1,417 research outputs found
Permutations generated by a depth 2 stack and an infinite stack in series are algebraic
© 2015, Australian National University. All rights reserved. We prove that the class of permutations generated by passing an ordered sequence 12... n through a stack of depth 2 and an in nite stack in series is in bi-jection with an unambiguous context-free language, where a permutation of length n is encoded by a string of length 3n. It follows that the sequence counting the number of permutations of each length has an algebraic generating function. We use the explicit context-free grammar to compute the generating function:(formula presented) where cn is the number of permutations of length n that can be generated, and (formula presented) is a simple variant of the Catalan generating function. This in turn implies that (formula presented
Quasi-Linear Cellular Automata
Simulating a cellular automaton (CA) for t time-steps into the future
requires t^2 serial computation steps or t parallel ones. However, certain CAs
based on an Abelian group, such as addition mod 2, are termed ``linear''
because they obey a principle of superposition. This allows them to be
predicted efficiently, in serial time O(t) or O(log t) in parallel.
In this paper, we generalize this by looking at CAs with a variety of
algebraic structures, including quasigroups, non-Abelian groups, Steiner
systems, and others. We show that in many cases, an efficient algorithm exists
even though these CAs are not linear in the previous sense; we term them
``quasilinear.'' We find examples which can be predicted in serial time
proportional to t, t log t, t log^2 t, and t^a for a < 2, and parallel time log
t, log t log log t and log^2 t.
We also discuss what algebraic properties are required or implied by the
existence of scaling relations and principles of superposition, and exhibit
several novel ``vector-valued'' CAs.Comment: 41 pages with figures, To appear in Physica
Two first-order logics of permutations
We consider two orthogonal points of view on finite permutations, seen as
pairs of linear orders (corresponding to the usual one line representation of
permutations as words) or seen as bijections (corresponding to the algebraic
point of view). For each of them, we define a corresponding first-order logical
theory, that we call (Theory Of Two Orders) and
(Theory Of One Bijection) respectively. We consider various expressibility
questions in these theories.
Our main results go in three different direction. First, we prove that, for
all , the set of -stack sortable permutations in the sense of West
is expressible in , and that a logical sentence describing this
set can be obtained automatically. Previously, descriptions of this set were
only known for . Next, we characterize permutation classes inside
which it is possible to express in that some given points form
a cycle. Lastly, we show that sets of permutations that can be described both
in and are in some sense trivial. This gives a
mathematical evidence that permutations-as-bijections and permutations-as-words
are somewhat different objects.Comment: v2: minor changes, following a referee repor
Transformers Learn Shortcuts to Automata
Algorithmic reasoning requires capabilities which are most naturally
understood through recurrent models of computation, like the Turing machine.
However, Transformer models, while lacking recurrence, are able to perform such
reasoning using far fewer layers than the number of reasoning steps. This
raises the question: what solutions are learned by these shallow and
non-recurrent models? We find that a low-depth Transformer can represent the
computations of any finite-state automaton (thus, any bounded-memory
algorithm), by hierarchically reparameterizing its recurrent dynamics. Our
theoretical results characterize shortcut solutions, whereby a Transformer with
layers can exactly replicate the computation of an automaton on an input
sequence of length . We find that polynomial-sized -depth
solutions always exist; furthermore, -depth simulators are surprisingly
common, and can be understood using tools from Krohn-Rhodes theory and circuit
complexity. Empirically, we perform synthetic experiments by training
Transformers to simulate a wide variety of automata, and show that shortcut
solutions can be learned via standard training. We further investigate the
brittleness of these solutions and propose potential mitigations
Enumeration of polyominoes defined in terms of pattern avoidance or convexity constraints
In this thesis, we consider the problem of characterizing and enumerating
sets of polyominoes described in terms of some constraints, defined either by
convexity or by pattern containment. We are interested in a well known subclass
of convex polyominoes, the k-convex polyominoes for which the enumeration
according to the semi-perimeter is known only for k=1,2. We obtain, from a
recursive decomposition, the generating function of the class of k-convex
parallelogram polyominoes, which turns out to be rational. Noting that this
generating function can be expressed in terms of the Fibonacci polynomials, we
describe a bijection between the class of k-parallelogram polyominoes and the
class of planted planar trees having height less than k+3. In the second part
of the thesis we examine the notion of pattern avoidance, which has been
extensively studied for permutations. We introduce the concept of pattern
avoidance in the context of matrices, more precisely permutation matrices and
polyomino matrices. We present definitions analogous to those given for
permutations and in particular we define polyomino classes, i.e. sets downward
closed with respect to the containment relation. So, the study of the old and
new properties of the redefined sets of objects has not only become
interesting, but it has also suggested the study of the associated poset. In
both approaches our results can be used to treat open problems related to
polyominoes as well as other combinatorial objects.Comment: PhD thesi
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