2,081 research outputs found
Undecidable Properties of Limit Set Dynamics of Cellular Automata
Cellular Automata (CA) are discrete dynamical systems and an abstract model
of parallel computation. The limit set of a cellular automaton is its maximal
topological attractor. A well know result, due to Kari, says that all
nontrivial properties of limit sets are undecidable. In this paper we consider
properties of limit set dynamics, i.e. properties of the dynamics of Cellular
Automata restricted to their limit sets. There can be no equivalent of Kari's
Theorem for limit set dynamics. Anyway we show that there is a large class of
undecidable properties of limit set dynamics, namely all properties of limit
set dynamics which imply stability or the existence of a unique subshift
attractor. As a consequence we have that it is undecidable whether the cellular
automaton map restricted to the limit set is the identity, closing, injective,
expansive, positively expansive, transitive
Mathematical open problems in Projected Entangled Pair States
Projected Entangled Pair States (PEPS) are used in practice as an efficient
parametrization of the set of ground states of quantum many body systems. The
aim of this paper is to present, for a broad mathematical audience, some
mathematical questions about PEPS.Comment: Notes associated to the Santal\'o Lecture 2017, Universidad
Complutense de Madrid (UCM), minor typos correcte
Weak Singular Hybrid Automata
The framework of Hybrid automata, introduced by Alur, Courcourbetis,
Henzinger, and Ho, provides a formal modeling and analysis environment to
analyze the interaction between the discrete and the continuous parts of
cyber-physical systems. Hybrid automata can be considered as generalizations of
finite state automata augmented with a finite set of real-valued variables
whose dynamics in each state is governed by a system of ordinary differential
equations. Moreover, the discrete transitions of hybrid automata are guarded by
constraints over the values of these real-valued variables, and enable
discontinuous jumps in the evolution of these variables. Singular hybrid
automata are a subclass of hybrid automata where dynamics is specified by
state-dependent constant vectors. Henzinger, Kopke, Puri, and Varaiya showed
that for even very restricted subclasses of singular hybrid automata, the
fundamental verification questions, like reachability and schedulability, are
undecidable. In this paper we present \emph{weak singular hybrid automata}
(WSHA), a previously unexplored subclass of singular hybrid automata, and show
the decidability (and the exact complexity) of various verification questions
for this class including reachability (NP-Complete) and LTL model-checking
(PSPACE-Complete). We further show that extending WSHA with a single
unrestricted clock or extending WSHA with unrestricted variable updates lead to
undecidability of reachability problem
Undecidability of the Spectral Gap in One Dimension
The spectral gap problem - determining whether the energy spectrum of a
system has an energy gap above ground state, or if there is a continuous range
of low-energy excitations - pervades quantum many-body physics. Recently, this
important problem was shown to be undecidable for quantum spin systems in two
(or more) spatial dimensions: there exists no algorithm that determines in
general whether a system is gapped or gapless, a result which has many
unexpected consequences for the physics of such systems. However, there are
many indications that one dimensional spin systems are simpler than their
higher-dimensional counterparts: for example, they cannot have thermal phase
transitions or topological order, and there exist highly-effective numerical
algorithms such as DMRG - and even provably polynomial-time ones - for gapped
1D systems, exploiting the fact that such systems obey an entropy area-law.
Furthermore, the spectral gap undecidability construction crucially relied on
aperiodic tilings, which are not possible in 1D.
So does the spectral gap problem become decidable in 1D? In this paper we
prove this is not the case, by constructing a family of 1D spin chains with
translationally-invariant nearest neighbour interactions for which no algorithm
can determine the presence of a spectral gap. This not only proves that the
spectral gap of 1D systems is just as intractable as in higher dimensions, but
also predicts the existence of qualitatively new types of complex physics in 1D
spin chains. In particular, it implies there are 1D systems with constant
spectral gap and non-degenerate classical ground state for all systems sizes up
to an uncomputably large size, whereupon they switch to a gapless behaviour
with dense spectrum.Comment: 7 figure
Generalized Satisfiability Problems via Operator Assignments
Schaefer introduced a framework for generalized satisfiability problems on
the Boolean domain and characterized the computational complexity of such
problems. We investigate an algebraization of Schaefer's framework in which the
Fourier transform is used to represent constraints by multilinear polynomials
in a unique way. The polynomial representation of constraints gives rise to a
relaxation of the notion of satisfiability in which the values to variables are
linear operators on some Hilbert space. For the case of constraints given by a
system of linear equations over the two-element field, this relaxation has
received considerable attention in the foundations of quantum mechanics, where
such constructions as the Mermin-Peres magic square show that there are systems
that have no solutions in the Boolean domain, but have solutions via operator
assignments on some finite-dimensional Hilbert space. We obtain a complete
characterization of the classes of Boolean relations for which there is a gap
between satisfiability in the Boolean domain and the relaxation of
satisfiability via operator assignments. To establish our main result, we adapt
the notion of primitive-positive definability (pp-definability) to our setting,
a notion that has been used extensively in the study of constraint satisfaction
problems. Here, we show that pp-definability gives rise to gadget reductions
that preserve satisfiability gaps. We also present several additional
applications of this method. In particular and perhaps surprisingly, we show
that the relaxed notion of pp-definability in which the quantified variables
are allowed to range over operator assignments gives no additional expressive
power in defining Boolean relations
Verifying Recursive Active Documents with Positive Data Tree Rewriting
This paper proposes a data tree-rewriting framework for modeling evolving
documents. The framework is close to Guarded Active XML, a platform used for
handling XML repositories evolving through web services. We focus on automatic
verification of properties of evolving documents that can contain data from an
infinite domain. We establish the boundaries of decidability, and show that
verification of a {\em positive} fragment that can handle recursive service
calls is decidable. We also consider bounded model-checking in our data
tree-rewriting framework and show that it is \nexptime-complete
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Petri net equivalence
Determining whether two Petri nets are equivalent is an interesting problem from both practical and theoretical standpoints. Although it is undecidable in the general case, for many interesting nets the equivalence problem is solvable. This paper explores, mostly from a theoretical point of view, some of the issues of Petri net equivalence, including both reachability sets and languages. Some new definitions of reachability set equivalence are described which allow the markings of some places to be treated identically or ignored, analogous to the Petri net languages in which multiple transitions may be labeled with the same symbol or with the empty string. The complexity of some decidable Petri net equivalence problems is analyzed
Algorithmic Verification of Asynchronous Programs
Asynchronous programming is a ubiquitous systems programming idiom to manage
concurrent interactions with the environment. In this style, instead of waiting
for time-consuming operations to complete, the programmer makes a non-blocking
call to the operation and posts a callback task to a task buffer that is
executed later when the time-consuming operation completes. A co-operative
scheduler mediates the interaction by picking and executing callback tasks from
the task buffer to completion (and these callbacks can post further callbacks
to be executed later). Writing correct asynchronous programs is hard because
the use of callbacks, while efficient, obscures program control flow.
We provide a formal model underlying asynchronous programs and study
verification problems for this model. We show that the safety verification
problem for finite-data asynchronous programs is expspace-complete. We show
that liveness verification for finite-data asynchronous programs is decidable
and polynomial-time equivalent to Petri Net reachability. Decidability is not
obvious, since even if the data is finite-state, asynchronous programs
constitute infinite-state transition systems: both the program stack and the
task buffer of pending asynchronous calls can be potentially unbounded.
Our main technical construction is a polynomial-time semantics-preserving
reduction from asynchronous programs to Petri Nets and conversely. The
reduction allows the use of algorithmic techniques on Petri Nets to the
verification of asynchronous programs.
We also study several extensions to the basic models of asynchronous programs
that are inspired by additional capabilities provided by implementations of
asynchronous libraries, and classify the decidability and undecidability of
verification questions on these extensions.Comment: 46 pages, 9 figure
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