8 research outputs found
On Problems Dual to Unification: The String-Rewriting Case
In this paper, we investigate problems which are dual to the unification
problem, namely the Fixed Point (FP) problem, Common Term (CT) problem and the
Common Equation (CE) problem for string rewriting systems. Our main motivation
is computing fixed points in systems, such as loop invariants in programming
languages. We show that the fixed point (FP) problem is reducible to the common
term problem. Our new results are: (i) the fixed point problem is undecidable
for finite convergent string rewriting systems whereas it is decidable in
polynomial time for finite, convergent and dwindling string rewriting systems,
(ii) the common term problem is undecidable for the class of dwindling string
rewriting systems, and (iii) for the class of finite, monadic and convergent
systems, the common equation problem is decidable in polynomial time but for
the class of dwindling string rewriting systems, common equation problem is
undecidable.Comment: 28 pages, 6 figures, will be submitted for LMCS journal. arXiv admin
note: substantial text overlap with arXiv:1706.0560
Estimation under group actions: recovering orbits from invariants
Motivated by geometric problems in signal processing, computer vision, and
structural biology, we study a class of orbit recovery problems where we
observe very noisy copies of an unknown signal, each acted upon by a random
element of some group (such as Z/p or SO(3)). The goal is to recover the orbit
of the signal under the group action in the high-noise regime. This generalizes
problems of interest such as multi-reference alignment (MRA) and the
reconstruction problem in cryo-electron microscopy (cryo-EM). We obtain
matching lower and upper bounds on the sample complexity of these problems in
high generality, showing that the statistical difficulty is intricately
determined by the invariant theory of the underlying symmetry group.
In particular, we determine that for cryo-EM with noise variance
and uniform viewing directions, the number of samples required scales as
. We match this bound with a novel algorithm for ab initio
reconstruction in cryo-EM, based on invariant features of degree at most 3. We
further discuss how to recover multiple molecular structures from heterogeneous
cryo-EM samples.Comment: 54 pages. This version contains a number of new result
Logic and Automata
Mathematical logic and automata theory are two scientific disciplines with a fundamentally close relationship. The authors of Logic and Automata take the occasion of the sixtieth birthday of Wolfgang Thomas to present a tour d'horizon of automata theory and logic. The twenty papers in this volume cover many different facets of logic and automata theory, emphasizing the connections to other disciplines such as games, algorithms, and semigroup theory, as well as discussing current challenges in the field