Möbius number systems based on interval covers


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1 Möbius number systems based on interval covers Petr Kůrka and Alexandr Kazda Center for Theoretical Study, Academy of Sciences and Charles University in Prague, Jilská, CZ Praha, Czechia Faculty of Mathematics and Physics of the Charles University in Prague, Sokolovská 83, CZ8675 Praha 8, Czechia Abstract. Given a finite alphabet A, a system of real orientationpreserving Möbius transformations (F a : R R) a A, a subshift Σ A N, and an interval cover W = {W a : a A} of R, we consider the expansion subshift Σ W Σ of all expansions of real numbers with respect to W. If the expansion quotient Q(Σ, W) is greater than then there exists a continuous and surjective symbolic mapping Φ : Σ W R and we say that (F,Σ W ) is a Möbius number system. We apply our theory to the system of binary continued fractions which is a combination of the binary signed system with the continued fractions, and to the binary square system whose transformations have stable fixed points,, and. AMS classification scheme numbers: 37B, 37F99. Introduction Möbius number systems introduced in Kůrka [5] and [8] are based on iterative systems of Möbius transformations (F a : R R) a A indexed by a finite alphabet A, acting on the extended real line R = R { }. The convergence space X F consists of all infinite words u A N such that F u F un (z) converge to a real number Φ(u) R whenever z is a complex number with positive imaginary part. Since the symbolic representation Φ : X F R is usually not continuous, we search for a subshift Σ X F such that Φ : Σ R is continuous and surjective. In this case we say that (F,Σ) is a Möbius number system. In Kůrka [8] we have developed a theory of Möbius number systems with sofic subshifts. In the present paper we consider expansion subshifts which are obtained when we expand real numbers x R into symbolic sequences u A N with Φ(u) = x. The expansion procedure uses an interval cover W = {W a : a A} of R and tests conditions x W u, Fu (x) W u, Fu u (x) W u2 etc. We limit the expansions to a predefined subshift Σ A N which may be the full shift or the subshift which forbids identities u with F u F un = Id. The expansion subshift Σ W consists of all expansions which belong to Σ and respect W. We define the expansion quotient Q(Σ,W) and show that if Q(Σ,W) >, then (F,Σ W ) is a Möbius number system (Corollary 2). There is a converse (Theorem 3) which uses an additional condition on images of cylinders of finite words. While the expansion subshifts Σ W are in general not sofic, the arithmetical algorithms which work with them are simpler than in the sofic case. If all transformations F a have rational entries, and if their expansion intervals W a overlap and have rational endpoints, then there exist arithmetical algorithms which generalize
2 the arithmetical algorithms for exact real computation described in Gosper [], Vuillemin [3], or Potts et al. []. We present algorithms for expansions of rational or algebraic numbers and for computation of rational functions and fractional bilinear functions such as the sum and product. Using the continued fractions of Gauss (see Wall [4]), algorithms for many transcendental functions can be obtained similarly as in Potts []. We apply our theory to several examples. We show that both the binary signed system and the system of continued fractions can be conceived as Möbius number systems. The system of binary continued fractions considered in Kůrka [8] is a combination of these two classical systems and corresponds to the continued logarithms of Gosper []. Then we treat polygonal number systems considered in Kůrka [5], in particular the binary square system consisting of transformations with stable fixed points,, and. Some of the present results have appeared in Kůrka [7] and Kazda [3]. 2. Möbius transformations The extended real line R = R { } can be regarded as a projective space, i.e., the space of onedimensional subspaces of the twodimensional vector space. On R we have homogenous coordinates x = (x,x ) R 2 \{(,)} with equality x = y iff x y = x y. We regard x R as a column vector, and write it usually as x = x /x, for example = /. The open and closed intervals with endpoints a,b R are defined by (a,b) = {x R : (a x a x )(x b x b )(b a b a ) > }, [a,b] = {x R : (a x a x )(x b x b )(b a b a ) }. For distinct a,b R, we have (a,b) = {x R : a < x or x < b} { } if a > b and (a,b) = {x R : a < x < b} if a < b. Moreover, (a,a) = and [a,a] = R. A real orientationpreserving Möbius transformation (MT) is a selfmap of R of the form M (a,b,c,d) (x) = (ax+b)/(cx+d) = (ax +bx )/(cx +dx ), where a,b,c,d R and ad bc >. An MTacts alsoon the complex spherec = C { } and on the upper halfplane U = {z C : I(z) > }: if z U then M(z) U. The map d(z) = (iz +)/(z +i) maps U conformally to the unit disc D = {z C : z < } and R to the unit circle D = {z C : z = }. Define the circle distance on R by (x,y) = 2arcsin x y x y (x 2 +x 2 )(y2 +y2 ) which is the length of the shortest arc joining d(x) and d(y) in D. The length of a closed interval B r (a) = {x R : (x,a) r} is B r (a) = min{2r,2π}. The length J of a set J R is the length of the shortest interval which contains J. For x R and < ε < 2π denote by x ε,x ε R the unique points for which [x,x ε] = ε and [x ε,x] = ε. On the closed disc D := D D we get disc Möbius transformations M : D D defined by M (a,b,c,d) (z) = d M (a,b,c,d) d (z) = αz +β βz +α, where α = (a + d) + (b c)i, β = (b + c) + (a d)i. Define the norm of a Möbius transformation M = M (a,b,c,d) by M := (a 2 + b 2 + c 2 + d 2 )/(ad bc). We have 2
3 M 2, and M = 2 iff M is a rotation, i.e., if M(z) = R α (z) = e iα z for some α R. Definition The circle derivation M : R (, ) and the expansion interval V(M) of M = M (a,b,c,d) are defined by M (x) := M (d(x)) = V(M) := {x R : (M ) (x) > }. (ad bc)(x 2 +x 2 ) (ax +bx ) 2 +(cx +dx ) 2, We have M (x) = lim y x (M(y),M(x))/ (x,y), so the circle derivation is the derivation with respect to, and (MN) (x) = M (N(x)) N (x). If M is a rotation, then V(M) =, otherwise V(M) is a nonempty open interval. The expansion interval V(M)isrelatedtothevalue M() DofthecorrespondingdiscMT.IfV(M) = (l,r), then M() is the middle of the line joining d(l) and d(r). This follows from Lemma 7 of Kazda [2]. Further usefull relations between M, M() and M (x) have been given in Kůrka [8]: M() 2 = ( M 2)/( M +2), 3. Möbius number systems min{m (x) : x R} = 2 ( M M 2 4), max{m (x) : x R} = 2 ( M + M 2 4). For a finite alphabet A denote by A := m Am the set of finite words and by A + := A \ {λ} the set of finite nonempty words. The length of a word u = u...u m A m is u := m. We denote by A N the Cantor space of infinite words u = u u... equipped with metric d(u,v) := 2 k, where k = min{i : u i v i }. We denote by u [i,j) = u i...u j and u [i,j] = u i...u j. We say that v A is a subword of u A A N and write v u, if v = u [i,j) for some i j u, where we put u = for u A N. Given u A n, v A m with n, m >, denote by u.v A N the periodic word with preperiod u and period v defined by (u.v) i = u i for i < n and (u.v) n+km+i = v i for i < m and for all k. Denote by [u] the cylinder {v A N : v [, u ) = u} of u A. The shift map σ : A N A N is defined by σ(u) i = u i+. A subshift is a nonempty setσ A N whichisclosedandσinvariant,i.e.,σ(σ) Σ. ForasubshiftΣthereexists asetd A + offorbidden wordssuchthatσ = S(D) := {x A N : u x,u D}. We denote by S = A N the full shift with the alphabet A. A subshift is uniquely determined by its language L(Σ) := {u A : x Σ,u x}. We denote by L n (Σ) = L(Σ) A n. A subshift is of finite type (SFT), if the set D of forbidden words is finite. If D A n, then we say that the order of Σ is at most n. A subshift is sofic, if its language is regular (Lind and Marcus [9]). An iterative system is a continuous map F : A X X, or a family of continuous maps (F u : X X) u A satisfyingf uv = F u F v, andf λ = Id. Itisdeterminedbygenerators(F a : X X) a A. Definition 2 We say that F : A R R, is a Möbius iterative system, if all F a : R R are orientationpreserving Möbius transformations. The convergence space X F A N and the symbolic representation Φ : X F R are defined by X F := {u A N : lim F u n [,n) (i) R}, Φ(u) := lim F u n [,n) (i), 3
4 where i U is the imaginary unit. If Σ X F is a subshift such that Φ : Σ R is continuous and surjective, then we say that (F, Σ) is a Möbius number system. We say that a Möbius number system is redundant, if for every continuous map g : R R there exists a continuous map f : Σ Σ such that Φf = gφ. If u X F then Φ(u) = lim n F u[,n) (z) for every z U (see Kazda [2]). For v A +, u A N we have vu X F iff u X F, and then Φ(vu) = F v (Φ(u)). Since Φ is usually not continuous on X F, we search for a subshift Σ X F such that Φ : Σ R is continuous and surjective, so that we get a symbolic representation of R by a Cantor space Σ. If the system is redundant, then continuous functions g : R R can be lifted to their continuous symbolic extensions f : Σ Σ. The continuity of a function f : Σ Σ is necessary for its computability (see e.g. Weihrauch [5]). We show that the system is redundant if the images of its cylinders overlap (Theorem ). We give some sufficient and some necessary conditions for Φ(u) = x, which will be needed in the proofs. Lemma 3 (Kůrka [8]) Let u A N and x R. Then Φ(u) = x iff there exists c > and a sequence of intervals I m x such that liminf n Fu [,n) (I m ) > c for each m, and lim m I m =. Lemma 4 Let u A N and x R. () If lim n (F u [,n) ) (x) =, then Φ(u) = x. (2) If Φ(u) = x, then for every r < there exists n such that (F u [,n) ) (x) r n. Proof: () Denote z = d(x), M n (z) = F u[,n) (z) = (α n z + β n )/(β n z + α n ), where α n 2 β n 2 =, so(m n ) (x) = ( M n ) (z) = α n β n z 2. Iflim n (M n ) (z) = then lim n α n β n z =. Since α n > for n sufficiently large, there exists < r < such that β n > r for all sufficiently large n. It follows α n lim = z = n β n z = lim β n = lim M n () Φ(u) = x. n α n n (2) Assume by contradiction that there exists r < such that (Fu [,n) ) (x) < r n for all n and choose s < with r < s. Since ln(fa ) (x) is uniformly continuous on R, there exists δ > such that if (x,y) < δ, then (Fa ) (y) < (Fa ) (x) s/r for each a A. Denote by I = (x δ,x δ). For y I denote by y n = Fu [,n) (y), x n = Fu [,n) (x). We prove by induction that (Fu [,n) ) (y) < s n and (y n,x n ) < δ. If the condition holds for all k n, then (F u [,n+) ) (y) = (F u ) (y ) (F u n ) (y n ) < (F u ) (x ) (F u n ) (x n ) (s/r) n+ = (F u [,n+) ) (x) (s/r) n+ s n+. Since F u [,n+) is a contraction on I, we get (x n+,y n+ ) < δ. Since each F u [,n) is a contraction on I, we cannot have Φ(u) = x by Lemma 3. 4
5 4. Interval systems We define the expansion subshift of all expansions of real numbers which are chosen from a predefined subshift Σ and respect an interval cover or almostcover {W a : a A} of R. A natural choice for Σ is either the full shift S = A N or the subshift with forbidden identities D = {u A + : F u = Id}. Definition 5 Let F : A R R be a Möbius iterative system and Σ A N a subshift. () We say that W = {W u : u L(Σ)} is an interval system for F and Σ, if each W u R is a finite union of open intervals, W u = R iff u = λ, and W uv = W u F u (W v ) for each uv L(Σ). (2) We say that W is almostcompatible with F and Σ if W u = {W ua : a A,ua L(Σ)} for each u L(Σ). (3) We say that W is compatible with F and Σ if W u = {W ua : a A,ua L(Σ)} for each u L(Σ). (4) Denote by Σ W := {u Σ : n,w u[,n) } the expansion subshift of Σ and W. (5) Denote by W n (Σ) := {W u : u L n (Σ W )}. (6) Denote by l(w) := sup{l > : ( I R)( I < l a A,I W a )} the Lebesgue number of W. WehaveW u = W u F u (W u ) F u[,2) (W u2 ) F u[,n) (W un )foreachu L n+ (Σ), so W is determined by W = {W a : a A}. In the examples, W a are usually open intervals, but their intersections need not be always intervals. A compatible interval system expresses the process of expansion of real numbers: Given x R, we construct nondeterministically its expansion u Σ W as follows: Choose u with x W u, choose u with Fu (x) W u and u u L(Σ), choose u 2 with Fu u (x) W u2 and u u u 2 L(Σ), etc. Then x W u[,n) for each n. In Theorem 9 we show that if W a = V(F a ) are the expansion intervals of F a (Definition ) and if u is the expansion of x, then Φ(u) = x. If W is almostcompatible with F and Σ then W (Σ) is an almostcover of R, i.e., a W a = R. If W is compatible with F and Σ then W (Σ) is a cover of W λ = R, i.e., a W a = R. If Σ = S = A N is the full shift, then we have equivalencies: W is almostcompatible iff W (S) is an almostcover, and W is compatible iff W (S) is a cover. This is a special case of Proposition 6. Proposition 6 Let Σ be a SFT of order at most n + and let W be an interval system. () If W u = {W ua : a A,ua L(Σ)} holds for all u L(Σ) with u n, then W is almostcompatible with F and Σ. (2) If W u = {W ua : a A,ua L(Σ)} holds for all u L(Σ) with u n, then W is compatible with F and Σ. Proof: () We use the fact that if U,V are finite unions of open intervals, then U V U V. Let v,u A + and u = n. If x W vu, then for some y x either 5
6 (x,y) W vu or (y,x) W vu. In the former case we have ( ) (x,y) W v F v (W u ) W v F v {Wua : a A,ua L(Σ)} = {W v F v (W ua ) : a A,vua L(Σ)} {W v F v (W ua ) : a A,vua L(Σ)} = {W vua : a A,vua L(Σ)}, so x {W vua : a A,vua L(Σ)}. The proof of (2) is analogous. Proposition 7 Let W be almostcompatible with F and Σ. Then each W n (Σ) is an almostcover of R. If W is compatible, then each W n (Σ) is a cover of R. Proof: Given x R there exists u A, y R such that (x,y ) W u. There exist u A, y R with u [,] L(Σ) and (x,y ) W u[,], and we continue in this way till we get (x,y n ) W u[,n), so x W u[,n). Proposition 8 Let W be an interval system for S = A N and assume that for each u A m, a A we have W a F a (W u ) F a (W u ) W a. Then S W is a SFT of order at most m+. Proof: Recall that a subshift Σ has order at most p, if u A N belongs to Σ whenever all subwords of u of length p belong to L(Σ). Assume that u A n+, and for all v u with v = m+ we have W v. Then and W u[n m,n], so W u. W u = W u F u (W u[,m] ) F u[,m] (W u[m+,n] ) = F u (W u[,m] ) F u[,m] (W u[m+,n] ) = F u (W u[,n] ) = = F u[,n m) (W u[n m,n] ) Theorem 9 Let F : A R R be a Möbius iterative system and assume that V = {V a = V(F a ) : a A} (see Definition ) is almostcompatible with S = A N, i.e., a A V a = R. Then (F,S V ) is a Möbius number system and for each u L(S V ) we have Φ([u] S V ) = V u. Proof: Denote by l a,r a the endpoints of V a = (l a,r a ) and by Va ε := (l a ε,r a ε) provided 2ε < V a. Since F a are contractions on Fa (V a ), there exists an increasing continuous function ψ : [,2π) [,2π) such that ψ() =, < ψ(t) < t for t >, and F a (I) ψ( I ) for each a A and I Fa (V a ). Given u S V and m n we have Fu [,m] (V u[,n] ) Fu [,m] F u[,m) (V um ) = Fu m (V um ), so V u[,n] = F u F u (V u[,n] ) ψ( F u (V u[,n] ) ) = ψ( F u F u [,] (V u[,n] ) ) ψ 2 ( F u [,] (V u[,n] ) ) ψ n ( F u [,n] (V u[,n] ) ) ψ n (2π). Since ψ(t) < t and the only fixed point of ψ is zero, we get lim n V u[,n) =, so there exists a unique point x n V u [,n). We show that Φ(u) = x. There exists 6
7 ε > such that whenever Fa ([l a,l a ε]) (V b \Vb ε ) then F a (l a ) {l b,r b } and similarly, if Fa ([r a ε,r a ]) (V b \Vb ε ) then F a (r a ) {l b,r b }. By choosing ε small enough, we can also guarantee that Fa ([l a,l a ε]) V b and Fa ([r a ε,r a ]) V b are intervals (as opposed to unions of two nonempty disjoint intervals) for all a, b. Denote by x n := Fu [,n) (x). Since V u[,n] F u[,n) (V un ) F u[,n) (V un ), we get x V u[,n] F u[,n) (V un ), so x n V un. We have (F u [,n) ) (x) = (F u ) (x ) (F u ) (x ) (F u n ) (x n ), and each factor in this product is at least. If x n Vu ε n for an infinite number of n, then lim n (Fu [,n) ) (x) = and Φ(u) = x by Lemma 4. Assume therefore that there exists n such that x n V un \ Vu ε n for each n n. Assume that x n [l un,l un ε] and observe that x n+ = Fu n (x n ). By the definition of ε, we then have that Fu n (l un ) {l un+,r un+ }. However, if Fu n (l un ) = r un+, then x n V un F un (V un+ ) and therefore x V u[,n+] which is a contradiction. Thus Fu n+ (l un ) = l un+ and x n [l un,l un ε] foralln > n. IfI V a then Fa (I) ψ ( I ). GivenanyintervalI δ = [x,x δ], where δ >, there exists n > n such that F u [,n) (I δ ) max{ V a : a A} ε, so Φ(u) = x by Lemma 3. If x n [r un ε,r un ] we proceed similarly. Therefore in all cases Φ(u) = x and S V X F. We show Φ([v] S V ) = V v for each v L(S V ). If u S V [v], then Φ(u) V v by the preceding proof, so Φ([v] S V ) V v. If x V v, then x n V vn for each n < v. If there are j < k < v with x j = l uj, x k = r uk, then x j V uj F u[j,k) (V uk ), which is a contradiction. Thus either x n r vn for each n < v or x n l vn for each n < v. We define u [v] S V by u n = v n for n < v and for n v by induction. If x k r vk for all k < n then there exists u n such that x n = F u [,n) (x) [l un,r un ). Then u S V [v] and Φ(u) = x so V v Φ([v] S V ). This works also for V λ = R, so Φ : S V R is surjective. Finally, since lim n Φ([u [,n) ] S V ) =, Φ : S V R is continuous. 5. Expansion quotients The condition of Theorem 9 is not necessary for the inclusion of Σ W in X F and much larger interval covers may yield Möbius number systems as well. We obtain a better condition with the concept of the expansion quotient Q(Σ,W) of Σ and W. Definition If W is almostcompatible with F and Σ, then we define q(u) := inf{(fu ) (x) : x W u }, u L(Σ W ) Q n (Σ,W) := min{q(u) : u L n (Σ W )}, n Q(Σ,W) := lim n Qn (Σ,W). For x W uv we have (F uv ) (x) = (F u ) (x) (F v ) (F u (x)) q(u) q(v), and therefore q(uv) q(u) q(v). It follows Q n+m (Σ,W) Q n (Σ,W) Q m (Σ,W), so the limit Q(Σ,W) exists and Q(Σ,W) n Q n (Σ,W) for each n. Theorem Let W be almostcompatible with Σ, and assume that for some n > we have Q n (Σ,W) and no F u with u L n (Σ) is a rotation. Then (F,Σ W ) is a Möbius number system and Φ([u] Σ W ) = W u for each u L(Σ W ). If W is compatible with F and Σ then (F,Σ W ) is redundant. 7
8 Proof: For each u L n (Σ W ) we have W u V(F u ). Consider the alphabet B = L n (Σ W ) and the Möbius number system G : B R R given by G u = F u. Then V = {V(F u ) : u B} is an almostcover, so (G,(B N ) V ) is a Möbius number system by Theorem 9. Given u A N, define ũ B N by ũ k = u [kn,(k+)n). If u Σ W, then ũ (B N ) V, so lim k F u[,kn) (z) = Φ G (ũ) for any z U. In particular the condition is satisfied for each z = F v (i), where v A +, v < n. It follows lim k F u[,k) (i) = Φ G (ũ), so Φ F (u) = Φ G (ũ). Thus Σ W X F and Φ : Σ W R is continuous, since lim n W u[,n) =. Since each W n (Σ) is an almostcover, Φ : Σ W R is surjective. By Theorem 9 we get Φ([u] Σ W ) = W u for each u L(Σ W ). We show that if W is compatible with F and Σ, then (F,Σ W ) is redundant. If g : R R is continuous, then gφ : Σ W R is uniformly continuous. Given u Σ W, we construct v = f(u) Σ W by induction so that for each n there exists k n such that gφ([u [,kn)]) W v[,n). If the condition holds for n, then there exists k n+ > k n such that gφ([u [,kn+)]) is a subset of W v[,n+) for some v n with v [,n+) L(Σ W ). Thus f : Σ W Σ W is continuous and Φf = gφ. Corollary 2 If W is almostcompatible with F and Σ, and if Q(Σ,W) >, then (F,Σ W ) is a Möbius number system and Φ([u] Σ W ) = W u for each u L(Σ W ). Proof: We get Q n (Σ,W) > for some n and it follows that no F u with u L n (Σ W ) is a rotation. Theorem 3 Assume that W is almostcompatible with F and Σ, (F,Σ W ) is a Möbius number system, and Φ([u] Σ W ) = W u for each u L(Σ W ). Then Q(Σ,W). Proof: Assume the contrary and choose r with Q(Σ,W) < r <. Then Q n (Σ,W) < r n for n larger than some n. For v L(Σ W ) denote by α(v) = q(v)/r v, and a n = min{α(v) : v L(Σ W ), v n}, so a n Q n (Σ,W)/r n (Q(Σ,W)/r) n. Denote by v (n) L(Σ W ) the word of minimal length such that α(v (n) ) = a n and by x n R the corresponding number for which the minimum is attained. We show that for each prefix u of v (n) we have (Fu ) (x n ) r u. Assume the contrary with v (n) = uw. Then α(uw) = (F uw) (x n ) = (F w ) (Fu (x n )) r uw r w > (F w ) (Fu (x n )) α(w), r w (F u ) (x n ) r u which is a contradiction since w is shorter than uw and therefore α(uw) α(w). Since Σ W is compact, there exist v Σ W such that (F v [,n) ) (x m ) r n for each n m, and x m W v[,m). By compactness of R there exists x R such that (F v [,n) ) (x) r n for each n m, so Φ(v) x by Lemma 4. On the other hand x n W v [,n) = Φ([v [,n) ] Σ W ), so x = Φ(v) by the continuity of Φ, and this is a contradiction. If (F,Σ) is a Möbius number system and W a = R is the trivial interval system then Σ W = Σ so (F,Σ W ) is a Möbius number system but Q(Σ,W) <. While 8
9 the trivial interval cover is excluded in Definition 5, the example shows that some condition like Φ([u] Σ W ) = W u is necessary in Theorem 3. There are some constraints on expansion subshifts of Möbius number systems. For example, no full shift is an expansion subshift of any Möbius number systems (see Kazda [3]). 6. Arithmetical algorithms In arithmetical algorithms we work with the extended rational numbers Q = Q { } with homogenous integer coordinates x = x /x Z 2 \ {/}. Denote by I the set of closed intervals I = [I,I ] with endpoints I I in Q. Denote by M the set of MT M = M (a,b,c,d) whose coefficients a,b,c,d Z are integers with ad bc >. We assume that Σ is a sofic subshift, whose language is accepted by a deterministic finite automaton (DFA). This is a triple A = (Y, δ, i), where Y is a finite set of states, δ : Y A Y is a partial transition function, and i Y is the initial state. The function δ is extended to a partial function δ : Y A Y by δ(p,λ) = p, δ(p,ua) = δ(δ(p,u),a), where the lefthandside is defined iff the righthandside is defined. The language L(A) of A consists of words u A which are accepted, i.e., for which δ(i, u) is defined. We assume that F : A R R is a Möbius iterative system, Σ A N is a sofic subshift whose language is accepted by a DFA A = (Y,δ,i), and W = {W u : u L(Σ)} is an interval system compatible with F and Σ, such that F a M and W a I for each a A. We also assume that for some n >, Q n (Σ,W) and no F u with u L n (Σ W ) is a rotation. Finally we assume that l(w) >, so (F,Σ W ) is a redundant Möbius number system. We generalize arithmetical algorithms of Gosper [] (see also Vuillemin [3], Kornerup and Matula [4] or Potts et al. []) using infinite labelled graphs. This is a structure G = (V,E,s,t,l) where V is the set of vertices, E is the set of edges, s,t : E V is the source and target map and l : E A {λ} is the labelling function. An infinite path with source s(u ) is any sequence of edges u E N such that t(u i ) = s(u i+ ). Its label is l(u )l(u )... A N A. Definition 4 The vertices of the rational expansion graph are pairs (x,q) a Q Y. Its labelled edges are (x,q) (Fa (x),δ(q,a)) such that δ(q,a) and x W a. Proposition 5 For each x Q there exists an infinite path with source (x,i). If u A N is its label, then u Σ W and Φ(u) = x. Proof: Set x = x, q = i. At step n choose u n such that x n W un and δ(q n,u n ). Set x n+ = F u n (x n ), q n+ = δ(q n,u n ) and continue. Then we have x W u[,n), and u [,n) L(Σ W ), so u Σ W and Φ(u) = x. The nondeterministic expansion algorithm based on the rational expansion graph ofdefinition4canberealizedbyamachinewhoseinnerstatesarepairs(x,q) Q Y. The machine is attached to an output tape where the expansion is written. The algorithm can be made deterministic by further specifications, like in the arithmetic expansion algorithm of Definition 2. Definition 6 The vertices of the linear graph are triples (M,q,u) where M M, q Y and u Σ W. Its edges are a (M,q,u) (Fa M,δ(q,a),u) if δ(q,a) & M(W u ) W a, (M,q,u) λ (MF u,q,σ(u)). 9
10 Proposition 7 If u Σ W and M M then there exists an infinite path with source (M,i,u) whose label w belongs to A N. For any such w we have w Σ W and Φ(w) = M(Φ(u)). Proof: Weshowbyinductionthatifw A isthelabelofapathwithsource(m,i,u), then its target is (Fw MF u[,k),δ(i,w),u [k, ) ) for some k, and M(W u[,k] ) W w. The statement holds for w = λ and k =. If the statement holds for w and some k then we have two possible continuations: (Fw λ MF u[,k),δ(i,w),u [k, ) ) (Fw MF u[,k],δ(i,w),u [k+, ) ), (Fw a MF u[,k),δ(i,w),u [k, ) ) (FwaMF u[,k),δ(i,wa),u [k, ) ). In the former case we get M(W u[,k+] ) M(W u[,k] ) W w. In the latter case we use the assumption Fw MF u[,k) (W uk ) W a, to get M(W u[,k] ) M(F u[,k) (W uk )) F w (W a ), and M(W u[,k] ) W w F w (W a ) = W wa. Since u Σ W, we have lim k W u[,k] =, so for each j there exists k and w j such that M(W u[,k) ) W w[,j). Thus there exists a path with label w Σ W and M(Φ(u)) = Φ(w). The algorithm based on the linear graph of Definition 6 can be realized by a machine whose inner states are (M,q) M Y. The machine is attached to an input and output tapes and can see the first letter of the input tape. 7. Fractional bilinear functions To obtain algorithms for functions of two variables like the sum or product, we consider orientationreversing MT M (a,b,c,d) with ad bc <, singular MT with ad bc = and a + b + c + d >, and the zero MT M (,,,) =. Each MT defines a closed relation M = {(x,y) R 2 : (ax +bx )y = (cx +dx )y }. We define the stable point s M R of a singular MT by s M { a c, b d } R. Note that a c or b d can be R, but not both. Similarly the unstable point u M R of a singular MT is defined by u M { b a, d c } R. A singular MT yields the relation M = (R {s M }) ({u M } R), and the zero MT yields the full relation M = R 2. The value M(I) = {y R : x I,(x,y) M} of M on a closed interval I = [I,I ] is [M(I ),M(I )] if ad bc >, [M(I ),M(I )] if ad bc <, M(I) = {s M } if ad bc = & M & u M I, R if ad bc = & M & u M I, R if M =. Denote by M (,) the set of fractional bilinear functions with integer coefficients. These are functions P : R R R { } of two variables x,y R of the form which represent closed relations P(x,y) = ax y +bx y +cx y +dx y ex y +fx y +gx y +hx y P = {(x,y,z) R 3 : (ax y +bx y +cx y +dx y )z = (ex y +fx y +gx y +hx y )z }.
11 For each x,y R, P(x, ), P(,y) are MT with matrices [ ] [ ax +cx P(x, ) = bx +dx ay +by, P(,y) = cy +dy ex +gx fx +hx ey +fy gy +hy ], so if x,y R and I,J are closed intervals, then P(x,J), P(I,y) are well defined intervals. Since P(x, ) and P(,y) are monotone, the image P(I,J) of closed intervals I = [I,I ], J = [J,J ] coincides with the image of the boundary of I J: P(I,J) := {z R : x I, y J,(x,y,z) P} = P(I,J) P(I,J ) P(I,J) P(I,J ). Given a (2 2)matrix M = M (a,b,c,d), define the (4 4)matrices M x and M y by M x = a b a b c d c d, My = a b c d a b c d Then both P(Mx,y) = PM x (x,y) and P(x,My) = PM y (x,y), as well as MP are fractional bilinear functions. In these formulas, P is regarded as a (2 4)matrix. Definition 8 The bilinear graph has vertices (P,q,u,v), where P M (,), q Y and u,v Σ W. The edges are a (P,q,u,v) (Fa P,δ(q,a),u,v) if δ(q,a) & P(W u,w v ) W a, λ (P,q,u,v) (PFu x,q,σ(u),v), (P,q,u,v) λ (PF y v,q,u,σ(v)). Proposition 9 If u,v Σ W and w A N is a label of a path with the source (P,i,u,v), then w Σ W and Φ(w) = P(Φ(u),Φ(v)). The proof is similar as in Proposition 7. However, for u,v Σ W there need not exist any path with source (P,i,u,v) whose label belongs to A N. The algorithm may give only a finite output, for example when we add words representing Rational functions Denote by M n the set of rational functions R(x) = a x n +a x x n + +a n x n b x n +b x x n + +b n x n of degree n with integer coefficients. If M M, then both composed functions RM and MR belong to M n. Denote by t(x) := argd(x) = 2arctanx the isomorphism of R with ( π,π). The circle derivation R : R R of R is defined by R (x) := (trt ) (t(x)) = R (x)(+x 2 ) +R 2. (x) We have R (x) = lim y x (R(y),R(x))/ (y,x). A monotone element is a pair (R,I) such that I is a closed interval, R M n is monotone on I, i.e., R (x) does.
12 not change sign on I, and R has no poles in I, i.e., R(x) for x I. We say that a monotone element (R,I) is signchanging, if either R(I ) < < R(I ) and R (x) > for x I, or R(I ) > > R(I ) and R (x) < for x I. A signchanging monotone element (R,I) has a unique root x I with R(x) =. Proposition 2 () There exists an algorithm which for R M n gives a list of signchanging elements (R,I ),...,(R k,i k ), such that each root of R is a root of some (R j,i j ). (2) It is decidable, whether (R, I) is a monotone element. (3) Given closed intervals I,J, it is decidable whether R(I) J. These algorithms can be obtained from the Sturm theorem (see e.g., van der Waerden [2]), which counts the number of roots of a polynomial P in an interval using the Euclidean algorithm applied to P and its derivative P. The condition (3) can be decided without evaluating the interval R(I), which may have irrational endpoints: We use the the Sturm theorem to test whether R(x) J and R(x) J have roots in I. Definition 2 The algebraic expansion graph is a graph whose vertices are (R,q,I), where (R,I) is a signchanging element, and q Y. Its edges are (R,q,I) a (RF a,δ(q,a),f a (I W a )), such that δ(q,a) and (R,I W a ) is a signchanging element. Proposition 22 For each signchanging element (R, I) there exists an infinite path with source (R,i,I). If w is its label, then w Σ W, Φ(w) I, and RΦ(w) =. Proof: We show by induction that if u is a label of a finite path with source (R,i,I), then its target is (RF u,δ(i,u),j), where F u (J) = I W u, (RF u,j) is a signchanging element, and there exists a path with source (R,i,I) and label ua for some a A. If this holds for u A, then u L(Σ W ), and (R,F u (J)) = (R,I W u ) is a signchanging element, so there exists a A such that ua L(Σ W ), and (R,I W ua ) = (R,I W u F u (W a )) = (R,F u (J W a )) is a signchanging element. Thus (RF u,j W a ) is a signchanging element, so there exists a path with label ua and target (RF ua,δ(i,ua),j ), where J = Fa (J W a ). It follows that (RF ua,j ) is a signchanging element and F ua (J ) = F u (J) F u (W a ) = I W ua. The graph for the computation of a rational function R M n at Φ(u) has the same structure as the linear graph from Definition 6. To test the condition R(W u ) W a, we use the Sturm theorem rather than the simple comparison of the endpoints of intervals. However, if R is monotone in a neighbourhood I of Φ(u), the test simplifies. Finally we have a conversion algorithm between two redundant Möbius number systems: Definition 23 The (G,Θ V ) to (F,Σ W ) conversion graph has vertices (M,q,u), where M M, q Y is the state of a DFA for Σ W and u Θ V. The labelled edges are a (M,q,u) (Fa M,δ(q,a),u) if δ(q,a) & V u W a, λ (M,q,u) (MG u,q,σ(u)). 2
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