619 research outputs found
Evaluating Matrix Circuits
The circuit evaluation problem (also known as the compressed word problem)
for finitely generated linear groups is studied. The best upper bound for this
problem is , which is shown by a reduction to polynomial
identity testing. Conversely, the compressed word problem for the linear group
is equivalent to polynomial identity testing. In
the paper, it is shown that the compressed word problem for every finitely
generated nilpotent group is in . Within
the larger class of polycyclic groups we find examples where the compressed
word problem is at least as hard as polynomial identity testing for skew
arithmetic circuits
Complexity of Equivalence and Learning for Multiplicity Tree Automata
We consider the complexity of equivalence and learning for multiplicity tree
automata, i.e., weighted tree automata over a field. We first show that the
equivalence problem is logspace equivalent to polynomial identity testing, the
complexity of which is a longstanding open problem. Secondly, we derive lower
bounds on the number of queries needed to learn multiplicity tree automata in
Angluin's exact learning model, over both arbitrary and fixed fields.
Habrard and Oncina (2006) give an exact learning algorithm for multiplicity
tree automata, in which the number of queries is proportional to the size of
the target automaton and the size of a largest counterexample, represented as a
tree, that is returned by the Teacher. However, the smallest
tree-counterexample may be exponential in the size of the target automaton.
Thus the above algorithm does not run in time polynomial in the size of the
target automaton, and has query complexity exponential in the lower bound.
Assuming a Teacher that returns minimal DAG representations of
counterexamples, we give a new exact learning algorithm whose query complexity
is quadratic in the target automaton size, almost matching the lower bound, and
improving the best previously-known algorithm by an exponential factor
Lessons from obesity prevention for the prevention of mental disorders: The primordial prevention approach
Background: Emerging evidence supports a relationship between risk factors for obesity and the genesis of the common mental disorders, depression and anxiety. This suggests common mental disorders should be considered as a form of non-communicable disease, preventable through the modification of lifestyle behaviours, particularly diet and physical activity.Discussion: Obesity prevention research since the 1970\u27s represents a considerable body of knowledge regarding strategies to modify diet and physical activity and so there may be clear lessons from obesity prevention that apply to the prevention of mental disorders. For obesity, as for common mental disorders, adolescence represents a key period of vulnerability. In this paper we briefly discuss relationships between modifiable lifestyle risk factors and mental health, lifestyle risk factor interventions in obesity prevention research, the current state of mental health prevention, and the implications of current applications of systems thinking in obesity prevention research for lifestyle interventions.Summary: We propose a potential focus for future mental health promotion interventions and emphasise the importance of lessons available from other lifestyle modification intervention programmes
The complexity of satisfaction problems in reverse mathematics
Satisfiability problems play a central role in computer science and
engineering as a general framework for studying the complexity of various
problems. Schaefer proved in 1978 that truth satisfaction of propositional
formulas given a language of relations is either NP-complete or tractable. We
classify the corresponding satisfying assignment construction problems in the
framework of reverse mathematics and show that the principles are either
provable over RCA or equivalent to WKL. We formulate also a Ramseyan version of
the problems and state a different dichotomy theorem. However, the different
classes arising from this classification are not known to be distinct.Comment: 19 page
On Resource-bounded versions of the van Lambalgen theorem
The van Lambalgen theorem is a surprising result in algorithmic information
theory concerning the symmetry of relative randomness. It establishes that for
any pair of infinite sequences and , is Martin-L\"of random and
is Martin-L\"of random relative to if and only if the interleaved sequence
is Martin-L\"of random. This implies that is relative random
to if and only if is random relative to \cite{vanLambalgen},
\cite{Nies09}, \cite{HirschfeldtBook}. This paper studies the validity of this
phenomenon for different notions of time-bounded relative randomness.
We prove the classical van Lambalgen theorem using martingales and Kolmogorov
compressibility. We establish the failure of relative randomness in these
settings, for both time-bounded martingales and time-bounded Kolmogorov
complexity. We adapt our classical proofs when applicable to the time-bounded
setting, and construct counterexamples when they fail. The mode of failure of
the theorem may depend on the notion of time-bounded randomness
Polynomial Time Algorithms for Branching Markov Decision Processes and Probabilistic Min(Max) Polynomial Bellman Equations
We show that one can approximate the least fixed point solution for a
multivariate system of monotone probabilistic max(min) polynomial equations,
referred to as maxPPSs (and minPPSs, respectively), in time polynomial in both
the encoding size of the system of equations and in log(1/epsilon), where
epsilon > 0 is the desired additive error bound of the solution. (The model of
computation is the standard Turing machine model.) We establish this result
using a generalization of Newton's method which applies to maxPPSs and minPPSs,
even though the underlying functions are only piecewise-differentiable. This
generalizes our recent work which provided a P-time algorithm for purely
probabilistic PPSs.
These equations form the Bellman optimality equations for several important
classes of infinite-state Markov Decision Processes (MDPs). Thus, as a
corollary, we obtain the first polynomial time algorithms for computing to
within arbitrary desired precision the optimal value vector for several classes
of infinite-state MDPs which arise as extensions of classic, and heavily
studied, purely stochastic processes. These include both the problem of
maximizing and mininizing the termination (extinction) probability of
multi-type branching MDPs, stochastic context-free MDPs, and 1-exit Recursive
MDPs.
Furthermore, we also show that we can compute in P-time an epsilon-optimal
policy for both maximizing and minimizing branching, context-free, and
1-exit-Recursive MDPs, for any given desired epsilon > 0. This is despite the
fact that actually computing optimal strategies is Sqrt-Sum-hard and
PosSLP-hard in this setting.
We also derive, as an easy consequence of these results, an FNP upper bound
on the complexity of computing the value (within arbitrary desired precision)
of branching simple stochastic games (BSSGs)
A parameterized halting problem, the linear time hierarchy, and the MRDP theorem
The complexity of the parameterized halting problem for nondeterministic Turing machines p-Halt is known to be related to the question of whether there are logics capturing various complexity classes [10]. Among others, if p-Halt is in para-AC0, the parameterized version of the circuit complexity class AC0, then AC0, or equivalently, (+, x)-invariant FO, has a logic. Although it is widely believed that p-Halt ∉. para-AC0, we show that the problem is hard to settle by establishing a connection to the question in classical complexity of whether NE ⊈ LINH. Here, LINH denotes the linear time hierarchy.
On the other hand, we suggest an approach toward proving NE ⊈ LINH using bounded arithmetic. More specifically, we demonstrate that if the much celebrated MRDP (for Matiyasevich-Robinson-Davis-Putnam) theorem can be proved in a certain fragment of arithmetic, then NE ⊈ LINH. Interestingly, central to this result is a para-AC0 lower bound for the parameterized model-checking problem for FO on arithmetical structures.Peer ReviewedPostprint (author's final draft
Improved bounds for reduction to depth 4 and depth 3
Koiran showed that if a -variate polynomial of degree (with
) is computed by a circuit of size , then it is also computed by
a homogeneous circuit of depth four and of size
. Using this result, Gupta, Kamath, Kayal and
Saptharishi gave a upper bound for the
size of the smallest depth three circuit computing a -variate polynomial of
degree given by a circuit of size .
We improve here Koiran's bound. Indeed, we show that if we reduce an
arithmetic circuit to depth four, then the size becomes
. Mimicking Gupta, Kamath, Kayal and
Saptharishi's proof, it also implies the same upper bound for depth three
circuits.
This new bound is not far from optimal in the sense that Gupta, Kamath, Kayal
and Saptharishi also showed a lower bound for the size
of homogeneous depth four circuits such that gates at the bottom have fan-in at
most . Finally, we show that this last lower bound also holds if the
fan-in is at least
Hardness of Sparse Sets and Minimal Circuit Size Problem
We develop a polynomial method on finite fields to amplify the hardness of
spare sets in nondeterministic time complexity classes on a randomized
streaming model. One of our results shows that if there exists a
-sparse set in that does not have any
randomized streaming algorithm with updating time, and
space, then , where a -sparse set is a language that has at
most strings of length . We also show that if MCSP is -hard
under polynomial time truth-table reductions, then
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