Green's Relations in Finite Transformation Semigroups

We consider the complexity of Green's relations when the semigroup is given by transformations on a finite set. Green's relations can be defined by reachability in the (right/left/two-sided) Cayley graph. The equivalence classes then correspond to the strongly connected components. It is not difficult to show that, in the worst case, the number of equivalence classes is in the same order of magnitude as the number of elements. Another important parameter is the maximal length of a chain of components. Our main contribution is an exponential lower bound for this parameter. There is a simple construction for an arbitrary set of generators. However, the proof for constant alphabet is rather involved. Our results also apply to automata and their syntactic semigroups.Comment: Full version of a paper submitted to CSR 2017 on 2016-12-1

Chain Reduction for Binary and Zero-Suppressed Decision Diagrams

Chain reduction enables reduced ordered binary decision diagrams (BDDs) and zero-suppressed binary decision diagrams (ZDDs) to each take advantage of the others' ability to symbolically represent Boolean functions in compact form. For any Boolean function, its chain-reduced ZDD (CZDD) representation will be no larger than its ZDD representation, and at most twice the size of its BDD representation. The chain-reduced BDD (CBDD) of a function will be no larger than its BDD representation, and at most three times the size of its CZDD representation. Extensions to the standard algorithms for operating on BDDs and ZDDs enable them to operate on the chain-reduced versions. Experimental evaluations on representative benchmarks for encoding word lists, solving combinatorial problems, and operating on digital circuits indicate that chain reduction can provide significant benefits in terms of both memory and execution time

Multiply Constant-Weight Codes and the Reliability of Loop Physically Unclonable Functions

We introduce the class of multiply constant-weight codes to improve the reliability of certain physically unclonable function (PUF) response. We extend classical coding methods to construct multiply constant-weight codes from known $q$-ary and constant-weight codes. Analogues of Johnson bounds are derived and are shown to be asymptotically tight to a constant factor under certain conditions. We also examine the rates of the multiply constant-weight codes and interestingly, demonstrate that these rates are the same as those of constant-weight codes of suitable parameters. Asymptotic analysis of our code constructions is provided