15,789 research outputs found
Solving Commutative Relaxations of Word Problems
We present an algebraic characterization of the standard commutative relaxation of the word problem in terms of a polynomial equality. We then consider a variant of the
commutative word problem, referred to as the “Zero-to-All
reachability” problem. We show that this problem is equivalent to a finite number of commutative word problems, and we use this insight to derive necessary conditions for Zero-to-All reachability. We conclude with a set of illustrative examples
Termination of Linear Programs with Nonlinear Constraints
Tiwari proved that termination of linear programs (loops with linear loop
conditions and updates) over the reals is decidable through Jordan forms and
eigenvectors computation. Braverman proved that it is also decidable over the
integers. In this paper, we consider the termination of loops with polynomial
loop conditions and linear updates over the reals and integers. First, we prove
that the termination of such loops over the integers is undecidable. Second,
with an assumption, we provide an complete algorithm to decide the termination
of a class of such programs over the reals. Our method is similar to that of
Tiwari in spirit but uses different techniques. Finally, we conjecture that the
termination of linear programs with polynomial loop conditions over the reals
is undecidable in general by %constructing a loop and reducing the problem to
another decision problem related to number theory and ergodic theory, which we
guess undecidable.Comment: 17pages, 0 figure
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
Symmetry groups, semidefinite programs, and sums of squares
We investigate the representation of symmetric polynomials as a sum of
squares. Since this task is solved using semidefinite programming tools we
explore the geometric, algebraic, and computational implications of the
presence of discrete symmetries in semidefinite programs. It is shown that
symmetry exploitation allows a significant reduction in both matrix size and
number of decision variables. This result is applied to semidefinite programs
arising from the computation of sum of squares decompositions for multivariate
polynomials. The results, reinterpreted from an invariant-theoretic viewpoint,
provide a novel representation of a class of nonnegative symmetric polynomials.
The main theorem states that an invariant sum of squares polynomial is a sum of
inner products of pairs of matrices, whose entries are invariant polynomials.
In these pairs, one of the matrices is computed based on the real irreducible
representations of the group, and the other is a sum of squares matrix. The
reduction techniques enable the numerical solution of large-scale instances,
otherwise computationally infeasible to solve.Comment: 38 pages, submitte
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