33 research outputs found
Positivity Problems for Low-Order Linear Recurrence Sequences
We consider two decision problems for linear recurrence sequences (LRS) over
the integers, namely the Positivity Problem (are all terms of a given LRS
positive?) and the Ultimate Positivity Problem} (are all but finitely many
terms of a given LRS positive?). We show decidability of both problems for LRS
of order 5 or less, with complexity in the Counting Hierarchy for Positivity,
and in polynomial time for Ultimate Positivity. Moreover, we show by way of
hardness that extending the decidability of either problem to LRS of order 6
would entail major breakthroughs in analytic number theory, more precisely in
the field of Diophantine approximation of transcendental numbers
On the Skolem Problem for Continuous Linear Dynamical Systems
The Continuous Skolem Problem asks whether a real-valued function satisfying
a linear differential equation has a zero in a given interval of real numbers.
This is a fundamental reachability problem for continuous linear dynamical
systems, such as linear hybrid automata and continuous-time Markov chains.
Decidability of the problem is currently open---indeed decidability is open
even for the sub-problem in which a zero is sought in a bounded interval. In
this paper we show decidability of the bounded problem subject to Schanuel's
Conjecture, a unifying conjecture in transcendental number theory. We
furthermore analyse the unbounded problem in terms of the frequencies of the
differential equation, that is, the imaginary parts of the characteristic
roots. We show that the unbounded problem can be reduced to the bounded problem
if there is at most one rationally linearly independent frequency, or if there
are two rationally linearly independent frequencies and all characteristic
roots are simple. We complete the picture by showing that decidability of the
unbounded problem in the case of two (or more) rationally linearly independent
frequencies would entail a major new effectiveness result in Diophantine
approximation, namely computability of the Diophantine-approximation types of
all real algebraic numbers.Comment: Full version of paper at ICALP'1
On the decidability and complexity of Metric Temporal Logic over finite words
Metric Temporal Logic (MTL) is a prominent specification formalism for
real-time systems. In this paper, we show that the satisfiability problem for
MTL over finite timed words is decidable, with non-primitive recursive
complexity. We also consider the model-checking problem for MTL: whether all
words accepted by a given Alur-Dill timed automaton satisfy a given MTL
formula. We show that this problem is decidable over finite words. Over
infinite words, we show that model checking the safety fragment of MTL--which
includes invariance and time-bounded response properties--is also decidable.
These results are quite surprising in that they contradict various claims to
the contrary that have appeared in the literature
Nonnegativity Problems for Matrix Semigroups
The matrix semigroup membership problem asks, given square matrices
of the same dimension, whether lies in the semigroup
generated by . It is classical that this problem is undecidable
in general but decidable in case commute. In this paper we
consider the problem of whether, given , the semigroup
generated by contains a non-negative matrix. We show that in
case commute, this problem is decidable subject to Schanuel's
Conjecture. We show also that the problem is undecidable if the commutativity
assumption is dropped. A key lemma in our decidability result is a procedure to
determine, given a matrix , whether the sequence of matrices is ultimately nonnegative. This answers a problem posed by S. Akshay
(arXiv:2205.09190). The latter result is in stark contrast to the notorious
fact that it is not known how to determine effectively whether for any specific
matrix index the sequence is ultimately nonnegative
(which is a formulation of the Ultimate Positivity Problem for linear
recurrence sequences)
A Static Analysis Framework for Livelock Freedom in CSP
In a process algebra with hiding and recursion it is possible to create
processes which compute internally without ever communicating with their
environment. Such processes are said to diverge or livelock. In this paper we
show how it is possible to conservatively classify processes as livelock-free
through a static analysis of their syntax. In particular, we present a
collection of rules, based on the inductive structure of terms, which guarantee
livelock-freedom of the denoted process. This gives rise to an algorithm which
conservatively flags processes that can potentially livelock. We illustrate our
approach by applying both BDD-based and SAT-based implementations of our
algorithm to a range of benchmarks, and show that our technique in general
substantially outperforms the model checker FDR whilst exhibiting a low rate of
inconclusive results.Comment: 53 pages, 20 figure
Parallel assignments in software model checking
In this paper we investigate how formal software verification systems can be improved by utilising parallel assignment in weakest precondition computations. We begin with an introduction to modern software verification systems. Specifically, we review the method in which software abstractions are built using counterexample-guided abstraction refinement (CEGAR). The classical NP-complete parallel assignment problem is first posed, and then an additional restriction is added to create a special case in which the problem is tractable with an algorithm. The parallel assignment problem is then discussed in the context of weakest precondition computations. In this special situation where statements can be assumed to execute truly concurrently, we show that any sequence of simple assignment statements without function calls can be transformed into an equivalent parallel assignment block. Results of compressing assignment statements into a parallel form with this algorithm are presented for a wide variety of software applications. The proposed algorithms were implemented in the ComFoRT reasoning framework [J. Ivers and N. Sharygina. Overview of ComFoRT: A model checking reasoning framework. Technical Report CMU/SEI-2004-TN-018, Carnegie Mellon Software Engineering Institute, 2004] and used to measure the improvement in the verification of real software systems. This improvement in time proved to be significant for many classes of software
On timed models and full abstraction
In this paper we study a denotational model for a discrete-time version of CSP. We give a compositional semantics for the language. The model records refusal information at the end of each time unit; we believe this model to be simpler than existing models. We also show that the model is fully abstract: equivalence in the model corresponds to the natural equivalence of may testing; and all members of the denotational model are syntactically expressible. We also consider a slightly weaker model, containing no refusal information; we show that this model corresponds to an alternative form of may testing. We briefly discuss the application of these models to the study of information flow in multi-level secure systems.</p
On the Language Inclusion Problem for Timed Automata: Closing a Decidability Gap
We consider the language inclusion problem for timed automata: given two timed automata A and B, are all the timed traces accepted by B also accepted by A? While this problem is known to be undecidable, we show here that it becomes decidable if A is restricted to having at most one clock. This is somewhat surprising, since it is well-known that there exist timed automata with a single clock that cannot be complemented. The crux of our proof consists in reducing the language inclusion problem to a reachability question on an infinite graph; we then construct a suitable well-quasi-order on the nodes of this graph, which ensures the termination of our search algorithm