WHY SOMETIMES PROBABILISTIC ALGORITHMS. CAN BE MORE EFFECTIVE. Farid M.Ablaev. Department of Mathematics. Kazan State University ul.Lenina ...
WHY SOMETIMES PROBABILISTIC
ALGORITHMS
CAN BE MORE EFFECTIVE
Farid M.Ablaev Department of Mathematics Kazan State University ul.Lenina 18 420000 Kazan USSR
For several problems effective
R ~ s i ~ Freivalds Computing Center Latvian State University Blvd. Rai~a 29 226250 Riga U S S R
there exist probabilistic
than any deterministic
lems probabilistic derstanding,
algorithms
algorithms which are more
solving these problems.
algorithms do not have such advantages.
For other prob-
We are interested
why it is so and how to tell one kind of the problems
in un-
from another.
Of course, we are not able to present a final answer to these questions. have just chosen a sequence of examples predictions
We
that can increase our ability to make right
whether or not the advantages
of probabilistic
algorithms
can be proved
for a given problem. Since probabilistic whether probabilistic
algorithms
algorithms
are widely used, the question arise naturally,
can solve a problem such that no deterministic
algo-
rithm can solve it. The very first answer to such a question was found by de Leeuw et al.
[20], and it was negative.
the probabilistic recursive
parameters
functions,
i.e. functions
tive answers were found as well Probabilistic counterparts
They proved that under reasonable
of the machine,
probabilistic
computable by deterministic
restrictions
on
can compute only
machines.
Later posi-
[28, 4, 7],
machines used in our paper are the same as their deterministic
but the probabilistic
machines
can in addition make use at each step of
a generator of random numbers with a finite alphabet, equal probabilities
machines
and in accordance with a Bernoulli
yielding
its output values with
distribution,
i.e. independently
of the values yielded at other instants. We say that the probabilistic
machine M recognizes
p if M, when working on an arbitrary x, yields a I if x a L with a probability
language L with probability and yields a 0 if x C Z
,
no less than p. We say in the case p>I/2 that L is recognized with
an isolated cut-point.
I. PROBLEMS ALLOWING MULTIPLE-VALUED
RESULTS
The very first results on advantages
of probabilistic
algorithms
over deter-
ministic ones were proved for problems where the result is not determined uniquely
but can be chosen from a certain set. Let us consider some examples. S.V.Yablonskij
[28] proved that a probabilistic Turing machine can produce
an infinite sequence of Boolean functions such that: a) the running time to produce the n-th element of the sequence is polynomial, and b) probability of the following event equals I: all but a finite number of the elemenes in the sequence are Boolean functions with nearly maximel circuit complexity.
(Different performances of the
probabi~istic machine produce different sequences of the Boolean functions). J.M. Barzdin [4] constructed a recursively enumerable set such that no deterministic Turing machine can enumerate an infinite subset of its complement but there is a probabilistic Turing machine which can enumerate with arbitrarily high probability |-e infinite subsets in the complement of the given set. (Different performances of the probabilistic machine produce different subsets). A.V.Vaiser
[27] proved that the function logloglx I can be approximated by a
probabilistic Turing machine in time Ixllogloglxl , i.e. in less time than any deterministic Turing machine can do this. R.Freivalds
[10] improved this upper bound and
showed that loglx I can be approximated by a probabilistic Turing machine in time Ixl loglog ~I , the function loglog Ix; can be approximated in time
Ix~ logloglog ~xl ,
etc. (Different performances of the probabilistic machine can produce different results but with high probability they are nearly the same). R.Freivalds
[13] considered probabilistic counterparts of the notion of m-
reducibility of sets. Let A C N ,
BeN.
We say that A is m-reducible to B (A ~m B) if there is a total
recursive function f of one argument such that x 6 A
~-~f(x) a B
holds for all x ~ N .
We say that A is probabilistically m-reducible to B (A ~m-prob B) if there is a probabilistic Turing machine such that for arbitrary x and i it produces results y so that the probability of the event ( x G A
~--~y~B)
exceeds I-I/i. (Different p e r -
formances of the probabilistic machine may produce different y's for the same x and i). Let z6{m,m-prob}.
We say that a set L is z-complete if: I) L is recursively
enumerable, and 2) A ~z L for arbitrary recursively enumerable set A. Since A ~ m B implies A ~m-prob B, every m-complete set is also m-prob-complete but not vice versa.
THEOREM 1.1. (R.Freivalds,
[13]). There is an m-prob-complete set which is not
m-complete. For readers familiar with the notions and the standard notation of the recursive function theory (see [24]) we additionally note a new result by R.Ereivalds: m-prob-complete sets always are tt-complete but they can be not btt-complete. Putting the matter in a highly imprecise way, we claim: if your kind of problems is such that the result is not determined uniquely and it can be chosen from an infinite set then there is a problem of your kind which can be solved by a probabilistic algorithm more easily than by any deterministic algorithm.
Of course, this claim is not a precise mathematical there are clear counterexamples,
statement. Moreover,
and we produce some of them below in this Section.
Nevertheless we are sure that precise theorems can be proved which say nearly the same thing as our claim. Now the promised counterexamples. E.M.Gold
Inductive inference machine as defined by
[19] is essentially any algorithmic device which attempts to infer rules
from examples.
In this model, a "rule" is any partial recursive function f, and an
"example" is a pair (x, f(x)) for some x in the domain of f. A predictive explanation for the rule f is simply a program p which computes f. Thus the machine takes as input the values of some partial recursive function f, and attempts to output a program p which computes f, based on the examples it has seen. Note that if after seeing some finite number of examples,
the machine gulsses the program p, the very next example
might be inconsistent with p. For this reason, the inference is seen as an infinite process, which occurs "in the limit".
It is required that the sequence of gulsses of
the machine converge to a single program computing f. We say that the set of functions U is identifiable if there is an inductive inference machine which identifies correctly in the limit every function from the set U. We say that the set of functions U is identifiable with probability r if there is a probabilis~ic
inductive inference machine which identifies correctly in the limit
every function from the set U with probability at least r (different correct programs are allowed for the function at different performances of the probabilistic
inductive
inference machine).
THEOREM 1.2. (R.Freivalds,
[12]). If U is a set of total recursive functions,
and U is identifiable with probability r > I/2 then U is identifiable by a deterministic inductive inference machine.
THEOREM 1.3. (L.Pitt,
[22]). If U is a set of partial recursive functions,
and U is identifiable with probability
r > 2/3 then U is identifiable by a determi-
nistic inductive inference machine.
THEOREM 1.4.
(R.Freivalds, new result).
(I) If
n+1 ~
n < rl < r2 < -i-~'
and
U is identifiable with probability r I then U is identifiable with probability r 2 as n (2) If r I < ~ - ~ < r 2 then there is a set of partial recursive functions
well.
identifiable with probability r I but not identifiable with probability r 2. A result similar to Theorem 1.4 was proved by R.Freivalds for so called finite identification of functions in [11].
2. THRESHOLD BOOLEAN OPERATIONS
The following theorem holds for all natural types of automata and machines
(e.g., for multitape
l-way or 2-way finite automata,
l-way or 2-way k-counter
THEOREM 2.1. by deterministic
(R.Freivalds,
l-way or 2-way pushdown machines,
real time machines,
etc.).
[8]). Let L I and L 2 he two languages
automata of a type W. Then there is a probabilistic
type W which recognizes Intersection in this Theorem.
or mulricounter
recognizable
automaton of the
the language L I N L 2 with isolated cut-point.
can be replaced by union and some other Boolean set operations
What operations namely?
Boolean function f(x I . . . . , x n) is called threshols
if there are real numbers
a I, ..., an, b such that I
I, if alx1+ ... +a n x n > b;
f(x I . . . . , x n) = 0, if otherwise. Boolean set operation
is called threshold
threshold function as intersection
corresponds
if it corresponds
to conjunction
to a Boolean
and union corresponds
to disjunction.
THEOREM 2.2. by deterministic
(R.Freivalds,
[16]). Let LI, ..., L n be languages recognizable
automata of a type W and f be a Boolean threshold set operation.
Then there is a probabilistic
automaton of the type W which recognizes
the language
f(L I, ..., Ln ) with isolated cut-point. Of course,
to make Theorems
2.1. and 2.2 to be precise mathematical
we need a formal notion of the type of automata.
statements
Such a notion is introduced
in [16].
For the sake of brevity we omit the definition here because the two proofs are merely combinatorial
constructions
valid for all the natural
needed to prove that only threshold contains
set operations
types. More formal notion is
can be used in Theorem 2.2.
such a proof but the theorem proved there has a rather unpleasant
restriction.
[16]
additional
It is still an open problem to prove that only threshold set operations
can be used in Theorem 2.2. We shall see in the subsequent language recognizable
Sections more complicated
by the probabilistic
automaton or machine
by a Boolean set operation from a constant number of languages ministic
automata.
Surprisingly
a bit more complicate operations
enough,
representation.
cases when the is not representable
recognizable
by deter-
in all these cases there is a similar though
One can informally
from a growing number of languages
say that Boolean threshold
are used. Unfortunately
we are not able
to make this assertion precise.
3. MORE SERIOUS ADVANTAGES
OF PROBABILISTIC
We consider multihead automata in this Section.
AUTOMATA
l-way finite automata and l-way real time multicounter
We define the following sequence of languages. A word x E ( 0 , 1 ~
is in D
b
if it is a palindrome, and it contains exactly 2b-I occurences of the symbol I.
THEOREM 3.1. (R.Freivalds,
[9]).
(I) Given any integer b and any e > 0, there
is a probabilistic 2-head l-way finite automaton which accepts every x ~ D b with probality I, and rejects every x E D b with probability 7-e. (2) No deterministic b-head l-way finite automaton can recognize D b. PROOF OF (I). Let n > b/e. We consider the following family of deterministic 2-head l-way finite automata. They differ only in the value of a 6 ( 1 , 2 ,
..., n}.
At first, the head h I goes to the (b+1)-th block of zeroes. Then the heads are performing .
.
s p e c i a I .
steps alternately.
.
.
If the automaton is processing a string
.
01110121 ... 10ZbloJbl .,, I0J210 jl, then the heads are performing a 0 steps on every symbol of the blocks 0 a
I
i2
steps on every symbol of 0
ib
and 0
J2
ab-1
, ...,
ii
and
0j
I,
steps on every symbol of the blocks
Jb
0
and
. The string is accepted if and only if b-1
I iI + a
i2
b-1 .
a 2 i3 + ... + a
+
ib
= a
2 .
]b
+
"'"
+
a
]3
+ a
J2
+
I
J1"
If a string is accepted by b distinct automata from this family, and these automata are characterized by values al, a2, ..., ab, then I (il-Jl) + a I (i2-J2) + a~ (i3-J3) + ... + a~ -I (ib-Jb) = 0, I (il-J I) + a 2 (i2-J2) + a22 (i3-J 3) + ... + a~ -I (ib-J b) = 0, .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
I (il-J I) + ab (i2-J2) + ab2 (i3_J3) + ... +a b-1 (ib-Jb) =
0.
The determinant of this system of equations is the Vandermonde determinant. For pairwise distinct al, a2, ..., ab it is not equal 0. Hence the system has only one solution il-Jl = ... = ib-Jb = 0 and the string is in D b. Thus if xlED b then no more than b distinct automata accept this string.
THEOREM 3.2. (R.Freivalds,
[17]). There is a language E such that: I) given
any e > 0, there is a probabilistic 3-head l-way finite automaton which accepts every x G E with probability I, and rejects every x ~ E
with probability l-e, 2) no deter-
ministic multihead l-way finite automaton can recognize E. IDEA OF PROOF. The language E is a modification of languages in the family D b from Theorem 3.1. Blocks Code(b), C(b, k, ik) are defined, and it is demanded that strings are of the form Code(b) * C(b, I, i I) * C(b, 2, i2) * ... * C(b, b, ib) ** C(b, b, jb ) * ... * C(D, 2, j2 ) * C(b, I, jl ), where b is an arbitrary natural number, and i I = J1' i2 = J2' "''' ib = Jb" OPEN PROBLEM. Can a probabilistic 2-head l-way finite automaton recognize
with isolated cut-point a language not recognizable by deterministic multihead l-way finite automata? The same language E can be used to prove that a probabilistic 3-counter l-way automaton can recognize in real time with arbitrarily high probability 1-e a language not recognizable in real time by any deterministic multicounter l-way automata. The language E can be modified to prove
THEOREM 3.3. (R.Freivalds,
[15]). There is a language F such that: I) given
any e > o, there is a probabilistic 2-counter l-way automaton which accepts every x~F
with probability I, and rejects every x G F
with probability l-e, 2) no deter-
ministic multicounter l-way automaton can recognize F. OPEN PROBLEM. Can a probabilistic l-counter l-way automaton recognize with isolated cut-point a language not recognizable by deterministic multicounter l-way automata? Why probabilistic multicounter or multihead automata are more powerful than their deterministic counterparts? A partial answer to this problem is given by the following theorems. F.M.Ablaev [2] proved several theorems on the number of equivalence classes of words modulo languages recognizable by deterministic, nondeterministic and probabilistic automata of rather general type. We formulate here only very much restricted cases of these theorems. Let L be a language, L ~ X * . v
For arbitrary n $ I we consider the equivalence
~v' which holds if vr and v'r are or are not in the language L simultaneously for
all possible r
X l, i ~ n. Let fL(n) denote the number of such equivalence classes.
THEOREM 3.4. (F.M.Ablaev, [2]). If a language L is recognizable in real time k by a deterministic k-counter l-way automaton then fL(n) ~ const.n .
THEOREM 3.5. (F.M.Ablaev,
[2]). If a language L is recognizable in real time
by a probabilist~c k-counter l-way automaton with isolated cut-point then fL (n) < (const) n .
THEOREM 3.6. (F.M.Ablaev,
[2]). If a language L is recognizable in real time nk .
by a nondeterministic k-counter t-way automaton then fL(n) ~ (const)
These theorems and examples of languages recognizable by probabilistic multicounter automata and having large functions fL(n) help to prove advantages of any type of automata over another type. For instance, the following hierarchy theorems are proved.
THEOREM 3.7. (F.M.Ablaev,
[2]). For arbitrary positive integer k there is a
language L k recognizable in real time by a probabilistic
(k+1)-counter l-way automaton
with non-isolated k-counter
cut-point but not recognizable
l-way automaton with non-isolated
Counterparts
of Theorems
in real time by a probabilistic
cut-point.
3.4, 3.5, 3.6, 3.7 are proved also for multihead
finite automata. OPEN PROBLEM.
Prove or disprove Theorem 3.7 for probabilistic
multicounter
automata with isolated cut-point. OPEN PROBLEM.
Prove or disprove Theorem 3.7 for probabilistic
multihead
finite automata with isolated cut-point.
4. PALINDROMES
AND SIMILAR LANGUAGES
More complicated probabilistic palindromes.
These algorithms
of the same kind.
can be modified
Strangely enough,
nevertheless
stay unaffected:
recognition
of these languages.
THEOREM 4.1.
algorithms
are invented £or recognition
to recognize
several other languages
some languages very much similar to palindromes
no effective probabilistic
(R.Freivalds,
of
algorithms
are known for
[7, 10]. For arbitrary e > o, there is a probabi-
listic off-line Turing machine recognizing
palindromes
with probability
1-e in
const n log n time. IDEA OF THE PROOF. The input word and its mirror image are compared modulo m where m is a specially chosen random number.
The number m contains
bits where n is the length of the input word and c is an absolute using the prime number theorem by Cebishev.
independence
constant
found by
The c log n random bits are generated
rather many times until the number m tuens out to be prime. to ensure statistical
c log n random
of different possible
(The primality
comparisons
is needed
modulo random
m).
A rather old result by J.Barzdin Turing machine recognizing probabilistic
[3] shows that every deterministic
off-line
palindromes
needs at least const n 2 running time. For
off-line Turing machines
Theorem 4.1 cannot be improved because of
the following
THEOREM 4.2. machine recognizing
(R.Freivalds, palindromes
[7, 10]). Every probabilistic
off-line Turing
with isolated cut-point uses at least const n log n
running time. Essentially language
~xbx~
the same results as for palindromes
can be obtained
where x is a word and b is a special separating
Let M denote the language
~xby~
for the
symbol.
where x and y are words of the same length
and it is required that y is a larger number,
i.e. there is a bit x(i) such that
x = x(1)x(2) = y(2),
... x(i)
..., x ( i - 1 )
THEOREM 4.3. off-line
... x(n), y = y(1)y(2)
= y(i-1),
x(i)
(R.Freivalds,
Turing machine
... y(n), x(1) = y(1), x(2) =
[10]). For arbitrary
recognizing
It is not known whether
... y(i)
< y(i).
M with probability
this upper bound
e > o, there
is a probabilistic
1-e in const n (logn) 2 time.
can be improved.
Most probably,
it
is. Let K denote the language and it is required
~xby}
where x and y are words of the same length
that for every i, y(i) > x(i).
No probabilistic
off-line
Turing machine
is known to recognize K in o(n 2)
time. Let S denote the language
{xby}
the y-th bit of the word x equals We do not k n o w a precise
where
Ixl = 2 I~ , and it is required
reference but rumours have reached us about a con-
ference paper by A.Yao with a lower bound for the running w h i c h exceeds
the upper bound
fective probabilistic
algorithm must use some global characteristics
On the other hand,
to a bit-to-bit
there is a type of machines
can have advantages
type S. We consider
time for this language
in T h e o r e m 4.1. This result would show that every ef-
under comparison which are not reduced
machines
that
I.
over deterministic
l-way finite automata
of the words
comparison. for which probabilistic
ones when recognizing
languages
of
and use the number of states as the measure
of complexity. Let S required
n
denote the language
where
that the y-th bit of the word x equals
T H E O R E M 4.4. automaton
(F.M.Ablaev,
n e w result).
IYl = n,
n lyl = 2 , and it is
I.
(I) Every deterministic
l-way finite
recognizing
a probabilistic bility
{xby)
S has at least 22n states; (2) For arbitrary e > o, there is n l-way finite automaton with o(22n) states recognizing S with proban
1-e. IDEA OF THE PROOF.
x. The probabilistic of the segment
The deterministic
automaton
is determined
remembers
automaton has to remember
all the word
only a segment of the word x where the length
by e and all possible
start- and end-points
are equi-
probable. Let T denote the language
T H E O R E M 4.5.
on~ n .
(R.Freivalds,
2-way finite automaton recognizing We note for the contrast 2-way finite automaton 2-way Turing machines
[14]). For arbitrary e > o, there is a probabilistic T with probability
that no deterministic,
can do this. Moreover,
1-e. nondeterministic
it is proved
need at least const log n tape to recognize
A further development
or alternating
in [25] that deterministic T.
of the proof of T h e o r e m 4.5 allows us to prove the
following
theorem.
We consider finite automata on binary trees. directions:
up, left-hand-down,
the ordinary vertices automaton
right-hand-down.
Such an automaton can move 3
The automaton
in the tree and the extremal
can distinguish between
ones (the root, the leaves).
The
starts at the root, moves up and down and finally stops either in accepting
or in rejecting
state.
THEOREM 4.6.
(R.Freivalds,
new result).
For arbitrary e > o, there is a proba-
bilistic 3-way finite automaton on binary trees recognizing with probability whether or not the given tree is complete,
1-e
i.e. whether all the leaves are equidistant
from the root. No deterministic,
nondeterministic
or alternating
automata can do this.
5. ~ - W O R D S
Infinite behaviour abstract
of automata and machines
is usually regarded as too
to have something to do with the real life. On the other hand, there are
algorithms which are intended never to stop, for instance, Automata on words.
W-words
operating
are rather similar to corresponding
This makes us believe
that their properties
systems.
automata on finite
and the most important results
are likewise similar for automata on K~-words and for automata on finite words. theless the properties Deterministic, cut-point)
nondeterministic,
finite automata
called regular languages. for automata on
alternating
(on finite words)
that a similar result can be true
ones, and this advantage
can be proved in a very strong form, namely,
(with isolated
accept the same class of languages,
This makes us believe
more powerful than deterministic
finite automata are
of probabilistic
automata
even in the case of probability
I of
(so called Las Vegas algorithms).
Deterministic
finite
W-automaton
differs from deterministic
on finite words only in the mechanism of acceptance. states it has the accepting automaton
and probabilistic
G>-words. However for ~>-words probabilistic
the right result
Never-
of the two kinds of automata differ strikingly.
finite automata
Instead of the set of accepting
set of sets of states. An U - w o r d
is accepted by the
if the set of those states which are entered infinitely many times is in
the accepting
set of sets of states.
We say that the automaton accepts the (p > I/2) if the automaton accepts every p, and the automaton accepts every
of sets of states
{{q|,
L with probability
in L with probability
K~-word in L with probability
Consider the following probabilistic O, 1 , the set of states
W-language
W-word
finite ~ - a u t o m a t o n
p
no less than
at most
1-p.
M with input alphaDet
{q1' q2' q3' q4~' the initial state q1' the accepting q3~'
{q1' q2' q3)'
{q2' q3~'
{q|' q3' q4)'
set
{q1'q2'q3'q4)'
10
{q2' q3' q4} } and the following transition probabilities.
t
input letter
l
1
to ql
0
q2
q3
q4
from
ql
I/2
I/2
0
0
q2
0
I
0
0
q3
I/2
I/2
0
0
q4
I/2
I/2
0
0
ql
q2
q3
q4
ql
0
0
~
0
q2
0
0
0
I
q3
0
0
0
I
q4
0
0
0
I
input
to
letter I from
THEOREM 5.1. (R.Freivalds, D.Taimi~a, new result). There is an V-language L such that: (I) the probabilistic automaton M accepts L with probability I, (2) there is no deterministic finite automaton which accepts L. The proof of this theorem is based on classical result by Borel and Cantelli. Let A t 1 A 2 ,
...
be an infinite sequence of independent Bernoulli events. Let
ak be the probability of A k.
LEMMA 5.2. (Borel, Cantelli,
[6]). If the sum ~ ' a k converges then only finite
number of events A k take place with probability I. If the sum 7" a k diverges then infinitely many events Ak take place with probability If the input ~ - w o r d
I.
has only finite number of ones then the state q3 can be
entered only finite number of times and the V - w o r d
is rejected. If the input V - w o r d
has only finite number of zeros then only the state q4 is repeated infinitely often
11
and again the 6#-word is rejected.
For all the other possible
K-words
let n. denote
--n.
the length of the i-th segment of zeros. rejected,
and if the sum diverges
nondeterministic
or alternating
If 3" 2
1
z
converges
then the ~ - w o r d
then the &)-word is accepted.
finite automaton
This theorem came rather unexpected
results which were interpreted When analyzing
nizable by probabilistic general reduction input alphabet
finite automata
theorem for automata
~" with
: S ~ ~- ~ S ,
as impossibility
the proof of Rabin's
and the initial
and automata
since there were rather general
of such theorem.
theorem on regularity
[23] R.G.Bukharaev A = < ~,
(possibly infinite)
or
can make this distinction.
Theorem 5.1 exposes a distinction between automata on W - w o r d s on finite words.
is
No deterministic
S,
~(s
set of states
of languages
recog-
proved the following , ~ ), s o
> over finite
S, transition
function
state s . o
THEOREM 5.3.
(R.G.Bukharaev,
[5]). Let automaton A have the propert-ies:
I. S is a compact subset of a metric 2. The transition function
where S! = o~(Si, ~ )
~(s 1 $2))
3. R ~ S is such that Then the language
i = 1
V(SI,
"{p g ~ *
space with metrics ~ ;
~ is such that
~($I,
$2, ~ )
!
(~($I,
S~)
2;
S 2) ((S I • R) & (S 2 ~ R) -~ j~(SI,S 2) > G
: ~(S
, p) E R
).
is regular.
o
The Rabin's theorem is a special case of this theorem. N.R.Nigmatullin
[21] and F.M.Ablaev
of Theorem 5.3 for &)-languages.
F.M.Ablaev's
[I] proved a bit different theorem provides
of the number of states of the minimal deterministic nizes the language. a probabilistic
Comparison
finite
counterparts
even an upper bound
finite automaton which recog-
of this theorem and Theorem 5.1 shows that there is
W-automaton
which is not a netric automaton
in the sense of
[1]. OPEN PROBLEM.
Does Turakainen's
theorem
[26] hold for V - a u t o m a t a ?
6. SINGLE-LETTER ALPHABET
A serious obstacle for advantages a restriction advantages
of probabilistic
of the size of the input alphabet.
of probabilistic
letter alphabet.
machines
machines
and automata
It is very difficult
in the case of recognition
is
to prove any
languages
in single-
On the other hand, until recently there were no results on such
difficulties. The single-letter
alphaber
is a serious obstacle
ministic machines as well. There is a well-known showing that context-free
languages
[18]. Hence nondeterministic deterministic
for advantages
result in the formal
in a single-letter
alphabet
of nondeter-
language
are regular languages
l-way pushdown machines are no more powerful
finite automata for single-letter
alphabet
theory
languages.
than
(For at least
12
2-1etter alphabet languages pushdown machines are much more powerful than finite automata).
THEOREM 6.1. (J.Ka~eps, new result).
If a language in single-letter alphabet
is recognized by a probabilistic l-way pushdown machine with an isolated cut-point then
the language is regular. ~This result can be mistakenly seen as trivial but it is not. The probabilistic
pushdown machines are very near to the ability to recognize a non-regular language. This can be shown by considering approximate computation of functions by probabilistic l-way pushdown machines. We consider l-way pushdown machines with output. The function is computed by transforming the input word in single-letter alphabet into the output word in singleletter alphabet using the work tape (being pushdown> in an arbitrary alphabet. If the machine is deterministic or nondeterministic then only linear functions can be computed.
THEOREM 6.2. (R.Freivalds, new result).
There is a function f(n) of the order
of magnitude log n which can be approximately computed by a probabilistic l-way pushdown machine with output.
THEOREM 6.3. (J.Ka~eps, new result).
If a language in single-letter alphabet
is recognized by a probabilistic 2-way finite automaton with an isolated cut-point then the language is regular. This latter result contrast nicely to the Theorem 4.5 above.
REFERENCES I.
F.M.Ablaev, On the problem of reduction of automata, Izvestija VUZ. Matematika, 1980, No.3, 75-77 (Russian).
2.
F.M.Ablaev, Capabilities of probabilistic machines to represent languages in real time, Izvestija VUZ. Matematika, 1985, No.7, 32-40 (Russian).
3. J.M.Barzdin, Complexity of recognition of palindromes by Turing machines, Problemy kibernetiki, v.15, Moscow, Nauka, 1965, 245-248 (Russian). 4. J.M.Barzdin, On computability by probabilistic machines, Doklady AN SSSR, 1969, v. 189, No.4, 699-702 (Russian). 5. R. GoBukharaev, Foundations of the theory of probabilistic automata, Moscow, Nauka, 1985. 6. W.Feller, An introduction to probability theory and its applications, v.1, New York et al., John Wiley, 1957. 7. R.Freivalds, Fast computation by probabilistic Turing machines, Ucenye Zapiski Latvijskogo Gosudarstvennogo Universiteta, 1975, v.233, 201-205 (Russian)
13
8.
R.Freivalds, Probabilistic machines can use less running time, in: Information Processing '77, IFIP (North Holland, 1977), 839-842.
9.
R.Freivalds, Language recognition with high probability by various classes of automata, Doklady AN SSSR, 1978, v.239, No.l, 60-62 (Russian).
10. R.Freivalds, Speeding of recognition of languages by usage of random number generators, Problemy kibernetiki, v.36, Moscow, Nauka, 1979, 209-224 (Russian). 11. R.Freivalds, Finite identification of general recursive functions by probabilistic strategies, Proc. Conference FCT, 1979, 138-145. 12. R.Freivalds, On principal capabilities of probabilistic algorithms in inductive inference, Semiotika i informatika, 1979, No.12, 137-140 (Russian). 13. R.Freivalds, A probabilistic reducibility of sets, Proc. USSR Conference on Mathematical Logics, 1979, 137 (Russian). 14. R.Freivalds, Probabilistic two-way machines, Lecture Notes in Computer Science, Springer, 1981, v°118, 33-45. 15. R.Freivalds, Capabilities of various models of probabilistic one-way automata, Izvestija VUZ. Matematika, 1981, No.5, 26-34 (Russian). 16. R.Freivalds, Characterization of capabilities of the simplest method for proving the advantages of probabilistic automata over deterministic ones, Latvijskij matematiceskij ezegodnik, 1983, v.27, 241-251 (Russian). 17. R.Freivalds, Advantages of probabilistic 3-head finite automata over deterministic multi-head ones, Latvijskij matematiceskij ezegodnik, 1985, v.29, 155-163 (Russian). 18. A.V.Gladkij, Formal grammars and languages, Moscow, Nauka, 1973. 19. E.M. Gold, Language identification in the limit, Information and Control, 1967, v. 10, 447-474. 20. K. de Leeuw, E.F.Moore, C.E.Shannon and N.Shapiro, Computability by probabilistic machines, Automata Studies, Princeton University Press, 1956, 183-212. 21. N.R.Nigmatullin, Towards the problem of reduction of W-automata, Verojatnostnye metodi i kibernetika, Kazan University Press, 1979, 61-67 (Russian). 22. L.Pitt, Probabilistic inductive inference, Yale University, YALEU /DCS/ TR-400, June 1985. 23. M.O.Rabin, Probabilistic automata, Information and Control, 1963, v.6, No.3, 230-245. 24. H.Rogers Jr., Theory of recursive functions and effective computability, New York, McGraw Hill, 1967. 25. R.E.Stearns, J.Hartmanis and P,M.Lewis II, Hierarchies of memory limited computation, Proc. IEEE Symposium on Switching Circuit Theory and Logical Design, 1965, 179-190. 26. P.Turakainen, On probabilistic automata and their generalizations, Suomalais, tiedenkat, toimituks., 1968, v°53. 27. A.V.Vaiser, Notes on complexity measures in probabilistic computations, in: Control systems, v.1, Tomsk University Press, 1975, 182-196 (Russian).
14
28. S.V.Yablonskiy, On lagorithmic difficulties in minimal circuit synthesis, Problemy kibernetiki, v.2, Moscow, Fizmatgiz, 75-121 (Russian).
