exact arithmetical algorithms in real number systems

187 Pages·2015·3.05 MB·English

EXACT ARITHMETICAL ALGORITHMS
IN REAL NUMBER SYSTEMS
Petr K(cid:23)urka
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Prague september 2015
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Contents
Preface 5
1 Basic number systems 7
1.1 The decadic system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Redundancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Symbolic spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Positional systems for bounded intervals . . . . . . . . . . . . . . . . . . . . . . 12
1.5 Positional systems for the extended real line . . . . . . . . . . . . . . . . . . . . 15
1.6 Continued fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2 Symbolic dynamics 25
2.1 Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 The Cantor space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3 Redundant symbolic extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.4 Subshifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.5 So(cid:12)c subshifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.6 Labelled graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3 Matrices and transformations 43
3.1 Projective geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2 The extended real line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.3 Projective metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.5 Conjugated transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.6 Complex transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.7 Hyperbolic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.8 Disc transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.9 Isometric circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.10 Singular transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.11 Representing sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4 M(cid:127)obius number systems 69
4.1 Iterative systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2 Interval number systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.3 Partition number systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.4 So(cid:12)c expansion subshifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.5 So(cid:12)c number systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.6 The contraction and length quotients . . . . . . . . . . . . . . . . . . . . . . . . 87
4.7 Polygonal number systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3
4 CONTENTS
4.8 Discrete groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5 Arithmetical algorithms 95
5.1 Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.2 The unary algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.3 Bilinear tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.4 The binary algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.5 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.6 Rational functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6 Integer vectors and matrices 117
6.1 Determinant, norm and length . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.2 Rational number systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.3 Modular systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.4 Finite state transducers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.5 Bimodular systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.6 Binary continued fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
7 Algebraic number (cid:12)elds 133
7.1 Polynomials with rational coe(cid:14)cients . . . . . . . . . . . . . . . . . . . . . . . . 133
7.2 Extension (cid:12)elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
7.3 Computable ordered (cid:12)elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
7.4 Algebraic integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
7.5 Pisot and Salem numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.6 Positional systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
7.7 Positional arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
8 Transcendent and iterative algorithms 159
8.1 Pad(cid:19)e approximants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
8.2 Algebraic tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
8.3 The transcendent algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
8.4 Arithmetical expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
8.5 Iterative algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
Bibliography 179
Notation 183
Index 184
Preface
Current computers work with real numbers in the (cid:13)oating point format and the numbers are
rounded up after each arithmetical operation. This usually works quite well but there are cases
in which the successive roundings yield wrong results. Exact arithmetical algorithms, on the
other hand, work with real numbers speci(cid:12)ed to an arbitrary precision. The precision of the
result depends on the precision of the operands. The theory of exact real computation is based
on the concept of on-line algorithms whose inputs and outputs are in(cid:12)nite expansions of real
numbers. The algorithms work in a loop in in(cid:12)nite time but each (cid:12)nite pre(cid:12)x of the output is
computed in (cid:12)nite time from (cid:12)nite pre(cid:12)xes of the inputs.
The theory of on-line algorithms has been developped by Weihrauch [68]. The idea of
on-line arithmetical algorithms has been suggested in an unpublished manuscript of Gosper
[21] and developped by Kornerup and Matula [34] and Vuillemin [66]. On-line arithmetical
algorithms are treated in the PhD thesis of Potts [55] and in the last chapter of the monograph
of Kornerup and Matula [33]. The on-line algorithms do not work in the standard decadic
or binary systems but they do work in redundant systems, for example in positional number
systems whose number of digits is larger than the base.
The present book is a theoretical treatment of arbitrary precision on-line arithmetical algo-
rithms in general M(cid:127)obius number systems. To specify a M(cid:127)obius number system, we start with
a (cid:12)nite alphabet A of digits and to each digit we associate a M(cid:127)obius transformation. This is
a mapping of the form M(x) = ax+b. For example in a positional number system with base
cx+d
(cid:12) > 1, the linear transformation F (x) = x+a is associated to the digit a. Then we specify a
a (cid:12)
subshift (cid:6) (cid:18) A! of admissible in(cid:12)nite sequences of digits and the value mapping (cid:8) : (cid:6) ! R,
where R = R[f1g is the extended real line. The value mapping (cid:8) should be surjective and
continuous. This means that each number x 2 R should have its symbolic representation (an
in(cid:12)nite sequence of digits) u 2 (cid:6) such that (cid:8)(u) = x. Continuity means that the pre(cid:12)xes u
[0;n)
of u of length n give with increasing n ever better approximations to (cid:8)(u) = x.
The (cid:12)rst chapter is introductory and treats classical positional number systems and number
systems based on continued fractions. On these examples it is shown how the M(cid:127)obius trans-
formations are assocoated to the digits, how the value mapping (cid:8) : (cid:6) ! R is constructed and
how symbolic representations of real numbers are obtained.
The second chapter treats redundancy as a topological concept and shows that for every
compactmetricspaceX (inparticularforthespaceX = R)thereexistsaredundantcontinuous
surjective mapping (cid:8) : (cid:6) ! X, where (cid:6) is a symbolic space. The property of redundancy
implies that each continuous mapping G : X ! X has a symbolic representation, which is a
continuous mapping F : (cid:6) ! (cid:6) such that (cid:8) (cid:14) F = G (cid:14) (cid:8). In arithmetical algorithms, the
symbolic space (cid:6) is supposed to be a so(cid:12)c subshift recognizable by a (cid:12)nite automaton, so the
rest of the chapter deals with so(cid:12)c subshifts.
The third chapter explains basic ideas of projective geometry which gives insight into the
spaces connected with a number system. The extended real line R is identi(cid:12)ed with the one-
dimensional projective space P(R2) and the space of M(cid:127)obius transformations M(R) is identi(cid:12)ed
5
6 CONTENTS
with the three-dimensional space of projective matrices P(R2(cid:2)2). The geometrical properties
of M(cid:127)obius transformations are exposed with the use of hyperbolic geometry. Then we explain
the concept of representation of real numbers by a sequence of transformations: A sequence
M of real M(cid:127)obius transformations represents a real number x i(cid:11) x = lim M (z) for every
n n!1 n
complex number z with a nonzero imaginary part. In particular, if (cid:8) : (cid:6) ! R is the value
mapping of a number system and u = u u (cid:1)(cid:1)(cid:1) 2 (cid:6), then (cid:8)(u) is represented by a sequence of
0 1
transformations F = F (cid:14)(cid:1)(cid:1)(cid:1)(cid:14)F .
u[0;n) u0 un(cid:0)1
ThefourthchapterexposesthetheoryofM(cid:127)obiusnumbersystemsandshowsseveralmethods
how to construct suitable subshifts (cid:6) (cid:18) A! and suitable value mappings (cid:8) : (cid:6) ! R. A special
treatment is given to so(cid:12)c M(cid:127)obius number systems for which arithmetical algorithms work.
Several examples of so(cid:12)c number systems are given.
The (cid:12)fth chapter develops the calculus of bilinear tensors which represent binary arithmeti-
cal operations. Intervals are represented by projective matrices and operations with tensors and
intervals are based on matrix calculus. Based on this calculus we describe the unary algorithm
which computes a M(cid:127)obius transformation and the binary algorithm which computes a bilinear
tensor.
The sixth chapter treats number systems whose matrices have integer entries. In particular,
modular systems have transformations with unit determinant. We show that if the unary
algorithm computes a transformation with integer entries in a modular number system, then
the norm of the state matrices is bounded, so the computation can be carried out by a (cid:12)nite
state transducer. On the other hand, M(cid:127)obius transformations are the only rational functions
which can be computed by a (cid:12)nite state transducer.
The seventh chapter treats number systems with matrices whose entries are algebraic num-
bers. We review the theory of algebraic extension (cid:12)elds, algebraic integers and integral bases
and give classical results of Parry and Schmidt on positional number systems with algebraic
base (cid:12) > 1 (so called (cid:12)-systems introduced by R(cid:19)enyi [58]). Then we treat the algorithms of
parallel addition in positional number systems.
Theeigthchaptertreatsalgorithmswhichcomputetranscendentfunctionslikeex,lnx,tanx
or arctanx. We review the theory of Pad(cid:19)e approximants and the representation of transcendent
functions by general continued fractions. We introduce the concept of algebraic tensor T(x;y),
which for a (cid:12)xed y is a rational function of x and for a (cid:12)xed x is a M(cid:127)obius transformation of y.
We de(cid:12)ne the transcendent algorithm which works with these algebraic tensors and we show
that it computes transcendent functions which can be expressed by general continued fractions.
Finally we treat algorithms which compute arithmetical expressions and iterative algorithms
which compute stable (cid:12)xed points of real functions.
The treatment is elementary and self-contained. The basic prerequisite is linear algebra and
matrix calculus.
Chapter 1
Basic number systems
Real numbers are de(cid:12)ned as cuts of rational numbers or as limits of Cauchy sequences of
rational numbers. Alternatively, the space of real numbers is characterized axiomatically by
the property of completeness: it is the smallest complete metric space which contains rational
numbers. A real number is usually given by its expansion in the decadic number system. But
a number should be distiguished from its representation in any number system. The concept of
number is geometrical or analytical, the representation of a number is a combinatorial concept.
A real number can have many representations in a number system.
1.1 The decadic system
In the decadic number system, a real number is represented by an in(cid:12)nite word (a string
of letters or characters) u = su u (cid:1)(cid:1)(cid:1)u :u u u (cid:1)(cid:1)(cid:1), where s is either the sign (cid:0) or empty,
n n+1 (cid:0)1 0 1 2
n (cid:20) 0 is an integer, u 2 f0;1;2;3;4;5;6;7;8;9g are digits and : is the positional decimal point.
i
Such a word u represents the number
∑1
x = (cid:6) u (cid:1)10(cid:0)i(cid:0)1:
i
i=n
We say that x is the value of u or that u is an expansion of x and we write (cid:8)(u) = x. We
admit as expansions also (cid:12)nite words u = su u (cid:1)(cid:1)(cid:1)u :u u (cid:1)(cid:1)(cid:1)u , which represent the same
n n+1 (cid:0)1 ∑0 1 k
numbers as in(cid:12)nite words with trailing zeros: (cid:8)(u) = (cid:6) k u (cid:1) 10(cid:0)i(cid:0)1, e.g., (cid:8)(:2) = 1 or
i=n i 5
(cid:8)((cid:0)1:5) = (cid:0)3. A (cid:12)nite pre(cid:12)x u = su u (cid:1)(cid:1)(cid:1)u :u u (cid:1)(cid:1)(cid:1)u of an expansion u of x with k
2 jk n n+1 (cid:0)1 0 1 k(cid:0)1
decimal places gives an approximation of x:
∑1
9
j(cid:8)(u)(cid:0)(cid:8)(u )j (cid:20) 9(cid:1)10(cid:0)i(cid:0)1 = = 10(cid:0)k:
jk 10k+1(1(cid:0) 1 )
i=k 10
This is essential, since neither people nor computers can handle in(cid:12)nite expansions but only
their (cid:12)nite pre(cid:12)xes. To determine a real number, we have to give a rule or an algorithm which
generates arbitrarily long pre(cid:12)xes of its expansion. Accordingly, we say that a real number
is an algorithmic number if there exists an algorithm which computes its expansion to an
arbitrary number of decimal places. Algorithmic numbers include all rational numbers, all
algebraic numbers, which are solutions of algebraic equations with rational coe(cid:14)cients, and
many transcendent numbers like (cid:25) or e, which can be computed by power series.
The expansions are in(cid:12)nite words in the alphabet A = f0;1;2;3;4;5;6;7;8;9;:;(cid:0)g, which
contains besides the decimal digits also the positional point : and the negative sign (cid:0). We
7
8 CHAPTER 1. BASIC NUMBER SYSTEMS
denote by A(cid:3) the set of (cid:12)nite words (sequences) of letters of A and by A! the set of in(cid:12)nite
words. Not every in(cid:12)nite word of A! represents a real number: the sign (cid:0) can appear only
at the beginning and the decimal point : must occur exactly once. Thus the expansions must
satisfy certain syntactic rules, which may be expressed by a set of forbidden words
D = fa(cid:0) : a 2 Ag[f:u: : u 2 A(cid:3)g:
This means that an expansion cannot contain as a subword any letter a 2 A followed by the
minus sign (cid:0) and it cannot contain two positional points. Denote by (cid:6) the set of in(cid:12)nite
D
words which do not contain as a subword any forbidden word. We say that (cid:6) is the subshift
D
with the forbidden set D. In the subshift (cid:6) there are also words which do not contain any
D
positional point at all. We cannot forbid them, since we cannot detect this property in (cid:12)nite
pre(cid:12)xes. We assign the value in(cid:12)nity to such words provided they contain at least one nonzero
digit, and the value zero otherwise. We therefore extend the real line R by a point 1 at in(cid:12)nity
and obtain the extended real line R = R[f1g. Then the value mapping (cid:8) : (cid:6) ! R is
D
de(cid:12)ned on the whole (cid:6) . Some arithmetical operations are extended to R. We have a = 1
D 0
for a ̸= 0 and a(cid:6)1 = 1, a = 0 for a ̸= 1. On the other hand, 0, 1, 1(cid:6)1 are unde(cid:12)ned
1 0 1
(indeterminate) expressions.
The value mapping (cid:8) is surjective, i.e., each x 2 R has an expansion u 2 (cid:6) with (cid:8)(u) = x,
D
but it is not one-to-one. There are in(cid:12)nitely many expansions of 1 and some (cid:12)nite numbers
have two di(cid:11)erent expansions, for example 0:999(cid:1)(cid:1)(cid:1) = 1:000(cid:1)(cid:1)(cid:1). In fact a real number has two
in(cid:12)nite expansions i(cid:11) it has a (cid:12)nite expansion:
u (cid:1)(cid:1)(cid:1)u :u (cid:1)(cid:1)(cid:1)u u = u (cid:1)(cid:1)(cid:1)u :u (cid:1)(cid:1)(cid:1)u u 000(cid:1)(cid:1)(cid:1)
n (cid:0)1 0 m(cid:0)1 m n (cid:0)1 0 m(cid:0)1 m
= u (cid:1)(cid:1)(cid:1)u :u (cid:1)(cid:1)(cid:1)u (u (cid:0)1)999(cid:1)(cid:1)(cid:1)
n (cid:0)1 0 m(cid:0)1 m
This duplicity can be felt as an inconvenience but cannot be detected in (cid:12)nite pre(cid:12)xes and
cannot be avoided by forbidding (cid:12)nite words. In fact, such a duplicity or redundancy is
necessary to perform arithmetical operations on the expansions. If we are able to determine
real numbers x and y to an arbitrary precision, we would like to determine to an arbitrary
precision also their sum x + y or the results of other algebraic operations. This means that
the pre(cid:12)x of a length k of (the expansion of) x+y should depend only on the pre(cid:12)xes of some
length n of (the expansions of) the operands x and y. In the standard decadic system this
k
is not possible since the system is not redundant enough: the carries to the left propagate
through arbitrarily long intervals. Imagine that we try to add numbers 1 = 0:33333::: and
3
2 = 0:66666:::, but we do not know in advance their exact values. We can only inspect
3
arbitrarily long pre(cid:12)xes of their expansions. Then we are unable to determine the (cid:12)rst digit of
the sum. The (cid:12)rst digit would be zero if u +v < 9 for some i or 1 if u +v > 9 for some i.
i i i i
In our case neither alternative ever happens so we are never able to determine the (cid:12)rst digit of
the result.
1.2 Redundancy
To perform arithmetic operations on the expansions of real numbers, we need redundant po-
sitional systems, in which the number of digits is greater than the base. For example, the
decadic system can be extended with a digit which represents 10. Another possibility is the
decadic signed system with digits A = f5;4;3;2;1;0;1;2;3;4;5g, where n stands for (cid:0)n.
This system has 11 digits - one more than the base 10, and it has an additional advantage that
1.3. SYMBOLIC SPACES 9
x = :001022222011110012102222:::
y = :022112221111200022221122:::
u = :023134443122310034323344:::
v = :11123221121111013121232:::
z = :1121210120111102120212:::
Table 1.1: Addition in the extended binary system
the negative numbers can be expressed without the (cid:0) sign. Many numbers have an in(cid:12)nite
number of expansions, e.g.,
4
= 0:5555(cid:1)(cid:1)(cid:1) = 0:55! = 0:455! = 0:4455! = 0:44455! = (cid:1)(cid:1)(cid:1)
9
In computer arithmetic, positional system with other bases than 10 are frequently used.
The standard binary system has base (cid:12) = 2 and digits A = f0;1g. Because the number
of digits is the same as the base, there is no properly working addition algorithm either. The
extended binary system has digits f0;1;2g and the binary signed system has digits
A = f1;0;1g representing (cid:0)1;0;1. In both these systems arithmetic operations are algorithmic.
The result can be evaluated to an arbitrary precision provided we know with su(cid:14)cient precision
the operands.
Denote by ⌊a⌋ 2 Z the integer part of a real number a 2 R, so a(cid:0) 1 < ⌊a⌋ (cid:20) a. Denote
by mod (n) 2 f0;1g the parity of an integer n 2 Z, so mod (n) = 0 i(cid:11) n is even. We have
2 2∑ ∑
n = 2⌊n⌋+mod (n) for each n 2 Z. To add two numbers x = 1 x 2(cid:0)i(cid:0)1, y = 1 y 2(cid:0)i(cid:0)1
2 2 i=n i i=n i
in the extended binary system, we (cid:12)rst add them componentwise, so we obtain u = x +y 2
i i i
f0;1;2;3;4g for i (cid:21) n and u = 0 for i < n. Then we perform the carries to the left and
i
determine v by ⌊ ⌋
u
i+1
v = +mod (u );
i 2 i
2
so v 2 f0;1;2;3g and v 2 f0;1;2;3g! represents the same number as u:
i
∑1 ∑1 ∑1
v (cid:1)2(cid:0)i(cid:0)1 = ⌊ui+1⌋(cid:1)2(cid:0)i(cid:0)1 + mod (u )(cid:1)2(cid:0)i(cid:0)1
i 2 2 i
i=n(cid:0)1 i=n(cid:0)1 i=n
∑1 ( ) ∑1
= 2(cid:1)⌊ui⌋+mod (u ) 2(cid:0)i(cid:0)1 = u (cid:1)2(cid:0)i(cid:0)1:
2 2 i i
i=n i=n
W∑e1perfozrm(cid:1) 2t(cid:0)hie(cid:0)1ca=rry∑op1er(axtio+n oyn)ce(cid:1) 2m(cid:0)oi(cid:0)re1 aannddozbtadienpzeind=s⌊ovni2+l1y⌋o+nmxod2(vi)=2xf0x;1;2xg. Tahnuds
i=n(cid:0)2 i i=n i i i [i;i+2] i i+1 i+2
y = y y y . The algorithm has an additional advantage that it may be performed in
[i;i+2] i i+1 i+2
parallel in all positions i simultaneously. This may be much faster than the serial addition. An
example can be seen in Table 1.1. Parallel addition is treated in more detail in Section 7.7.
1.3 Symbolic spaces
The principle that (cid:12)nite pre(cid:12)xes of the expansions approximate the expanded numbers can be
expressed by the concept of continuity. We regard the set of in(cid:12)nite expansions as a symbolic
metric space. An alphabet A is a (cid:12)nite set with at least two elements, which are referred to
as lettres. Words of A are (cid:12)nite or in(cid:12)nite sequences u = u u ::: of letters of A. We denote by
0 1
An = fu = u (cid:1)(cid:1)(cid:1)u : u 2 Ag
0 n(cid:0)1 i
10 CHAPTER 1. BASIC NUMBER SYSTEMS
the set of words of length n. In particular, A0 = f(cid:21)g consists only of the empty word (cid:21). Denote
∪ ∪
by A(cid:3) = An the set of (cid:12)nite words, by A+ = An the set of nonempty (cid:12)nite words,
n(cid:21)0 n>0
and by
A! = fu = u u (cid:1)(cid:1)(cid:1) : u 2 Ag
0 1 i
the set of in(cid:12)nite words. The length of a word u = u :::u 2 An is denoted by juj = n
0 n(cid:0)1
and juj = 1 for u 2 A!. We say that v 2 A(cid:3) is a subword of u 2 A(cid:3) [A! and write v ⊑ u,
if v = u = u (cid:1)(cid:1)(cid:1)u for some 0 (cid:20) i (cid:20) j (cid:20) juj. The concatenation of words u;v 2 A(cid:3) is
[i;j) i j(cid:0)1
written as uv, so (uv) = u for i < juj and (uv) = v for i < jvj. The concatenation of
i i juj+i i
u 2 A+ with itself n times is written as un and the in(cid:12)nite concatenation of u with itself as
u! 2 A!. We say that u 2 A! is a periodic word if u = vw! for some preperiod v 2 A(cid:3) and
period w 2 A+. Given a set of forbidden words D (cid:18) A+, we denote by
(cid:6) = fu 2 A! : 8v ⊑ u;v ̸2 Dg
D
L = fu 2 A(cid:3) : 8v ⊑ u;v ̸2 Dg
D
the subshift and language of D, and by Ln = L \An. The distance of words u;v 2 A!
D D
is de(cid:12)ned by
d(u;v) = 2(cid:0)n; where n = minfk (cid:21) 0 : u ̸= v g:
k k
Then d is a metric on A!. For example, in the binary alphabet we have d(0100:::;0110:::) =
2(cid:0)2 = 1 and d(0:::;1:::) = 2(cid:0)0 = 1. Thus u;v 2 A! are close, if they have a long common
4
pre(cid:12)x:
d(u;v) (cid:20) 2(cid:0)n , u = v , d(u;v) < 2(cid:0)n+1
[0;n) [0;n)
0
/
1 1 1
/ /
3 3
1/2 - 2 / 1
-
- - 1/1-- - 1/1
3_ 2_ 1_ 1_ 0_ 1_ 1_ 2_ 3_
1 1 1 2 1 2 1 1 1
2 / 1 - 1/2
0
/
1
Figure 1.1: The stereographic projection
With the metric d, A! and its subspaces (cid:6) are turned into metric spaces. A mapping
D
(cid:8) : (cid:6) ! R is continuous at u 2 (cid:6) , if for every " > 0 there exists (cid:14) = 2(cid:0)k such that for
D D
every v 2 (cid:6) with d(u;v) (cid:20) (cid:14) we have j(cid:8)(u) (cid:0) (cid:8)(v)j (cid:20) ". If the range of (cid:8) is the extended
D
real line R = R [ f1g, then the Euclidean metric d (x;y) = jx (cid:0) yj does not work. Since
e
the extended real line does not distinguish positive and negative in(cid:12)nity, it is topologically
equivalent to a circle. Consider the unit circle
√
S = fz = x+iy 2 C : jzj = x2 +y2 = 1g
in the complex plane C and project the point z 2 R on the real line to S by the ray from the
imaginary unit i (see Figure 1.1). This mapping is called the one-dimensional stereographic
projection. The line which joins z 2 R with i has parametric equation x(t) = tz+(1(cid:0)t)i. The