8,140 research outputs found
Generating all polynomial invariants in simple loops
AbstractThis paper presents a method for automatically generating all polynomial invariants in simple loops. It is first shown that the set of polynomials serving as loop invariants has the algebraic structure of an ideal. Based on this connection, a fixpoint procedure using operations on ideals and Gröbner basis constructions is proposed for finding all polynomial invariants. Most importantly, it is proved that the procedure terminates in at most m+1 iterations, where m is the number of program variables. The proof relies on showing that the irreducible components of the varieties associated with the ideals generated by the procedure either remain the same or increase their dimension at every iteration of the fixpoint procedure. This yields a correct and complete algorithm for inferring conjunctions of polynomial equalities as invariants. The method has been implemented in Maple using the Groebner package. The implementation has been used to automatically discover non-trivial invariants for several examples to illustrate the power of the technique
Invariant Generation for Multi-Path Loops with Polynomial Assignments
Program analysis requires the generation of program properties expressing
conditions to hold at intermediate program locations. When it comes to programs
with loops, these properties are typically expressed as loop invariants. In
this paper we study a class of multi-path program loops with numeric variables,
in particular nested loops with conditionals, where assignments to program
variables are polynomial expressions over program variables. We call this class
of loops extended P-solvable and introduce an algorithm for generating all
polynomial invariants of such loops. By an iterative procedure employing
Gr\"obner basis computation, our approach computes the polynomial ideal of the
polynomial invariants of each program path and combines these ideals
sequentially until a fixed point is reached. This fixed point represents the
polynomial ideal of all polynomial invariants of the given extended P-solvable
loop. We prove termination of our method and show that the maximal number of
iterations for reaching the fixed point depends linearly on the number of
program variables and the number of inner loops. In particular, for a loop with
m program variables and r conditional branches we prove an upper bound of m*r
iterations. We implemented our approach in the Aligator software package.
Furthermore, we evaluated it on 18 programs with polynomial arithmetic and
compared it to existing methods in invariant generation. The results show the
efficiency of our approach
Aligator.jl - A Julia Package for Loop Invariant Generation
We describe the Aligator.jl software package for automatically generating all
polynomial invariants of the rich class of extended P-solvable loops with
nested conditionals. Aligator.jl is written in the programming language Julia
and is open-source. Aligator.jl transforms program loops into a system of
algebraic recurrences and implements techniques from symbolic computation to
solve recurrences, derive closed form solutions of loop variables and infer the
ideal of polynomial invariants by variable elimination based on Gr\"obner basis
computation
All degree six local unitary invariants of k qudits
We give explicit index-free formulae for all the degree six (and also degree
four and two) algebraically independent local unitary invariant polynomials for
finite dimensional k-partite pure and mixed quantum states. We carry out this
by the use of graph-technical methods, which provides illustrations for this
abstract topic.Comment: 18 pages, 6 figures, extended version. Comments are welcom
Cabled Wilson Loops in BF Theories
A generating function for cabled Wilson loops in three-dimensional BF
theories is defined, and a careful study of its behavior for vanishing
cosmological constant is performed. This allows an exhaustive description of
the unframed knot invariants coming from the pure BF theory based on SU(2), and
in particular, it proves a conjecture relating them to the Alexander-Conway
polynomial.Comment: 30 pages, LaTe
Topological strings, strips and quivers
We find a direct relation between quiver representation theory and open
topological string theory on a class of toric Calabi-Yau manifolds without
compact four-cycles, also referred to as strip geometries. We show that various
quantities that characterize open topological string theory on these manifolds,
such as partition functions, Gromov-Witten invariants, or open BPS invariants,
can be expressed in terms of characteristics of the moduli space of
representations of the corresponding quiver. This has various deep
consequences; in particular, expressing open BPS invariants in terms of motivic
Donaldson-Thomas invariants, immediately proves integrality of the former ones.
Taking advantage of the relation to quivers we also derive explicit expressions
for classical open BPS invariants for an arbitrary strip geometry, which lead
to a large set of number theoretic integrality statements. Furthermore, for a
specific framing, open topological string partition functions for strip
geometries take form of generalized -hypergeometric functions, which leads
to a novel representation of these functions in terms of quantum dilogarithms
and integral invariants. We also study quantum curves and A-polynomials
associated to quivers, various limits thereof, and their specializations
relevant for strip geometries. The relation between toric manifolds and quivers
can be regarded as a generalization of the knots-quivers correspondence to more
general Calabi-Yau geometries.Comment: 47 pages, 9 figure
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