600 research outputs found
Model-Checking of Ordered Multi-Pushdown Automata
We address the verification problem of ordered multi-pushdown automata: A
multi-stack extension of pushdown automata that comes with a constraint on
stack transitions such that a pop can only be performed on the first non-empty
stack. First, we show that the emptiness problem for ordered multi-pushdown
automata is in 2ETIME. Then, we prove that, for an ordered multi-pushdown
automata, the set of all predecessors of a regular set of configurations is an
effectively constructible regular set. We exploit this result to solve the
global model-checking which consists in computing the set of all configurations
of an ordered multi-pushdown automaton that satisfy a given w-regular property
(expressible in linear-time temporal logics or the linear-time \mu-calculus).
As an immediate consequence, we obtain an 2ETIME upper bound for the
model-checking problem of w-regular properties for ordered multi-pushdown
automata (matching its lower-bound).Comment: 31 page
One Theorem to Rule Them All: A Unified Translation of LTL into {\omega}-Automata
We present a unified translation of LTL formulas into deterministic Rabin
automata, limit-deterministic B\"uchi automata, and nondeterministic B\"uchi
automata. The translations yield automata of asymptotically optimal size
(double or single exponential, respectively). All three translations are
derived from one single Master Theorem of purely logical nature. The Master
Theorem decomposes the language of a formula into a positive boolean
combination of languages that can be translated into {\omega}-automata by
elementary means. In particular, Safra's, ranking, and breakpoint constructions
used in other translations are not needed
Computing the Least Fixed Point of Positive Polynomial Systems
We consider equation systems of the form X_1 = f_1(X_1, ..., X_n), ..., X_n =
f_n(X_1, ..., X_n) where f_1, ..., f_n are polynomials with positive real
coefficients. In vector form we denote such an equation system by X = f(X) and
call f a system of positive polynomials, short SPP. Equation systems of this
kind appear naturally in the analysis of stochastic models like stochastic
context-free grammars (with numerous applications to natural language
processing and computational biology), probabilistic programs with procedures,
web-surfing models with back buttons, and branching processes. The least
nonnegative solution mu f of an SPP equation X = f(X) is of central interest
for these models. Etessami and Yannakakis have suggested a particular version
of Newton's method to approximate mu f.
We extend a result of Etessami and Yannakakis and show that Newton's method
starting at 0 always converges to mu f. We obtain lower bounds on the
convergence speed of the method. For so-called strongly connected SPPs we prove
the existence of a threshold k_f such that for every i >= 0 the (k_f+i)-th
iteration of Newton's method has at least i valid bits of mu f. The proof
yields an explicit bound for k_f depending only on syntactic parameters of f.
We further show that for arbitrary SPP equations Newton's method still
converges linearly: there are k_f>=0 and alpha_f>0 such that for every i>=0 the
(k_f+alpha_f i)-th iteration of Newton's method has at least i valid bits of mu
f. The proof yields an explicit bound for alpha_f; the bound is exponential in
the number of equations, but we also show that it is essentially optimal.
Constructing a bound for k_f is still an open problem. Finally, we also provide
a geometric interpretation of Newton's method for SPPs.Comment: This is a technical report that goes along with an article to appear
in SIAM Journal on Computing
Static Analysis of Deterministic Negotiations
Negotiation diagrams are a model of concurrent computation akin to workflow
Petri nets. Deterministic negotiation diagrams, equivalent to the much studied
and used free-choice workflow Petri nets, are surprisingly amenable to
verification. Soundness (a property close to deadlock-freedom) can be decided
in PTIME. Further, other fundamental questions like computing summaries or the
expected cost, can also be solved in PTIME for sound deterministic negotiation
diagrams, while they are PSPACE-complete in the general case.
In this paper we generalize and explain these results. We extend the
classical "meet-over-all-paths" (MOP) formulation of static analysis problems
to our concurrent setting, and introduce Mazurkiewicz-invariant analysis
problems, which encompass the questions above and new ones. We show that any
Mazurkiewicz-invariant analysis problem can be solved in PTIME for sound
deterministic negotiations whenever it is in PTIME for sequential
flow-graphs---even though the flow-graph of a deterministic negotiation diagram
can be exponentially larger than the diagram itself. This gives a common
explanation to the low-complexity of all the analysis questions studied so far.
Finally, we show that classical gen/kill analyses are also an instance of our
framework, and obtain a PTIME algorithm for detecting anti-patterns in
free-choice workflow Petri nets.
Our result is based on a novel decomposition theorem, of independent
interest, showing that sound deterministic negotiation diagrams can be
hierarchically decomposed into (possibly overlapping) smaller sound diagrams.Comment: To appear in the Proceedings of LICS 2017, IEEE Computer Societ
Computing Least Fixed Points of Probabilistic Systems of Polynomials
We study systems of equations of the form X1 = f1(X1, ..., Xn), ..., Xn =
fn(X1, ..., Xn), where each fi is a polynomial with nonnegative coefficients
that add up to 1. The least nonnegative solution, say mu, of such equation
systems is central to problems from various areas, like physics, biology,
computational linguistics and probabilistic program verification. We give a
simple and strongly polynomial algorithm to decide whether mu=(1, ..., 1)
holds. Furthermore, we present an algorithm that computes reliable sequences of
lower and upper bounds on mu, converging linearly to mu. Our algorithm has
these features despite using inexact arithmetic for efficiency. We report on
experiments that show the performance of our algorithms.Comment: Published in the Proceedings of the 27th International Symposium on
Theoretical Aspects of Computer Science (STACS). Technical Report is also
available via arxiv.or
An Effective Tableau System for the Linear Time µ-Calculus
We present a tableau system for the model checking problem of the linear time µ-calculus. It improves the system of Stirling and Walker by simplifying the success condition for a tableau. In our system success for a leaf is determined by the path leading to it, whereas Stirling and Walker's method requires the examination of a potentially infinite number of paths extending over the whole tableau
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