph G =(Q,E ) with A A (p,q)∈E iff ∃a∈Σ,r∈Q: δ(a,p,r)=q or δ(a,r,p)=q. (1) A Clearly,thisgraphiscomputablefromA.Thelemmabelowisshownbyapplying the idea of pumping to tree automata. Therein, the height h(t) of a tree t∈T Σ is the number h(t)=max{|u||u∈dom(t)}∈N. Lemma 3.3. For every q ∈Q the following are equivalent: (1) there are infinitely many t∈T satisfying A(t)=q, Σ (2) there is a tree t∈T satisfying h(t)≥n and A(t)=q, where n=|Q|, Σ (3) G contains a cycle from which q is reachable. A An edge (p,q) ∈ E is special if in the definition of E in Eq. (1) the state A A r ∈Q can be chosen such that it satisfies the conditions of Lemma 3.3 (for r in place of q). Since condition (3) is decidable, it is decidable whether an edge is special. The key idea for provingTheorem3.2 is stated by the following lemma: Lemma 3.4. The following are equivalent: (1) the tree language L recognised by A is fat, (2) there is a tree t∈L satisfying (cid:31)(t)>2n−1, where n=|Q|, (3) G contains a cycle including a special edge and from which F is reachable. A The proof of this lemma works similar to the one of Lemma 3.3. Since condition (3) is decidable given A as input, Theorem 3.2 follows. Title Suppressed Dueto Excessive Length 5 4 Slim Tree Automatic Structures Are Word Automatic This section is devoted to the proof of the following theorem: Theorem 4.1. Let A be a tree automatic structure such that dom(A) is slim. Then, A is already word automatic and one can compute a word automatic presentation of A from a tree automatic presentation of A. The idea of the proof is the following. Let K ≥1 be such that dom(A)⊆T . Σ,K We give an alphabet Σ and a one-to-one map C: T →Σ⋆, the encoding, Σ,K such that C(L) is regulbar for all regular L ⊆ T (Propositiobn 4.4) and C(R) Σ,K is automatic for all automatic relations R ⊆ (T )n (Proposition 4.6). Thus, Σ,K the structure C(A) is word automatic. A word automatic presentation of C(A) is computable since both propositions are effective and Theorem 3.2 allows for computingasuitableK.Althoughitispossibletoshowbothpropositionsusing automata, it is much more convenient to accomplish this by means of logic. 4.1 Monadic Second Order Logic Monadic second order logic MSO extends FO by set variables, which range τ τ over subsets of the domain and are denoted by capital letters, quantifiers for these variables,andthe formula“x∈X”(cf.[7]).Letτ =(R,ar)andτ′ be two signatures. An (MSO-)interpretation of a τ-structure A in a τ′-structure B is a pair hf,Ii comprised of a one-to-one map f: dom(A)→dom(B) and a tuple I = ∆;(ΦR)R∈R of MSOτ′-formulae with free FO-variables only such that (cid:0) (cid:1) f dom(A) = ∆B and f RA = ΦB for each R ∈ R. In fact, f induces an (cid:0) (cid:1) (cid:0) (cid:1) R isomorphism between A and I(B) = ∆B;(ΦB) . Replacing in an MSO - (cid:0) R R∈R(cid:1) τ formulaϕ(x¯)allsymbolsR∈RwithΦ andrelativisingquantifiersto ∆yields R an MSOτ′-formula ϕI(x¯) satisfying A|=ϕ(a¯) iff B|=ϕI f(a¯) for all a¯∈An. (cid:0) (cid:1) ForanalphabetΣ thesignatureWΣ consistsofonebinaryrelationsymbol≤ and a unary symbol P for each a ∈ Σ. Every word w = a a ...a ∈ Σ⋆ is a 1 2 |w| regarded as a WΣ-structure with domain dom(w)={1,...,|w|}, ≤w being the natural order on dom(w), and i ∈ Pw iff a = a. For fixed numbers m,r ∈ N, a i relations like x = y +m and x ≡ r(modm) are expressible in MSOWΣ. The languagedefined by anMSOWΣ-sentenceΦ isthe setofallw ∈Σ⋆ with w |=Φ. The signature TΣ is similarto WΣ but contains two binary symbols S and 0 S instead of ≤. Each tree t∈T is considered as a TΣ-structure with domain 1 Σ dom(t), (u,v)∈ St iff ud= v (d = 0,1), and u ∈ Pt iff t(u)= a. The language d a defined by some MSOTΣ-sentence Φ is the set of all t∈TΣ with t|=Φ. Thefollowingtheoremholdsforwordlanguagesaswellasfortreelanguages: Theorem 4.2 (cf. [7]). A language L is regular iff it is definable in MSO, and both conversions, from automata to formulae and vice versa, are effective. 6 Martin Huschenbett 4.2 The Encoding and Preservation of Regularity For the rest of this section fix the K ≥1 from above. The first a objective is to give the encoding C: T → Σ⋆, where $ is b c Σ,K a new symbol and Σ =Σ×{0,1}∪{$}. For abtree t ∈ T c b b a Σ,K of height m = h(t)bits encoding C(t) = σ0σ1...σm is made a c up of m+1 blocks σ ,...,σ ∈ΣK describing the individual 0 m Figure1. The levelsoft.Morespecifically,σ conbsistsofthelabelsoftheℓ-th ℓ level from left to right, each enriched by a bit stating whether tree tex. the corresponding node possesses children, and is padded up to length K by $ symbols. For example, the tree tex ∈ T{a,b,c} in Figure 1 on the right satisfies (cid:31)(t )=4 and is, under the assumption K =5, encoded by the word ex C(t )=ha,1i$$$$hb,1ihc,1i$$$hc,0ihb,1ihb,0iha,0i$ha,0ihc,0i$$$. ex Formally, for each ℓ∈[0,m] let u ,...,u be the lexicographic enumeration ℓ,1 ℓ,sℓ (w.r.t. 0<1) of dom(t)∩{0,1}ℓ. For r∈[1,s ] we let c =1 if u is an inner ℓ ℓ,r ℓ,r node, i.e. u {0,1}⊆dom(t), and c =0 if u is a leaf. Finally, we put ℓ,r ℓ,r ℓ,r σ =ht(u ),c iht(u ),c i...ht(u ),c i$K−sℓ. ℓ ℓ,1 ℓ,1 ℓ,2 ℓ,2 ℓ,sℓ ℓ,sℓ The main tool for studying the map C: T →Σ⋆ is the following lemma: Σ,K b Lemma 4.3. For all t ∈ T there is an MSO-interpretation hf ,I i of t Σ,K C C in C(t) such that I does not depend on t. C Proof. Observethatforeachinnernodeuoftthechildrenofuarethe(2s−1)-th and2s-thnode onthe nextlevel,where s is the number ofinner nodes fromleft uptouonitslevel.Formally,foraninnernodeu wehaveu d=u , ℓ,r ℓ,r ℓ+1,2s−1+d where d ∈ {0,1} and s = c +···+c . Based on this observation, one can ℓ,1 ℓ,r give an interpretation hf ,I i of t in C(t) such that f (u )=ℓ·K+r. ⊓⊔ C C C ℓ,r As a first consequence, we obtain t ∼= I (t) = I (t′) ∼= t′, and hence t = t′, C C for all t,t′ ∈ T with C(t) = C(t′). Thus, the encoding C is one-to-one. The Σ,K proof of Proposition 4.4 is mainly based on Lemma 4.3 and Lemma 4.5 below. Proposition 4.4. Let L ⊆ T be a regular language. Then, the language Σ,K C(L) ⊆ Σ⋆ is also regular and one can compute a finite automaton recognising C(L) fromb a tree automaton recognising L. Lemma 4.5. Let σ ∈ Σ⋆. There exists a tree t ∈ T with C(t) = σ iff Σ,K σ =σ σ ...σ for somebn≥0 and σ ,...,σ ∈ΣK satisfying (a) and (b): 0 1 n 0 n b (a) σℓ =αℓ,1...αℓ,sℓ$K−sℓ for some sℓ ≥1 and αℓ,1,...,αℓ,sℓ ∈Σ×{0,1} and for each ℓ∈[0,n], (b) s =1, s =2·(c +···+c ) for 0≤ℓ<n, and c +···+c =0, 0 ℓ+1 ℓ,1 ℓ,sℓ m,1 m,sm where α =ha ,c i. ℓ,r ℓ,r ℓ,r Title Suppressed Dueto Excessive Length 7 Proof. ToseethatC(t)hastherequiredshape,noticethat(b)mainlyreflectsthe relationshipbetweenthe numbersofnodesontwoadjacentlevels.Conversely,if σ ∈Σ⋆ is of the required shape, then there is a tree t∈T with t∼=I (σ) and Σ C it turbns out that (cid:31)(t)≤K and C(t)=σ.2 ⊓⊔ Proof (of Proposition 4.4). Let ΓC be an MSOWΣb-sentence which expresses the requirement on the shape of σ from Lemma 4.5. By Theorem 4.2, there is an MSOTΣ-sentence Φ defining L ⊆ TΣ,K. Then, the MSOWΣb-sentence ΓC ∧ΦIC defines C(L) and, again by Theorem 4.2, this language is regular. Finally, all employed constructions are effective. ⊓⊔ 4.3 Preservation of Automaticity The purpose of this subsection is to complete the proof of Theorem 4.1. Proposition 4.6. Let R ⊆ (T )n be an automatic relation. Then, the rela- Σ,K tion C(R) ⊆ (Σ⋆)n is also automatic and one can compute a finite automaton recognising ⊗Cb(R) from a tree automaton recognising ⊗R. Basically,thekeyideabehindtheproofisthesameasforProposition4.4though it is more involved. Let t¯= (t ,...,t ) ∈ (T )n. Due to cardinality reasons, 1 n Σ,K ⊗t¯is commonly not directly interpretable in ⊗C(t¯) but only in an n-fold copy of ⊗C(t¯). This is formalised by means of the one-to-one monoid morphism H: (Σn)⋆ →(Σn)⋆,α¯ ...α¯ 7→α¯n...α¯n . 2 2 1 m 1 m b b The interpretation of ⊗t¯in H ⊗C(t¯) embraces two aspects which are better (cid:0) (cid:1) considered separately. Thus, we define an intermediate structure ∐t¯which extends the disjoint union of the t ’s on domain dom(∐t¯)= {i}×dom(t ) i Si∈[1,n] i by a binary relation L∐t¯, relating all (i,u) and (j,v) with |u| = |v|, and unary relations Q∐t¯ = {i}×dom(t ) for each i ∈ [1,n]. Altogether, we give several i i interpretations whose formulae naturally do not depend on the specific choice of t¯. An overview of the whole setting is depicted in Figure 2. hfC,i, ICi C(ti) hf⊗,i,I⊗,ii ⊗(t1,...,tn) hf∐,I∐i ∐(t1,...,ti,...,tn) hfH,IHi H(cid:0)⊗(cid:0)C(t1),...,C(tn)(cid:1)(cid:1) Figure2. Interpretationsinvolved in provingProposition 4.6. The Interpretation hf ,I i. The main idea is to construct an MSO-formula ∐ ∐ E(x,y) with ∐t¯|= E (i,u),(j,v) iff u = v. To achieve this, consider for each (cid:0) (cid:1) (i,u) ∈ dom(∐t¯) the set Pre(i,u) of all (i,u′) ∈ dom(∐t¯) where u′ is a prefix of u. For (i,u),(j,v) ∈ dom(∐t¯) we have u = v iff |u| = |v| and for all (i,u′) ∈ 2 More details on this can be found in AppendixB.2. 8 Martin Huschenbett Pre(i,u) and (j,v′) ∈ Pre(j,v) with |u′| = |v′| > 0 the last symbols of u′ and v′ coincide.Since the setPre (i,u),X is definable in MSO, we can express this characterisationin MSO as w(cid:0)ell. Heav(cid:1)ily using the resulting formula E, one can construct an interpretationhf ,I i of ⊗t¯in ∐t¯such that f (u)=(i,u), where ∐ ∐ ∐ i is minimal with u∈dom(t ). i The Interpretations hf ,I i. For all i ∈[1,n] and w¯ ∈ (Σ⋆)n one can easily ⊗,i ⊗,i giveaninterpretationhf ,I iofw inH(⊗w¯)suchthatf b (p)=(p−1)·n+i. ⊗,i ⊗,i i ⊗,i The Interpretation hf ,I i. For i ∈ [1,n] let hf ,I i be the interpretation H H C,i C of t in C(t ) from Lemma 4.3. Since the f ’s have mutually disjoint images, i i ⊗,i the map f : dom(∐t¯)→dom H(⊗C(t¯)) with f (i,u)=f (f (u)) is one- H H ⊗,i C,i (cid:0) (cid:1) to-one. For (i,u)∈dom(∐t¯) we get |u|·K <f (u)≤ |u|+1 ·K and hence C,i (cid:0) (cid:1) |u|·K·n<f (i,u)≤ |u|+1 ·K·n. H (cid:0) (cid:1) Exploiting this observationfor the formula LIH and using IC and I⊗,i, one can constructformulaeI suchthathf ,I iisaninterpretationof∐t¯inH ⊗C(t¯) . H H H (cid:0) (cid:1) Proof (of Proposition 4.6). Let ΓH be an MSOWΣb2n-sentence defining the language H ⊗(Σ⋆)n ⊆(Σn)⋆. If Φ defines ⊗R, then 2 (cid:0) (cid:1) b b Γ ∧ ΓI⊗,i ∧(ΦI∐)IH H î∈[1,n] C definesH ⊗C(R) .SinceH isaone-to-onemonoidmorphism,⊗C(R)isregular (cid:0) (cid:1) as well. Finally, all employed constructions are effective. ⊓⊔ 5 Fat Tree Automatic Ordinals Are Not Word Automatic The goal of this section is to give the last missing piece for the proof of The- orem 1.1, namely the following theorem: Theorem 5.1. Let L be a tree automatic scattered linear ordering such that dom(L) is fat. Then, L is not word automatic. The theorem below states the necessary condition on word automatic linear orderings we use to show non-automaticity: Theorem 5.2 (Khoussainov,Rubin,Stephan[5]).IfLisawordautomatic linear ordering, then its FC-rank is finite. Actually, we do not need any details on the FC-rank (finite condensation rank) besides the factthat everyscatteredlinearorderingL, havingthe propertythat for each r ≥1 at least one linear ordering from N = A ·A ···A A ,...,A ∈ (N;<),(N;>) r (cid:8) 1 2 r (cid:12) 1 r (cid:8) (cid:9)(cid:9) (cid:12) can be embedded into L, has infinite FC-rank. The main idea of the proof is as follows: Title Suppressed Dueto Excessive Length 9 Lemma 5.3. Let L = (L;<) be a tree automatic scattered linear ordering, (A;A )an automatic presentation of L,n thenumberof states ofA, andr ≥1. < If there exists some tree t ∈ L with (cid:31)(t) ≥ r·2n, then there are infinite linear orderings A ,...,A such that A ·A ···A can be embedded into L. 1 r 1 2 r For any linear ordering A and all a ,a ∈ dom(A) we define cmp (a ,a ) ∈ 1 2 A 1 2 {−1,0,1}tobe−1ifa <A a ,0ifa =a ,and1ifa <A a .Tosimplifynota- 1 2 1 2 2 1 tion,weput s,t =A ⊗(s,t) foralls,t∈T .Moreover,weassumew.l.o.g. < < Σ that from s,Jt Kone can(cid:0)deduce(cid:1)whether s = t holds true. Then, cmp (s,t) is < L determinedJbyK s,t for all s,t ∈ L, i.e., there is a map f from the state set < of A to {−1,0J,1}Ksuch that cmp (s,t)=f s,t for all s,t∈L. < L < (cid:0)J K (cid:1) Proof. LetT∈Lbeatreeandℓ≥nsuchthat dom(T)∩{0,1}ℓ ≥r·2n.Thus, (cid:12) (cid:12) thereexistatleastr mutually distinctu∈dom((cid:12)T)∩{0,1}ℓ−n fo(cid:12)rwhichthereis a v ∈ {0,1}n with uv ∈ dom(T), say u ,...,u . For t¯= (t ,...,t ) ∈ (T )r let 1 r 1 r Σ T[t¯]∈T be thetreeobtainedfromTbyreplacingforeachi∈[1,r]thesubtree Σ rootedatu witht .Then,A T[t¯] isdeterminedbytherstatesA(t ),...,A(t ) i i 1 r (cid:0) (cid:1) for all t¯∈(T )r. Moreover,for s¯∈(T )r the tree ⊗ T[s¯],T[t¯] is obtainedfrom Σ Σ ⊗(T,T) by replacing for each i ∈ [1,r] the subtree(cid:0)rooted at(cid:1)u with ⊗(s ,t ). i i i fCoornaslelqs¯u,et¯n∈tly(,TqT)r[s¯.],T[t¯]y< is determined by the r states Js1,t1K<,...,Jsr,trK< Σ Observe that h(T↾u ) ≥ n for each i ∈ [1,r]. Therefore, by Lemma 3.3 and i Ramsey’s theorem for infinite, undirected, finitely coloured graphs, there exists an infinite set A ⊆T of trees t∈T with A(t)=A(T↾u ) such that i Σ Σ i c(s,t)= s,s , t,t , s,t , t,s < < < < (cid:8)J K J K J K J K (cid:9) isthesamesetQ foralldistincts,t∈A .ItturnsoutthatQ hasexactlythree i i i elements and s,s = t,t for all s,t∈A . < < i Now, put AJ =KA ×J···K×A . For each t¯∈ A we have A T[t¯] = A(T) and 1 r (cid:0) (cid:1) henceT[t¯]∈L.WedefinealinearorderingA= A;<A bys¯<A t¯iffT[s¯]<T[t¯]. By definition, A can be embedded into L. (cid:0) (cid:1) For i ∈ [1,r], a¯ ∈ A, and t ∈ A we let a¯ ∈ A be the tuple a¯ with i i/t the i-th component replaced by t. Then, for all a¯,¯b and s,t ∈ A we obtain i qT[a¯ ],T[a¯ ]y = qT[¯b ],T[¯b ]y and hence a <A a iff b <A b . i/s i/t < i/s i/t < i/s i/t i/s i/t Thus, defining a linear ordering Ai = (cid:0)Ai;<Ai(cid:1) by s <Ai t iff a¯i/s <A a¯i/t is independentfromthespecificchoiceofa¯∈A.Clearly,cmp (s,t)isdetermined A i by s,t for all s,t ∈ A . Since Q contains exactly three elements, s,t is < i i < deteJrmiKned by cmp (s,t) for all s,t ∈ A as well. Hence, the linear oJrdeKrings A i A and A ,...,A satiisfy the condition of Lemma 5.4 below and consequently 1 r A ···A can be embedded into L. ⊓⊔ π(1) π(r) Lemma 5.4. Let A and A ,...,A be infinite linear orderings with dom(A) = 1 r dom(A )×···×dom(A ) and satisfying the following two conditions: 1 r (1) cmp (a¯,¯b)isdeterminedbycmp (a ,b ),...,cmp (a ,b )foralla¯,¯b∈A, A A 1 1 A r r (2) ifa¯,¯b∈Adifferonlyinthei-thco1mponent,thencmpr (a¯,¯b)=cmp (a ,b ). A A i i i 10 Martin Huschenbett Then, there exists a permutation π of {1,...,r} such that A is isomorphic to A ·A ···A . π(1) π(2) π(r) Finally, we are in a position to prove Theorem 5.1. Proof (of Theorem 5.1). Let (A;A ) be an automatic presentation of L and n < thenumberofstatesofA.Sincedom(L)isfat,foranyr ≥1thereisat∈dom(L) with(cid:31)(t)≥r·2n.LetA ,...,A betheinfinitelinearorderingsfromLemma5.3. 1 r For each i ∈ [1,r] some B ∈ (N;<),(N;>) can be embedded into A . Then, i i B ·B ···B ∈N canbeemb(cid:8)eddedintoA (cid:9)·A ···A andconsequentlyintoL. 1 2 r r 1 2 r Hence, L has infinite FC-rank and is, by Theorem 5.2, not wordautomatic. ⊓⊔ 6 Conclusions Altogether,weprovedthatisdecidablewhetheragiventreeautomaticscattered linear ordering is already word automatic. Taking a closer look at the proof reveals that the problem is solvable nondeterministically in logarithmic space, provided the tree automaton recognising the domain is reduced. The restriction to scattered linear orderings naturally rises the question whether this result holds true for general linear orderings. Unfortunately, this problem cannot be solved by means of our technique since the ordering (Q;<) oftherationalsadmitsawordautomaticaswellasafattreeautomaticpresent- ation. As the Boolean algebra of finite and co-finite subsets of N shares this feature, the same pertains to an analogue of Theorem 1.1 for Boolean algeb- ras. In spite of that, we suggest trying to apply the technique to other classes of structures, such as groups, for which a necessary condition on its automatic members is known. Finally, Theorem 1.1 provides a decidable characterisation of all tree automatic ordinalsα≥ωω. Finding sucha characterisationfor eachωωk with k ∈N possibly turns out to be the main ingredient for showing that the isomorphism problem for tree automatic ordinals is decidable. References 1. Bárány, V., Grädel, E., Rubin, S.: Automata-based presentations of infinite structures. In Esparza, J., Michaux, C., Steinhorn, C., eds.: Finite and Algorithmic Model Theory. Cambridge UniversityPress (2011) 1–76 2. Blumensath, A.: Automatic structures. Diploma thesis, RWTH Aachen (1999) 3. Gécseg, F., Steinby, M.: Tree languages. In Rozenberg, G., Salomaa, A., eds.: Handbook of Formal Languages. Volume3. Springer(1997) 1–68 4. Khoussainov,B.,Nerode,A.: Automaticpresentationsofstructures. In:LCC1994. Volume 960 of LNCS,Springer (1995) 367–392 5. Khoussainov, B., Rubin, S., Stephan, F.: On automatic partial orders. In: LICS 2003. (2003) 168–177 6. Rubin,S.: Automata presentingstructures: A surveyof thefinitestring case. Bul- letin of SymbolicLogic 14(2) (2008) 169–209 7. Thomas,W.: Languages,automata,andlogic. InRozenberg,G.,Salomaa,A.,eds.: Handbook of Formal Languages. Volume3. Springer(1997) 384–455