1,548 research outputs found
Alternation-Trading Proofs, Linear Programming, and Lower Bounds
A fertile area of recent research has demonstrated concrete polynomial time
lower bounds for solving natural hard problems on restricted computational
models. Among these problems are Satisfiability, Vertex Cover, Hamilton Path,
Mod6-SAT, Majority-of-Majority-SAT, and Tautologies, to name a few. The proofs
of these lower bounds follow a certain proof-by-contradiction strategy that we
call alternation-trading. An important open problem is to determine how
powerful such proofs can possibly be.
We propose a methodology for studying these proofs that makes them amenable
to both formal analysis and automated theorem proving. We prove that the search
for better lower bounds can often be turned into a problem of solving a large
series of linear programming instances. Implementing a small-scale theorem
prover based on this result, we extract new human-readable time lower bounds
for several problems. This framework can also be used to prove concrete
limitations on the current techniques.Comment: To appear in STACS 2010, 12 page
Defect Particle Kinematics in One-Dimensional Cellular Automata
Let A^Z be the Cantor space of bi-infinite sequences in a finite alphabet A,
and let sigma be the shift map on A^Z. A `cellular automaton' is a continuous,
sigma-commuting self-map Phi of A^Z, and a `Phi-invariant subshift' is a
closed, (Phi,sigma)-invariant subset X of A^Z. Suppose x is a sequence in A^Z
which is X-admissible everywhere except for some small region we call a
`defect'. It has been empirically observed that such defects persist under
iteration of Phi, and often propagate like `particles'. We characterize the
motion of these particles, and show that it falls into several regimes, ranging
from simple deterministic motion, to generalized random walks, to complex
motion emulating Turing machines or pushdown automata. One consequence is that
some questions about defect behaviour are formally undecidable.Comment: 37 pages, 9 figures, 3 table
Coding-theorem Like Behaviour and Emergence of the Universal Distribution from Resource-bounded Algorithmic Probability
Previously referred to as `miraculous' in the scientific literature because
of its powerful properties and its wide application as optimal solution to the
problem of induction/inference, (approximations to) Algorithmic Probability
(AP) and the associated Universal Distribution are (or should be) of the
greatest importance in science. Here we investigate the emergence, the rates of
emergence and convergence, and the Coding-theorem like behaviour of AP in
Turing-subuniversal models of computation. We investigate empirical
distributions of computing models in the Chomsky hierarchy. We introduce
measures of algorithmic probability and algorithmic complexity based upon
resource-bounded computation, in contrast to previously thoroughly investigated
distributions produced from the output distribution of Turing machines. This
approach allows for numerical approximations to algorithmic
(Kolmogorov-Chaitin) complexity-based estimations at each of the levels of a
computational hierarchy. We demonstrate that all these estimations are
correlated in rank and that they converge both in rank and values as a function
of computational power, despite fundamental differences between computational
models. In the context of natural processes that operate below the Turing
universal level because of finite resources and physical degradation, the
investigation of natural biases stemming from algorithmic rules may shed light
on the distribution of outcomes. We show that up to 60\% of the
simplicity/complexity bias in distributions produced even by the weakest of the
computational models can be accounted for by Algorithmic Probability in its
approximation to the Universal Distribution.Comment: 27 pages main text, 39 pages including supplement. Online complexity
calculator: http://complexitycalculator.com
Bounded Counter Languages
We show that deterministic finite automata equipped with two-way heads
are equivalent to deterministic machines with a single two-way input head and
linearly bounded counters if the accepted language is strictly bounded,
i.e., a subset of for a fixed sequence of symbols . Then we investigate linear speed-up for counter machines. Lower
and upper time bounds for concrete recognition problems are shown, implying
that in general linear speed-up does not hold for counter machines. For bounded
languages we develop a technique for speeding up computations by any constant
factor at the expense of adding a fixed number of counters
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