2,126 research outputs found
Lower Bounds for Alternating Online State Complexity
The notion of Online State Complexity, introduced by Karp in 1967, quantifies
the amount of states required to solve a given problem using an online
algorithm, which is represented by a deterministic machine scanning the input
from left to right in one pass. In this paper, we extend the setting to
alternating machines as introduced by Chandra, Kozen and Stockmeyer in 1976:
such machines run independent passes scanning the input from left to right and
gather their answers through boolean combinations. We devise a lower bound
technique relying on boundedly generated lattices of languages, and give two
applications of this technique. The first is a hierarchy theorem , stating that
the polynomial hierarchy of alternating online state complexity is infinite,
and the second is a linear lower bound on the alternating online state
complexity of the prime numbers written in binary. This second result
strengthens a result of Hartmanis and Shank from 1968, which implies an
exponentially worse lower bound for the same model
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
Two-Way Automata Making Choices Only at the Endmarkers
The question of the state-size cost for simulation of two-way
nondeterministic automata (2NFAs) by two-way deterministic automata (2DFAs) was
raised in 1978 and, despite many attempts, it is still open. Subsequently, the
problem was attacked by restricting the power of 2DFAs (e.g., using a
restricted input head movement) to the degree for which it was already possible
to derive some exponential gaps between the weaker model and the standard
2NFAs. Here we use an opposite approach, increasing the power of 2DFAs to the
degree for which it is still possible to obtain a subexponential conversion
from the stronger model to the standard 2DFAs. In particular, it turns out that
subexponential conversion is possible for two-way automata that make
nondeterministic choices only when the input head scans one of the input tape
endmarkers. However, there is no restriction on the input head movement. This
implies that an exponential gap between 2NFAs and 2DFAs can be obtained only
for unrestricted 2NFAs using capabilities beyond the proposed new model. As an
additional bonus, conversion into a machine for the complement of the original
language is polynomial in this model. The same holds for making such machines
self-verifying, halting, or unambiguous. Finally, any superpolynomial lower
bound for the simulation of such machines by standard 2DFAs would imply LNL.
In the same way, the alternating version of these machines is related to L =?
NL =? P, the classical computational complexity problems.Comment: 23 page
How Much Lookahead is Needed to Win Infinite Games?
Delay games are two-player games of infinite duration in which one player may
delay her moves to obtain a lookahead on her opponent's moves. For
-regular winning conditions it is known that such games can be solved
in doubly-exponential time and that doubly-exponential lookahead is sufficient.
We improve upon both results by giving an exponential time algorithm and an
exponential upper bound on the necessary lookahead. This is complemented by
showing EXPTIME-hardness of the solution problem and tight exponential lower
bounds on the lookahead. Both lower bounds already hold for safety conditions.
Furthermore, solving delay games with reachability conditions is shown to be
PSPACE-complete.
This is a corrected version of the paper https://arxiv.org/abs/1412.3701v4
published originally on August 26, 2016
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