Behavioural Differential Equations and Coinduction for Binary Trees

Transcription

1 Behviourl Differentil Equtions nd Coinduction for Binry Trees Alexndr Silv nd Jn Rutten,2 Centrum voor Wiskunde en Informtic (CWI) 2 Vrije Universiteit Amsterdm (VUA) {ms,jnr}@cwi.nl Abstrct. We study the set T A of infinite binry trees with nodes lbelled in semiring A from colgebric perspective. We present coinductive definition nd proof principles bsed on the fct tht T A crries finl colgebr structure. By viewing trees s forml power series, we develop clculus where definitions re presented s behviourl differentil equtions. We present generl formt for these equtions tht gurntees the existence nd uniqueness of solutions. Although techniclly not very difficult, the resulting frmework hs surprisingly nice pplictions, which is illustrted by vrious concrete exmples. Introduction Infinite dt structures re often used to model problems nd computing solutions for them. Therefore, resoning tools for such structures hve become more nd more relevnt. Colgebric techniques turned out to be suited for proving nd deriving properties of infinite systems. In [6], coinductive clculus of forml power series ws developed. In close nlogy to clssicl nlysis, the definitions were presented s behviourl differentil equtions nd properties were proved in clcultionl (nd very nturl) wy. This pproch hs shown to be quite effective for resoning bout strems [6,7] nd it seems worthwhile to explore its effectiveness for other dt structures s well. In this pper, we shll tke colgebric perspective on clssicl dt structure infinite binry trees, nd develop similr clculus, using the fct tht binry trees re prticulr cse of forml power series. The contributions of the present pper re: coinductive clculus, bsed on the notion of derivtive, to define nd to reson bout trees nd functions on trees; set of illustrtive exmples nd properties tht show the usefulness nd expressiveness of such clculus; the formultion of generl formt tht gurntees the existence nd uniqueness of solutions of behviourl differentil equtions. Infinite trees rise in severl forms in other res. Forml tree series (functions from trees to n rbitrry semiring) hve been studied in [3], closely relted to distributive Σ-lgebrs. The work presented in this pper is completely different since we re concerned with infinite binry trees nd not with forml series over trees. In [5], infinite trees pper s generlistions of infinite words nd n extensive study of tree utomt nd topologicl spects of trees is mde. We hve not yet ddressed the reltion of our work with utomt theory. Here we emphsize coinductive definition nd proof principles for defining nd resoning bout (functions on) trees. At the end of the pper, in Section 6, the novelty of our pproch is discussed further. Also severl directions for further pplictions re mentioned there. 2 Trees nd coinduction We introduce the set T A of infinite node-lbelled binry trees, show tht T A stisfies coinduction proof principle nd illustrte its usefulness. Prtilly supported by the Fundção pr Ciênci e Tecnologi, Portugl, under grnt number SFRH/BD/27482/26.

2 The set T A of infinite node-lbelled binry trees, where to ech node is ssigned vlue in A, is the finl colgebr for the functor F X = X A X nd cn be formlly defined by: T A = {t t : {L, R} A} The set T A crries finl colgebr structure consisting of the following function: l, i, r : T A T A A T A t λw.t(lw), t(ε), λw.t(rw) where l nd r return the left nd right subtrees, respectively, nd i gives the lbel of the root node of the tree. Here, ε denotes the empty word nd, for b {L, R}, bw denotes the word resulting from prefixing the word w with the letter b. These definitions of the set T A nd the respective colgebr mp my not seem obvious. The follow resoning justifies its correctness: It is well known from the literture [4] tht the finl system for the functor G(X) = A X B is (A B, π), where π : A B A (A B ) B π(φ) = φ(ε), λb v. φ(bv) The functor F is isomorphic to H(X) = A X 2. Therefore, the set A 2 is the finl colgebr for the functor F. Considering 2 = {L, R} we cn derive the definition of l, i, r from the one presented bove for π. The fct tht T A is finl colgebr mens tht for ny rbitrry colgebr lt, o, rt : U U A U, there exists unique f : U T A, such tht the following digrm commutes: lt,o,rt U U A U!f f id A f T A l,i,r T A A T A The existence prt of this sttement gives us coinductive definition principle. Every triplet of functions lt : U U, o : U A nd rt : U U defines function h : U T A, such tht: i(h(x)) = o(x) l(h(x)) = h(lt(x)) r(h(x)) = h(rt(x)) We will see more generl formultion of this principle in section 3, where the right hnd side of the bove equtions will be more generl. Tking A = R we present the definition of the elementwise sum s n exmple. d b e f c g + u s v r w t By the definition principle presented bove, tking o( σ, τ ) = i(σ) + i(τ), lt( σ, τ ) = l(σ), l(τ) nd rt( σ, τ ) = r(σ), r(τ) there is unique function + : T R T R T R stisfying: i(σ + τ) = i(σ) + i(τ) l(σ + τ) = l(σ) + l(τ) r(σ + τ) = r(σ) + r(τ) Note tht in the first eqution bove we re using + to represent both the sum of trees nd the sum of rel numbers. Now tht we hve explined the forml definition for the set T A nd how one cn uniquely define functions into T A, nother importnt question is still to be nswered: how do we prove equlity on T A? In order to prove tht two trees σ nd τ re equl it is necessry nd sufficient to prove x = d+u b+s e+v +r w {L,R} σ(w) = τ(w) () The use of induction on w (prove tht σ(ε) = τ(ε) nd tht whenever σ(w) = τ(w) holds then σ(w) = τ(w) lso holds, for {L, R}) clerly is correct method to estblish the vlidity of (). However, we will often encounter exmples where there is not generl formul for σ(w) nd τ(w). Insted, we tke colgebric perspective on T A nd use the coinduction proof principle in order to estblish equlities. This proof principle is bsed on f+w c+t g+x

3 the notion of bisimultion. A bisimultion on T A is reltion S T A T A such tht, for ll σ nd τ in T A, (σ, τ) S σ(ε) = τ(ε) (l(σ), l(τ)) S (r(σ), r(τ)) S We will write σ τ whenever there exists bisimultion tht contins (σ, τ). The reltion, clled the bisimilrity reltion, is the union of ll bisimultions (one cn esily check tht the union of bisimultion is itself bisimultion). The following theorem expresses the proof principle mentioned bove. Theorem (Coinduction). For ll trees σ nd τ in T A, if σ τ then σ = τ. Proof. Consider two trees σ nd τ in T A nd let S T A T A be bisimultion reltion which contins the pir (σ, τ). The equlity σ(w) = τ(w) now follows by induction on the length of w. We hve tht σ(ε) = τ(ε), becuse S is bisimultion. If w = Lw, then σ(lw ) = l(σ)(w ) (Definition of l) = l(τ)(w ) (S is bisimultion nd induction hypothesis) = τ(lw ) (Definition of l) Similrly, if w = Rw, then σ(rw ) = r(σ)(w ) = r(τ)(w ) = τ(rw ). Therefore, for ll w {L, R}, σ(w) = τ(w). This proves σ = τ. Thus, in order to prove tht two trees re equl, it is sufficient to show tht they re bisimilr. We shll see severl exmples of proofs by coinduction below. As first simple exmple, let us prove tht the pointwise sum for trees of rel numbers defined before is commuttive. Let S = { σ + τ, τ + σ σ, τ T R }. Since i(σ + τ) = i(σ) + i(τ) = i(τ + σ) nd l(σ + τ) = l(σ) + l(τ) S l(τ) + l(σ) = l(τ + σ) r(σ + τ) = r(σ) + r(τ) S r(τ) + r(σ) = r(τ + σ) for ny σ nd τ, S is bisimultion reltion on T R. The commuttivity property follows by coinduction. 3 Behviourl differentil equtions In this section, we shll view trees s forml power series. Following [6], coinductive definitions of opertors into T A nd constnt trees then tke the shpe of so-clled behviourl differentil equtions. We shll prove theorem gurnteeing the existence of unique solution for lrge fmily of systems of behviourl differentil equtions. Forml power series re functions σ : X k from the set of words over n lphbet X to semiring k. If A is semiring, T A, s defined in section 2, is the set of ll forml power series over the lphbet {L, R} with coefficients in A. In ccordnce with the generl notion of derivtive of forml power series [6] we shll write σ L for l(σ) nd σ R for r(σ). We will often refer to σ L, σ R nd σ(ε) s the left derivtive, right derivtive nd initil vlue of σ. Following [6], we will develop coinductive clculus of infinite binry trees. From now on coinductive definitions will hve the shpe of behviourl differentil equtions. Let us illustrte this pproch by simple exmple the coinductive definition of tree, clled one, decorted with s in every node. A forml definition of this tree consists the following behviourl differentil equtions: one L = one one(ε) = one R = one

4 The fct tht there exists unique tree tht stisfies the bove equtions will follow from the theorem below, which presents generl formt for behviourl differentil equtions gurnteeing the existence nd uniqueness of solutions. Behviourl differentil equtions will be used not just to define single constnt trees but lso functions on trees. We shll see exmples below. Before we present the min result of this section, we need one more definition. We wnt to be ble to view ny element n A s tree (which we will denote by [n]): n The tree [n] is formlly defined s [n](ε) = n [n](w) = w ε Next we present syntx describing the formt of behviourl differentil equtions tht we will consider. Let Σ be set of function symbols, ech with n rity r(f) for f Σ. (As usul we cll f constnt if r(f) =.) Let X = {x, x 2,...} be set of (syntctic) vribles, nd let X L = {x L x X}, X R = {x R x X}, [X(ε)] = {[x(ε)] x X} nd X(ε) = {x(ε) x X} be sets of nottionl vrints of them. The vribles x X will ply the role of plce holders for trees τ T A. Vribles x L, x R, nd [x(ε)] will then ct s plce holders for the corresponding trees τ L, τ R nd [τ(ε)] in T A, while x(ε) (without the squre brckets) will correspond to τ s initil vlue τ(ε) A. We cll behviourl differentil eqution for function symbol f Σ with rity r = r(f) well-formed if it is of the form f ( x,..., x r ) L = t ( f ( x f ( x,..., x r ) R = t,..., x r ) ) (ε) = c(x (ε),..., x r(ε)) 2 where c : A r A is given function, nd where t nd t 2 re terms built out of function symbols in Σ nd vribles in {x,..., x r} nd their corresponding nottionl vrints in X L, X R nd [X(ε)]. A well-formed system of equtions for Σ will then consist of one wellformed eqution for ech f Σ. A solution of such system of equtions is set of tree functions Σ = { f : (TA) r T A f Σ} stisfying, for ll f Σ with rity r nd for ll τ,..., τ r T A, f(τ,..., τ r) (ε) = c(τ (ε),..., τ r(ε)) nd f(τ,..., τ r) L = t nd f(τ,..., τ r) where the tree t T A (nd similrly for t 2) is obtined from the term t by replcing (ll occurrences of) x i by τ i, (x i) L by (τ i) L, (x i) R by (τ i) R, nd [x i(ε)] by [τ i(ε)], for ll i =,..., r, nd ll function symbols g Σ by their corresponding function g. Theorem 2. Let Σ be set of function symbols. Every well-formed system of behviourl differentil equtions for Σ hs precisely one solution of tree functions Σ. Proof. Plese see ppendix A. Let us illustrte the generlity of this theorem by mentioning few exmples of systems of differentil equtions tht stisfy the formt bove. As first exmple, tke Σ = {one} consisting of single constnt symbol (with rity ) nd X =. We observe tht the differentil equtions for one mentioned t the beginning of this section stisfy the formt of the theorem. For second exmple, let Σ = {+, } with rities r(+) = r( ) = 2 nd let X = {σ, τ}. Consider the following equtions: (σ + τ) L = σ L + τ L (σ + τ) R = σ R + τ R (σ + τ)(ε) = σ(ε) + τ(ε) R = t 2 differentil equtions initil vlue (σ τ) L = (σ L τ) + ([σ(ε)] τ L) (σ τ)(ε) = σ(ε) τ(ε) (σ τ) R = (σ R τ) + ([σ(ε)] τ R)

5 These equtions define the opertions of sum nd convolution product of trees, to be further discussed in Section 4. Note tht the right-hnd side of the eqution for (σ τ) L (nd similrly for (σ τ) R) is good illustrtion of the generl formt: it is built from the functions + nd, pplied to ( subset of) the vribles on the left (τ), their derivtives (σ L nd τ L), nd their initil vlues viewed s trees ([σ(ε)]). Clerly there re mny interesting interesting instnces of well-formed differentil equtions. Note, however, tht the formt does impose certin restrictions. The min point is tht in the right-hnd sides of the equtions, only single L nd R derivtives re llowed. The following is n exmple of system of equtions tht is not well-formed nd tht does not hve unique solution. Let Σ = {f}, with rity r(f) =, nd let X = {σ}. The equtions for f re f(σ) L = f(f(σ LL)) f(σ)(ε) = σ(ε) f(σ) R = [] Both g(σ) = [σ(ε)] + (L [σ LL(ε)]) nd h(σ) = [σ(ε)] + (L [σ LL(ε)] + L 2 ( L) ) re solutions. All the exmples of systems of behviourl differentil equtions tht will pper in the rest of this document fit into the formt of Theorem 2. Therefore, we will omit proofs of the existence nd uniqueness of solutions for those systems. In the next section we will define opertors on trees, bsed on some generl opertors on forml power series [6]. 4 Tree clculus In this section, we present opertors on trees, nmely sum, convolution product nd inverse, nd stte some elementry properties, which we will prove using coinduction. The sum of two trees is defined s the unique opertor stisfying: (σ + τ) L = σ L + τ L (σ + τ) R = σ R + τ R (σ + τ)(ε) = σ(ε) + τ(ε) Note tht this is generlistion of the sum on trees of rel numbers defined in section 2 nd tht gin we re overloding the use of + to represent both sum on trees nd sum on the elements of the semiring. Sum stisfies some desired properties, esily proved by coinduction, such s commuttivity or ssocitivity: Theorem 3. For ll σ, τ nd ρ in T A, σ + = σ, σ +τ = τ +σ nd σ +(τ +ρ) = (σ +τ)+ρ. Here, we re using s shorthnd for []. We shll use this convention (for ll n A) throughout this document. We define the convolution product of two trees s the unique opertion stisfying: (σ τ) L = (σ L τ) + (σ(ε) τ L) (σ τ)(ε) = σ(ε) τ(ε) (σ τ) R = (σ R τ) + (σ(ε) τ R) Note tht in the bove definition we use for representing both multipliction on trees nd on the elements of the semiring. Following the convention mentioned bove σ(ε) τ L nd σ(ε) τ R re shorthnds for [σ(ε)] τ L nd [σ(ε)] τ R. We shll lso use the stndrd convention of writing στ for σ τ. The generl formul to compute the vlue of σ ccording to pth given by the word w {L, R} is: (σ τ)(w) = X σ(u) τ(v) w=u v where denotes word conctention. To give the reder some intuition bout this opertion we will give concrete exmple. Tke A to be the Boolen semiring B = {, }, with opertions + = nd =. Then, T A corresponds to the lnguges over two letter lphbet, nd in this cse the tree product opertor coincides with lnguge conctention. The following theorem sttes some fmilir properties of this opertor.

6 Theorem 4. For ll σ, τ, ρ in T A nd, b in A, σ = σ = σ, σ = σ =, σ (τ + ρ) = (σ τ) + (σ ρ), σ (τ ρ) = (σ τ) ρ nd [] [b] = [ b]. Proof. An exercise in coinduction. In [7], these properties re proved for strems. Note tht the convolution product is not commuttive. Before we present the inverse opertion, let us introduce two (very useful) constnts, which we shll cll left constnt L nd right constnt R. They will hve n importnt role in the tree clculus tht we re bout to develop nd will turn out to hve interesting properties when intercting with the product opertion. The left constnt L is tree with s in every node except in the root of the left brnch where it hs : L = It is formlly defined s L(w) = if w = L L(w) = otherwise Similrly, the right constnt R is only different from in the root of its right brnch: R = nd is defined s R(w) = if w = R R(w) = otherwise These constnts hve interesting properties when multiplied by n rbitrry tree. L σ produces tree whose root nd right subtrees re null nd the left brnch is σ: X d b c = e f g d f b c Dully, R σ produces tree whose root nd left subtrees re null nd the right brnch is σ: X s p q r = t u As before, if we see L nd σ s lnguges nd the product s conctention, we cn gin some intuition on the mening of this opertion. L σ will prefix every word of σ with the letter L, mening tht no word strting by R will be n element of L σ, nd thus, L σ hs null right brnch. Similr for R σ. As we pointed out before, the product opertion is not commuttive. For exmple, σ L L σ nd σ R R σ. In fct, multiplying tree σ on the right with L or R is interesting in itself. For instnce, σ L stisfies j σ(u) w = ul (σ L)(w) = otherwise which corresponds to the following trnsformtion: b c X = d e f g Similrly, σ R produces the following tree: b c X = d e f g Agin, if you interpret these opertions in the lnguge setting, wht is being constructed is the lnguge tht hs ll words of the form ul nd ur, respectively, such tht σ(u). v b d d e e b e s q g c t f f p u r v g c g

7 We define the inverse of tree s the unique opertor stisfying: (σ ) L = σ(ε) ( σ L (σ )) (σ ) R = σ(ε) ( σ R (σ )) σ (ε) = σ(ε) We re using σ L nd σ R s shorthnds for [ ] σ L nd [ ] σ R, respectively. In this definition, we need to require A to be ring, in order to hve dditive inverses. Moreover, the tree σ is supposed to hve lso multiplictive inverse for its initil vlue. The inverse of tree hs the usul properties: Theorem 5. For ll σ nd τ in T A: σ is the unique tree s.t. σ σ = (2) (σ τ) = τ σ (3) Proof. For the existence prt of (2), note tht. (σ σ )(ε) = σ(ε) σ(ε) = 2. (σ σ ) L = (σ L σ ) + (σ(ε) (σ(ε) ( σ L σ ))) = 3. (σ σ ) R = (σ R σ ) + (σ(ε) (σ(ε) ( σ R σ ))) = So, by uniqueness (using the behviourl differentil equtions tht define ) we hve proved tht σ σ =. Now, for the uniqueness prt of (2), suppose tht there is tree τ such tht σ τ =. We shll prove tht τ = σ. Note tht from the equlity σ τ = we derive tht. τ(ε) = σ(ε) 2. τ L = σ(ε) ( σ L τ) 3. τ R = σ(ε) ( σ R τ) Thus, by uniqueness of solutions for systems of behviourl differentil equtions, τ = σ. For (3), note tht (σ τ) τ σ = σ (τ τ ) σ =. Therefore, using the uniqueness property of (2), (σ τ) = τ σ. 5 Applictions of tree clculus We will illustrte the usefulness of our clculus by looking t series of interesting exmples. In order to compute closed formule for trees we will be using the following identity: σ TA σ = σ(ε) + (L σ L) + (R σ R) (4) which cn be esily proved by coinduction. We will now show how to use this identity to construct the closed formul for tree. Recll our first system of behviourl differentil equtions: one L = one one(ε) = one R = one There we sw tht the unique solution for this system ws the tree with s in every node. Alterntively, we cn compute the solution using (4) s follows. one = one(ε) + (L one L) + (R one R) one = + (L one) + (R one) ( L R)one = one = ( L R) Therefore, one cn be represented by the (very compct) closed formul 3 ( L R). Let us see few more exmples. From now on we will work with A = R, where we hve the extr property: [n] σ = σ [n], for ll n R nd σ T R. 3 Note the similrity of this closed formul with the one obtined for the strem (,,...) in [7]: ( X).

8 The tree where every node t level k is lbelled with the vlue 2 k, clled pow, is defined by the following system: We proceed s before, pplying (4): pow L = 2 pow pow(ε) = pow R = 2 pow pow = pow(ε) + (L pow L) + (R pow R) pow = + (2L pow) + (2R pow) ( 2L 2R)pow = pow = ( 2L 2R) which gives us nice closed formul for pow 4. The tree with the nturl numbers is represented by the following system of differentil equtions: Applying identity (4): nt L = nt + pow nt R = nt + (2 pow) nt(ε) = nt = nt(ε) + (L nt L) + (R nt R) nt = + (L (nt + pow)) + (R (nt + 2pow)) ( L R)nt = + L( 2L 2R) + 2R( 2L 2R) ( L R)nt = ( L) ( 2L 2R) nt = ( L R) ( L) ( 2L 2R) The Thue-Morse sequence [] cn be obtined by tking the prities of the counts of s in the binry representtion of non-negtive integers. Alterntively, it cn be defined by the repeted ppliction of the substitution mp { ; }:... We cn encode this substitution mp in binry tree, clled thue, which t ech level k will hve the first 2 k digits of the Thue-Morse sequence: In this exmple, we tke for A the Boolen ring 2 = {, } (where + = ). The following system of differentil equtions defines thue: thue L = thue thue R = thue + one thue(ε) = 4 Agin, there is strong similrity with strems: the closed formul for the strem (, 2, 4, 8,...) is ( 2X)

9 Note tht thue + one equls the (elementwise) complement of thue. Applying (4) to thue, we clculte: thue = (L thue) + (R (thue + one)) ( L R) thue = R one thue = ( L R) R one which then leds to the following pretty formul for thue: thue = one R one Let us present nother exmple substitution opertion, which given two trees σ nd τ, replces the left subtree of σ by τ. σ(ε), τ σ σ L R = subst ( ) τ It is esy to see tht the equtions tht define this opertion re: Then, we pply (4) nd we reson: σ(ε) subst(σ, τ) L = τ subst(σ, τ)(ε) = σ(ε) subst(σ, τ) R = σ R subst(σ, τ) = σ(ε) + (L τ) + (R σ R) subst(σ, τ) = σ (L σ L) + (L τ) subst(σ, τ) = σ L(σ L τ) Note tht in the second step, we pplied identity (4) to σ. Moreover, remrk tht the finl closed formul for subst(σ, τ) gives us the lgorithm to compute the substitution: σ(ε), τ σ σ L R = subst ( ) τ σ(ε) σ R σ R - σ L σ(ε) σ R + τ We cn now wonder how to define more generl substitution opertion tht hs n rbitrry pth P {L, R} + s n extr rgument nd replces the subtree of σ given by this pth by τ. It seems obvious to define it s subst(σ, τ, P ) = σ P (σ P τ) where, in the right hnd side, P = 2... n is interpreted s 2... n nd the derivtive σ P is defined s j σδ P = δ σ P = (σ δ ) P P = δ.p with δ being either L or R. Let us check tht our intuition is correct. First, we present the definition for this opertion: differentil equtions 8 initil vlue < τ P = δ subst(σ, τ, P ) δ = subst(σ δ, τ, P ) P = δ.p subst(σ, τ, P )(ε) = σ(ε) : σ δ P = δ.p where δ δ. Now, observe tht R = { subst(σ, τ, P ), σ P (σ P τ) σ, τ T R, P {L, R} + } { σ, σ σ T R } is bisimultion reltion becuse:. (σ P (σ P τ))(ε) = σ(ε) = subst(σ, τ, P )(ε)

10 2. For δ {L, R}, (σ P (σ P τ)) δ = σ δ P δ (σ P τ) 8 < τ P = δ = σ δ P ((σ δ ) P τ) P = δ.p : σ δ P = δ.p 8 < τ P = δ R subst(σ δ, τ, P ) P = δ.p : σ δ P = δ.p = subst(σ, τ, P ) δ Therefore, by Theorem, subst(σ, τ, P ) = σ P (σ P τ). Using this formul we cn now prove properties bout this opertion. For instnce, one would expect tht subst(σ, σ P, P ) = σ nd lso subst(subst(σ, τ, P ), σ P, P ) = σ. The first equlity follows esily: subst(σ, σ P, P ) = σ P (σ P σ P ) = σ. For the second one we hve: subst(subst(σ, τ, P ), σ P, P ) = subst(σ P (σ P τ), σ P, P ) (Definition of subst) = σ P (σ P τ) P ((σ P (σ P τ)) P σ P ) (Definition of subst) = σ P (σ P τ) P (τ σ P ) ((σ P (σ P τ)) P = σ P σ P + τ = τ) = σ Remrk tht this opertion is stndrd exmple in introductory courses on lgorithms nd dt structures. It is often presented either s recursive expression (very much in the style of our differentil equtions) or s contrived itertive procedure. This exmple shows tht our compct formule constitute cler wy of presenting lgorithms nd tht they cn be used to eliminte recursion. Moreover, the differentil equtions re directly implementble lgorithms (in functionl progrmming) nd our clculus provides systemtic wy of resoning bout such progrms. 6 Discussion We hve modelled binry trees s forml power series nd, using the fct tht the ltter constitute finl colgebr, this hs enbled us to pply some colgebric resoning. Techniclly, none of this is very difficult. Rther, it is n ppliction of well known colgebric insights. As is the cse with mny of such pplictions, it hs the flvour of n exercise. At the sme time, the result contins severl new elements tht hve surprised us. Although techniclly Theorem 2 is n esy extension of similr such theorem for strems, the resulting formt for differentil equtions for trees is surprisingly generl nd useful. It hs llowed us to define vrious non-trivil trees by mens of simple differentil equtions, nd to compute rther plesnt closed formule for them. We hve lso illustrted tht bsed on this, coinduction is convenient proof method for trees. As n ppliction, ll of this is new, to the best of our knowledge. (Forml tree series, which hve been studied extensively, my seem to be closely relted but re not: here we re deling with differentil equtions tht chrcterise single trees.) In ddition to the illustrtions of the present differentil clculus for trees, we see vrious directions for further pplictions: (i) The connection with (vrious types of) utomt nd the finl colgebr T A of binry trees needs further study. For instnce, every Moore utomton with input in 2 = {L, R} nd output in A hs miniml representtion in T A. Detils of these utomt (nd on weighted vrints of them) re not complicted but hve not been worked out here becuse of lck of spce. (ii) The closed formul tht we hve obtined for the (binry tree representing the) Thue-Morse sequence suggests possible use of coinduction nd differentil equtions in the re of utomtic sequences [2]. Typiclly, utomtic sequences re represented by utomt. The present clculus seems n interesting lterntive, in which properties such s lgebricity of sequences cn be derived from the tree differentil equtions tht define them. (iii) Finlly, the closed formule tht we obtin for tree substitution suggest mny further pplictions of our tree clculus to (functionl) progrms on trees, including the nlysis of their complexity.

11 References. J.-P. Allouche nd J. Shllit. The ubiquitous Prouhet-Thue-Morse sequence. In C. Ding, T. Helleseth, nd N. H., editors, Sequences nd their pplictions, Proceedings of SETA 98, pges 6. Springer Verlg, J.-P. Allouche nd J. Shllit. Automtic sequences: theory, pplictions, generliztions. Cmbridge University Press, Z. Ésik nd W. Kuich. Forml tree series. Journl of Automt, Lnguges nd Combintorics, 8(2):29 285, E. G. Mnes nd M. A. Arbib. Algebric pproches to progrm semntics. Springer- Verlg New York, Inc., New York, NY, USA, D. Perrin nd J.-E. Pin. Infinite Words, volume 4 of Pure nd Applied Mthemtics. Elsevier, 24. ISBN J. J. M. M. Rutten. Behviourl differentil equtions: coinductive clculus of strems, utomt, nd power series. Theor. Comput. Sci., 38(-3): 53, J. J. M. M. Rutten. A coinductive clculus of strems. Mthemticl Structures in Computer Science, 5():93 47, 25. A Proof of theorem 2 Proof (Proof of theorem 2). Consider well-formed system of differentil equtions for Σ, s defined bove. We define set T of terms t by the following syntx: t ::= τ (τ T A) f(t,..., t r(f) ) (f Σ) where for every tree τ T A the set T contins corresponding term, denoted by τ, nd where new terms re constructed by (syntctic) composition of function symbols from Σ with the pproprite number of rgument terms. Next we turn T into n F -colgebr by defining function l, o, r : T (T A T ) by induction on the structure of terms, s follows. First we define o : T A by o(τ) = τ(ε) o ` f(t,... t r(f) ) = c ` o(t ),..., o(t r(f) ) (where c is the function used in the equtions for f). Next we define l : T T nd r : T T by l(τ) = τ L nd r(τ) = τ R, nd by l ( f(t,... t r) ) = t nd l ( f(t,... t r) ) = t 2 Here the terms t nd t 2 re obtined from the terms t nd t 2 used in the equtions for f, by replcing (every occurrence of) x i by t i, (x i) L by l(t i), (x i) R by r(t i), nd [x i(ε)] by [o(t)], for i =,..., r. Becuse T A is finl F -colgebr, there exists unique homomorphism h : T T A. We cn use it to define tree functions f : (T A) r T A, for every f Σ, by putting, for ll τ,..., τ r T A, f(τ,..., τ r) = h ` f(τ,..., τ r) This gives us set Σ of tree functions. One cn prove tht it is solution of our system of differentil equtions by coinduction, using the fcts tht h(τ) = τ, for ll τ T A, nd h ( f(t,..., t r) ) = f ( h(t ),..., h(t r) ) for ll f Σ nd t i T. This solution is unique becuse, by finlity of T A, the homomorphism h is.