51,487 research outputs found
The polynomial-time hierarchy
AbstractThe polynomial-time hierarchy is that subrecursive analog of the Kleene arithmetical hierarchy in which deterministic (nondeterministic) polynomial time plays the role of recursive (recursively enumerable) time. Known properties of the polynomial-time hierarchy are summarized. A word problem which is complete in the second stage of the hierarchy is exhibited. In the analogy between the polynomial-time hierarchy and the arithmetical hierarchy, the first order theory of equality plays the role of elementary arithmetic (as the ω-jump of the hierarchy). The problem of deciding validity in the theory of equality is shown to be complete in polynomial-space, and close upper and lower bounds on the space complexity of this problem are established
On space efficiency of algorithms working on structural decompositions of graphs
Dynamic programming on path and tree decompositions of graphs is a technique
that is ubiquitous in the field of parameterized and exponential-time
algorithms. However, one of its drawbacks is that the space usage is
exponential in the decomposition's width. Following the work of Allender et al.
[Theory of Computing, '14], we investigate whether this space complexity
explosion is unavoidable. Using the idea of reparameterization of Cai and
Juedes [J. Comput. Syst. Sci., '03], we prove that the question is closely
related to a conjecture that the Longest Common Subsequence problem
parameterized by the number of input strings does not admit an algorithm that
simultaneously uses XP time and FPT space. Moreover, we complete the complexity
landscape sketched for pathwidth and treewidth by Allender et al. by
considering the parameter tree-depth. We prove that computations on tree-depth
decompositions correspond to a model of non-deterministic machines that work in
polynomial time and logarithmic space, with access to an auxiliary stack of
maximum height equal to the decomposition's depth. Together with the results of
Allender et al., this describes a hierarchy of complexity classes for
polynomial-time non-deterministic machines with different restrictions on the
access to working space, which mirrors the classic relations between treewidth,
pathwidth, and tree-depth.Comment: An extended abstract appeared in the proceedings of STACS'16. The new
version is augmented with a space-efficient algorithm for Dominating Set
using the Chinese remainder theore
Computational complexity of ecological and evolutionary spatial dynamics
There are deep, yet largely unexplored, connections between computer science and biology. Both disciplines examine how information proliferates in time and space. Central results in computer science describe the complexity of algorithms that solve certain classes of problems. An algorithm is deemed efficient if it can solve a problem in polynomial time, which means the running time of the algorithm is a polynomial function of the length of the input. There are classes of harder problems for which the fastest possible algorithm requires exponential time. Another criterion is the space requirement of the algorithm. There is a crucial distinction between algorithms that can find a solution, verify a solution, or list several distinct solutions in given time and space. The complexity hierarchy that is generated in this way is the foundation of theoretical computer science. Precise complexity results can be notoriously difficult. The famous question whether polynomial time equals nondeterministic polynomial time (i.e., P = NP) is one of the hardest open problems in computer science and all of mathematics. Here, we consider simple processes of ecological and evolutionary spatial dynamics. The basic question is: What is the probability that a new invader (or a new mutant) will take over a resident population? We derive precise complexity results for a variety of scenarios. We therefore show that some fundamental questions in this area cannot be answered by simple equations (assuming that P is not equal to NP)
Robust Simulations and Significant Separations
We define and study a new notion of "robust simulations" between complexity
classes which is intermediate between the traditional notions of
infinitely-often and almost-everywhere, as well as a corresponding notion of
"significant separations". A language L has a robust simulation in a complexity
class C if there is a language in C which agrees with L on arbitrarily large
polynomial stretches of input lengths. There is a significant separation of L
from C if there is no robust simulation of L in C. The new notion of simulation
is a cleaner and more natural notion of simulation than the infinitely-often
notion. We show that various implications in complexity theory such as the
collapse of PH if NP = P and the Karp-Lipton theorem have analogues for robust
simulations. We then use these results to prove that most known separations in
complexity theory, such as hierarchy theorems, fixed polynomial circuit lower
bounds, time-space tradeoffs, and the theorems of Allender and Williams, can be
strengthened to significant separations, though in each case, an almost
everywhere separation is unknown.
Proving our results requires several new ideas, including a completely
different proof of the hierarchy theorem for non-deterministic polynomial time
than the ones previously known
The Descriptive Complexity of the Deterministic Exponential Time Hierarchy
AbstractIn Descriptive Complexity, we investigate the use of logics to characterize computational complexity classes. Since 1974, when Fagin proved that the class NP is captured by existential second-order logic, considered the first result in this area, other relations between logics and complexity classes have been established. Well-known results usually involve first-order logic and its extensions, and complexity classes in polynomial time or space. Some examples are that the first-order logic extended by the least fixed-point operator captures the class P and the second-order logic extended by the transitive closure operator captures the class PSPACE. In this paper, we will analyze the combined use of higher-order logics of order i, HOi, for i⩾2, extended by the least fixed-point operator, and we will prove that each level of this hierarchy captures each level of the deterministic exponential time hierarchy. As a corollary, we will prove that the hierarchy of HOi(LFP), for i⩾2, does not collapse, that is, HOi(LFP)⊂HOi+1(LFP)
The word problem for omega-terms over the Trotter-Weil hierarchy [extended abstract]
© Springer International Publishing Switzerland 2016. Over finitewords, there is a tight connection between the quantifier alternation hierarchy inside two-variable first-order logic FO 2 and a hierarchy of finite monoids: theTrotter-Weil Hierarchy. The variousways of climbing up this hierarchy include Mal’cev products, deterministic and codeterministic concatenation as well as identities of ω-terms.We show that the word problem for ω-terms over each level of the Trotter-Weil Hierarchy is decidable; this means, for every variety V of the hierarchy and every identity u = v of ω-terms, one can decide whether all monoids in V satisfy u = v. More precisely, for every fixed variety V, our approach yields nondeterministic logarithmic space (NL) and deterministic polynomial time algorithms, which are more efficient than straightforward translations of the NL-algorithms. From a language perspective, the word problem for ω- terms is the following: for every language variety V in theTrotter-Weil Hierarchy and every language varietyWgivenbyan identity of ω-terms, one can decide whether V ⊆ W. This includes the case where V is some level of the FO 2 quantifier alternation hierarchy. As an application of our results, we show that the separation problems for the so-called corners of the Trotter- Weil Hierarchy are decidable
Stochastic Schroedinger equation from optimal observable evolution
In this article, we consider a set of trial wave-functions denoted by | Q
\right> and an associated set of operators which generate
transformations connecting those trial states. Using variational principles, we
show that we can always obtain a quantum Monte-Carlo method where the quantum
evolution of a system is replaced by jumps between density matrices of the form
, and where the average evolutions of the moments of
up to a given order , i.e. , $< A_{\alpha_1}
A_{\alpha_2} >$,..., , are constrained to
follow the exact Ehrenfest evolution at each time step along each stochastic
trajectory. Then, a set of more and more elaborated stochastic approximations
of a quantum problem is obtained which approach the exact solution when more
and more constraints are imposed, i.e. when increases. The Monte-Carlo
process might even become exact if the Hamiltonian applied on the trial
state can be written as a polynomial of . The formalism makes a
natural connection between quantum jumps in Hilbert space and phase-space
dynamics. We show that the derivation of stochastic Schroedinger equations can
be greatly simplified by taking advantage of the existence of this hierarchy of
approximations and its connection to the Ehrenfest theorem. Several examples
are illustrated: the free wave-packet expansion, the Kerr oscillator, a
generalized version of the Kerr oscillator, as well as interacting bosons or
fermions.Comment: 13 pages, 1 figur
Space Efficiency of Propositional Knowledge Representation Formalisms
We investigate the space efficiency of a Propositional Knowledge
Representation (PKR) formalism. Intuitively, the space efficiency of a
formalism F in representing a certain piece of knowledge A, is the size of the
shortest formula of F that represents A. In this paper we assume that knowledge
is either a set of propositional interpretations (models) or a set of
propositional formulae (theorems). We provide a formal way of talking about the
relative ability of PKR formalisms to compactly represent a set of models or a
set of theorems. We introduce two new compactness measures, the corresponding
classes, and show that the relative space efficiency of a PKR formalism in
representing models/theorems is directly related to such classes. In
particular, we consider formalisms for nonmonotonic reasoning, such as
circumscription and default logic, as well as belief revision operators and the
stable model semantics for logic programs with negation. One interesting result
is that formalisms with the same time complexity do not necessarily belong to
the same space efficiency class
- …