Class Of Symmetric Functions Research Articles

Exact ordered binary decision diagram size when representing classes of symmetric functions

An ordered binary decision diagram (OBDD) is a canonical, graphical representation of a switching function. The space complexity of this representation as well as the time complexity for manipulating functions in this form is determained by the number of vertices in the OBDD. Symmetric functions are a class of functions which include the basic Boolean gates such as NOT, AND, NAND, NOR, XOR, etc., as well as less basic functions, such as voter logic for redundant circuit implementations. Symmetric functions exploit the most powerful properties of OBDDs to a very great extent. OBDDs have been shown to have size of O(n2), where n is the number of switching variables. However, this says little of the actual performance of OBDDs in practice. Exact equations of OBDD size are derived for the common classes of symmetric functions, as well as an exact equation for the largest OBDD that can exist for any arbitrary symmetric function. It is shown that OBDDs are Ω(n 2) for the majority of functions from each common class of symmetric functions beyond the simplest Boolean gates. Since most functions can be expected to be more complex to represent than symmetric functions, this result has profound implications to the straightforward application of OBDDs to large functional problems.

Chern classes of the Grassmannians and Schubert calculus

IN THIS note we continue the theme of [3,4] by using the zeros of a holomorphic vector field to give a geometric demonstration that the ring structure of the cohomology of the Grassman manifolds (the multiplicative aspect of Schubert calculus) is a consequence of general properties of symmetric functions. More precisely, this amounts to showing that certain properties of Schubert cycles such as independence; multiplicative structure etc. are formally deducible from similar properties of a class of symmetric functions called Schur functions provided certain so called meaningless Schubert conditions are ignoied. This principle seems to have originated with Giambelli and was formulated precisely by Lesieur[9]. More recently, Horrocks[6], Lascoux[8], Porteous [ 1 l] and Stanley [ 131 have discussed various aspects of the principle. Recall that if, X is a compact complex manifold and V a holomorphic vector field on X with zeros 2, one may view 2 as the variety defined by the sheaf of ideals i(V)R’ C 6 where i(V):W+fP-’ is the contraction from the sheaf of germs of holomorphic p-forms to (p l)-forms and 6 = R”. The structure sheaf fY2 = Q/i( V)Q’ of 2 is a coherent possibly unreduced sheaf of rings on Z. The main result of [4] is that if X is in addition compact Kaehler and Z finite but nontrivial, then H”(X, I!?~), the ring of functions on Z, has a decreasing filtration F having the property that fiFj C F;+j such that there is an isomorphism of graded rings

The recognition and identification of symmetric switching functions

IN the design of switching circuits based on relay-contact networks, it is known that if the function which describes the network is of the symmetric type it leads to a network which is much more economical of switching elements than the best series-parallel circuit. C. E. Shannon1 first described this type of function and its associated relay contact network in 1938. His definition of the symmetric function was general, but most of the generality of concept has been lost in subsequent treatment of the subject. For example, S. H. Washburn2 defines the symmetric relay-contact network as ?one in which the conditions for closing a particular input-to-output path are given & in terms of the number of relays operated and un-operated.? This particular type of network is derived from a restricted class of symmetric functions ? a class of functions which may be realized by means of iterative networks.3 There is a much larger class of symmetric functions contained within Shannon's definition, and these can be realized only by means of non-iterative networks.