This paper presents a framework for describing information and information flow. We show that information can be represented as a lattice. We will motivate the idea that this framework is applicable for demonstrating security properties of systems. In particular, we show the relationship between the lattice representing information and the unwinding theorem. We will also demonstrate the relationship between properties of this lattice and the aggregation problem. IntroductionConsider a system as a black box that allows users to query its internal state. For example, an airline database can be queried for different kinds of information, e.g., estimated time of arrival. The queries can be ordered by the amount of information returned. For example, if a query returns the complete flight information, then one can deduce the estimated time of arrival. In this example, the first query that requests ETA is "less" than the query that requests the flight information.Users may also be ordered by the type of information that they can access. For example, the president of an airline company may be able to make more detailed queries than a random customer. In particular, a customer may not be able to obtain the passenger manifest whereas this information is available to the president.In this paper, we formalize the notion of information as a complete lattice. The queries described in the above example will define elements of this lattice. The information determined by a query, q1, is greater than the information determined by a query, 42, if the result of qz can be explicitly determined from the result of the query q1. The information obtained by making two queries at once will be the join of the information obtained by making each of the queries individually.We will then provide a necessary and sufficient condition for non-interference in terms of the information lattice. The condition is the existence of a sensitivity labelling of the information lattice in such a way that e instructions with a high sensitivity label do not modify information with a low sensitivity label e the flow of information in the system is from information with low sensitivity labels to information with high sensitivity labels. the output at a sensitivity level can be determined by the information at that sensitivity level.We will also show a possible connection of some properties of the lattice with the aggregation problem. This may point to a way of dealing with the aggregation problem in an algebraic manner. The Information LatticeFor the purposes of this section, we will fix a set C, representing a state space of a system. In this paper we will show how information about elements of the set C can be regarded as a lattice. This information lattice can be described in two equivalent manners. First, we can view the lattice as the set of equivalence relations on the set C. The equivalence classes represent sets of states that cannot be distinguished with the information being described. Second, we can view information in terms of functions from C. The...

This paper begins with a brief discussion of a class of polynomial Riemann hypotheses, which leads to the consideration of sequences of orthogonal polynomials and 3-term recursions. The discussion further leads to higher order polynomial recursions, including 4-term recursions where orthogonality is lost. Nevertheless, we show that classical results on the nature of zeros of real orthogonal polynomials (i. e., that the zeros of $p_n$ are real and those of $p_{n+1}$ interleave those of $p_n$) may be extended to polynomial sequences satisfying certain 4-term recursions. We identify specific polynomial sequences satisfying higher order recursions that should also satisfy this classical result. As with the 3-term recursions, the 4-term recursions give rise naturally to a linear functional. In the case of 3-term recursions the zeros fall nicely into place when it is known that the functional is positive, but in the case of our 4-term recursions, we show that the functional can be positive even when there are non-real zeros among some of the polynomials. It is interesting, however, that for our 4-term recursions positivity is guaranteed when a certain real parameter $C$ satisfies $C\ge 3$, and this is exactly the condition of our result that guarantees the zeros have the aforementioned interleaving property. We conjecture the condition $C\ge 3$ is also necessary. Next we used a classical determinant criterion to find exactly when the associated linear functional is positive, and we found that the Hankel determinants $\Delta_n$ formed from the sequence of moments of the functional when $C = 3$ give rise to the initial values of the integer sequence $1, 3, 26, 646, 45885, \cdots,$ of Alternating Sign Matrices (ASMs) with vertical symmetry. This spurred an intense interest in these moments, and we give 9 diverse characterizations of this sequence of moments. We then specify these Hankel determinants as Macdonald-type integrals. We also provide an an infinite class of integer sequences, each sequence of which gives the Hankel determinants $\Delta_n$ of the moments. Finally we show that certain $n$-tuples of non-intersecting lattice paths are evaluated by a related class of special Hankel determinants. This class includes the $\Delta_n$. At the same time, ASMs with vertical symmetry can readily be identified with certain $n$-tuples of osculating paths. These two lattice path models appear as a natural bridge from the ASMs with vertical symmetry to Hankel determinants.

An extensive literature exists describing various techniques for the evaluation of Hankel determinants. The prevailing methods such as Dodgson condensation, continued fraction expansion, LU decomposition, all produce product formulas when they are applicable. We mention the classic case of the Hankel determinants with binomial entries 3k+2 k and those with entries 3k k ; both of these classes of Hankel determinants have product form evaluations. The intermediate case, 3k+1 k has not been evaluated. There is a good reason for this: these latter determinants do not have product form evaluations.In this paper we evaluate the Hankel determinant of 3k+1 k . The evaluation is a sum of a small number of products, an almost product. The method actually provides more, and as applications, we present the salient points for the evaluation of a number of other Hankel determinants with polynomial entries, along with product and almost product form evaluations at special points.

In a recent paper we have presented a method to evaluate certain Hankel determinants as almost products; i.e. as a sum of a small number of products. The technique to find the explicit form of the almost product relies on differential-convolution equations and trace calculations. In the trace calculations a number of intermediate nonlinear terms involving determinants occur, but only to cancel out in the end.In this paper, we introduce a class of multilinear operators γ acting on tuples of matrices as an alternative to the trace method. These operators do not produce extraneous nonlinear terms, and can be combined easily with differentiation. The paper is self contained. An example of an almost product evaluation using γ -operators is worked out in detail and tables of the γ -operator values on various forms of matrices are provided.We also present an explicit evaluation of a new class of Hankel determinants and conjectures.

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