11 research outputs found
Complexity of Two-Dimensional Patterns
In dynamical systems such as cellular automata and iterated maps, it is often
useful to look at a language or set of symbol sequences produced by the system.
There are well-established classification schemes, such as the Chomsky
hierarchy, with which we can measure the complexity of these sets of sequences,
and thus the complexity of the systems which produce them.
In this paper, we look at the first few levels of a hierarchy of complexity
for two-or-more-dimensional patterns. We show that several definitions of
``regular language'' or ``local rule'' that are equivalent in d=1 lead to
distinct classes in d >= 2. We explore the closure properties and computational
complexity of these classes, including undecidability and L-, NL- and
NP-completeness results.
We apply these classes to cellular automata, in particular to their sets of
fixed and periodic points, finite-time images, and limit sets. We show that it
is undecidable whether a CA in d >= 2 has a periodic point of a given period,
and that certain ``local lattice languages'' are not finite-time images or
limit sets of any CA. We also show that the entropy of a d-dimensional CA's
finite-time image cannot decrease faster than t^{-d} unless it maps every
initial condition to a single homogeneous state.Comment: To appear in J. Stat. Phy
q-Plane Zeros of the Potts Partition Function on Diamond Hierarchical Graphs
We report exact results concerning the zeros of the partition function of the Potts model in the complex q-plane, as a function of a temperature-like Boltzmann variable v, for the m-th iterate graphs Dm of the diamond hierarchical lattice, including the limit m → ∞. In this limit, we denote the continuous accumulation locus of zeros in the q-planes at fixed v = v0 as ℬ(0). We apply theorems from complex dynamics to establish the properties of ℬ(0). For v = −1 (the zero-temperature Potts antiferromagnet or, equivalently, chromatic polynomial), we prove that ℬ(−1) crosses the real q-axis at (i) a minimal point q = 0, (ii) a maximal point q = 3, (iii) q = 32/27, (iv) a cubic root that we give, with the value q = q1 = 1.638 896 9…, and (v) an infinite number of points smaller than q1, converging to 32/27 from above. Similar results hold for ℬ(0) for any −1 0 (Potts ferromagnet). We also provide the computer-generated plots of ℬ(0) at various values of v0 in both the antiferromagnetic and ferromagnetic regimes and compare them to the numerically computed zeros of Z(D4, q, v0)
Multiclass Learnability Beyond the PAC Framework: Universal Rates and Partial Concept Classes
In this paper we study the problem of multiclass classification with a
bounded number of different labels , in the realizable setting. We extend
the traditional PAC model to a) distribution-dependent learning rates, and b)
learning rates under data-dependent assumptions. First, we consider the
universal learning setting (Bousquet, Hanneke, Moran, van Handel and
Yehudayoff, STOC '21), for which we provide a complete characterization of the
achievable learning rates that holds for every fixed distribution. In
particular, we show the following trichotomy: for any concept class, the
optimal learning rate is either exponential, linear or arbitrarily slow.
Additionally, we provide complexity measures of the underlying hypothesis class
that characterize when these rates occur. Second, we consider the problem of
multiclass classification with structured data (such as data lying on a low
dimensional manifold or satisfying margin conditions), a setting which is
captured by partial concept classes (Alon, Hanneke, Holzman and Moran, FOCS
'21). Partial concepts are functions that can be undefined in certain parts of
the input space. We extend the traditional PAC learnability of total concept
classes to partial concept classes in the multiclass setting and investigate
differences between partial and total concepts
Nonstandard Methods in Ramsey Theory and Combinatorial Number Theory
The goal of this present manuscript is to introduce the reader to the
nonstandard method and to provide an overview of its most prominent
applications in Ramsey theory and combinatorial number theory.Comment: 126 pages. Comments welcom