217 research outputs found
Reachability in Vector Addition Systems is Primitive-Recursive in Fixed Dimension
The reachability problem in vector addition systems is a central question,
not only for the static verification of these systems, but also for many
inter-reducible decision problems occurring in various fields. The currently
best known upper bound on this problem is not primitive-recursive, even when
considering systems of fixed dimension. We provide significant refinements to
the classical decomposition algorithm of Mayr, Kosaraju, and Lambert and to its
termination proof, which yield an ACKERMANN upper bound in the general case,
and primitive-recursive upper bounds in fixed dimension. While this does not
match the currently best known TOWER lower bound for reachability, it is
optimal for related problems
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
Reasoning about Data Repetitions with Counter Systems
We study linear-time temporal logics interpreted over data words with
multiple attributes. We restrict the atomic formulas to equalities of attribute
values in successive positions and to repetitions of attribute values in the
future or past. We demonstrate correspondences between satisfiability problems
for logics and reachability-like decision problems for counter systems. We show
that allowing/disallowing atomic formulas expressing repetitions of values in
the past corresponds to the reachability/coverability problem in Petri nets.
This gives us 2EXPSPACE upper bounds for several satisfiability problems. We
prove matching lower bounds by reduction from a reachability problem for a
newly introduced class of counter systems. This new class is a succinct version
of vector addition systems with states in which counters are accessed via
pointers, a potentially useful feature in other contexts. We strengthen further
the correspondences between data logics and counter systems by characterizing
the complexity of fragments, extensions and variants of the logic. For
instance, we precisely characterize the relationship between the number of
attributes allowed in the logic and the number of counters needed in the
counter system.Comment: 54 page
KReach : a tool for reachability in petri nets
We present KReach, a tool for deciding reachability in general Petri nets. The tool is a full implementation of Kosaraju’s original 1982 decision procedure for reachability in VASS. We believe this to be the first implementation of its kind. We include a comprehensive suite of libraries for development with Vector Addition Systems (with States) in the Haskell programming language. KReach serves as a practical tool, and acts as an effective teaching aid for the theory behind the algorithm. Preliminary tests suggest that there are some classes of Petri nets for which we can quickly show unreachability. In particular, using KReach for coverability problems, by reduction to reachability, is competitive even against state-of-the-art coverability checkers
History-Register Automata
Programs with dynamic allocation are able to create and use an unbounded
number of fresh resources, such as references, objects, files, etc. We propose
History-Register Automata (HRA), a new automata-theoretic formalism for
modelling such programs. HRAs extend the expressiveness of previous approaches
and bring us to the limits of decidability for reachability checks. The
distinctive feature of our machines is their use of unbounded memory sets
(histories) where input symbols can be selectively stored and compared with
symbols to follow. In addition, stored symbols can be consumed or deleted by
reset. We show that the combination of consumption and reset capabilities
renders the automata powerful enough to imitate counter machines, and yields
closure under all regular operations apart from complementation. We moreover
examine weaker notions of HRAs which strike different balances between
expressiveness and effectiveness.Comment: LMCS (improved version of FoSSaCS
Efficient Algorithms for Asymptotic Bounds on Termination Time in VASS
Vector Addition Systems with States (VASS) provide a well-known and
fundamental model for the analysis of concurrent processes, parameterized
systems, and are also used as abstract models of programs in resource bound
analysis. In this paper we study the problem of obtaining asymptotic bounds on
the termination time of a given VASS. In particular, we focus on the
practically important case of obtaining polynomial bounds on termination time.
Our main contributions are as follows: First, we present a polynomial-time
algorithm for deciding whether a given VASS has a linear asymptotic complexity.
We also show that if the complexity of a VASS is not linear, it is at least
quadratic. Second, we classify VASS according to quantitative properties of
their cycles. We show that certain singularities in these properties are the
key reason for non-polynomial asymptotic complexity of VASS. In absence of
singularities, we show that the asymptotic complexity is always polynomial and
of the form , for some integer , where is the
dimension of the VASS. We present a polynomial-time algorithm computing the
optimal . For general VASS, the same algorithm, which is based on a complete
technique for the construction of ranking functions in VASS, produces a valid
lower bound, i.e., a such that the termination complexity is .
Our results are based on new insights into the geometry of VASS dynamics, which
hold the potential for further applicability to VASS analysis.Comment: arXiv admin note: text overlap with arXiv:1708.0925
Rewriting Systems for Reachability in Vector Addition Systems with Pairs
15 pagesInternational audienceWe adapt hypergraph rewriting system to a generalization of Vector Addition Systems with States (VASS) that we call vector addition systems with pairs (VASP). We give rewriting systems and strategies, that allow us to obtain reachability equivalence results between some classes of VASP and VASS. Reachability for the later is well known be equivalent to reachability in Petri nets. VASP generalize also Branching Extension of VASS (BVASS) for which it is unknown if they are more expressive than VASS. We consider here a more restricted notion of reachability for VASP than that for BVASS. However the reachability decision problem corresponding is already equivalent to decidability of the provability in Multiplicative and Exponential Linear Logic (MELL), a question left open for more than 20 years
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