14 research outputs found
On Nonnegative Integer Matrices and Short Killing Words
Let be a natural number and a set of -matrices
over the nonnegative integers such that the joint spectral radius of
is at most one. We show that if the zero matrix is a product
of matrices in , then there are with . This result has applications in
automata theory and the theory of codes. Specifically, if
is a finite incomplete code, then there exists a word of
length polynomial in such that is not a factor of any
word in . This proves a weak version of Restivo's conjecture.Comment: This version is a journal submission based on a STACS'19 paper. It
extends the conference version as follows. (1) The main result has been
generalized to apply to monoids generated by finite sets whose joint spectral
radius is at most 1. (2) The use of Carpi's theorem is avoided to make the
paper more self-contained. (3) A more precise result is offered on Restivo's
conjecture for finite code
Controlling a Random Population is EXPTIME-hard
Bertrand et al. [1] (LMCS 2019) describe two-player zero-sum games in which one player tries to achieve a reachability objective in games (on the same finite arena) simultaneously by broadcasting actions, and where the opponent has full control of resolving non-deterministic choices. They show EXPTIME completeness for the question if such games can be won for every number of games. We consider the almost-sure variant in which the opponent randomizes their actions, and where the player tries to achieve the reachability objective eventually with probability one. The lower bound construction in [1] does not directly carry over to this randomized setting. In this note we show EXPTIME hardness for the almost-sure problem by reduction from Countdown Games
The Keys to Decidable HyperLTL Satisfiability: Small Models or Very Simple Formulas
HyperLTL, the extension of Linear Temporal Logic by trace quantifiers, is a uniform framework for expressing information flow policies by relating multiple traces of a security-critical system. HyperLTL has been successfully applied to express fundamental security policies like noninterference and observational determinism, but has also found applications beyond security, e.g., distributed protocols and coding theory. However, HyperLTL satisfiability is undecidable as soon as there are existential quantifiers in the scope of a universal one. To overcome this severe limitation to applicability, we investigate here restricted variants of the satisfiability problem to pinpoint the decidability border. First, we restrict the space of admissible models and show decidability when restricting the search space to models of bounded size or to finitely representable ones. Second, we consider formulas with restricted nesting of temporal operators and show that nesting depth one yields decidability for a slightly larger class of quantifier prefixes. We provide tight complexity bounds in almost all cases
Model-Checking Parametric Lock-Sharing Systems Against Regular Constraints
In parametric lock-sharing systems processes can spawn new processes to run in parallel, and can create new locks. The behavior of every process is given by a pushdown automaton. We consider infinite behaviors of such systems under strong process fairness condition. A result of a potentially infinite execution of a system is a limit configuration, that is a potentially infinite tree. The verification problem is to determine if a given system has a limit configuration satisfying a given regular property. This formulation of the problem encompasses verification of reachability as well as of many liveness properties. We show that this verification problem, while undecidable in general, is decidable for nested lock usage.
We show Exptime-completeness of the verification problem. The main source of complexity is the number of parameters in the spawn operation. If the number of parameters is bounded, our algorithm works in Ptime for properties expressed by parity automata with a fixed number of ranks
Parameterized Broadcast Networks with Registers: from NP to the Frontiers of Decidability
We consider the parameterized verification of arbitrarily large networks of
agents which communicate by broadcasting and receiving messages. In our model,
the broadcast topology is reconfigurable so that a sent message can be received
by any set of agents. In addition, agents have local registers which are
initially distinct and may therefore be thought of as identifiers. When an
agent broadcasts a message, it appends to the message the value stored in one
of its registers. Upon reception, an agent can store the received value or test
this value for equality with one of its own registers. We consider the
coverability problem, where one asks whether a given state of the system may be
reached by at least one agent. We establish that this problem is decidable;
however, it is as hard as coverability in lossy channel systems, which is
non-primitive recursive. This model lies at the frontier of decidability as
other classical problems on this model are undecidable; this is in particular
true for the target problem where all processes must synchronize on a given
state. By contrast, we show that the coverability problem is NP-complete when
each agent has only one register
Keyboards as a New Model of Computation
We introduce a new formalisation of language computation, called keyboards. We consider a set of atomic operations (writing a letter, erasing a letter, going to the right or to the left) and we define a keyboard as a set of finite sequences of such operations, called keys. The generated language is the set of words obtained by applying some non-empty sequence of those keys. Unlike classical models of computation, every key can be applied anytime. We define various classes of languages based on different sets of atomic operations, and compare their expressive powers. We also compare them to rational, context-free and context-sensitive languages. We obtain a strict hierarchy of classes, whose expressiveness is orthogonal to the one of the aforementioned classical models. We also study closure properties of those classes, as well as fundamental complexity problems on keyboards
Responsibility and verification: Importance value in temporal logics
We aim at measuring the influence of the nondeterministic choices of a part
of a system on its ability to satisfy a specification. For this purpose, we
apply the concept of Shapley values to verification as a means to evaluate how
important a part of a system is. The importance of a component is measured by
giving its control to an adversary, alone or along with other components, and
testing whether the system can still fulfill the specification. We study this
idea in the framework of model-checking with various classical types of
linear-time specification, and propose several ways to transpose it to
branching ones. We also provide tight complexity bounds in almost every case.Comment: 22 pages, 12 figure
From LTL to rLTL monitoring
Runtime monitoring is commonly used to detect the violation of desired properties in safety critical systems by observing run prefixes of the system. Bauer et al. introduced an influential framework for monitoring Linear Temporal Logic (LTL) properties, which is based on a three-valued semantics: the formula is already satisfied by the given prefix, it is already violated, or it is still undetermined, i.e., it can be satisfied and violated. However, a wide range of formulas are not monitorable under this approach, meaning that every prefix is undetermined. In particular, Bauer et al. report that 44% of the formulas they consider in their experiments fall into this category. Recently, robust semantics for LTL were introduced to capture degrees of violation of universal properties. Here, we define robust semantics for run prefixes and show its potential in monitoring: every formula considered by Bauer et al. is monitorable under our approach. Furthermore, we show that properties expressed with the robust semantics can be monitored by deterministic automata
On Finite Monoids over Nonnegative Integer Matrices and Short Killing Words
Let n be a natural number and M a set of n x n-matrices over the nonnegative integers such that M generates a finite multiplicative monoid. We show that if the zero matrix 0 is a product of matrices in M, then there are M_1, ..., M_{n^5} in M with M_1 *s M_{n^5} = 0. This result has applications in automata theory and the theory of codes. Specifically, if X subset Sigma^* is a finite incomplete code, then there exists a word w in Sigma^* of length polynomial in sum_{x in X} |x| such that w is not a factor of any word in X^*. This proves a weak version of Restivo\u27s conjecture