4,370 research outputs found
2-State 2-Symbol Turing Machines with Periodic Support Produce Regular Sets
We say that a Turing machine has periodic support if there is an infinitely repeated word to the left of the input and another infinitely repeated word to the right. In the search for the smallest universal Turing machines, machines that use periodic support have been significantly smaller than those for the standard model (i.e. machines with the usual blank tape on either side of the input). While generalising the model allows us to construct smaller universal machines it makes proving decidability results for the various state-symbol products that restrict program size more difficult. Here we show that given an arbitrary 2-state 2-symbol Turing machine and a configuration with periodic support the set of reachable configurations is regular. Unlike previous decidability results for 2-state 2-symbol machines, here we include in our consideration machines that do not reserve a transition rule for halting, which further adds to the difficulty of giving decidability results
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
On the boundaries of solvability and unsolvability in tag systems. Theoretical and Experimental Results
Several older and more recent results on the boundaries of solvability and
unsolvability in tag systems are surveyed. Emphasis will be put on the
significance of computer experiments in research on very small tag systems
Decidability and Universality in Symbolic Dynamical Systems
Many different definitions of computational universality for various types of
dynamical systems have flourished since Turing's work. We propose a general
definition of universality that applies to arbitrary discrete time symbolic
dynamical systems. Universality of a system is defined as undecidability of a
model-checking problem. For Turing machines, counter machines and tag systems,
our definition coincides with the classical one. It yields, however, a new
definition for cellular automata and subshifts. Our definition is robust with
respect to initial condition, which is a desirable feature for physical
realizability.
We derive necessary conditions for undecidability and universality. For
instance, a universal system must have a sensitive point and a proper
subsystem. We conjecture that universal systems have infinite number of
subsystems. We also discuss the thesis according to which computation should
occur at the `edge of chaos' and we exhibit a universal chaotic system.Comment: 23 pages; a shorter version is submitted to conference MCU 2004 v2:
minor orthographic changes v3: section 5.2 (collatz functions) mathematically
improved v4: orthographic corrections, one reference added v5:27 pages.
Important modifications. The formalism is strengthened: temporal logic
replaced by finite automata. New results. Submitte
Undecidability of the Spectral Gap (full version)
We show that the spectral gap problem is undecidable. Specifically, we
construct families of translationally-invariant, nearest-neighbour Hamiltonians
on a 2D square lattice of d-level quantum systems (d constant), for which
determining whether the system is gapped or gapless is an undecidable problem.
This is true even with the promise that each Hamiltonian is either gapped or
gapless in the strongest sense: it is promised to either have continuous
spectrum above the ground state in the thermodynamic limit, or its spectral gap
is lower-bounded by a constant in the thermodynamic limit. Moreover, this
constant can be taken equal to the local interaction strength of the
Hamiltonian.Comment: v1: 146 pages, 56 theorems etc., 15 figures. See shorter companion
paper arXiv:1502.04135 (same title and authors) for a short version omitting
technical details. v2: Small but important fix to wording of abstract. v3:
Simplified and shortened some parts of the proof; minor fixes to other parts.
Now only 127 pages, 55 theorems etc., 10 figures. v4: Minor updates to
introductio
Calibrating Generative Models: The Probabilistic Chomsky-SchĂĽtzenberger Hierarchy
A probabilistic Chomsky–Schützenberger hierarchy of grammars is introduced and studied, with the aim of understanding the expressive power of generative models. We offer characterizations of the distributions definable at each level of the hierarchy, including probabilistic regular, context-free, (linear) indexed, context-sensitive, and unrestricted grammars, each corresponding to familiar probabilistic machine classes. Special attention is given to distributions on (unary notations for) positive integers. Unlike in the classical case where the "semi-linear" languages all collapse into the regular languages, using analytic tools adapted from the classical setting we show there is no collapse in the probabilistic hierarchy: more distributions become definable at each level. We also address related issues such as closure under probabilistic conditioning
A Computation in a Cellular Automaton Collider Rule 110
A cellular automaton collider is a finite state machine build of rings of
one-dimensional cellular automata. We show how a computation can be performed
on the collider by exploiting interactions between gliders (particles,
localisations). The constructions proposed are based on universality of
elementary cellular automaton rule 110, cyclic tag systems, supercolliders, and
computing on rings.Comment: 39 pages, 32 figures, 3 table
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