110,595 research outputs found
Computing Runs on a General Alphabet
We describe a RAM algorithm computing all runs (maximal repetitions) of a
given string of length over a general ordered alphabet in
time and linear space. Our algorithm outperforms all
known solutions working in time provided , where is the alphabet size. We conjecture that there
exists a linear time RAM algorithm finding all runs.Comment: 4 pages, 2 figure
Towards the Jacquet conjecture on the local converse problem for p-adic GL(n)
The Local Converse Problem is to determine how the family of twisted local gamma factors characterizes the isomorphism class of an irreducible admissible generic representation of GL(n,F), with F a non-archimedean local field, where the twists run through all irreducible supercuspidal representations of GL(r,F) and r runs through positive integers. The Jacquet conjecture asserts that it is enough to take r = 1,2,...,[n/2]. Based on arguments in the work of Henniart and of Chen giving preliminary steps towards the Jacquet conjecture, we formulate a general approach to prove the Jacquet conjecture. With this approach, the Jacquet conjecture is proved under an assumption which is then verified in several cases, including the case of level zero representations
The Reachability Problem for Petri Nets is Not Elementary
Petri nets, also known as vector addition systems, are a long established
model of concurrency with extensive applications in modelling and analysis of
hardware, software and database systems, as well as chemical, biological and
business processes. The central algorithmic problem for Petri nets is
reachability: whether from the given initial configuration there exists a
sequence of valid execution steps that reaches the given final configuration.
The complexity of the problem has remained unsettled since the 1960s, and it is
one of the most prominent open questions in the theory of verification.
Decidability was proved by Mayr in his seminal STOC 1981 work, and the
currently best published upper bound is non-primitive recursive Ackermannian of
Leroux and Schmitz from LICS 2019. We establish a non-elementary lower bound,
i.e. that the reachability problem needs a tower of exponentials of time and
space. Until this work, the best lower bound has been exponential space, due to
Lipton in 1976. The new lower bound is a major breakthrough for several
reasons. Firstly, it shows that the reachability problem is much harder than
the coverability (i.e., state reachability) problem, which is also ubiquitous
but has been known to be complete for exponential space since the late 1970s.
Secondly, it implies that a plethora of problems from formal languages, logic,
concurrent systems, process calculi and other areas, that are known to admit
reductions from the Petri nets reachability problem, are also not elementary.
Thirdly, it makes obsolete the currently best lower bounds for the reachability
problems for two key extensions of Petri nets: with branching and with a
pushdown stack.Comment: Final version of STOC'1
Lang-Trotter and Sato-Tate Distributions in Single and Double Parametric Families of Elliptic Curves
We obtain new results concerning Lang-Trotter conjecture on Frobenius traces
and Frobenius fields over single and double parametric families of elliptic
curves. We also obtain similar results with respect to the Sato-Tate
conjecture. In particular, we improve a result of A.C. Cojocaru and the second
author (2008) towards the Lang-Trotter conjecture on average for polynomially
parameterized families of elliptic curves when the parameter runs through a set
of rational numbers of bounded height. Some of the families we consider are
much thinner than the ones previously studied
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