531 research outputs found
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
Does Church-Turing thesis apply outside computer science?
We analyze whether Church-Turing thesis can be applied to mathematical and physical systems. We find the factors that allow to a class of systems to reach a Turing or a super-Turing computational power. We illustrate our general statements by some more concrete theorems on hybrid and stochastic systems
An algorithmic complexity interpretation of Lin's third law of information theory
Instead of static entropy we assert that the Kolmogorov complexity of a
static structure such as a solid is the proper measure of disorder (or
chaoticity). A static structure in a surrounding perfectly-random universe acts
as an interfering entity which introduces local disruption in randomness. This
is modeled by a selection rule which selects a subsequence of the random
input sequence that hits the structure. Through the inequality that relates
stochasticity and chaoticity of random binary sequences we maintain that Lin's
notion of stability corresponds to the stability of the frequency of 1s in the
selected subsequence. This explains why more complex static structures are less
stable. Lin's third law is represented as the inevitable change that static
structure undergo towards conforming to the universe's perfect randomness
Deciding Reachability for Piecewise Constant Derivative Systems on Orientable Manifolds
Β© 2019 Springer-Verlag. This is a post-peer-review, pre-copyedit version of a paper published in Reachability Problems: 13th International Conference, RP 2019, Brussels, Belgium, September 11β13, 2019, Proceedings. The final authenticated version is available online at: http://dx.doi.org/10.1007/978-3-030-30806-3_14A hybrid automaton is a finite state machine combined with some k real-valued continuous variables, where k determines the number of the automaton dimensions. This formalism is widely used for modelling safety-critical systems, and verification tasks for such systems can often be expressed as the reachability problem for hybrid automata. Asarin, Mysore, Pnueli and Schneider defined classes of hybrid automata lying on the boundary between decidability and undecidability in their seminal paper βLow dimensional hybrid systems - decidable, undecidable, donβt knowβ [9]. They proved that certain decidable classes become undecidable when given a little additional computational power, and showed that the reachability question remains unsolved for some 2-dimensional systems. Piecewise Constant Derivative Systems on 2-dimensional manifolds (or PCD2m) constitute a class of hybrid automata for which decidability of the reachability problem is unknown. In this paper we show that the reachability problem becomes decidable for PCD2m if we slightly limit their dynamics, and thus we partially answer the open question of Asarin, Mysore, Pnueli and Schneider posed in [9]
Reachability problems for PAMs
Piecewise affine maps (PAMs) are frequently used as a reference model to show
the openness of the reachability questions in other systems. The reachability
problem for one-dimentional PAM is still open even if we define it with only
two intervals. As the main contribution of this paper we introduce new
techniques for solving reachability problems based on p-adic norms and weights
as well as showing decidability for two classes of maps. Then we show the
connections between topological properties for PAM's orbits, reachability
problems and representation of numbers in a rational base system. Finally we
show a particular instance where the uniform distribution of the original orbit
may not remain uniform or even dense after making regular shifts and taking a
fractional part in that sequence.Comment: 16 page
Entropy Games and Matrix Multiplication Games
Two intimately related new classes of games are introduced and studied:
entropy games (EGs) and matrix multiplication games (MMGs). An EG is played on
a finite arena by two-and-a-half players: Despot, Tribune and the
non-deterministic People. Despot wants to make the set of possible People's
behaviors as small as possible, while Tribune wants to make it as large as
possible.An MMG is played by two players that alternately write matrices from
some predefined finite sets. One wants to maximize the growth rate of the
product, and the other to minimize it. We show that in general MMGs are
undecidable in quite a strong sense.On the positive side, EGs correspond to a
subclass of MMGs, and we prove that such MMGs and EGs are determined, and that
the optimal strategies are simple. The complexity of solving such games is in
NP\&coNP.Comment: Accepted to STACS 201
Simple Algorithm for Simple Timed Games
version 1.1We propose a subclass of timed game automata (TGA), called Task TGA, representing networks of communicating tasks where the system can choose when to start the task and the environment can choose the duration of the task. We search to solve finite-horizon reachability games on Task TGA by building strategies in the form of Simple Temporal Networks with Uncertainty (STNU). Such strategies have the advantage of being very succinct due to the partial order reduction of independent tasks. We show that the existence of such strategies is an NP-complete problem. A practical consequence of this result is a fully forward algorithm for building STNU strategies. Potential applications of this work are planning and scheduling under temporal uncertainty
On the decidability and complexity of problems for restricted hierarchical hybrid systems
We study variants of a recently introduced hybrid system model, called a Hierarchical Piecewise Constant Derivative (HPCD). These variants (loosely called Restricted HPCDs) form a class of natural models with similarities to many other well known hybrid system models in the literature such as Stopwatch Automata, Rectangular Automata and PCDs. We study the complexity of reachability and mortality problems for variants of RHPCDs and show a variety of results, depending upon the allowed powers. These models form a useful tool for the study of the complexity of such problems for hybrid systems, due to their connections with existing models.
We show that the reachability problem and the mortality problem are co-NP-hard for bounded 3-dimensional RHPCDs (3-RHPCDs). Reachability is shown to be in PSPACE, even for n-dimensional RHPCDs. We show that for an unbounded 3-RHPCD, the reachability and mortality problems become undecidable. For a nondeterministic variant of 2-RHPCDs, the reachability problem is shown to be PSPACE-complete
ΠΠ΅ΡΠΎΠ΄ ΡΠ°ΡΡΠ΅ΡΠ° ΡΠ΅ΠΏΠ»ΠΎΠΎΠ±ΠΌΠ΅Π½Π° ΠΈΠ·Π»ΡΡΠ΅Π½ΠΈΠ΅ΠΌ Π² ΡΠΎΠΏΠΊΠ΅ ΠΎΡΠ΅ΡΠΈΠΌΠΌΠ΅ΡΡΠΈΡΠ½ΠΎΠΉ ΠΊΠΎΠ½ΡΠΈΠ³ΡΡΠ°ΡΠΈΠΈ Π½Π° ΠΎΡΠ½ΠΎΠ²Π΅ ΡΡΠ°Π²Π½Π΅Π½ΠΈΠΉ Π΄Π»Ρ ΠΊΠΎΠΌΠΏΠΎΠ½Π΅Π½Ρ ΡΡΠΌΠΌΠ°ΡΠ½ΠΎΠ³ΠΎ Π²Π΅ΠΊΡΠΎΡΠ° ΠΏΠΎΡΠΎΠΊΠ° Π»ΡΡΠΈΡΡΠΎΠΉ ΡΠ½Π΅ΡΠ³ΠΈΠΈ. ΠΠ½ΠΆΠ΅Π½Π΅ΡΠ½Π°Ρ ΠΌΠ΅ΡΠΎΠ΄ΠΈΠΊΠ°
ΠΠ° ΠΎΡΠ½ΠΎΠ²Π΅ ΡΠΈΡΡΠ΅ΠΌΡ ΡΡΠ°Π²Π½Π΅Π½ΠΈΠΉ ΠΏΠ΅ΡΠ²ΠΎΠΉ ΡΠ°ΡΡΠΈ ΡΡΠ°ΡΡΠΈ, ΠΏΠΎΠ»ΡΡΠ΅Π½Ρ ΡΠ°Π·Π½ΠΎΡΡΠ½ΡΠ΅ ΡΡΠ°Π²Π½Π΅Π½ΠΈΡ ΠΊΠ²Π°Π·ΠΈΠΎΠ΄Π½ΠΎΠΌΠ΅ΡΠ½ΠΎΠ³ΠΎ ΠΌΠ΅ΡΠΎΠ΄Π° ΡΠ°ΡΡΠ΅ΡΠ° ΡΠ΅ΠΏΠ»ΠΎΠΎΠ±ΠΌΠ΅Π½Π° ΠΈΠ·Π»ΡΡΠ΅Π½ΠΈΠ΅ΠΌ Π² ΡΠΎΠΏΠΊΠ΅ ΠΊΠΎΡΠ»Π° ΠΎΡΠ΅ΡΠΈΠΌΠΌΠ΅ΡΡΠΈΡΠ½ΠΎΠΉ ΠΊΠΎΠ½ΡΠΈΠ³ΡΡΠ°ΡΠΈΠΈ. ΠΠ°ΠΏΠΈΡΠ°Π½Ρ ΡΡΠ°Π²Π½Π΅Π½ΠΈΡ ΠΈ ΡΠ°Π·Π½ΠΎΡΡΠ½ΡΠ΅ ΡΠΎΡΠΌΡΠ»Ρ ΠΌΠ΅ΡΠΎΠ΄Π° ΠΈΠ½ΠΆΠ΅Π½Π΅ΡΠ½ΠΎΠ³ΠΎ ΡΠ°ΡΡΠ΅ΡΠ° ΡΠ΅ΠΏΠ»ΠΎΠΎΠ±ΠΌΠ΅Π½Π° ΠΈΠ·Π»ΡΡΠ΅Π½ΠΈΠ΅ΠΌ ΠΏΡΠΈ Π½Π°Π»ΠΈΡΠΈΠΈ ΡΠΊΡΠ°Π½Π½ΠΎΠΉ ΡΠ΅ΡΠΊΠΈ ΠΎΠΊΠΎΠ»ΠΎ ΠΏΠΎΠ²Π΅ΡΡ
Π½ΠΎΡΡΠΈ Π³ΠΎΡΠ΅Π»ΠΊΠΈ. ΠΠ΅ΡΠΎΠ΄ΠΈΠΊΠ° ΠΏΡΠΈΠ³ΠΎΠ΄Π½Π° Π΄Π»Ρ ΠΈΡΠΏΠΎΠ»ΡΠ·ΠΎΠ²Π°Π½ΠΈΡ Π² ΠΈΠ½ΠΆΠ΅Π½Π΅ΡΠ½ΡΡ
ΡΠ°ΡΡΠ΅ΡΠ°Ρ
ΠΏΡΠΈ ΠΎΠΏΡΠ΅Π΄Π΅Π»Π΅Π½ΠΈΠΈ ΠΎΠΏΡΠΈΠΌΠ°Π»ΡΠ½ΠΎΠΉ ΠΊΠΎΠ½ΡΠΈΠ³ΡΡΠ°ΡΠΈΠΈ ΡΠΎΠΏΠΊΠΈ ΠΈ ΠΎΠΏΡΠ΅Π΄Π΅Π»Π΅Π½ΠΈΠΈ ΠΎΠΏΡΠΈΠΌΠ°Π»ΡΠ½ΠΎΠ³ΠΎ ΡΠ°ΡΡΡΠΎΡΠ½ΠΈΡ ΡΠΊΡΠ°Π½Π½ΠΎΠΉ ΡΠ΅ΡΠΊΠΈ Π΄ΠΎ ΠΏΠΎΠ²Π΅ΡΡ
Π½ΠΎΡΡΠΈ Π³ΠΎΡΠ΅Π»ΠΊΠΈ
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