625 research outputs found
Towards Verifying Nonlinear Integer Arithmetic
We eliminate a key roadblock to efficient verification of nonlinear integer
arithmetic using CDCL SAT solvers, by showing how to construct short resolution
proofs for many properties of the most widely used multiplier circuits. Such
short proofs were conjectured not to exist. More precisely, we give n^{O(1)}
size regular resolution proofs for arbitrary degree 2 identities on array,
diagonal, and Booth multipliers and quasipolynomial- n^{O(\log n)} size proofs
for these identities on Wallace tree multipliers.Comment: Expanded and simplified with improved result
Carries, shuffling, and symmetric functions
The "carries" when n random numbers are added base b form a Markov chain with
an "amazing" transition matrix determined by Holte. This same Markov chain
occurs in following the number of descents or rising sequences when n cards are
repeatedly riffle shuffled. We give generating and symmetric function proofs
and determine the rate of convergence of this Markov chain to stationarity.
Similar results are given for type B shuffles. We also develop connections with
Gaussian autoregressive processes and the Veronese mapping of commutative
algebra.Comment: 23 page
A Polynomial Time Algorithm for Deciding Branching Bisimilarity on Totally Normed BPA
Strong bisimilarity on normed BPA is polynomial-time decidable, while weak
bisimilarity on totally normed BPA is NP-hard. It is natural to ask where the
computational complexity of branching bisimilarity on totally normed BPA lies.
This paper confirms that this problem is polynomial-time decidable. To our
knowledge, in the presence of silent transitions, this is the first
bisimilarity checking algorithm on infinite state systems which runs in
polynomial time. This result spots an instance in which branching bisimilarity
and weak bisimilarity are both decidable but lie in different complexity
classes (unless NP=P), which is not known before.
The algorithm takes the partition refinement approach and the final
implementation can be thought of as a generalization of the previous algorithm
of Czerwi\'{n}ski and Lasota. However, unexpectedly, the correctness of the
algorithm cannot be directly generalized from previous works, and the
correctness proof turns out to be subtle. The proof depends on the existence of
a carefully defined refinement operation fitted for our algorithm and the
proposal of elaborately developed techniques, which are quite different from
previous works.Comment: 32 page
Cylindric versions of specialised Macdonald functions and a deformed Verlinde algebra
We define cylindric generalisations of skew Macdonald functions when one of
their parameters is set to zero. We define these functions as weighted sums
over cylindric skew tableaux: fixing two integers n>2 and k>0 we shift an
ordinary skew diagram of two partitions, viewed as a subset of the
two-dimensional integer lattice, by the period vector (n,-k). Imposing a
periodicity condition one defines cylindric skew tableaux as a map from the
periodically continued skew diagram into the integers. The resulting cylindric
Macdonald functions appear in the coproduct of a commutative Frobenius algebra,
which is a particular quotient of the spherical Hecke algebra. We realise this
Frobenius algebra as a commutative subalgebra in the endomorphisms over a
Kirillov-Reshetikhin module of the quantum affine sl(n) algebra. Acting with
special elements of this subalgebra, which are noncommutative analogues of
Macdonald polynomials, on a highest weight vector, one obtains Lusztig's
canonical basis. In the limit q=0, one recovers the sl(n) Verlinde algebra,
i.e. the structure constants of the Frobenius algebra become the WZNW fusion
coefficients which are known to be dimensions of moduli spaces of generalized
theta-functions and multiplicities of tilting modules of quantum groups at
roots of unity. Further motivation comes from exactly solvable lattice models
in statistical mechanics: the cylindric Macdonald functions arise as partition
functions of so-called vertex models obtained from solutions to the quantum
Yang-Baxter equation. We show this by stating explicit bijections between
cylindric tableaux and lattice configurations of non-intersecting paths. Using
the algebraic Bethe ansatz the idempotents of the Frobenius algebra are
computed.Comment: 77 pages, 12 figures; v3: some minor typos corrected and title
slightly changed. Version to appear in Comm. Math. Phy
Seminormal forms and Gram determinants for cellular algebras
This paper develops an abstract framework for constructing ``seminormal
forms'' for cellular algebras. That is, given a cellular R-algebra A which is
equipped with a family of JM-elements we give a general technique for
constructing orthogonal bases for A, and for all of its irreducible
representations, when the JM-elements separate A. The seminormal forms for A
are defined over the field of fractions of R. Significantly, we show that the
Gram determinant of each irreducible A-module is equal to a product of certain
structure constants coming from the seminormal basis of A. In the non-separated
case we use our seminormal forms to give an explicit basis for a block
decomposition of A.
The appendix, by Marcos Soriano, gives a general construction of a complete
set of orthogonal idempotents for an algera starting from a set of elements
which act on the algebra in an upper triangular fashion. The appendix shows
that constructions with "Jucys-Murphy elements"depend, ultimately, on the
Cayley-Hamilton theorem.Comment: Final version. To appear J. Reine Angew. Math. Appendix by Marcos
Sorian
Bethe ansatz at q=0 and periodic box-ball systems
A class of periodic soliton cellular automata is introduced associated with
crystals of non-exceptional quantum affine algebras. Based on the Bethe ansatz
at q=0, we propose explicit formulas for the dynamical period and the size of
certain orbits under the time evolution in A^{(1)}_n case.Comment: 12 pages, Introduction expanded, Summary added and minor
modifications mad
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