Comment on "Detection of Totally Symmetric Boolean Functions''

A method is presented for determining whether a Boolean function possesses any group invariance; that is, whether there are any permutations or primings of the independent variables which leave the function unchanged This method is then extended to the detection of functions which are totally symmetric

http://ieeexplore.ieee.org/iel7/6731005/6769977/06769984.pdf

Detection of group invariance or total symmetry of a Boolean function

Detection of Total or Partial Symmetry of a Switching Function with the Use of Decomposition Charts

The Detection and Identification of Symmetric Switching Functions with the Use of Tables of Combinations

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.

http://ieeexplore.ieee.org/iel5/6367273/6372118/06372129.pdf

The recognition and identification of symmetric switching functions

A new method for detection and identification of totally symmetric Boolean functions is developed. Instead of using maps, charts, or truth tables, with permutations and complementations of variables, this method is based on the number of vertices contained in symmetric functions and the concept of distance between two vertices. Since the distance between any two vertices remains invariant under any number of permutations and complementations of the variables, this method lends itself equally conveniently to the detection of total symmetry of Boolean functions with respect to purely uncomplemented, purely complemented, or mixed variables. A procedure is presented, and an example is worked out to illustrate the application of the method.

Comment on "Detection of Totally Symmetric Boolean Functions''

References

Detection of group invariance or total symmetry of a Boolean function

Detection of Total or Partial Symmetry of a Switching Function with the Use of Decomposition Charts

The Detection and Identification of Symmetric Switching Functions with the Use of Tables of Combinations

The recognition and identification of symmetric switching functions

Detection of Totally Symmetric Boolean Functions

Related Papers (5)

Synthesis of Electronic Circuits for Symmetric Functions

Algebra of switching networks

Antisymmetries in the realization of Boolean functions

Reducing quantum cost of reversible circuits for homogeneous boolean functions

Folding of Symmetric Functions