76,907 research outputs found
Ultrafilters maximal for finite embeddability
In [1] the authors showed some basic properties of a pre-order that arose in
combinatorial number theory, namely the finite embeddability between sets of
natural numbers, and they presented its generalization to ultrafilters, which
is related to the algebraical and topological structure of the Stone-\v{C}ech
compactification of the discrete space of natural numbers. In this present
paper we continue the study of these pre-orders. In particular, we prove that
there exist ultrafilters maximal for finite embeddability, and we show that the
set of such ultrafilters is the closure of the minimal bilateral ideal in the
semigroup (\bN,\oplus), namely \overline{K(\bN,\oplus)}. As a consequence,
we easily derive many combinatorial properties of ultrafilters in
\overline{K(\bN,\oplus)}. We also give an alternative proof of our main
result based on nonstandard models of arithmetic
Incremental, Inductive Coverability
We give an incremental, inductive (IC3) procedure to check coverability of
well-structured transition systems. Our procedure generalizes the IC3 procedure
for safety verification that has been successfully applied in finite-state
hardware verification to infinite-state well-structured transition systems. We
show that our procedure is sound, complete, and terminating for downward-finite
well-structured transition systems---where each state has a finite number of
states below it---a class that contains extensions of Petri nets, broadcast
protocols, and lossy channel systems.
We have implemented our algorithm for checking coverability of Petri nets. We
describe how the algorithm can be efficiently implemented without the use of
SMT solvers. Our experiments on standard Petri net benchmarks show that IC3 is
competitive with state-of-the-art implementations for coverability based on
symbolic backward analysis or expand-enlarge-and-check algorithms both in time
taken and space usage.Comment: Non-reviewed version, original version submitted to CAV 2013; this is
a revised version, containing more experimental results and some correction
Area-Universal Rectangular Layouts
A rectangular layout is a partition of a rectangle into a finite set of
interior-disjoint rectangles. Rectangular layouts appear in various
applications: as rectangular cartograms in cartography, as floorplans in
building architecture and VLSI design, and as graph drawings. Often areas are
associated with the rectangles of a rectangular layout and it might hence be
desirable if one rectangular layout can represent several area assignments. A
layout is area-universal if any assignment of areas to rectangles can be
realized by a combinatorially equivalent rectangular layout. We identify a
simple necessary and sufficient condition for a rectangular layout to be
area-universal: a rectangular layout is area-universal if and only if it is
one-sided. More generally, given any rectangular layout L and any assignment of
areas to its regions, we show that there can be at most one layout (up to
horizontal and vertical scaling) which is combinatorially equivalent to L and
achieves a given area assignment. We also investigate similar questions for
perimeter assignments. The adjacency requirements for the rectangles of a
rectangular layout can be specified in various ways, most commonly via the dual
graph of the layout. We show how to find an area-universal layout for a given
set of adjacency requirements whenever such a layout exists.Comment: 19 pages, 16 figure
Analysis of Probabilistic Basic Parallel Processes
Basic Parallel Processes (BPPs) are a well-known subclass of Petri Nets. They
are the simplest common model of concurrent programs that allows unbounded
spawning of processes. In the probabilistic version of BPPs, every process
generates other processes according to a probability distribution. We study the
decidability and complexity of fundamental qualitative problems over
probabilistic BPPs -- in particular reachability with probability 1 of
different classes of target sets (e.g. upward-closed sets). Our results concern
both the Markov-chain model, where processes are scheduled randomly, and the
MDP model, where processes are picked by a scheduler.Comment: This is the technical report for a FoSSaCS'14 pape
Lossy Channel Games under Incomplete Information
In this paper we investigate lossy channel games under incomplete
information, where two players operate on a finite set of unbounded FIFO
channels and one player, representing a system component under consideration
operates under incomplete information, while the other player, representing the
component's environment is allowed to lose messages from the channels. We argue
that these games are a suitable model for synthesis of communication protocols
where processes communicate over unreliable channels. We show that in the case
of finite message alphabets, games with safety and reachability winning
conditions are decidable and finite-state observation-based strategies for the
component can be effectively computed. Undecidability for (weak) parity
objectives follows from the undecidability of (weak) parity perfect information
games where only one player can lose messages.Comment: In Proceedings SR 2013, arXiv:1303.007
Approaching the Coverability Problem Continuously
The coverability problem for Petri nets plays a central role in the
verification of concurrent shared-memory programs. However, its high
EXPSPACE-complete complexity poses a challenge when encountered in real-world
instances. In this paper, we develop a new approach to this problem which is
primarily based on applying forward coverability in continuous Petri nets as a
pruning criterion inside a backward coverability framework. A cornerstone of
our approach is the efficient encoding of a recently developed polynomial-time
algorithm for reachability in continuous Petri nets into SMT. We demonstrate
the effectiveness of our approach on standard benchmarks from the literature,
which shows that our approach decides significantly more instances than any
existing tool and is in addition often much faster, in particular on large
instances.Comment: 18 pages, 4 figure
Weak Alternating Timed Automata
Alternating timed automata on infinite words are considered. The main result
is a characterization of acceptance conditions for which the emptiness problem
for these automata is decidable. This result implies new decidability results
for fragments of timed temporal logics. It is also shown that, unlike for MITL,
the characterisation remains the same even if no punctual constraints are
allowed
Decisive Markov Chains
We consider qualitative and quantitative verification problems for
infinite-state Markov chains. We call a Markov chain decisive w.r.t. a given
set of target states F if it almost certainly eventually reaches either F or a
state from which F can no longer be reached. While all finite Markov chains are
trivially decisive (for every set F), this also holds for many classes of
infinite Markov chains. Infinite Markov chains which contain a finite attractor
are decisive w.r.t. every set F. In particular, this holds for probabilistic
lossy channel systems (PLCS). Furthermore, all globally coarse Markov chains
are decisive. This class includes probabilistic vector addition systems (PVASS)
and probabilistic noisy Turing machines (PNTM). We consider both safety and
liveness problems for decisive Markov chains, i.e., the probabilities that a
given set of states F is eventually reached or reached infinitely often,
respectively. 1. We express the qualitative problems in abstract terms for
decisive Markov chains, and show an almost complete picture of its decidability
for PLCS, PVASS and PNTM. 2. We also show that the path enumeration algorithm
of Iyer and Narasimha terminates for decisive Markov chains and can thus be
used to solve the approximate quantitative safety problem. A modified variant
of this algorithm solves the approximate quantitative liveness problem. 3.
Finally, we show that the exact probability of (repeatedly) reaching F cannot
be effectively expressed (in a uniform way) in Tarski-algebra for either PLCS,
PVASS or (P)NTM.Comment: 32 pages, 0 figure
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