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Journal ArticleDOI

Parallel Time and Quantifier Prefixes

01 Dec 2009-Computational Complexity (Birkhäuser-Verlag)-Vol. 18, Iss: 4, pp 527-550

TL;DR: A real version of Savitch’s Theorem is proved about the amount of alternation between blocks of Boolean quantifiers (having both existential and universal), blocks ofreal existential quantifiers, and blocks of real universal quantifiers that can be decided in parallel polynomial time over the reals.
Abstract: We characterize the amount of alternation between blocks of Boolean quantifiers (having both existential and universal), blocks of real existential quantifiers, and blocks of real universal quantifiers that can be decided in parallel polynomial time over the reals. We do so under the assumption that blocks have a uniform bound on their size, both for the case of this bound to be polynomial and constant. On the way towards this characterization we prove a real version of Savitch’s Theorem.
Topics: Complexity class (53%), Polynomial (52%), Quantifier (logic) (52%)

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Parallel Time and Quantier Prxes
Felipe Cucker, Paulin Jacobé de Naurois
To cite this version:
Felipe Cucker, Paulin Jacobé de Naurois. Parallel Time and Quantier Prxes. 2007. �hal-00273889�

Parallel Time and Quantifier Prefixes
Felipe Cucker
Dept. of Mathematics
City University of Hong Kong
83 Tat Chee Avenue, Kowloon
Hong Kong
e-mail: macucker@cityu.edu.hk
Paulin Jacob´e de Naurois
CNRS UMR 7030 - LIPN
Universit´e Paris Nord
99 av. J.B. Clement
93430 Villetaneuse cedex
France
e-mail: denaurois@lipn.univ-paris13.fr
Abstract. We characterize the amount of alternation between blocks
of digital quantifiers (having both existential and universal), blocks of
real existential quantifiers, and blocks of real universal quantifiers that
can be decided in parallel polynomial time over the reals. We do so
under the assumption that blocks have a uniform bound in their size,
both for the case of this bound to be polynomial and constant. As a
result of this characterization (and as a stepping stone towards it) we
prove a real version of Savitch Theorem.
1 Introduction
In classical complexity theory there is a neat relationship between complex-
ity classes and quantifier prefixes preceding a predicate decidable in polyno-
mial time. A prefix made of existential quantifiers only corresponds to the
class NP, one made of universal quantifiers only to the class coNP and, more
generally, one alternating k blocks of quantifiers to the class Σ
k
(if the first
block is of existential quantifiers) and to the class Π
k
(if the first block is of

universal quantifiers). Furthermore, if one allows the (polynomial number
of) quantifiers in the prefix to arbitrarily vary we obtain the class PSPACE of
sets decidable in polynomial space (or, equivalently, in polynomial parallel
time).
In the complexity theory over the reals developed by Blum, Shub, and
Smale [3] some differences with the situation above stand out. Firtsly, space
in itself is not such a meaningful resource [8] and the role of the class PSPACE
is played over the reals by the class PAR
R
of sets decidable in polynomial
parallel time. Secondly, no quantifier prefix appears to correspond with this
complexity class.
Indeed, while the alternation of k blocks of quantifiers leads to the classes
Σ
k
R
and Π
k
R
of the polynomial hierarchy PH
R
over the reals [2] which is
included in PAR
R
[1], the unrestricted alternation of polynomially many
quantifiers yields a class PAT
R
(from polynomial alternating time) which
strictly includes PAR
R
[6]. But there is more. Call a quantifier digital if its
argument is restricted to take values in {0, 1}. Then, it is easy to see, the
class DPAT
R
obtained via the unrestricted alternation of digital quantifiers
is included in PAR
R
. A natural question arising at this moment is
Which quantifier prefixes, allowing for both digital and real quan-
tifiers, can be solved within PAR
R
?
While a complete answer to this question seems elusive since sequences of
quantifier prefixes can be very unstructured one can nevertheless restrict
attention to quantifier prefixes possessing a certain regularity. Some re-
sults within this framework are shown in [4]. For instance, it is shown
that DPAT
PH
R
R
PAR
R
. Furthermore, define the classes MA
R
(Mixed
Alternation with real Existentials) and MA
R
(Mixed Alternation with real
Universals) consisting of all the sets decidable alternating digital univer-
sal and real existential (respectively, digital existential and real universal)
guesses in polynomial time. It is also shown in [4] that PAR
R
( MA
R
and
PAR
R
( MA
R
.
These results shed some light on the relations between quantifier prefixes
and computations in PAR
R
. For, on the one hand, the class DPAT
PH
R
R
can be characterized by a form of alternation where one first alternates a
polynomial number of digital quantifiers and then a polynomial number of
real quantifiers (but these ones with only a bounded number of alternations).
And, on the other hand, the classes MA
R
and MA
R
allow real quantifiers
to alternate with digital ones provided all the real quantifiers are of the same
kind.
2

A first result in this paper extends the results above by showing that
PH
DPAT
R
R
PAR
R
. Together with the results in [4] this allows to build a
whole hierarchy of complexity classes within PAR
R
. Define
Θ
0
= DPAT
R
and Υ
0
= PH
R
and, for k > 1,
Θ
k
= DPAT
Υ
k1
R
and Υ
0
= PH
Θ
k1
R
.
Finally, let the Quantifier Hierarchy be QH
R
=
S
k0
Θ
k
=
S
k0
Υ
k
. We
show that QH
R
PAR
R
. This gives a complete answer on how much
alternation can be decided in PAR
R
if we allow both the digital blocks
(themselves alternating existential and universal quantifiers), the existential
real blocks, and the universal real blocks to have polynomial size.
We further extend this result to a characterization of the amount of
alternation decidable in PAR
R
when the size of the (three kinds of) blocks
above is bounded. In this case the number of block alternations has to be
at most (log(n))
O(1)
.
The power of quantification is also related with a well-known result in
classical complexity. In [9] Savitch proved that NPSPACE = PSPACE. To
extend this result to the real setting (besides replacing PSPACE by PAR
R
)
requires to agree on how much nondeterminism we want to endow parallelism
with. The obvious definition for a set A to be decidable in nondeterministic
parallel polynomial time requires the existence of a set B deciding pairs
(x, y) in parallel time polynomial in the size of x and of a function g such
that, for x R
n
,
x A y R
g(n)
s.t. (x, y) B.
The issue is how ‘big’ should g be. Denote by NPAR
R
and NPAR
R
the
classes obtained by taking g to be a polynomial and an exponential function
respectively. Using a similar notation, Savitch result shows that PSPACE =
NPSPACE = NPSPACE
. A main result in this paper shows that over the
reals the situation differs once more since we actually have (using obvious
notations)
DNPAR
R
= NPAR
R
= PAR
R
( DNPAR
R
NPAR
R
.
We can summarize the relationship between complexity classes emerging
from our results in the following diagram (where a line means inclusion of
the left-hand side class in the right-hand side one and the expressions EXP
R
and PAREXP
R
denote the classes of sets decidable in exponential time and
parallel exponential time, respectively).
3

QH
R
. . .
DPAT
R
PH
R
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PAR
R
pp
NPAR
R
PAT
R
PAREXP
R
...........................................
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MA
R
MA
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6=
6=
6=
6=
...........................................
EXP
R
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DNPAR
R
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NPAR
R
....................................
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2 Preliminaries
We denote by R
the disjoint union of the Euclidean spaces R
n
, for n 1.
Given x R
we denote by |x| its size, i.e., the only n 1 such that x R
n
.
For a set S R
we write S
n
= S R
n
.
We consider sequential machines over R as originally defined in [3] (see
also [2]). As a model of parallel machine we consider P-uniform families of
algebraic circuits (see [2, §18.4]). Actually, we endow the sign gates of a
circuit with the function sign : R {−1, 0, 1} where, for a R,
sign(a) =
1 if a > 0
0 if a = 0
1 if a < 0
instead of the two-valued sign function in [2, §18.4]. These machine mod-
els allow one to define the classes P
R
, NP
R
, NC
R
, and PAR
R
of subsets
S R
decidable in polynomial (resp. nondeterministic polynomial, paral-
lel polylogarithmic, and parallel polynomial) time, see [2] for details. Other
complexity classes may be defined from these ones using relativized compu-
tations. If C and D are complexity classes and A R
, we denote by C
A
the class of subsets decided by machines in C using A as an oracle and we
denote
C
D
=
[
S∈D
C
S
.
Definition 2.1 Let A R
. We define inductively the class PH
A
R
as fol-
lows:
Σ
0
R
A
= Π
0
R
A
= P
A
R
Σ
i+1
R
A
= NP
i
R
A
)
R
= NP
i
R
A
)
R
Π
i+1
R
A
= coNP
i
R
A
)
R
= coNP
i
R
A
)
R
,
4

References
More filters

Book
30 Oct 1997
TL;DR: This chapter discusses decision problems and Complexity over a Ring and the Fundamental Theorem of Algebra: Complexity Aspects.
Abstract: 1 Introduction.- 2 Definitions and First Properties of Computation.- 3 Computation over a Ring.- 4 Decision Problems and Complexity over a Ring.- 5 The Class NP and NP-Complete Problems.- 6 Integer Machines.- 7 Algebraic Settings for the Problem "P ? NP?".- 8 Newton's Method.- 9 Fundamental Theorem of Algebra: Complexity Aspects.- 10 Bezout's Theorem.- 11 Condition Numbers and the Loss of Precision of Linear Equations.- 12 The Condition Number for Nonlinear Problems.- 13 The Condition Number in ?(H(d).- 14 Complexity and the Condition Number.- 15 Linear Programming.- 16 Deterministic Lower Bounds.- 17 Probabilistic Machines.- 18 Parallel Computations.- 19 Some Separations of Complexity Classes.- 20 Weak Machines.- 21 Additive Machines.- 22 Nonuniform Complexity Classes.- 23 Descriptive Complexity.- References.

1,542 citations


"Parallel Time and Quantifier Prefix..." refers background or methods in this paper

  • ...These machine models allow one to define the classes PR and PARR of subsets S ⊆ R∞ decidable, respectively, in polynomial and parallel polynomial time (see [2] for details) as well as their exponential versions EXPR and PAREXPR....

    [...]

  • ...Indeed, while the alternation of k blocks of quantifiers leads to the classes Σ R and Π R of the polynomial hierarchy PHR over the reals [2] which is included in PARR [1], the unrestricted alternation of polynomially many quantifiers yields a class PATR (from polynomial alternating time) which strictly includes PARR [6]....

    [...]

  • ...We consider sequential machines over R as originally defined in [3] (see also [2])....

    [...]


Journal ArticleDOI
Walter John Savitch1Institutions (1)
TL;DR: The amount of storage needed to simulate a nondeterministic tape bounded Turingmachine on a deterministic Turing machine is investigated and a specific set is produced, namely the set of all codings of threadable mazes, such that, if there is any set which distinguishes nondeter microscopic complexity classes from deterministic tape complexity classes, then this is one such set.
Abstract: The amount of storage needed to simulate a nondeterministic tape bounded Turingmachine on a deterministic Turing machine is investigated. Results include the following: Theorem. A nondeterministic L(n)-tape bounded Turing machine can be simulated by a deterministic [L(n)]^2-tape bounded Turing machine, provided L(n)>=log"2n. Computations of nondeterministic machines are shown to correspond to threadings of certain mazes. This correspondence is used to produce a specific set, namely the set of all codings of threadable mazes, such that, if there is any set which distinguishes nondeterministic tape complexity classes from deterministic tape complexity classes, then this is one such set.

1,340 citations


"Parallel Time and Quantifier Prefix..." refers background in this paper

  • ...The following result can be seen as a real version of Savitch’s Theorem [9]....

    [...]

  • ...In [9] Savitch proved that NPSPACE = PSPACE....

    [...]



Journal ArticleDOI
TL;DR: This algorithm improves the complexity of the asymptotically fastest algorithm for this problem, known to this data, and new and improved algorithms for deciding a sentence in the first order theory over real closed fields, are obtained.
Abstract: In this paper, a new algorithm for performing quantifier elimination from first order formulas over real closed fields in given. This algorithm improves the complexity of the asymptotically fastest algorithm for this problem, known to this data. A new feature of this algorithm is that the role of the algebraic part (the dependence on the degrees of the imput polynomials) and the combinatorial part (the dependence on the number of polynomials) are sparated. Another new feature is that the degrees of the polynomials in the equivalent quantifier-free formula that is output, are independent of the number of input polynomials. As special cases of this algorithm new and improved algorithms for deciding a sentence in the first order theory over real closed fields, and also for solving the existential problem in the first order theory over real closed fields, are obtained.

383 citations


"Parallel Time and Quantifier Prefix..." refers background or methods in this paper

  • ...On the other hand, the first term in the upper bound for the parallel time of the algorithm in [1] vanishes (since t = 1) and its second term becomes ( ω ∏...

    [...]

  • ...Indeed, while the alternation of k blocks of quantifiers leads to the classes Σ R and Π R of the polynomial hierarchy PHR over the reals [2] which is included in PARR [1], the unrestricted alternation of polynomially many quantifiers yields a class PATR (from polynomial alternating time) which strictly includes PARR [6]....

    [...]

  • ...14 with bounds of similar order would require a thorough reworking of the algorithms of [1] allowing for a recursive computation of nested lists of sign conditions for real and Boolean quantifier elimination altogether, with tighter size and computation time bounds when dealing with Boolean quantifiers....

    [...]

  • ...(iii) One of the main results in [1] shows that quantified sentences over R can be decided in parallel time ( ω ∏...

    [...]


Journal ArticleDOI
James H. Davenport1, Joos Heintz2Institutions (2)
TL;DR: It is shown that quantifier elimination over real closed fields can require doubly exponential space (and hence time) and time, and is done by explicitly constructing a sequence of expressions whose length is linear in the number of quantifiers, but whose quantifier-free expression has length doubly exponentiable.
Abstract: We show that quantifier elimination over real closed fields can require doubly exponential space (and hence time). This is done by explicitly constructing a sequence of expressions whose length is linear in the number of quantifiers, but whose quantifier-free expression has length doubly exponential in the number of quantifiers. The results can be applied to cylindrical algebraic decomposition, showing that this can be doubly exponential. The double exponents of our lower bounds are about one fifth of the double exponents of the best-known upper bounds.

351 citations


"Parallel Time and Quantifier Prefix..." refers background in this paper

  • ...[7] J. H. Davenport and J. Heintz....

    [...]

  • ...The following lemma has its origin in a paper by Davenport and Heintz [7]....

    [...]


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