Abstract
Research on symmetry detection focuses on identifying and detecting new types of symmetry. We present an algorithm that is capable of detecting any type of permutation-based symmetry, including many types for which there are no existing algorithms. General symmetry detection is library-based, but symmetries that can be parameterized, (i.e. total, partial, rotational, and dihedral symmetry), can be detected without using libraries. In many cases it is faster than existing techniques. Furthermore, it is simpler than most existing techniques, and can easily be incorporated into existing software.
Highlights
A symmetric Boolean function is a function whose inputs can be rearranged in some fashion without changing the output of the function
Some progress has been made in detecting symmetries beyond partial and total symmetry (Chrzanowska-Jeske 2001; Tsai and Marek-Sadowska 1994; Kravets and Sakallah 2000), but the problem of universal symmetry detection has remained open since 1949
Let G be a permutation group that is compatible with a Boolean function f, and let X be the set of input variables of f
Summary
A symmetric Boolean function is a function whose inputs can be rearranged in some fashion without changing the output of the function. This paper presents an entirely new approach which, effectively, considers all inputs simultaneously instead of in pairs This approach allows the algorithm to detect virtually any type of symmetry, including some types that go beyond permutations. When no libraries exist for a particular number of inputs, this makes it possible to detect any partial, total, multi-phase, anti, and Kronecker symmetry using the conventional approach. The function f is said to be invariant with respect to p This terminology is extended in the obvious way to subgroups of Sn. The symmetry group Gf is the set of all permutations that leave f invariant. Most existing symmetry-detection algorithms use symmetric variable pairs, which are detected by comparing the cofactors of a function (Chrzanowska-Jeske 2001). The concept of Boolean orbits can be extended to virtually all types of known symmetry, allowing these types of symmetry to be detected by the algorithm described here
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