ON DETERMINISM VERSUS NON-DETERMINISM AND RELATED ...
ON DETERMINISM VERSUS NON-DETERMINISM AND RELATED PROBLEMS (Preliminary Version)
Nicholas Pippenger Endre Szemeredi William T. Trotter IBM Research Laboratory University of South Carolina University of South Carolina San Jose, CA 95193 Columbia, SC 29208 Columbia, SC 29208
Wolfgang J. Paul IBM Research Laboratory San Jose, CA 95193
ABSTRACT:
We
show
that,
for
multi-tape
deterministic Turing machines (that
Turing
receive
their
2 Q(n )
time
is
more
input on their work tape) require
powerful than deterministic linear time.
We
also
recognize non-palindromes of length n (it is easy to
machines, non-deterministic
linear
discuss the prospects for extending this result to
see that time
more general Turing machines.
non-deterministic machine).
O(n
log
n)
time
is. sufficient
to
for
a
Among subsequent attempts to extend this result 1. Introduction to multi-tape Turing
machines,
we
may note
two
results of Kannan. The first (Kannan [9]) shows that
Our main result in this paper states that,
for non-deterministic
multi-tape
Turing
machines,
non-deterministic
powerful
than deterministic
two-tape
machines
are
more
powerful than deterministic one-tape machines (both linear time is
more
machines receive their input on a read-only
input
linear time. More specifically, we show that there tape). is
a
language
recognized
by
a
The
second
(Kannan
[10])
shows
that
are
more
two-tape non-deterministic
multi-tape machines
non-deterministic Turing machine in linear time, but powerful than deterministic multi-tape machines with not
recognized
by
any
multi-tape
deterministic an additional space bound
(growing
strictly more
Turing machine in linear time (both machines receive slowly than their time bound). In
both
of
these
their input on a read-only input tape). This result results, (which will be proved in Section 4 below) shows not
the
deterministic machines
suffer
an
additional handicap, and it is not clear that this
only that non-determinism adds power, but that this handicap alone does not account for additional power cannot be compensated for by any
the
observed
difference in power.
number of additional tapes.
We should also note a paper of Paul and Reischuk The
first
evidence
that
non-deterministic [15], which proved a result similar to ours on the
time-bounded Turing machines are more powerful than deterministic
time-bounded Turing
provided by Hennie [6], who showed
machines that
assumption of a certain graph-theoretic hypothesis.
was
This
one-tape
429 0272-5428/83/0000/0429$01.00
©
1983 IEEE
hypothesis
was
subseq~ent1y
disproved
by
Schnitger [18,19], but the present paper owes both
Analogously, the simulation of Dymond and Tompa [3],
its overall strategy and many 9f its tactics to the
as well as the simualtion of
paper of Paul and Reischuk [15].
involves
a
certain
the
present
paper,
"two-person
pebble
game" .
Additionally, in the present paper we must exploit certain constraints satisfied
The overall strategy may be described briefly as follows.
If
non-deterministic linear time, linear time
equals
computation
equals
graphs of deterministic multi-tape Turing machines.
then deterministic
These constraiI;lts imply a "segregator theorem" for
linear
deterministic
by the
alternating
time
linear
time
thes'e graphs, which is proved in Section 2.
for
machines making a bounded number of alternations. By a padding argument,
this
implication extends
to
2.
~
Segregator Theorem
non-linear time bounds. This much is as in Paul and Reischuk [15]. The key to the proof is a simulation (which will be given language
recognized
in
Section 3)
by a
whereby
deterministic
In
any
Turing
alternations. As
in
machine Paul
making
and
simulation
mentioned
alternating time with powerful
than
four
above
[15],
shows
alternations
deterministic
time.
This
graph-theoretic
vertices {I, ... , N} in which every
a
is may
edge
(i,
j)
satisfies i<j (so that these graphs are acyclic). We
that
shall say that two edges (i, j) and (i', j')
~
i
Nicholas Pippenger Endre Szemeredi William T. Trotter IBM Research Laboratory University of South Carolina University of South Carolina San Jose, CA 95193 Columbia, SC 29208 Columbia, SC 29208
Wolfgang J. Paul IBM Research Laboratory San Jose, CA 95193
ABSTRACT:
We
show
that,
for
multi-tape
deterministic Turing machines (that
Turing
receive
their
2 Q(n )
time
is
more
input on their work tape) require
powerful than deterministic linear time.
We
also
recognize non-palindromes of length n (it is easy to
machines, non-deterministic
linear
discuss the prospects for extending this result to
see that time
more general Turing machines.
non-deterministic machine).
O(n
log
n)
time
is. sufficient
to
for
a
Among subsequent attempts to extend this result 1. Introduction to multi-tape Turing
machines,
we
may note
two
results of Kannan. The first (Kannan [9]) shows that
Our main result in this paper states that,
for non-deterministic
multi-tape
Turing
machines,
non-deterministic
powerful
than deterministic
two-tape
machines
are
more
powerful than deterministic one-tape machines (both linear time is
more
machines receive their input on a read-only
input
linear time. More specifically, we show that there tape). is
a
language
recognized
by
a
The
second
(Kannan
[10])
shows
that
are
more
two-tape non-deterministic
multi-tape machines
non-deterministic Turing machine in linear time, but powerful than deterministic multi-tape machines with not
recognized
by
any
multi-tape
deterministic an additional space bound
(growing
strictly more
Turing machine in linear time (both machines receive slowly than their time bound). In
both
of
these
their input on a read-only input tape). This result results, (which will be proved in Section 4 below) shows not
the
deterministic machines
suffer
an
additional handicap, and it is not clear that this
only that non-determinism adds power, but that this handicap alone does not account for additional power cannot be compensated for by any
the
observed
difference in power.
number of additional tapes.
We should also note a paper of Paul and Reischuk The
first
evidence
that
non-deterministic [15], which proved a result similar to ours on the
time-bounded Turing machines are more powerful than deterministic
time-bounded Turing
provided by Hennie [6], who showed
machines that
assumption of a certain graph-theoretic hypothesis.
was
This
one-tape
429 0272-5428/83/0000/0429$01.00
©
1983 IEEE
hypothesis
was
subseq~ent1y
disproved
by
Schnitger [18,19], but the present paper owes both
Analogously, the simulation of Dymond and Tompa [3],
its overall strategy and many 9f its tactics to the
as well as the simualtion of
paper of Paul and Reischuk [15].
involves
a
certain
the
present
paper,
"two-person
pebble
game" .
Additionally, in the present paper we must exploit certain constraints satisfied
The overall strategy may be described briefly as follows.
If
non-deterministic linear time, linear time
equals
computation
equals
graphs of deterministic multi-tape Turing machines.
then deterministic
These constraiI;lts imply a "segregator theorem" for
linear
deterministic
by the
alternating
time
linear
time
thes'e graphs, which is proved in Section 2.
for
machines making a bounded number of alternations. By a padding argument,
this
implication extends
to
2.
~
Segregator Theorem
non-linear time bounds. This much is as in Paul and Reischuk [15]. The key to the proof is a simulation (which will be given language
recognized
in
Section 3)
by a
whereby
deterministic
In
any
Turing
alternations. As
in
machine Paul
making
and
simulation
mentioned
alternating time with powerful
than
four
above
[15],
shows
alternations
deterministic
time.
This
graph-theoretic
vertices {I, ... , N} in which every
a
is may
edge
(i,
j)
satisfies i<j (so that these graphs are acyclic). We
that
shall say that two edges (i, j) and (i', j')
~
i
