1,159 research outputs found
Efficient Parallel Path Checking for Linear-Time Temporal Logic With Past and Bounds
Path checking, the special case of the model checking problem where the model
under consideration is a single path, plays an important role in monitoring,
testing, and verification. We prove that for linear-time temporal logic (LTL),
path checking can be efficiently parallelized. In addition to the core logic,
we consider the extensions of LTL with bounded-future (BLTL) and past-time
(LTL+Past) operators. Even though both extensions improve the succinctness of
the logic exponentially, path checking remains efficiently parallelizable: Our
algorithm for LTL, LTL+Past, and BLTL+Past is in AC^1(logDCFL) \subseteq NC
On the Complexity of Temporal-Logic Path Checking
Given a formula in a temporal logic such as LTL or MTL, a fundamental problem
is the complexity of evaluating the formula on a given finite word. For LTL,
the complexity of this task was recently shown to be in NC. In this paper, we
present an NC algorithm for MTL, a quantitative (or metric) extension of LTL,
and give an NCC algorithm for UTL, the unary fragment of LTL. At the time of
writing, MTL is the most expressive logic with an NC path-checking algorithm,
and UTL is the most expressive fragment of LTL with a more efficient
path-checking algorithm than for full LTL (subject to standard
complexity-theoretic assumptions). We then establish a connection between LTL
path checking and planar circuits, which we exploit to show that any further
progress in determining the precise complexity of LTL path checking would
immediately entail more efficient evaluation algorithms than are known for a
certain class of planar circuits. The connection further implies that the
complexity of LTL path checking depends on the Boolean connectives allowed:
adding Boolean exclusive or yields a temporal logic with P-complete
path-checking problem
Majority-Vote Cellular Automata, Ising Dynamics, and P-Completeness
We study cellular automata where the state at each site is decided by a
majority vote of the sites in its neighborhood. These are equivalent, for a
restricted set of initial conditions, to non-zero probability transitions in
single spin-flip dynamics of the Ising model at zero temperature.
We show that in three or more dimensions these systems can simulate Boolean
circuits of AND and OR gates, and are therefore P-complete. That is, predicting
their state t time-steps in the future is at least as hard as any other problem
that takes polynomial time on a serial computer.
Therefore, unless a widely believed conjecture in computer science is false,
it is impossible even with parallel computation to predict majority-vote
cellular automata, or zero-temperature single spin-flip Ising dynamics,
qualitatively faster than by explicit simulation.Comment: 10 pages with figure
The Computational Complexity of Generating Random Fractals
In this paper we examine a number of models that generate random fractals.
The models are studied using the tools of computational complexity theory from
the perspective of parallel computation. Diffusion limited aggregation and
several widely used algorithms for equilibrating the Ising model are shown to
be highly sequential; it is unlikely they can be simulated efficiently in
parallel. This is in contrast to Mandelbrot percolation that can be simulated
in constant parallel time. Our research helps shed light on the intrinsic
complexity of these models relative to each other and to different growth
processes that have been recently studied using complexity theory. In addition,
the results may serve as a guide to simulation physics.Comment: 28 pages, LATEX, 8 Postscript figures available from
[email protected]
Parallel dynamics and computational complexity of the Bak-Sneppen model
The parallel computational complexity of the Bak-Sneppen evolution model is
studied. It is shown that Bak-Sneppen histories can be generated by a massively
parallel computer in a time that is polylogarithmic in the length of the
history. In this parallel dynamics, histories are built up via a nested
hierarchy of avalanches. Stated in another way, the main result is that the
logical depth of producing a Bak-Sneppen history is exponentially less than the
length of the history. This finding is surprising because the self-organized
critical state of the Bak-Sneppen model has long range correlations in time and
space that appear to imply that the dynamics is sequential and history
dependent. The parallel dynamics for generating Bak-Sneppen histories is
contrasted to standard Bak-Sneppen dynamics. Standard dynamics and an alternate
method for generating histories, conditional dynamics, are both shown to be
related to P-complete natural decision problems implying that they cannot be
efficiently implemented in parallel.Comment: 37 pages, 12 figure
Three Puzzles on Mathematics, Computation, and Games
In this lecture I will talk about three mathematical puzzles involving
mathematics and computation that have preoccupied me over the years. The first
puzzle is to understand the amazing success of the simplex algorithm for linear
programming. The second puzzle is about errors made when votes are counted
during elections. The third puzzle is: are quantum computers possible?Comment: ICM 2018 plenary lecture, Rio de Janeiro, 36 pages, 7 Figure
Parameterized complexity of the MINCCA problem on graphs of bounded decomposability
In an edge-colored graph, the cost incurred at a vertex on a path when two
incident edges with different colors are traversed is called reload or
changeover cost. The "Minimum Changeover Cost Arborescence" (MINCCA) problem
consists in finding an arborescence with a given root vertex such that the
total changeover cost of the internal vertices is minimized. It has been
recently proved by G\"oz\"upek et al. [TCS 2016] that the problem is FPT when
parameterized by the treewidth and the maximum degree of the input graph. In
this article we present the following results for the MINCCA problem:
- the problem is W[1]-hard parameterized by the treedepth of the input graph,
even on graphs of average degree at most 8. In particular, it is W[1]-hard
parameterized by the treewidth of the input graph, which answers the main open
problem of G\"oz\"upek et al. [TCS 2016];
- it is W[1]-hard on multigraphs parameterized by the tree-cutwidth of the
input multigraph;
- it is FPT parameterized by the star tree-cutwidth of the input graph, which
is a slightly restricted version of tree-cutwidth. This result strictly
generalizes the FPT result given in G\"oz\"upek et al. [TCS 2016];
- it remains NP-hard on planar graphs even when restricted to instances with
at most 6 colors and 0/1 symmetric costs, or when restricted to instances with
at most 8 colors, maximum degree bounded by 4, and 0/1 symmetric costs.Comment: 25 pages, 11 figure
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