13 research outputs found
Two-Player Perfect-Information Shift-Invariant Submixing Stochastic Games Are Half-Positional
We consider zero-sum stochastic games with perfect information and finitely
many states and actions. The payoff is computed by a payoff function which
associates to each infinite sequence of states and actions a real number. We
prove that if the the payoff function is both shift-invariant and submixing,
then the game is half-positional, i.e. the first player has an optimal strategy
which is both deterministic and stationary. This result relies on the existence
of -subgame-perfect equilibria in shift-invariant games, a second
contribution of the paper
Computing the Density of the Positivity Set for Linear Recurrence Sequences
The set of indices that correspond to the positive entries of a sequence of
numbers is called its positivity set. In this paper, we study the density of
the positivity set of a given linear recurrence sequence, that is the question
of how much more frequent are the positive entries compared to the non-positive
ones. We show that one can compute this density to arbitrary precision, as well
as decide whether it is equal to zero (or one). If the sequence is
diagonalisable, we prove that its positivity set is finite if and only if its
density is zero. Lastly, arithmetic properties of densities are treated, in
particular we prove that it is decidable whether the density is a rational
number, given that the recurrence sequence has at most one pair of dominant
complex roots
Deciding the value 1 problem for probabilistic leaktight automata
The value 1 problem is a decision problem for probabilistic automata over
finite words: given a probabilistic automaton, are there words accepted with
probability arbitrarily close to 1? This problem was proved undecidable
recently; to overcome this, several classes of probabilistic automata of
different nature were proposed, for which the value 1 problem has been shown
decidable. In this paper, we introduce yet another class of probabilistic
automata, called leaktight automata, which strictly subsumes all classes of
probabilistic automata whose value 1 problem is known to be decidable. We prove
that for leaktight automata, the value 1 problem is decidable (in fact,
PSPACE-complete) by constructing a saturation algorithm based on the
computation of a monoid abstracting the behaviours of the automaton. We rely on
algebraic techniques developed by Simon to prove that this abstraction is
complete. Furthermore, we adapt this saturation algorithm to decide whether an
automaton is leaktight. Finally, we show a reduction allowing to extend our
decidability results from finite words to infinite ones, implying that the
value 1 problem for probabilistic leaktight parity automata is decidable
Invariants for Continuous Linear Dynamical Systems
Continuous linear dynamical systems are used extensively in mathematics,
computer science, physics, and engineering to model the evolution of a system
over time. A central technique for certifying safety properties of such systems
is by synthesising inductive invariants. This is the task of finding a set of
states that is closed under the dynamics of the system and is disjoint from a
given set of error states. In this paper we study the problem of synthesising
inductive invariants that are definable in o-minimal expansions of the ordered
field of real numbers. In particular, assuming Schanuel's conjecture in
transcendental number theory, we establish effective synthesis of o-minimal
invariants in the case of semi-algebraic error sets. Without using Schanuel's
conjecture, we give a procedure for synthesizing o-minimal invariants that
contain all but a bounded initial segment of the orbit and are disjoint from a
given semi-algebraic error set. We further prove that effective synthesis of
semi-algebraic invariants that contain the whole orbit, is at least as hard as
a certain open problem in transcendental number theory.Comment: Full version of a ICALP 2020 pape
Emptiness of Zero Automata Is Decidable
Zero automata are a probabilistic extension of parity automata on infinite trees. The satisfiability of a certain probabilistic variant of MSO, called TMSO+zero, reduces to the emptiness problem for zero automata. We introduce a variant of zero automata called nonzero automata. We prove that for every zero automaton there is an equivalent nonzero automaton of quadratic size
and the emptiness problem of nonzero automata is decidable, with complexity co-NP. These results imply that TMSO+zero has decidable satisfiability
Stamina : stabilisation nonoids in automata theory
We present Stamina, a tool solving three algorithmic problems in automata theory. First, compute the star height of a regular language, i.e. the minimal number of nested Kleene stars needed for expressing the language with a complement-free regular expression. Second, decide limitedness for regular cost functions. Third, decide whether a probabilistic leaktight automaton has value 1, i.e. whether a probabilistic leaktight automaton accepts words with probability arbitrarily close to 1.
All three problems reduce to the computation of the stabilisation monoid associated with an automaton, which is computationally challenging because the monoid is exponentially larger than the automaton. The compact data structures used in Stamina, together with optimisations and heuristics, allow us to handle automata with several hundreds of states.
This radically improves upon the performances of ACME, a similar tool solving a subset of these problems
What's Decidable about Discrete Linear Dynamical Systems?
We survey the state of the art on the algorithmic analysis of discrete linear
dynamical systems, focussing in particular on reachability, model-checking, and
invariant-generation questions, both unconditionally as well as relative to
oracles for the Skolem Problem
Creating and analysing the Digital Terrain Model of the Slivovo area using QGIS software
The aim of the paper is developing the Digital Terrain Model (DTM in the further text) through QGIS software. In order to accomplish intention of the paper will test some of the methods and techniques that are widely known in nowadays and those are supported by QGIS software – an open source software. And those methods named TIN and GRID. For showing complexity on the study area will analyse some features or characteristics of terrain in the created DTM. All of these methods and techniques will be applied in QGIS. In general, the QGIS software has rich methodology for creation, intepretation, visualization and analysing the geo-spatial data and the DTM in particular