On the computational complexity of algebraic numbers: the Hartmanis--Stearns problem revisited

0.32 MB·

by  Boris Adamczewski

ON THE COMPUTATIONAL COMPLEXITY OF
ALGEBRAIC NUMBERS: THE HARTMANIS–STEARNS
PROBLEM REVISITED
6 by
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Boris Adamczewski, Julien Cassaigne & Marion Le Gonidec
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n
a
J

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1 Abstract. — Weconsiderthecomplexityofintegerbase expansionsof alge-
braic irrational numbers from a computational point of view. We show that

] the Hartmanis–Stearns problem can be solved in a satisfactory way for the
T
class of multistack machines. In this direction, our main result is that the
N
base-b expansion of an algebraic irrational real number cannot be generated
h. by a deterministic pushdown automaton. We also confirm an old claim of
t Cobham proving that such numbers cannot be generated by a tag machine
a
with dilation factor larger than one.
m
[

1
v
1 1. Introduction
7
7 An old source of frustration for mathematicians arises from the study of
2
integer base expansions of classical constants like
0
. √2 = 1.414213562373095048801688724209698078569
1
···
0 or
6 π = 3.141592653589793238462643383279502884197
1 ···
: While these numbers admit very simple geometric descriptions, a close look
v
at their digital expansion suggest highly complex phenomena. Over the years,
i
X
different ways have been envisaged to formalize this old problem. This reoc-
r curring theme appeared in particular in three fundamental papers using: the
a
language of probability after E´. Borel [17], the language of dynamical systems
after Morse and Hedlund [35], and the language of Turing machines after
Hartmanis and Stearns [29]. Each of these points of view leads to a different
This project has received funding from the European Research Council (ERC) under
the European Union’s Horizon 2020 research and innovation programme under the Grant
Agreement No648132.
2 BORIS ADAMCZEWSKI, JULIEN CASSAIGNE & MARION LE GONIDEC
assortment of challenging conjectures. As its title suggests, the present paper
will focus on the latter approach. It is addressed to researchers interested
both in Number Theory and Theoretical Computer Science. In this respect,
we took care to make the paper as self-contained as possible and hopefully
readable by members from these different communities.
After the seminal work of Turing [43], real numbers can be rudely divided
into two classes. On one side we find computable real numbers, those whose
base-bexpansioncanbeproducedbyaTuringmachine,whileontheotherside
lie uncomputablerealnumberswhichwillbelong for everbeyondtheability of
computers. Note that, though most real numbers belong to the second class,
classical mathematical constants are usually computable. This is in particular
thecaseofanyalgebraicnumber. However,amongcomputablenumbers,some
are quite easy to compute while others seem to have an inherent complexity
that make them difficult to compute. In 1965, Hartmanis and Stearns [29]
investigated the fundamental question of how hard a real number may be to
compute, introducing the now classical time complexity classes. The notion of
time complexity takes into account the number T(n) of operations needed by
a multitape deterministic Turing machine to produce the first n digits of the
expansion. In this regard, a real number is considered all the more simple as
its base-b expansion can be produced very fast by a Turing machine. At the
end of their paper, Hartmanis and Stearns suggested the following problem.
Problem HS. — Do there exist irrational algebraic numbers for which the
first n binary digits can be computed in O(n) operation by a multitape deter-
ministic Turing machine?
Let us briefly recall why Problem HS is still open and likely uneasy to
solve. On the one hand, all known approaches to compute efficiently the
base-b expansion of algebraic irrational numbers intimately relate on the cost
of the multiplication M(n) of two n-digits numbers (see for instance [18]).
This operation is computable in quasilinear time(1) but to determine whether
one may have M(n) = O(n) or not remains a famous open problem in this
area. On the other hand, a negative answer to Problem HS(2) would contain
a powerful transcendental statement, a very special instance of which is the
transcendence of the following three simple irrational real-time computable
numbers:
∞ ∞ ∞
1 1 1
, and
2n! 2n2 2n3 ·
nX=1 nX=1 nX=1
1. This means computablein O(nlog1+εn) operations for some ε.
2. Asobserved in [26], thismay be theless surprising issue.
3
Of course, for the first one, Liouville’s inequality easily does the job. But the
transcendance of the second number only dates back to 1996 [16, 28] and
its proof requires the deep work of Nesterenko about algebraic independence
of values of Eisenstein’s series [37]. Finally, the transcendence of the third
number remains unknown.
In 1968, Cobham [26] (see also [24, 25]) was the first to consider the re-
strictionoftheHartmanis-StearnsproblemtosomeclassesofTuringmachines.
The model of computation he investigated is the so-called Tag machine. The
outputs of such machines correspond to the class of morphic sequences, a
well-known object of study in combinatorics on words and symbolic dynamics
(see for instance [12, 40, 41]). In his paper, Cobham stated two main theo-
rems without proof and only gave some hints that these statements should be
deducedfrom a general transcendencemethod based on some functionalequa-
tions, now knownas Mahler’s method. His firstclaim was finally confirmedby
the first author and Bugeaud [3], but using a totally different approach based
on a p-adic version of the subspace Theorem (see [3, 8])(3).
Theorem AB (Cobham’s first claim). — The base-b expansion of an
algebraic irrational number cannot be generated by a uniform tag machine or,
equivalently, by a finite automaton.
Remark 1.1. — TheoremABactuallyreferstotwoconceptuallyquitediffer-
ent models of computation: uniform tag machines and finite automata. There
are twonatural waysa multitape deterministic Turingmachinecan be used to
define computable numbers. First, it can be considered as enumerator, which
means that the machine will produce one by one all the digits, separated by
a special symbol, on its output tape. Problem HS originally referred to the
model of enumerators. The other way, referred to as Turing transducer, con-
sists in feeding to the machine some input representing a positive integer n
and asking that the machine compute the n-th digit on its output tape. In
TheoremAB,uniformtagmachinesareenumeratorswhilefiniteautomata are
used as transducers(4). The fact that these two models are equivalent is due
to Cobham [27].
TheoremAB is themain contributionupto date towardanegative solution
to Problem HS. In this paper, we show that the approach developed in [3, 8]
leads to two interesting generalizations of this result. First, we revisit the
3. Veryrecently,someadvancesinMahler’smethod[39, 10]allowtocompletetheproof
originally envisaged by Cobham.
4. Note that, used as enumerators, finite automata can only produce eventually peri-
odic sequences of digits and thus rational numbers. In contrast, used as transducers, finite
automata outputtheinteresting class of automatic sequences.
4 BORIS ADAMCZEWSKI, JULIEN CASSAIGNE & MARION LE GONIDEC
Hartmanis–Stearns problem as follows: instead of some time constraint, we
put some restriction based on the way the memory may be stored by Turing
machines. This leads us to consider a classical computation model called mul-
tistack machines. The classical finite automaton of Theorem AB corresponds
to a stack machine with no stack, that is with a strictly finite memory only
stored in the finite state control. The one-stack model is actually equivalent
to another famous device: the deterministic pushdown automaton. It is of
great importance on the one hand for theoretical aspects because of Chom-
sky’s hierarchy [23] in formal language theory and on the other for practical
applications, especially in parsing (see [30]). In this direction, our main re-
sult is Theorem 2.3: no algebraic irrational can be generated by a one-stack
machine. Incidentally, this result turns out to provide a complete picture con-
cerning multistack machines (see the discussion at the end of Section 2.1).
Our second generalization of Theorem AB concerns the model of tag machine.
In this direction, Theorem 2.6 confirms Cobham’s second claim: no algebraic
irrational can be generated by a tag machine with dilation factor larger than
one. The problem for tag machines with dilation factor equal to one remains
open.
This paper is organized as follows. Our two main results are stated in
Section 2, where multistack machines and tag machines are also introduced.
The useful combinatorial transcendence criterion of [8], on which our results
are based, is recalled in Section 3. Sections 4 and 5 are then respectively
devoted totheproofs of Theorems2.3 and2.6. We also recall thelinkbetween
uniform tag machines, morphisms, and finite automata in Section 5. Though
writtenindifferentterms,westressthattheTh`esedeDoctoratofJulienAlbert
[11] contains results closely related to Theorem 2.6. In order to providea self-
contained proof of Theorem 2.6, we complete and reprove with permission
some content of [11] in Section 5. Finally, Section 6 is devoted to concluding
remarksregardingfactorcomplexity,somequantitativeaspectsofthismethod,
and continued fractions.
2. Main results
In this section, we introduce multistack machines and tag machines and
state our two main results: Theorems 2.3 and 2.6.
All along the paper, we will use the following notation. An alphabet A is
a finite set of symbols, also called letters. A finite word over A is a finite
sequence of letters in A or equivalently an element of A∗, the free monoid
generated by A. The length of a finite word W, that is the numberof symbols
composing W, is denoted by W . We will denote by ǫ the empty word, that
| |
is the unique word of length 0. If a is a letter and W a finite word, then W
a
| |
5
stands for the number of occurrences of the letter a in W. Let k 2 be a
≥
natural number. We let Σ denote the alphabet 0,1,...,k 1 . Given a
k
{ − }
positive integer n, we set n := w w w w for the base-k expansion of
k r r−1 1 0
n, which means that n =h ir w ki with·w·· Σ and w = 0. Note that by
i=0 i i ∈ k r 6
convention 0 k := ǫ. ConPversely, if w := w1 wr is a finite word over the
alphabet Σh, iwe set [w] := r w ki. The··u·sual notations x , x , and
k k i=0 r−i { } ⌊ ⌋
x respectively stand for thePfractional part, the floor, and the ceil of the real
⌈ ⌉
number x.
2.1. Multistack machines. — For a formal definition of Turing machines
the reader is referred to any of the classical references such as [30, 34, 42].
We will content ourself with the following informal definition of multistack
machines but a formal definition of the k-pushdown automaton (equivalent to
the model of one-stack machine) will be provided in Section 4.
Whenusedas atransducer,amultitapeTuringmachinecan bedividedinto
three parts:
The input tape, on which there is a read-only head.
•
The internal part, which consists in a finite control and the mem-
•
ory/working tapes (several tapes with one head per tape).
The output tape on which there is a write-only head and from which
•
nothing can be erased.
w1 w2 w3 w4 w5
Input tape
ol s1s2s3s4s5s6s7
r
t
n
co Q q1 q2 q3 q4 q5 q6
e
t
Fini r1 r2 r3 r4
Working tapes
a(w)
Output tape
Figure 2.1. A multitape Turing machine
Furthermore, the machineis said to be one-way or on-line if the head of the
input tape cannot go to the left. A (multi)stack machine is a one-way multi-
tape deterministic Turing machine in which the memory is simply organized
by stacks. This means that the head of each working/memory tape is always
6 BORIS ADAMCZEWSKI, JULIEN CASSAIGNE & MARION LE GONIDEC
located on the rightmost symbol so that the tape can be though of simply as
a stack with a head on the topmost symbol.
w1 w2 w3 w4 w5
Input tape
ol q1 q2 q3 q4 q5 q6
r
t
n
co Q s1s2s3s4s5s6s7
e
t
Fini r1 r2 r3r4
Working tapes
a(w)
Output tape
Figure 2.2. A stack machine
Letusbrieflydescribehowsuchamachineoperates. Amoveonamultistack
machine is based on:
M
The current state of the finite control.
•
The input symbol read.
•
The top stack symbol on each of its stacks.
•
Based on these data, a move of the multistack machine consists in:
Change the finite state control to a new state.
•
For each stack, replace the top symbol by a (possibly empty) string of
•
stack symbols. The choice of this string of symbols only depends on the
input symbol read, the state of the finite control and on the top symbol
of each stack.
Move the head of the input tape to the right.
•
can also possibly perform an ǫ-move: a move for which the head of the
M
input tape does not move. The possibility of such a move depends only on
the current state of the finite control and the top stack symbol on each of the
stacks. To make the machine deterministic, a move is uniquely determined by
the knowledge of the input symbol read, the state of the finite control and on
the top symbol of each stack.
Remark 2.1. — After reading a symbol of an input word w, the finite state
controlof couldhavereachedastateq fromwhichǫ-movesarestillpossible.
M
In that case, we ask to perform all possible ǫ-moves before reading the
M
7
next input symbol. Also, a multistack machine is not allowed to stop its
computation in a state from which an ǫ-move is possible.
After reading an input word w, produces a output symbol a(w) that
M
belongs to a finite output alphabet. The symbol a(w) depends only on the
state of the finite control and the top symbol of each stack. Given an integer
k 2, a k-multistack machine is a multistack machine that takes as input the
≥
base-k expansion of an integer (that is, for which the input alphabet is Σ ).
k
In that case, the sequence a( n ) is called the output sequence produced
k n≥0
h i
by .
M
Definition 2.2. — A real number ξ can be generated by a k-multistack ma-
chine if, for some integer b 2, one has ξ = 0.a a , where (a )
b 1 2 n n≥1
M ≥ h{ }i ···
corresponds to the output sequence produced by . A real number can be
M
generated by a multistack machine if it can be generated by a k-multistack
machine for some k.
A stack machine with no stack has a strictly finite memory which is all
contained in the finite state control. Such a machine simply corresponds to a
finite automaton used as a transducer. Theorem AB can thus be rephrased as
follows: an algebraic irrational number cannot be generated by a zero-stack
machine. A stack machine with one stack corresponds to another famous
device: the deterministic pushdown automaton. We prove here the following
generalization of Theorem AB.
Theorem 2.3. — An algebraic irrational real number cannot be generated by
a one-stack machine, or equivalently, by a determistic pushdown automaton.
For instance, this gives the transcendence of binary number
ξ := 1.110111001101000011111101110100000010110
1
···
whosen-thbinarydigitis1ifthedifferencebetweenthenumberofoccurrences
of the digits 0 and 1 in the binary expansion of n is at most 1, and is 0
otherwise.
Theorem2.3actuallyprovidesacompletepictureconcerningtheHartmanis-
Stearns problem for multistack machines. Indeed, it is well-known (see for
instance [30]) that stack machines with two stacks or more have the same
power as general Turing machines. In consequence, any computable number
can be generated by a two-stack machine, while, following Theorem 2.3, at
least two stacks are needed to generate an algebraic irrational number.
8 BORIS ADAMCZEWSKI, JULIEN CASSAIGNE & MARION LE GONIDEC
2.2. Tag machines. — As already mentioned in the introduction, Cobham
suggested in 1968 to restrict Problem HS to a special class of Turingmachines
calledtagmachines. Theyformaclassofinterestingtwo-taperestrictedTuring
machineswhoseoutputs,as described inSection 5, precisely correspondto the
well-known class of morphic sequences. Contrary to the model of multistack
machines described before, tag machines are enumerators. Furthermore, it is
not hard to see that they compute their output sequence in real-time.
A tag machine is a two-tape enumerator that can be described as follows.
In internal structure, a tag machine has:
T
A finite state control.
•
A tape on which operate a read-only head R and a write-only head W.
•
In external structure, has:
M
An output tape on which operates a write-only head W′ and from which
•
nothing can be erased.
Input tape
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
b (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)σ(cid:0)(cid:0)(cid:1)(cid:1)((cid:0)(cid:0)(cid:1)(cid:1)b)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
R W
Q
Finite control
W′
ϕ(b)
Output tape
Figure 2.3. A tag machine
Let us briefly describe how a tag machine operates. The finite state control
of contains some basic information: a finite set of symbols A together with
T
a special starting symbol a, so that with every element b of A is associated
a finite word σ(b) over A and a symbol ϕ(b) that belongs to a finite set of
symbols B. When the computation starts, R and W are both positioned on
theleftmostsquareof the(blank)tape andWproceeds writingthewordσ(a),
one symbol per square. Then both head R and W move one square right, R
scans the symbol written in the corresponding square, say b, and W proceeds
writing the word σ(b). Again both heads move one square to the right and
the process keeps on for ever unless R eventually catches W in which case the
machinestops. Meanwhile, each time R readsa symbolb on the internaltape,
W′ writes the symbol ϕ(b) on the output tape and moves one square right.
Each symbol written on the output tape is thus irrevocable and cannot be
erased in the process of computation. The output sequence produced by is
T
9
the sequence of symbols written on its output tape. As we will see in Section
5, a practical and formal definition of Tag machine can be given in terms of
morphisms of free monoids.
Definition 2.4. — A real number ξ can be generated by a tag machine if,
T
forsomeinteger b 2, onehas ξ = 0.a a , where(a ) corresponds
b 1 2 n n≥1
≥ h{ }i ···
to the output sequence produced by .
T
Cobham [26] introduced the following interesting quantity which measures
the rate of production of symbols by a tag machine.
Definition 2.5. — The (minimum) dilation factor of a tag machine is
T
defined by
W(n)
d( )= liminf ,
T n→∞ n
where W(n) denotes the position of the write-only head W of when the
read-only head R occupies the n-th square of the internal tape. T
Atag machineis called uniformif for each symbolb of its internalalphabet,
thewordσ(b)hasthesamelength. Itiseasy toseethatuniformtagmachines,
or equivalently finite automata used as transducers (see Section 5), all have
dilation factor at least two. The following generalization of Theorem AB was
conjectured by Cobham in 1968.
Theorem 2.6 (Cobham’s second claim). — The base-b expansion of an
algebraic irrational number cannot be generated by a tag machine with dilation
factor larger than one.
For instance, this implies the transcendence of the ternary number
ξ := 0.021201220210122202120120210122220212122
2
···
whose ternary expansion is generated by the tag machine defined by σ(a) =
acb, σ(b) = abc, σ(c) = c, ϕ(a) = 0, ϕ(b) = 1, and ϕ(c) = 2.
Cobham claimed that Mahler’s method only applied when d( ) > 1. We
T
note that our approach, which follows a totally different way, suffers from the
same limitation. In particular, it does not imply the transcendence of the
∞
1
binary number . We recall that this numbercan be generated by a tag
2n2
nX=1
machine with dilation factor 1. Such a tag machine is defined by: σ(a) = ab,
σ(b) = ccb, σ(c) = c, ϕ(a) = ϕ(c) = 0, and ϕ(b) = 1.
10 BORIS ADAMCZEWSKI, JULIEN CASSAIGNE & MARION LE GONIDEC
3. A combinatorial transcendence criterion
In this section, we recall the fundamental relation between Diophantine
approximation and repetitive patterns occurring in integer base expansions of
real numbers.
Let A be an alphabet and W be a finite word over A. For any positive
integer k, we write Wk for the word
W W
···
k times
| {z }
(the concatenation of the word W repeated k times). More generally, for any
positive real number x, Wx denotes the word W⌊x⌋W′, where W′ is the prefix
of W of length x W . The following natural measure of periodicity for
⌈{ }| |⌉
infinite words was introduced in [4] (see also [1, 9]).
Definition 3.1. — TheDiophantine exponentofaninfinitewordaisdefined
as the supremum of the real numbers ρ for which there exist arbitrarily long
prefixes of a that can be factorized as UVα, where U and V are two finite
words (U possibly empty) and α is a real number such that
UVα
| | ρ.
UV ≥
| |
The Diophantine exponent of a is denoted by dio(a).
Of course, for any infinite word a one has the following relation
1 dio(a) + .
≤ ≤ ∞
Furthermore, dio(a) = + for an eventually periodic word a, but the con-
∞
verse is not true. There is some interesting interplay between the Diophantine
exponent and Diophantine approximation, which is actually reponsible for the
name of the exponent. Let ξ be a real number whose base-b expansion is
0.a a . Set a := a a . Let us assume that the word a begins with a
1 2 1 2
··· ···
prefix of the form UVα. Set q = b|U|(b|V| 1). A simple computation shows
−
that there exists an integer p such that
p/q = 0.UVVV .
b
h i ···
Since ξ and p/q have the same first UVα digits in their base-b expansion, we
| |
obtain that
p 1
ξ <
(cid:12) − q(cid:12) b|UVα|
(cid:12) (cid:12)
and thus (cid:12) (cid:12)
(cid:12) (cid:12)
p 1
(3.1) ξ < ,
(cid:12) − q(cid:12) qρ
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)