Abstract Numeration Systems or Decimation of Languages

Transcription

1 Abstract Numeration Systems or Decimation of Languages Emilie Charlier University of Waterloo, School of Computer Science Languages and Automata Theory Seminar Waterloo, April 1 and 8, 211

2 Positional numeration systems A positional numeration system (PNS) is given by a sequence of integers U = (U i ) i such that U = 1 i U i < U i+1 (U i+1 /U i ) i is bounded C U = sup i U i+1 /U i The greedy U-representation of a positive integer n is the unique word rep U (n) = c l 1 c over Σ U = {,...,C U 1} satisfying l 1 n = c i U i, c l 1 and t i= t c i U i < U t+1. i= A set X N is U-recognizable or U-automatic if rep U (X) Σ U is regular.

3 Integer base b 2 U = (b i ) i rep U, Σ U rep b, Σ b Σ b = {,,b 1} L b = rep b (N) = Σ b \Σ b,1,2 1,2 rep 3 (N) N is 3-recognizable ε

4 Integer base b 2 U = (b i ) i rep U, Σ U rep b, Σ b Σ b = {,,b 1} L b = rep b (N) = Σ b \Σ b,2, rep U3 (2N) 2 N is 3-recognizable ε

5 Fibonacci (or Zeckendorf) numeration system Let F = (F i ) i = (1,2,3,5,8,13,21,...) be defined by F = 1, F 1 = 2 and i N, F i+2 = F i+1 +F i. Σ F = {,1} The factor 11 is forbidden : 1 1 rep F (N) = 1{,1} {ε} N is F-recognizable ε

6 Fibonacci (or Zeckendorf) numeration system Let F = (F i ) i = (1,2,3,5,8,13,21,...) be defined by F = 1, F 1 = 2 and i N, F i+2 = F i+1 +F i rep F (2N) 2N is F-recognizable ε

7 U-recognizability of N Is the set N U-recognizable? Otherwise stated, is the numeration language regular? Not necessarily: Theorem (Shallit 1994) Let U be a PNS. If N is U-recognizable, then U is linear, i.e., it satisfies a linear recurrence relation over Z. Loraud (1995) and Hollander (1998) gave sufficient conditions for the numeration language to be regular : The characteristic polynomial of the recurrence relation has a particular form.

8 Abstract numeration systems An abstract numeration system (ANS) is a triple S = (L,Σ,<) where L is an infinite regular language over a totally ordered alphabet (Σ, <). By enumerating the words of L w.r.t. the radix order < rad induced by <, we define a bijection : rep S : N L val S = rep 1 S : L N. A set X N is S-recognizable if rep S (X) is regular.

9 L = {a,b} Σ = {a,b} a < b n rep(n) ε a b aa ab ba bb aaa L = a b Σ = {a,b} a < b n rep(n) ε a b aa ab bb aaa

10 A generalization ANS generalize PNS having a regular numeration language: Let U be a PNS and let x,y N. We have x < y rep U (x) < rad rep U (y). Example (Fibonacci) rep F (N) = 1{,1} {ε} 6 < 7 and 11 < rad 11 (same length) 6 < 8 and 11 < rad 1 (different lengths) ε

11 A generalization ANS generalize PNS having a regular numeration language: Let U be a PNS and let x,y N. We have x < y rep U (x) < rad rep U (y). Example (Fibonacci) rep F (N) = 1{,1} {ε} 6 < 7 and 11 < rad 11 (same length) 6 < 8 and 11 < rad 1 (different lengths) ε

12 A generalization ANS generalize PNS having a regular numeration language: Let U be a PNS and let x,y N. We have x < y rep U (x) < rad rep U (y). Example (Fibonacci) rep F (N) = 1{,1} {ε} rep F (n) n ε

13 Decimation of languages Let L be a language ordered w.r.t. the radix order. If w < w 1 < are the elements of L and X N, then L[X] = {w n : n X}. If S = (L,Σ,<), then L[X] = rep S (X). If L[X] is accepted by a finite automaton, what does it imply on X? What conditions on X insures that L[X] is regular?

14 Motivation for ANS ANS are a generalization of all usual PNS like integer base numeration systems or linear numeration systems, and even rational numeration systems. Thanks to this general point of view on numeration systems, we try to distinguish results that deeply depend on the algorithm used to represent the integers from results that only depend on the set of representations. Due to the general setting of ANS, some new questions concerning languages arise naturally from this numeration point of view.

15 Some questions around ANS Rec. sets in a given ANS? Rec. sets in all ANS? Are there subsets of N that are never recognizable? Given a subset of N can we build an ANS for which it is rec.? How do rec. depend on the choice of the numeration? For which ANS do arithmetic operations preserve rec.? Operations preserving rec. in a given ANS? How to represent real numbers? Can we define automatic sequences in that context? Logical characterization of rec. sets? Extensions to the multidimensional setting?...

16 Some questions around ANS Rec. sets in a given ANS? Rec. sets in all ANS? Are there subsets of N that are never recognizable? Given a subset of N can we build an ANS for which it is rec.? How do rec. depend on the choice of the numeration? For which ANS do arithmetic operations preserve rec.? Operations preserving rec. in a given ANS? How to represent real numbers? Can we define automatic sequences in that context? Logical characterization of rec. sets? Extensions to the multidimensional setting?...

17 Some questions around ANS Rec. sets in a given ANS? Rec. sets in all ANS? Are there subsets of N that are never recognizable? Given a subset of N can we build an ANS for which it is rec.? How do rec. depend on the choice of the numeration? For which ANS do arithmetic operations preserve rec.? Operations preserving rec. in a given ANS? How to represent real numbers? Can we define automatic sequences in that context? Logical characterization of rec. sets? Extensions to the multidimensional setting?...

18 Some questions around ANS Rec. sets in a given ANS? Rec. sets in all ANS? Are there subsets of N that are never recognizable? Given a subset of N can we build an ANS for which it is rec.? How do rec. depend on the choice of the numeration? For which ANS do arithmetic operations preserve rec.? Operations preserving rec. in a given ANS? How to represent real numbers? Can we define automatic sequences in that context? Logical characterization of rec. sets? Extensions to the multidimensional setting?...

19 Some questions around ANS Rec. sets in a given ANS? Rec. sets in all ANS? Are there subsets of N that are never recognizable? Given a subset of N can we build an ANS for which it is rec.? How do rec. depend on the choice of the numeration? For which ANS do arithmetic operations preserve rec.? Operations preserving rec. in a given ANS? How to represent real numbers? Can we define automatic sequences in that context? Logical characterization of rec. sets? Extensions to the multidimensional setting?...

20 Some questions around ANS Rec. sets in a given ANS? Rec. sets in all ANS? Are there subsets of N that are never recognizable? Given a subset of N can we build an ANS for which it is rec.? How do rec. depend on the choice of the numeration? For which ANS do arithmetic operations preserve rec.? Operations preserving rec. in a given ANS? How to represent real numbers? Can we define automatic sequences in that context? Logical characterization of rec. sets? Extensions to the multidimensional setting?...

21 Some questions around ANS Rec. sets in a given ANS? Rec. sets in all ANS? Are there subsets of N that are never recognizable? Given a subset of N can we build an ANS for which it is rec.? How do rec. depend on the choice of the numeration? For which ANS do arithmetic operations preserve rec.? Operations preserving rec. in a given ANS? How to represent real numbers? Can we define automatic sequences in that context? Logical characterization of rec. sets? Extensions to the multidimensional setting?...

22 Some questions around ANS Rec. sets in a given ANS? Rec. sets in all ANS? Are there subsets of N that are never recognizable? Given a subset of N can we build an ANS for which it is rec.? How do rec. depend on the choice of the numeration? For which ANS do arithmetic operations preserve rec.? Operations preserving rec. in a given ANS? How to represent real numbers? Can we define automatic sequences in that context? Logical characterization of rec. sets? Extensions to the multidimensional setting?...

23 Some questions around ANS Rec. sets in a given ANS? Rec. sets in all ANS? Are there subsets of N that are never recognizable? Given a subset of N can we build an ANS for which it is rec.? How do rec. depend on the choice of the numeration? For which ANS do arithmetic operations preserve rec.? Operations preserving rec. in a given ANS? How to represent real numbers? Can we define automatic sequences in that context? Logical characterization of rec. sets? Extensions to the multidimensional setting?...

24 Some questions around ANS Rec. sets in a given ANS? Rec. sets in all ANS? Are there subsets of N that are never recognizable? Given a subset of N can we build an ANS for which it is rec.? How do rec. depend on the choice of the numeration? For which ANS do arithmetic operations preserve rec.? Operations preserving rec. in a given ANS? How to represent real numbers? Can we define automatic sequences in that context? Logical characterization of rec. sets? Extensions to the multidimensional setting?...

25 Some questions around ANS Rec. sets in a given ANS? Rec. sets in all ANS? Are there subsets of N that are never recognizable? Given a subset of N can we build an ANS for which it is rec.? How do rec. depend on the choice of the numeration? For which ANS do arithmetic operations preserve rec.? Operations preserving rec. in a given ANS? How to represent real numbers? Can we define automatic sequences in that context? Logical characterization of rec. sets? Extensions to the multidimensional setting?...

26 Some questions around ANS Rec. sets in a given ANS? Rec. sets in all ANS? Are there subsets of N that are never recognizable? Given a subset of N can we build an ANS for which it is rec.? How do rec. depend on the choice of the numeration? For which ANS do arithmetic operations preserve rec.? Operations preserving rec. in a given ANS? How to represent real numbers? Can we define automatic sequences in that context? Logical characterization of rec. sets? Extensions to the multidimensional setting?...

27 S-automatic words

28 b-automatic words An infinite word x = (x n ) n is b-automatic if there exists a DFAO A = (Q,q,Σ b,δ,γ,τ) s.t. for all n, x n = τ(δ(q,rep b (n))). Theorem (Cobham 1972) Let b 2. An infinite word is b-automatic iff it is the image under a coding of an infinite fixed point of a b-uniform morphism.

29 S-automatic words Let S = (L,Σ,<) be an ANS. An infinite word x = (x n ) n is S-automatic if there exists a DFAO A = (Q,q,Σ,δ,Γ,τ) s.t. for all n, x n = τ(δ(q,rep S (n))). Theorem (Rigo-Maes 22) An infinite word is S-automatic for some ANS S iff it is the image under a coding of an infinite fixed point of a morphism, i.e. a morphic word.

30 Corollary The factor complexity of an S-automatic word is O(n 2 ). Corollary The set of primes is never S-recognizable. Its characteristic sequence is not morphic (Mauduit 1988).

31 Idea of the proof Example (Morphic S-Automatic) Consider the morphism µ defined by a abc ; b bc ; c aac. We have µ ω (a) = abcbcaacbcaacabcabcaacbcaacabcabc. One canonically associates the DFA A µ,a a 1 b,1 2 1 c 2 L µ,a = {ε,1,2,1,11,2,21,22,1,11,11,111,112,2,...} If S = (L µ,a,{,1,2}, < 1 < 2), then (µ ω (a)) n = δ µ (a,rep S (n)) for all n.

32 Idea of the proof Example (S-Automatic Morphic) S = (L,{,1,2}, < 1 < 2) where L = {w Σ : w 1 is odd} minimal automaton of L DFAO generating x,2 I 1 1 F,2 2 a,1,1 b 2 n rep S (n) x b a a b b b b a a

33 Example (Continued) I a F b 1 f: α αi a F a F b I b F a g: α,i a,i b ε I a I b F b I a F b F a I a F b F a a I b I a F a I b 1 I b F a F b b L Σ ε f ω (α) α I a I b F b I a I a F a I b F a I a F b x b a a b g(f ω (α)) = x

34 Multidimensional Case A d-dimensional infinite word over an alphabet Σ is a map x : N d Σ. We use notation like x n1,...,n d or x(n 1,...,n d ) to denote the value of x at (n 1,...,n d ). If w 1,...,w d are finite words over the alphabet Σ, (w 1,...,w d ) # := (# m w 1 w 1,...,# m w d w d ) where m = max{ w 1,..., w d }. Example (ab,bbaa) # = (##ab,bbaa) = (#,b)(#,b)(a,a)(b,a)

35 A d-dimensional infinite word over an alphabet Γ is b-automatic if there exists a DFAO A = (Q,q,(Σ b ) d,δ,γ,τ) s.t. for all n 1,...,n d, ( ( τ δ q,(rep b (n 1 ),...,rep b (n d )) )) = x n1,...,n d. Theorem (Salon 1987) Let b 2 and d 1. A d-dimensional infinite word is b-automatic iff it is the image under a coding of a fixed point of a b-uniform d-dimensional morphism.

36 Theorem (C-Kärki-Rigo 21) Let d 1. The d-dimensional infinite word is S-automatic for some ANS S = (L,Σ,<) where ε L iff it is the image under a coding of a shape-symmetric infinite d-dimensional word.

37 Shape-symmetric µ(a) = µ(f) = a b c d ; µ(b) = e c ; µ(c) = e b ;µ(d) = f µ(e) = e b g d ; µ(g) = h b ; µ(h) = h b c d. µ ω (a) = a b e e b e b e c d c g d g d c e b f e b h b f e b e a b e b e g d c c d g d c e b e e b a b e g d c g d c d c h b f e b e b f....

38 Consider the morphism µ 1 defined by a ab ;b e ;e eb. We have µ ω 1 (a) = abeebebeebeebebeebebeebeeb. One canonically associates the DFA A µ1,a a 1 b e 1 L µ1,a = {ε,1,1,1,11,1,11,11,1,...}

39 Open question If S and T are two ANS, (S,T)-automatic words are bidimensional infinite words (x m,n ) m,n for which there exists a DFAO A = (Q,(Σ {#}) d,δ,q,γ,τ) s.t. m,n N, x m,n = τ(δ(q,(rep S (m),rep T (n)) # )). Can these (S, T)-automatic words be characterized by iterating morphisms?

40 b-kernel An infinite word (x n ) n is b-automatic iff its b-kernel {(x b e n+r) n : e,r N, r < b e } is finite. The b-kernel can be rewritten {(x b w n+val b (w) ) n : w Σ b } ε NB: b w n+val b (w) is the base-b value of the (n+1)-th word in L b having w as a suffix.

41 Open question The S-kernel of (x n ) n is {(x fw(n)) n : w Σ } where f w (n) is the S-value of the (n+1)-th word in L having w as a suffix. Theorem (Rigo-Maes 22) An infinite word is S-automatic iff its S-kernel is finite. Does a similar characterization hold in the multidimensional setting?

42 Sets S-recognizable for all S

43 Ultimately periodic sets It is an exercise to show that all ultimately periodic set are b-recognizable for all b 2. Theorem (Cobham 1969) Let k, l 2 be two multiplicatively independent integers. A subset of N is both k-recognizable and l-recognizable iff it is ultimately periodic. Two numbers k and l are multiplicatively independent if k m = l n and m,n N implies m = n =. Corollary A subset of N is b-recognizable for all b 2 iff it is ultimately periodic.

44 Generalization to ANS Theorem (Lecomte-Rigo 21, Krieger et al. 29) Ultimately periodic sets are S-recognizable for all ANS S. Corollary A subset of N is S-recognizable for all ANS S iff it is ultimately periodic. Theorem (Krieger et al. 29, Angrand-Sakarovitch 21) Let m,r N with m 2 and r m 1 and let S = (L,Σ,<) be an ANS. If L is accepted by a n-state DFA, then the minimal DFA of rep S (mn+r) has at most nm n states.

45 Semi-linear sets A subset X of N d is b-recognizable if the language (rep b (X)) # over ({,1,...,b 1} {#}) d is regular, where rep b (X) = {(rep b (n 1 ),...,rep b (n d )): (n 1,...,n d ) X}. Theorem (Cobham Semenov, Semenov 1977) Let k, l 2 be two multiplicatively independent integers. A subset of N d is both k-recognizable and l-recognizable iff it is semi-linear. A set X N d is linear if there exist v,v 1,,v t N d such that X = v +Nv 1 +Nv 2 + +Nv t. A set X N d is semi-linear if it is a finite union of linear sets.

46 {(n,m) : n,m N and n m} = N(1,) +N(1,1)

47 Semi-linear sets: a good generalization? Corollary A subset of N d is b-recognizable for all b 2 iff it is semi-linear. In the one-dimensional case, we have the following equivalences: semi-linear ultimately periodic 1-recognizable.

48 Multidimensional case for ANS One might therefore expect that the semi-linear sets are recognizable in all ANS. However, this fails to be the case, as the following example shows. Example The semi-linear set X = {n(1,2) : n N} = {(n,2n) n N} is not 1-recognizable. Consider the language {(a n # n,a 2n ) n N}, consisting of the unary representations of the elements of X. Use the pumping lemma to show that this is not accepted by a finite automaton.

49 Let S = (L,Σ,<) be an ANS. A subset X of N d is S-recognizable if the language (rep S (X)) # over (Σ {#})) d is regular, where rep S (X) = {(rep S (n 1 ),...,rep S (n d )): (n 1,...,n d ) X}. It is 1-automatic if it is S-automatic for the ANS S built on a.

50 Multidimensional 1-recognizable sets Theorem (C-Lacroix-Rampersad 21) A subset of N d is S-recognizable for all ANS S iff it is 1-recognizable. Theorem (C-Lacroix-Rampersad 21) The multidimensional 1-recognizable sets are the finite unions of sets of the form where Nv 1 +a 1 +Nv 2 +a 2 + +Nv t +a t, i Supp(v i ) = Supp(a i ) Supp(v 1 ) Supp(v 2 ) Supp(v t ) All v i and a i are multiples of vectors all of whose components are or 1.

51 Recognizable sets Another well-studied subclass of the class of semi-linear sets is the class of recognizable sets. A subset X of N d is recognizable if the right congruence X has finite index (x X y if z N d (x+z X y +z X)). When d = 1, we have again the following equivalences: recognizable ultimately periodic 1-recognizable. However, for d > 1 these equivalences no longer hold.

52 Multidimensional recognizable sets: a characterization Theorem (Mezei) The recognizable subsets of N 2 are precisely finite unions of sets of the form Y Z, where Y and Z are ultimately periodic subsets of N. In particular, the diagonal set D = {(n,n) n N} is not recognizable. However, the set D is clearly a 1-recognizable subset of N 2. So we see that for d > 1, the class of 1-recognizable sets corresponds neither to the class of semi-linear sets, nor to the class of recognizable sets.

53 A description of S-recognizable sets

54 Which growth functions for recognizable sets? Let L be a language over an alphabet Σ. Define u L (n) = L Σ n and v L (n) = n u L (i) = L Σ n i= The maps u L : N N and v L : N N are the growth functions of L. If X N, we let t X (n) denote the (n+1)-th term of X. The map t X : N N is the growth function of X. What do the growth functions of S-recognizable sets look like?

55 Theorem (C-Rampersad 21) Let S = (L,Σ,<) be an ANS built on a regular language and let X N be an infinite S-recognizable set. Suppose i {,...,p 1}, v L (np+i) a i n c α n (n + ), for some p,c N with p 1, some α 1 and some positive constants a,...,a p 1, and j {,...,q 1}, v reps (X)(nq +j) b j n d β n (n + ), for some q,d N with q 1, some β 1 and some positive constants b,...,b q 1. Then we have ( p α) t X (n) = Θ (log(n)) c dlog( log( q β)n log( p α) ) log( q β) if β > 1; ( ) t X (n) = Θ nd( c p α) Θ(n1 d) if β = 1.

56 Exponential numeration language Proposition (C-Rampersad 21) For all k,l N with l >, there exists an ANS S built on an exponential regular language and an infinite S-recognizable set X N s.t. t X (n) = Θ((log(n)) k n l ). For all k,l N with l > 1, there exists an ANS S built on an exponential regular language ( and an infinite S-recognizable set X N s.t. t X (n) = Θ n l (log(n)) k ). For all k N with k > and for all ANS S, ( there is no S-recognizable set X N s.t. t X (n) = Θ. ) n (log(n)) k

57 Polynomial numeration language Corollary Let S = (L,Σ,<) be an ANS built on a polynomial regular language and let X N be an infinite S-recognizable set. Then we have t X (n) = Θ(n r ) for some rational number r 1. Proposition (C-Rampersad 21) For every rational number r 1, there exists an ANS S built on a polynomial regular language and an infinite S-recognizable set X N such that t X (n) = Θ(n r ).

58 Theorem (Eilenberg 1974) A b-recognizable set X = {x n : n N}, where x < x 1 <, satisfies either limsup n + (x n+1 x n ) < + or limsup n + x n+1 x n > 1. Corollary The set of squares {n 2 n N} is not b-automatic for all b 2. But it is S-automatic for the ANS built on a b a c with a < b < c since we have rep S ({n 2 n N}) = a. Proposition (Rigo 22) For all k N, the set {n k n N} is S-automatic for some S. In those constructions, the exhibited ANS are built on polynomial languages.

59 An example Consider the base 4 numeration system, that is, the ANS built on L 4 = {ε} {1,2,3}{,1,2,3} with the natural order on the digits. Let X = val 4 ({1,3} ) = {1,3,5,7,13,15,21,23,29,31,...}. It is clearly 4-automatic. We have v L4 (n) = 4 n and v {1,3} (n) = 2 n+1 1. We obtain t X (n) = Θ(n 2 ).

60 Theorem (Durand-Rigo 29) Let S be an ANS built on a polynomial regular language and let T be an ANS built on an exponential regular language. If a subset of N is both S-recognizable and T-recognizable, then it is ultimately periodic. Corollary Let S be an ANS built on an exponential regular language. If f Q[x] is a polynomial of degree greater than 1 such that f(n) N, then the set f(n) is not S-recognizable. Open problem (hard) Find a Cobham-style theorem for ANS.

61 The periodicity problem

62 The periodicity problem Open problem Given an ANS S and a DFA accepting an S-recognizable set, decide if this set it ultimately periodic. Theorem (Honkala 1985) The periodicity problem is decidable for integer bases. Theorem (Muchnik 1991) The periodicity problem is decidable for linear PNS U s.t. N is U-recognizable and addition is computable by a finite automaton.

63 Results for ANS Theorem (Honkala-Rigo 24) The periodicity problem for all ANS is equivalent to the HDL periodicity problem. Theorem (C-Rigo 28, Bell-C-Fraenkel-Rigo 29) The periodicity problem is decidable for a large class of linear PNS. NB: Our class contains PNS for which addition is not computable by a finite automaton. Intermediate open problem Given a PNS U s.t. N is U-recognizable and a DFA accepting an U-recognizable set, decide if this set it ultimately periodic.

64 Automata recognizing languages arising from linear PNS

65 Even numbers in Fibonacci system rep F (2N) ε

66 Motivation What is the best automaton we can get? DFAs accepting the binary representations of 4N + 3. Question The general algorithm doesn t provide a minimal automaton. What is the state complexity of rep U (pn+r)?

67 Information we are looking for Consider a linear PNS U such that N is U-recognizable. How many states does the minimal automaton recognizing rep U (mn) contain? 1. Give upper/lower bounds. 2. Study special cases, e.g., Fibonacci numeration system. 3. Get information on the minimal automaton A U recognizing rep U (N).

68 Information we are looking for Consider a linear PNS U such that N is U-recognizable. How many states does the minimal automaton recognizing rep U (mn) contain? 1. Give upper/lower bounds. 2. Study special cases, e.g., Fibonacci numeration system. 3. Get information on the minimal automaton A U recognizing rep U (N).

69 First results Theorem (C-Rampersad-Rigo-Waxweiler 21) Let U be a linear PNS such that rep U (N) is regular. (i) The automaton A U has a non-trivial strongly connected component C U containing the initial state. (ii) If p is a state in C U, then there exists N N such that δ U (p, n ) = q U, for all n N. In particular, one cannot leave C U by reading a.

70 Theorem (cont d.) (iii) If C U is the only non-trivial strongly connected component of A U, then lim n + U n+1 U n = +. (iv) If lim n + U n+1 U n = +, then δ U (q U,,1) is in C U.

71 Dominant root condition U satisfies the dominant root condition if lim n + U n+1/u n = β for some real β > 1. β is the dominant root of the recurrence. E.g., Fibonacci: dominant root β = (1+ 5)/2 Theorem (cont d.) Suppose U has a dominant root β > 1. (v) If A U has more than one non-trivial strongly connected component, then any such component other than C U is a cycle all of whose edges are labeled. (vi) If lim n + U n+1/u n = β, then there is only one non-trivial strongly connected component.

72 An example with two components Let t 1. Let U = 1, U tn+1 = 2U tn +1, and U tn+r = 2U tn+r 1, for 1 < r t. E.g., for t = 2 we have U = (1,3,6,13,26,53,...). Then rep U (N) = {,1} {,1} 2( t ). The second component is a cycle of t s.,1 2

73 Theorem (Hollander 1998) If U is a linear PNS has a dominant root β and if rep U (N) is regular, then β is a Parry number. With any Parry number β is associated a canonical finite automaton A β. We will study the relationship between A U and A β.

74 β-expansions Let β > 1 be a real number. The β-expansion of a real number x [,1] is the lexicographically greatest sequence d β (x) := (x i ) i 1 over {,..., β } satisfying x = x i β i. i=1

75 Parry numbers If d β (1) = t 1 t m ω, with t m, then d β (1) is finite. In this case d β (1) := (t 1 t m 1 (t m 1)) ω. For instance, d 2 (1) = 2 ω and d 2 (1) = 1ω. Otherwise d β (1) := d β(1). If d β (1) is ultimately periodic, then β is a Parry number.

76 The Parry automaton Theorem (Parry 196) A sequence (s i ) i 1 over N is the β-expansion of a real number in [,1) iff (s n+i ) i 1 is lexicographically less than d β (1) for all n N. Define D β to be the set of all β-expansions of real numbers in [,1). So, for β Parry, the language Fact(D β ) is regular. For β Parry, let A β be the minimal DFA accepting Fact(D β ).

77 An example of the automaton A β,1 2 Let β be the largest root of X 3 2X 2 1. d β (1) = 21 ω and d β (1) = (2)ω. This automaton also accepts rep U (N) for U defined by U n+3 = 2U n+2 +U n,(u,u 1,U 2 ) = (1,3,7). A U = A β

78 Bertrand numeration systems Bertrand numeration system: w rep U (N) w rep U (N). Example (The l-bonacci system is Bertrand.) U n+l = U n+l 1 +U n+l 2 + +U n U i = 2 i, i {,...,l 1} A U accepts all words that do not contain 1 l.

79 A non-bertrand system 1 2 U n+2 = U n+1 +U n,(u = 1,U 1 = 3) (U n ) n = 1,3,4,7,11,18,29,47,... 2 is a greedy representation but 2 is not.

80 Theorem (Bertrand 1989) A PNS U is Bertrand iff there is a β > 1 such that rep U (N) = Fact(D β ). Moreover, the system is derived from the β-development of 1. If β is a Parry number, then U is linear and we have a minimal finite automaton A β accepting Fact(D β ). Consequently, rep U (N) is regular and A U = A β.

81 Back to a previous example,1 2 Let β be the largest root of X 3 2X 2 1. d β (1) = 21 ω and d β (1) = (2)ω. This automaton accepts rep U (N) for U defined by U n+3 = 2U n+2 +U n,(u,u 1,U 2 ) = (1,3,7). A U = A β

82 Changing the initial conditions 3,4 a 2 d f g 1 b 1 c e U n+3 = 2U n+2 +U n,(u,u 1,U 2 ) = (1,3,7) We change the initial values to (U,U 1,U 2 ) = (1,5,6). A U A β

83 Relationship with A β Theorem (cont d.) Suppose U has a dominant root β > 1. There is a morphism of automata Φ from C U to A β. Φ maps the states of C U onto the states of A β so that Φ(q U, ) = q β,, for all states q and all letters σ s.t. q and δ U (q,σ) are in C U, we have Φ(δ U (q,σ)) = δ β (Φ(q),σ).

84 3,4 a 2 d f g 1 b 1 c e,1 2

85 Other results When U has a dominant root β > 1, we can say more. E.g., if A U has more than one non-trivial strongly connected component, then d β (1) is finite. We can also give sufficient conditions for A U to have more than one non-trivial strongly connected component. In addition, we can give an upper bound on the number of non-trivial strongly connected components. When U has no dominant root, the situation is more complicated.

86 A system with no dominant root,1,2,3, ,1,2, ,3 2,1 2,1 2,1,2 U n+3 = 24U n,(u,u 1,U 2 ) = (1,2,6) 3 non-trivial strongly connected components

87 A system with no dominant root 2 1, U n+4 = 3U n+2 +U n,(u,u 1,U 2,U 3 ) = (1,2,3,7) U n+1 /U n doesn t converge, but lim U 2n+2/U 2n = lim U 2n+3/U 2n+1 = (3+ 13)/2 n + n + M. Hollander, Greedy numeration systems and regularity, Theory Comput. Systems 31 (1998).

88 Application to the l-bonacci system Corollary For U the l-bonacci numeration system, the number of states of the trim minimal automaton accepting rep U (mn) is lm l.

89 1 1 1 rep F (2N) ε

90 Further work in this area Analyze the structure of A U for systems with no dominant root. Remove the assumption that (U i mod m) i is purely periodic in the state complexity result that we have. Get analogous syntactic complexity or radius complexity results.

91 Real numbers in ANS

92 The decimal representation of is.(846153)ω : 8 1, 84 1, 846 1, , ,... n-th fraction = val 1(prefix of length n of (846153) ω ) 1 n

93 L Σ, u L (n) = Card(L Σ n ); v L (n) = Card(L Σ n ) = n u L (i). i= For the integer base b 2: L b = {ε} {1,...,b 1}{,...,b 1} n v Lb (n) = u Lb (i) = b n. i=

94 The decimal representation of is.(846153)ω : 8 1, 84 1, 846 1, , ,... n-th fraction = val 1(prefix of length n of (846153) ω ) v L1 (n)

95 The binary representation of is.(111111)ω : 1 2, 3 4 = 6 8, 13 16, = = = , 433 = , n-th fraction = val 2(prefix of length n of (111111) ω ) v L2 (n) 7-th fraction: 18 = = val 2 (1111) 128 = 2 7 = v L2 (7).

96 Lecomte and Rigo S = (L,Σ,<) w Σ ω (w (n) ) n L N, w (n) w as n + POINT: To show that, under certain hypotheses, the limit val S (w (n) ) lim n + v L ( w (n) ) exists and only depends on w. In that case, w is an S-representation of the corresponding real.

97 QUESTION: And when L is not regular? Example The 3 2-number system introduced by Akiyama, Frougny and Sakarovitch (28) has a numeration language which is not context-free. AIM: To provide a unified approach for representing real numbers

98 Generalization to non-regular languages Arbitrary infinite language L (not necessarily regular) Minimal automaton of L: A = (Q,Σ,δ,q,F) Generalized ANS: S = (L,Σ,<) For all x L, the numerical value val S (x) of x is given by x 1 v L ( x 1)+ i= a<x[i] where x[,i 1] = prefix of length i of x and L q = language accepted from q in A. u Lδ(q,x[,i 1]a) ( x i 1),

99 w = limit of words in L Pref(w) Pref(L) w Adh(L) Since Adh(L) = Adh(Pref(L)), there is no new representation if we assume that L is prefix-closed.

100 Example: L = {w {a,b} w a w b 1} = {ε,a,b,ab,ba,aab,aba,abb,baa,bab,bba,aabb,...} a b -3 a a a a a a a b b b b b b b For S = (L,{a,b},a < b), we can compute val S ((ab) n ) lim = 3 n + v L (2n) 4 val S ((ab) n a) and lim n + v L (2n+1) = 3 5 val S ((ab) ω [,n 1]) which shows that lim n + v L (n) does not exist. L not prefix-closed: Pref(L) = {a,b}

101 Hypotheses needed? (H1) L is prefix-closed (H2) Adh(L) is uncountable QUESTION: What conditions must L satisfy so that val S (w[,n 1]) the limits lim n + v L (n) exist for all w Adh(L)?

102 Hypotheses needed? AIM: Define some approximation intervals of reals. Their length should decrease as the prefix that is read becomes larger and larger. x L Σ n, v L (n 1) val S(x) v L (n) }{{} v L (n) vl(n) v L (n) }{{} =1 u L (n) =1 v L (n) u L (n) If lim n + v L (n) exists, then it is denoted by r ε and the represented interval is I ε = [1 r ε,1].

103 Hypotheses needed? Recall that, for all x L, x 1 val S (x) = v L ( x 1)+ i= a<x[i] u Lδ(q,x[,i 1]a)( x i 1) (H3) x Σ, r x, lim n + u Lδ(q,x) (n x ) v L (n) = r x

104 Hypotheses needed? In general, I x = r x (H4) w Adh(L), lim l + r w[,l 1] = Let Center(L) = Pref(Adh(L)). Then x Center(L) r x =.

105 4 hypotheses Theorem (C.-Le Gonidec-Rigo 21) val S (w[,n 1]) The limits lim n + v L (n) the following conditions: exist when L satisfies (H1) L prefix-closed (H2) Adh(L) uncountable (H3) x Σ u Lδ(q,x), r x, lim (n x ) n + v L (n) (H4) w Adh(L), lim l + r w[,l 1] = = r x

106 val S (w[,n 1]) For all w Adh(L), val S (w) = lim n + v L (n) is the numerical value of w. The infinite word w is an S-representation of val S (w).

107 Proposition (C.-Le Gonidec-Rigo 21) Let L Σ, S = (Pref(L),Σ,<) be a (generalized) ANS. If Pref(L) satisfies (H1), (H2) and (H3), then for all sequences (w (n) ) n L N converging to a word w Adh(L), we have lim n + val S (w (n) ) v Pref(L) ( w (n) ) = val S(w).

108 Example: Prefixes of Dyck words D = {w {a,b} w a = w b and u Pref(w), u a u b } not prefix-closed we consider S = (Pref(D),{a,b},a < b) Pref(D) = {w {a,b} u Pref(w), u a u b } = {ε,a,aa,ab,aaa,aab,aba,aaaa,aaab,aaba,aabb,...}. a,b -1 b a a a b b b a b

109 Example: Prefixes of Dyck words (continued) val S ((aab) ω [,n 1]) lim = 39 n + v L (n) 49 = x val S (x) v L ( x ) val S (x) v L ( x ) a aa aab aaba aabaa aabaab aabaaba aabaabaa aabaabaab aabaabaaba aabaabaabaa

110 Example: Prefixes of Dyck words (continued) v L (n 1) Since lim = 1 n + v L (n) 2, we represent the interval I ε = [ 1 2,1] Center(Pref(D)) = Pref(D): I a = [1/2,1] I aa = [1/2,7/8] I ab = [7/8,1] I aaa = [1/2,3/4] I aab = [3/4,7/8] I aba = [7/8,1]...

111 Example: Prefixes of Dyck words (continued) x [ 1 2,1], Q x designates the set of representations of x. We have Q 1/2 = {a ω } and Q 1 = {(ab) ω }. If x ]1/2,1[ and x = supi w = infi z then Q x = { w(ab) ω,za ω }, where w = the least Dyck word having w as a prefix. Proposition (C.-Le Gonidec-Rigo 21) If L is context-free, then the representations of the endpoints of the intervals are ultimately periodic.

112 Open problems Characterize the automata recognizing a language L such that the corresponding ω-language Adh(L) is uncountable.

113 Open problems Theorem (Boasson-Nivat 198) For every context-free language L, there exists a sequential mapping f such that f(adh(d 2 )) = Adh(L), where D 2 is the Dyck language over two kinds of parentheses. Let S and T be abstract numeration systems built respectively on Pref(D 2 ) and Pref(L). Give a mapping g such that the following diagram commutes. Adh(D 2 ) f Adh(L) val S val T g [s,1] [t,1]