107 research outputs found
The Polyhedron-Hitting Problem
We consider polyhedral versions of Kannan and Lipton's Orbit Problem (STOC
'80 and JACM '86)---determining whether a target polyhedron V may be reached
from a starting point x under repeated applications of a linear transformation
A in an ambient vector space Q^m. In the context of program verification, very
similar reachability questions were also considered and left open by Lee and
Yannakakis in (STOC '92). We present what amounts to a complete
characterisation of the decidability landscape for the Polyhedron-Hitting
Problem, expressed as a function of the dimension m of the ambient space,
together with the dimension of the polyhedral target V: more precisely, for
each pair of dimensions, we either establish decidability, or show hardness for
longstanding number-theoretic open problems
On the Polytope Escape Problem for Continuous Linear Dynamical Systems
The Polyhedral Escape Problem for continuous linear dynamical systems
consists of deciding, given an affine function and a convex polyhedron ,
whether, for some initial point in , the
trajectory of the unique solution to the differential equation
,
, is entirely contained in .
We show that this problem is decidable, by reducing it in polynomial time to
the decision version of linear programming with real algebraic coefficients,
thus placing it in , which lies between NP and PSPACE. Our
algorithm makes use of spectral techniques and relies among others on tools
from Diophantine approximation.Comment: Accepted to HSCC 201
On Termination of Integer Linear Loops
A fundamental problem in program verification concerns the termination of
simple linear loops of the form x := u ; while Bx >= b do {x := Ax + a} where x
is a vector of variables, u, a, and c are integer vectors, and A and B are
integer matrices. Assuming the matrix A is diagonalisable, we give a decision
procedure for the problem of whether, for all initial integer vectors u, such a
loop terminates. The correctness of our algorithm relies on sophisticated tools
from algebraic and analytic number theory, Diophantine geometry, and real
algebraic geometry. To the best of our knowledge, this is the first substantial
advance on a 10-year-old open problem of Tiwari (2004) and Braverman (2006).Comment: Accepted to SODA1
Polynomial Invariants for Affine Programs
We exhibit an algorithm to compute the strongest polynomial (or algebraic)
invariants that hold at each location of a given affine program (i.e., a
program having only non-deterministic (as opposed to conditional) branching and
all of whose assignments are given by affine expressions). Our main tool is an
algebraic result of independent interest: given a finite set of rational square
matrices of the same dimension, we show how to compute the Zariski closure of
the semigroup that they generate
On Termination for Faulty Channel Machines
A channel machine consists of a finite controller together with several fifo
channels; the controller can read messages from the head of a channel and write
messages to the tail of a channel. In this paper, we focus on channel machines
with insertion errors, i.e., machines in whose channels messages can
spontaneously appear. Such devices have been previously introduced in the study
of Metric Temporal Logic. We consider the termination problem: are all the
computations of a given insertion channel machine finite? We show that this
problem has non-elementary, yet primitive recursive complexity
On the Complexity of the Equivalence Problem for Probabilistic Automata
Checking two probabilistic automata for equivalence has been shown to be a
key problem for efficiently establishing various behavioural and anonymity
properties of probabilistic systems. In recent experiments a randomised
equivalence test based on polynomial identity testing outperformed
deterministic algorithms. In this paper we show that polynomial identity
testing yields efficient algorithms for various generalisations of the
equivalence problem. First, we provide a randomized NC procedure that also
outputs a counterexample trace in case of inequivalence. Second, we show how to
check for equivalence two probabilistic automata with (cumulative) rewards. Our
algorithm runs in deterministic polynomial time, if the number of reward
counters is fixed. Finally we show that the equivalence problem for
probabilistic visibly pushdown automata is logspace equivalent to the
Arithmetic Circuit Identity Testing problem, which is to decide whether a
polynomial represented by an arithmetic circuit is identically zero.Comment: technical report for a FoSSaCS'12 pape
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