(ABSTRACT) NUMERATION SYSTEMS

Transcription

1 (ABSTRACT) NUMERATION SYSTEMS Michel Rigo Department of Mathematics, University of Liège

2 OUTLINE OF THE TALK WHAT IS A NUMERATION SYSTEM? CONNECTION WITH FORMAL LANGUAGES THEORY BASE DEPENDENCE CHARACTERIZATIONS OF k-recognizable SETS SOME PUZZLING PROPERTIES OF P.-T.-M. SEQUENCE LET S COME BACK TO U-RECOGNIZABILITY MOTIVATION FOR A GENERALIZATION ABSTRACT NUMERATION SYSTEMS FIRST RESULTS REPRESENTING REAL NUMBERS

3 WHAT IS A NUMERATION SYSTEM? INTEGER BASE NUMERATION SYSTEM, k 2 l n = c i k i, with c i Σ k = {0,..., k 1}, c l 0 i=0 Any integer n corresponds to a word rep k (n) = c l c 0 over Σ k. (NON-STANDARD) SYSTEM BUILT UPON A SEQUENCE U = (U i ) i 0 OF INTEGERS l n = c i U i, with c l 0 greedy expansion i=0 Any integer n corresponds to a word rep U (n) = c l c 0.

4 WHAT IS A NUMERATION SYSTEM? INTEGER BASE NUMERATION SYSTEM, k 2 l n = c i k i, with c i Σ k = {0,..., k 1}, c l 0 i=0 Any integer n corresponds to a word rep k (n) = c l c 0 over Σ k. (NON-STANDARD) SYSTEM BUILT UPON A SEQUENCE U = (U i ) i 0 OF INTEGERS l n = c i U i, with c l 0 greedy expansion i=0 Any integer n corresponds to a word rep U (n) = c l c 0.

5 SOME CONDITIONS ON U = (U i ) i 0 U i+1 < U i, non-ambiguity U 0 = 1, any integer can be represented U i+1 U i is bounded, finite alphabet of digits A U EXAMPLE (U i = 2 i+1 : 2, 4, 8, 16, 32,...) you cannot represent odd integers! EXAMPLE (U i = (i + 1)! : 1, 2, 6, 24,...) Any integer n can be uniquely written as n = l c i i! with 0 c i i i=0 Fraenkel 85, Lenstra 06 (EMS Newsletter, profinite numbers)

6 SOME CONDITIONS ON U = (U i ) i 0 U i+1 < U i, non-ambiguity U 0 = 1, any integer can be represented U i+1 U i is bounded, finite alphabet of digits A U EXAMPLE (U i = 2 i+1 : 2, 4, 8, 16, 32,...) you cannot represent odd integers! EXAMPLE (U i = (i + 1)! : 1, 2, 6, 24,...) Any integer n can be uniquely written as n = l c i i! with 0 c i i i=0 Fraenkel 85, Lenstra 06 (EMS Newsletter, profinite numbers)

7 SOME CONDITIONS ON U = (U i ) i 0 U i+1 < U i, non-ambiguity U 0 = 1, any integer can be represented U i+1 U i is bounded, finite alphabet of digits A U EXAMPLE (U i = 2 i+1 : 2, 4, 8, 16, 32,...) you cannot represent odd integers! EXAMPLE (U i = (i + 1)! : 1, 2, 6, 24,...) Any integer n can be uniquely written as n = l c i i! with 0 c i i i=0 Fraenkel 85, Lenstra 06 (EMS Newsletter, profinite numbers)

8 A NICE SETTING Take (U i ) i 0 satisfying a linear recurrence equation, U i+k = a k 1 U i+k a 0 U i, a j Z, a 0 0. EXAMPLE (U i+2 = U i+1 + U i, U 0 = 1, U 1 = 2) Use greedy expansion,..., 21, 13, 8, 5, 3, 2, The pattern 11 is forbidden, A U = {0, 1}.

9 WHAT USE FOR NUMERATION SYSTEMS? Number theory, Transcendence Combinatorics on words Automatic sequences, Digital sequences Sloane and Plouff s encyclopedia of integer sequences Theoretical computer science, Automata theory Formal languages theory Logic, Complexity theory Symbolic dynamics Ergodic theory Graph theory, Game theory Physics, Quasi-crystals, Fractal geometry

10 CONNECTION WITH FORMAL LANGUAGES THEORY FACT Any n 0 is represented by a word rep U (n) over A U. A set X N corresponds to a set of words, i.e., a language. The Chomsky s hierarchy : Recursively enumerable languages (Turing Machine) Context-sensitive languages (linear bounded T.M.) Context-free languages (pushdown automaton) Regular (or rational) languages (finite automaton)

11 CONNECTION WITH FORMAL LANGUAGES THEORY FACT Any n 0 is represented by a word rep U (n) over A U. A set X N corresponds to a set of words, i.e., a language. The Chomsky s hierarchy : Recursively enumerable languages (Turing Machine) Context-sensitive languages (linear bounded T.M.) Context-free languages (pushdown automaton) Regular (or rational) languages (finite automaton)

12 EASIEST MODEL OF COMPUTATION DETERMINISTIC FINITE AUTOMATON A = (Q, q 0, Σ, δ, F) Q finite set of states, q 0 Q initial state δ : Q Σ Q transition function F Q set of final (or accepting) states EXAMPLE (FIBONACCI) ,1

13 EXAMPLE (USE IN BIO-INFORMATICS, DNA: a,c,g,t) g,c,t g,c,t c,t a g,c,t a c g a g a a t g a agata c,t EXAMPLE (USE IN COMPUTER SCIENCE) Complete algorithmic solution for model checking, program verification,...

14 WHAT ARE WE LOOKING FOR? The simplest sets X of integers are the ones such that rep U (X) is regular, i.e., representations are accepted by some finite automaton. Such sets are said to be U-recognizable. DIVISIBILITY CRITERION IN BASE k Let k 2. The set X = {n n r (mod s)} is k-recognizable. EXAMPLE 1,2,3,4,6,7,8,9 0,5 0,5 0,3,6,9 1,4,7 0 2,5,8 1 2,5,8 1,4,7 2,5,8 1,4,7 2 0,3,6,9 1,2,3,4,6,7,8,9 0,3,6,9

15 WHAT ARE WE LOOKING FOR? The simplest sets X of integers are the ones such that rep U (X) is regular, i.e., representations are accepted by some finite automaton. Such sets are said to be U-recognizable. DIVISIBILITY CRITERION IN BASE k Let k 2. The set X = {n n r (mod s)} is k-recognizable. EXAMPLE 1,2,3,4,6,7,8,9 0,5 0,5 0,3,6,9 1,4,7 0 2,5,8 1 2,5,8 1,4,7 2,5,8 1,4,7 2 0,3,6,9 1,2,3,4,6,7,8,9 0,3,6,9

16 WHAT ARE WE LOOKING FOR? The simplest sets X of integers are the ones such that rep U (X) is regular, i.e., representations are accepted by some finite automaton. Such sets are said to be U-recognizable. DIVISIBILITY CRITERION IN BASE k Let k 2. The set X = {n n r (mod s)} is k-recognizable. EXAMPLE 1,2,3,4,6,7,8,9 0,5 0,5 0,3,6,9 1,4,7 0 2,5,8 1 2,5,8 1,4,7 2,5,8 1,4,7 2 0,3,6,9 1,2,3,4,6,7,8,9 0,3,6,9

17 BASE DEPENDENCE A NATURAL QUESTION If X N is p-recognizable, is it also q-recognizable? p, q are multiplicatively independent if p k = q l k = l = 0, i.e., log p/ log q is irrational. being multiplicatively dependent is an equivalence relation, PROPOSITION If p, q 2 are multiplicatively dependent, then X N is p-recognizable iff it is q-recognizable.

18 BASE DEPENDENCE A NATURAL QUESTION If X N is p-recognizable, is it also q-recognizable? p, q are multiplicatively independent if p k = q l k = l = 0, i.e., log p/ log q is irrational. being multiplicatively dependent is an equivalence relation, PROPOSITION If p, q 2 are multiplicatively dependent, then X N is p-recognizable iff it is q-recognizable.

19 BASE DEPENDENCE A NATURAL QUESTION If X N is p-recognizable, is it also q-recognizable? p, q are multiplicatively independent if p k = q l k = l = 0, i.e., log p/ log q is irrational. being multiplicatively dependent is an equivalence relation, PROPOSITION If p, q 2 are multiplicatively dependent, then X N is p-recognizable iff it is q-recognizable.

20 BASE DEPENDENCE A NATURAL QUESTION If X N is p-recognizable, is it also q-recognizable? p, q are multiplicatively independent if p k = q l k = l = 0, i.e., log p/ log q is irrational. being multiplicatively dependent is an equivalence relation, PROPOSITION If p, q 2 are multiplicatively dependent, then X N is p-recognizable iff it is q-recognizable.

21 THEOREM (COBHAM 69) Let p, q 2 be two multiplicatively independent integers. If X N is both p- and q-recognizable, then X is ultimately periodic (finite union of A. P.). COROLLARY There exists sets which are k-recognizable for any k (ultimately periodic sets), k-recognizable for some (minimal) k and exactly all the k m, k-recognizable for no k. EXAMPLE The set of even integers is k-recognizable for any k. The set {2 n n 0} is 2-recognizable but not 3-recognizable. The set of primes is never k-recognizable.

22 THEOREM (COBHAM 69) Let p, q 2 be two multiplicatively independent integers. If X N is both p- and q-recognizable, then X is ultimately periodic (finite union of A. P.). COROLLARY There exists sets which are k-recognizable for any k (ultimately periodic sets), k-recognizable for some (minimal) k and exactly all the k m, k-recognizable for no k. EXAMPLE The set of even integers is k-recognizable for any k. The set {2 n n 0} is 2-recognizable but not 3-recognizable. The set of primes is never k-recognizable.

23 S. Eilenberg 74 (p.118) The proof is correct, long and hard. It is a challenge to find a more reasonable proof of this fine theorem {p m /q n m, n 0} is dense in [0, + ) VARIOUS PROOF SIMPLIFICATIONS AND GENERALIZATIONS G. Hansel 82, D. Perrin 90, F. Durand 05, V. Bruyère 97, F. Point, C. Michaux 96, R. Villemaire, A. Bès 00, J. Bell 05, J. Honkala, S. Fabre, C. Reutenauer, A.L. Semenov 77, L. Waxweiler 06, M.R

24 S. Eilenberg 74 (p.118) The proof is correct, long and hard. It is a challenge to find a more reasonable proof of this fine theorem {p m /q n m, n 0} is dense in [0, + ) VARIOUS PROOF SIMPLIFICATIONS AND GENERALIZATIONS G. Hansel 82, D. Perrin 90, F. Durand 05, V. Bruyère 97, F. Point, C. Michaux 96, R. Villemaire, A. Bès 00, J. Bell 05, J. Honkala, S. Fabre, C. Reutenauer, A.L. Semenov 77, L. Waxweiler 06, M.R

25 CHARACTERIZATIONS OF k-recognizable SETS THEOREM (J.R. BÜCHI 60) k-recognizable sets are exactly the sets definable by first order formula in the extended Presburger arithmetic N, +, V k. EXAMPLE φ(x) ( y)(x = y + y), X = {x N N, +, V k = φ(x)}. THEOREM (G. CHRISTOL, T. KAMAE, M. MENDÈS FRANCE, G. RAUZY 80) Let p prime. A set S is p-recognizable iff the formal power series S(X) = n 0 χ S (n) X n where χ S (n) = 1 n S, is algebraic over F p (X) (i.e., root of a polynomial over F p [X]).

26 CHARACTERIZATIONS OF k-recognizable SETS THEOREM (J.R. BÜCHI 60) k-recognizable sets are exactly the sets definable by first order formula in the extended Presburger arithmetic N, +, V k. EXAMPLE φ(x) ( y)(x = y + y), X = {x N N, +, V k = φ(x)}. THEOREM (G. CHRISTOL, T. KAMAE, M. MENDÈS FRANCE, G. RAUZY 80) Let p prime. A set S is p-recognizable iff the formal power series S(X) = n 0 χ S (n) X n where χ S (n) = 1 n S, is algebraic over F p (X) (i.e., root of a polynomial over F p [X]).

27 THEOREM (COBHAM 72) k-recognizable sets = k-automatic sets EXAMPLE ((PROUHET)-THUE-MORSE SEQUENCE) f : a ab, b ba, g : a 0, b 1. f 0 (a) f 1 (a) f 2 (a) f 3 (a) f 4 (a). a ab abba abbabaab abbabaabbaababba. t = is 2-automatic : generated by an iterated morphism of constant length 2.

28 EXAMPLE (CONT.) t n = 1 iff rep 2 (n) contains an odd number of 1 s, ε This set is 2-recognizable,

29 Jean-Paul Allouche, Jeffrey Shallit 04

30 SOME PUZZLING PROPERTIES OF P.-T.-M. SEQUENCE PROUHET S PROBLEM 1851 Find a partition of A n = {0, 1, 2, 3,..., 2 n 1} into two sets I = {i 1,..., i 2 n 1} and J = {j 1,..., j 2 n 1} such that (Multi-grades) : i I i = j J j i I i2 = j J j2 i I in 1. = j J jn 1 n = 3 : , I = {0, 3, 5, 6} and J = {1, 2, 4, 7} = 14 = and = 70 =

31 QUESTION What is the sign of f n (x) = sin x sin 2x sin 4x sin 2 n x over [0, π]?

32 QUESTION What is the sign of f n (x) = sin x sin 2x sin 4x sin 2 n x over [0, π]?

33 QUESTION What is the sign of f n (x) = sin x sin 2x sin 4x sin 2 n x over [0, π]?

34 QUESTION What is the sign of f n (x) = sin x sin 2x sin 4x sin 2 n x over [0, π]?

35 n = 0 + n = 1 + n = n = A word w is an overlap if w = avava, a is a letter, w = avava = avava. THEOREM (MORSE-HEDLUND) The infinite word t is overlap-free. Recurrent geodesics on a surface of negative curvature (Morse 1921, Morse-Hedlund ).

36 n = 0 + n = 1 + n = n = A word w is an overlap if w = avava, a is a letter, w = avava = avava. THEOREM (MORSE-HEDLUND) The infinite word t is overlap-free. Recurrent geodesics on a surface of negative curvature (Morse 1921, Morse-Hedlund ).

37 LET S COME BACK TO U-RECOGNIZABILITY QUESTION Let U = (U i ) i 0 be a strictly increasing sequence of integers, is the whole set N U-recognizable? i.e., is L U = rep U (N) regular? Even if U is linear, the answer is not completely known... THEOREM (SHALLIT 94) If L U is regular, then (U i ) i 0 satisfies a linear recurrent equation. THEOREM (N. LORAUD 95, M. HOLLANDER 98) They give (technical) sufficient conditions for L U to be regular: the characteristic polynomial of the recurrence has a special form.

38 LET S COME BACK TO U-RECOGNIZABILITY QUESTION Let U = (U i ) i 0 be a strictly increasing sequence of integers, is the whole set N U-recognizable? i.e., is L U = rep U (N) regular? Even if U is linear, the answer is not completely known... THEOREM (SHALLIT 94) If L U is regular, then (U i ) i 0 satisfies a linear recurrent equation. THEOREM (N. LORAUD 95, M. HOLLANDER 98) They give (technical) sufficient conditions for L U to be regular: the characteristic polynomial of the recurrence has a special form.

39 LET S COME BACK TO U-RECOGNIZABILITY QUESTION Let U = (U i ) i 0 be a strictly increasing sequence of integers, is the whole set N U-recognizable? i.e., is L U = rep U (N) regular? Even if U is linear, the answer is not completely known... THEOREM (SHALLIT 94) If L U is regular, then (U i ) i 0 satisfies a linear recurrent equation. THEOREM (N. LORAUD 95, M. HOLLANDER 98) They give (technical) sufficient conditions for L U to be regular: the characteristic polynomial of the recurrence has a special form.

40 BEST KNOWN CASE : LINEAR PISOT SYSTEMS If the characteristic polynomial of (U i ) i 0 is the minimal polynomial of a Pisot number θ then everything is fine: L U is regular, addition preserves recognizability, logical first order characterization of recognizable sets,... Just like in the integer case : U i θ i. A. Bertrand 89, C. Frougny, B. Solomyak, D. Berend, J. Sakarovitch, V. Bruyère and G. Hansel 97,... DEFINITION A Pisot (resp. Salem, Perron) number is an algebraic integer α > 1 such that its Galois conjugates have modulus < 1 (resp. 1, < α).

41 BEST KNOWN CASE : LINEAR PISOT SYSTEMS If the characteristic polynomial of (U i ) i 0 is the minimal polynomial of a Pisot number θ then everything is fine: L U is regular, addition preserves recognizability, logical first order characterization of recognizable sets,... Just like in the integer case : U i θ i. A. Bertrand 89, C. Frougny, B. Solomyak, D. Berend, J. Sakarovitch, V. Bruyère and G. Hansel 97,... DEFINITION A Pisot (resp. Salem, Perron) number is an algebraic integer α > 1 such that its Galois conjugates have modulus < 1 (resp. 1, < α).

42 MOTIVATION FOR A GENERALIZATION A QUESTION After G. Hansel s talk during JM 94 in Mons and knowing Shallit s results, P. Lecomte has the following question : Everybody takes first a sequence (U k ) k 0 then ask for the language L U of the numeration to be regular and play with recognizable sets Why not proceed backwards? REMARK Let x, y N, x < y rep U (x) < gen rep U (y). EXAMPLE (FIBONACCI) 6 < 7 and 1001 < gen 1010 (same length) 6 < 8 and 1001 < gen (different lengths).

43 MOTIVATION FOR A GENERALIZATION A QUESTION After G. Hansel s talk during JM 94 in Mons and knowing Shallit s results, P. Lecomte has the following question : Everybody takes first a sequence (U k ) k 0 then ask for the language L U of the numeration to be regular and play with recognizable sets Why not proceed backwards? REMARK Let x, y N, x < y rep U (x) < gen rep U (y). EXAMPLE (FIBONACCI) 6 < 7 and 1001 < gen 1010 (same length) 6 < 8 and 1001 < gen (different lengths).

44 MOTIVATION FOR A GENERALIZATION A QUESTION After G. Hansel s talk during JM 94 in Mons and knowing Shallit s results, P. Lecomte has the following question : Everybody takes first a sequence (U k ) k 0 then ask for the language L U of the numeration to be regular and play with recognizable sets Why not proceed backwards? REMARK Let x, y N, x < y rep U (x) < gen rep U (y). EXAMPLE (FIBONACCI) 6 < 7 and 1001 < gen 1010 (same length) 6 < 8 and 1001 < gen (different lengths).

45 MOTIVATION FOR A GENERALIZATION A QUESTION After G. Hansel s talk during JM 94 in Mons and knowing Shallit s results, P. Lecomte has the following question : Everybody takes first a sequence (U k ) k 0 then ask for the language L U of the numeration to be regular and play with recognizable sets Why not proceed backwards? REMARK Let x, y N, x < y rep U (x) < gen rep U (y). EXAMPLE (FIBONACCI) 6 < 7 and 1001 < gen 1010 (same length) 6 < 8 and 1001 < gen (different lengths).

46 MOTIVATION FOR A GENERALIZATION A QUESTION After G. Hansel s talk during JM 94 in Mons and knowing Shallit s results, P. Lecomte has the following question : Everybody takes first a sequence (U k ) k 0 then ask for the language L U of the numeration to be regular and play with recognizable sets Why not proceed backwards? REMARK Let x, y N, x < y rep U (x) < gen rep U (y). EXAMPLE (FIBONACCI) 6 < 7 and 1001 < gen 1010 (same length) 6 < 8 and 1001 < gen (different lengths).

47 MOTIVATION FOR A GENERALIZATION A QUESTION After G. Hansel s talk during JM 94 in Mons and knowing Shallit s results, P. Lecomte has the following question : Everybody takes first a sequence (U k ) k 0 then ask for the language L U of the numeration to be regular and play with recognizable sets Why not proceed backwards? REMARK Let x, y N, x < y rep U (x) < gen rep U (y). EXAMPLE (FIBONACCI) 6 < 7 and 1001 < gen 1010 (same length) 6 < 8 and 1001 < gen (different lengths).

48 ABSTRACT NUMERATION SYSTEMS DEFINITION (P. LECOMTE, M.R. 01) An abstract numeration system is a triple S = (L, Σ, <) where L is a regular language over a totally ordered alphabet (Σ, <). Enumerating the words of L with respect to the genealogical ordering induced by < gives a one-to-one correspondence rep S : N L val S = rep 1 S : L N.

49 FIRST RESULTS REMARK This generalizes classical Pisot systems like integer base systems or Fibonacci system. EXAMPLE (POSITIONAL) L = {ε} {1,..., k 1}{0,..., k 1} or L = {ε} 1{0, 01} EXAMPLE (NON POSITIONAL) Non positional numeration system : L = a b Σ = {a < b} n rep(n) ε a b aa ab bb aaa val(a p b q ) = 1 2 (p + q)(p + q + 1) + q

50 FIRST RESULTS REMARK This generalizes classical Pisot systems like integer base systems or Fibonacci system. EXAMPLE (POSITIONAL) L = {ε} {1,..., k 1}{0,..., k 1} or L = {ε} 1{0, 01} EXAMPLE (NON POSITIONAL) Non positional numeration system : L = a b Σ = {a < b} n rep(n) ε a b aa ab bb aaa val(a p b q ) = 1 2 (p + q)(p + q + 1) + q

51 EXAMPLE (CONTINUED...) #b #a

52 EXAMPLE (CONTINUED...) #b #a

53 EXAMPLE (CONTINUED...) #b #a

54 EXAMPLE (CONTINUED...) #b #a

55 EXAMPLE (CONTINUED...) #b #a

56 EXAMPLE (CONTINUED...) #b #a

57 EXAMPLE (CONTINUED...) #b #a

58 EXAMPLE (CONTINUED...) #b #a

59 MANY NATURAL QUESTIONS... What about S-recognizable sets? Are ultimately periodic sets S-recognizable for any S? For a given X N, can we find S s.t. X is S-recognizable? For a given S, what are the S-recognizable sets? Can we compute easily in these systems? Addition, multiplication by a constant,... Are these systems equivalent to something else? Any hope for a Cobham s theorem? Can we also represent real numbers? Number theoretic problems like additive functions? Dynamics, odometer, tilings, logic...

60 THEOREM Let S = (L, Σ, <) be an abstract numeration system. Any ultimately periodic set is S-recognizable. EXAMPLE (FOR a b MOD 3, 5, 6 AND 8)

61 WELL-KNOWN FACT (SEE EILENBERG S BOOK) The set of squares is never recognizable in any integer base system. EXAMPLE Let L = a b a c, a < b < c ε a b c aa ab ac bb cc aaa THEOREM If P Q[X] is such that P(N) N then there exists an abstract system S such that P(N) is S-recognizable.

62 WELL-KNOWN FACT (SEE EILENBERG S BOOK) The set of squares is never recognizable in any integer base system. EXAMPLE Let L = a b a c, a < b < c ε a b c aa ab ac bb cc aaa THEOREM If P Q[X] is such that P(N) N then there exists an abstract system S such that P(N) is S-recognizable.

63 WELL-KNOWN FACT (SEE EILENBERG S BOOK) The set of squares is never recognizable in any integer base system. EXAMPLE Let L = a b a c, a < b < c ε a b c aa ab ac bb cc aaa THEOREM If P Q[X] is such that P(N) N then there exists an abstract system S such that P(N) is S-recognizable.

64 Consider multiplication by a constant... THEOREM Let S = (a b, {a < b}). Multiplication by λ N preserves S-recognizability iff λ is an odd square. EXAMPLE There exists X 3 N such that X 3 is S-recognizable but such that 3X 3 is not S-recognizable. (3 is not a square) There exists X 4 N such that X 4 is S-recognizable but such that 4X 4 is not S-recognizable. (4 is an even square) For any S-recognizable set X N, 9X or 25X is also S-recognizable.

65 BOUNDED LANGUAGES a 1 a l Let S = (a 1 a l, {a 1 < < a l }). We have val(a n 1 1 an l l ) = l ( ) ni + + n l + l i. l i + 1 i=1 COROLLARY (KATONA S EXPANSION 66) Let l N \ {0}. Any integer n can be uniquely written as ( ) ( ) ( ) zl zl 1 z1 n = l l 1 1 with z l > z l 1 > > z 1 0.

66 THEOREM For the abstract numeration system S = (a b c, {a < b < c}), if β N \ {0, 1} is such that β ±1 (mod 6) then the multiplication by β 3 does not preserve the S-recognizability. CONJECTURE Multiplication by β l preserves S-recognizability for the abstract numeration system S = (a 1 a l, {a 1 < < a l }) iff β = where p 1,..., p k are prime numbers strictly greater than l. k i=1 p θ i i

67 Why just looking at multiplication by β l? DEFINITION OF COMPLEXITY Let A = (Q, q 0, F, Σ, δ), u q (n) = #{w Σ n δ(q, w) F} i.e., number of words of length n accepted from q in A. u q0 (n) = #(L Σ n ). THEOREM If u q0 (n) = #(L Σ n ) is in Θ(n k ) and if λ β k+1 then there exists X which is S-recognizable and such that λx is not.

68 THEOREM ( MULTIPLICATION BY A CONSTANT ) slender language u q0 (n) O(1) OK polynomial language u q0 (n) O(n k ) NOT OK exponential language with polynomial complement u q0 (n) 2 Ω(n) NOT OK exponential language with exponential complement u q0 (n) 2 Ω(n) OK? EXAMPLE Pisot systems belong to the last class.

69 EQUIVALENCE WITH NON CONSTANT LENGTH MORPHISMS EXAMPLE (CHARACTERISTIC SEQUENCE OF SQUARES) f : a abcd, b b, c cdd, d d, g : a, b 1, c, d 0. f ω (a) = abcdbcdddbcdddddbcdddddddbc g(f ω (a)) = Analogous to the Cobham s result from 72 THEOREM (A. MAES, M.R. 02) A set is morphic iff it is S-recognizable for some abstract system S.

70 EQUIVALENCE WITH NON CONSTANT LENGTH MORPHISMS EXAMPLE (CHARACTERISTIC SEQUENCE OF SQUARES) f : a abcd, b b, c cdd, d d, g : a, b 1, c, d 0. f ω (a) = abcdbcdddbcdddddbcdddddddbc g(f ω (a)) = Analogous to the Cobham s result from 72 THEOREM (A. MAES, M.R. 02) A set is morphic iff it is S-recognizable for some abstract system S.

71 REPRESENTING REAL NUMBERS EXAMPLE (BASE 10) π 3 = , , ,..., val(w n ) 10 n,... val(w) #{words of length w } THIS DESERVES NOTATION v q0 (n) = #(L Σ n ) = n u q0 (i). i=0

72 REPRESENTING REAL NUMBERS EXAMPLE (BASE 10) π 3 = , , ,..., val(w n ) 10 n,... val(w) #{words of length w } THIS DESERVES NOTATION v q0 (n) = #(L Σ n ) = n u q0 (i). i=0

73 REPRESENTING REAL NUMBERS EXAMPLE (BASE 10) π 3 = , , ,..., val(w n ) 10 n,... val(w) #{words of length w } THIS DESERVES NOTATION v q0 (n) = #(L Σ n ) = n u q0 (i). i=0

74 EXAMPLE (AVOID aa ON THREE LETTERS) b,c s a t a p b,c a,b,c w val(w) v q0 ( w ) val(w)/v q0 ( w ) bc bac babc babac bababc bababac babababc lim n val((ba) n c) v q0 (2n + 1) =

75 NUMERICAL VALUE OF A WORD w = w 1 w l L val(w) = l (θ q,i (w) + δ q,s ) u q ( w i) i=1 q Q where θ q,i (w) = #{σ < w i s.w 1 w i 1 σ = q} HYPOTHESES: FOR ALL STATE q OF M L, EITHER (i) N q N : n > N q, u q (n) = 0, or (ii) β q 1, P q (x) R[x], b q > 0 : lim n From automata theory, we have u q(n) P q(n)β n q = b q. β q0 β q and β q = β q0 deg(p q ) deg(p q0 )

76 NUMERICAL VALUE OF A WORD w = w 1 w l L val(w) = l (θ q,i (w) + δ q,s ) u q ( w i) i=1 q Q where θ q,i (w) = #{σ < w i s.w 1 w i 1 σ = q} HYPOTHESES: FOR ALL STATE q OF M L, EITHER (i) N q N : n > N q, u q (n) = 0, or (ii) β q 1, P q (x) R[x], b q > 0 : lim n From automata theory, we have u q(n) P q(n)β n q = b q. β q0 β q and β q = β q0 deg(p q ) deg(p q0 )

77 Let β = β q0 and for any state q, define lim n u q (n) P q0 (n)β n = a q Q(β), a q0 > 0 and a q could be zero. IF (w n ) n N IS CONVERGING TO W = W 1 W 2 THEN lim n val(w n ) v q0 ( w n ) = β 1 β 2 j=0 q Q a q a q0 (θ q,j+1 (W ) + δ q,s ) β j. REMARK [W. STEINER, M.R. 05] By normalizing we can specify the value of a q0, why not take a q0 = 1 1 β? Doing so, we obtain something close to β-expansion...

78 Let β = β q0 and for any state q, define lim n u q (n) P q0 (n)β n = a q Q(β), a q0 > 0 and a q could be zero. IF (w n ) n N IS CONVERGING TO W = W 1 W 2 THEN lim n val(w n ) v q0 ( w n ) = β 1 β 2 j=0 q Q a q a q0 (θ q,j+1 (W ) + δ q,s ) β j. REMARK [W. STEINER, M.R. 05] By normalizing we can specify the value of a q0, why not take a q0 = 1 1 β? Doing so, we obtain something close to β-expansion...

79 Let β = β q0 and for any state q, define lim n u q (n) P q0 (n)β n = a q Q(β), a q0 > 0 and a q could be zero. IF (w n ) n N IS CONVERGING TO W = W 1 W 2 THEN lim n val(w n ) v q0 ( w n ) = β 1 β 2 j=0 q Q a q a q0 (θ q,j+1 (W ) + δ q,s ) β j. REMARK [W. STEINER, M.R. 05] By normalizing we can specify the value of a q0, why not take a q0 = 1 1 β? Doing so, we obtain something close to β-expansion...

80 W = W 1 W 2 x = 1 β + α q0.w 1 W j 1 (W j ) β j j=1 where α q (σ) = τ<σ a q.τ. 1/ β 1 /β /β /β a s.d /β a s.a a s.b a s.c a s.ca a s.cb a s.cd β 2 β 2 β 2

81 This generalizes classical base 10 system : I I 1 2 I 9 1/10 2/10 3/10 4/10 8/10 9/10 1 I 30 I 31 4/10 39/100 31/100 3/10

82 This gives rises to several question... Which real number have an ultimately periodic representation?