On the efficiency of normal form systems for representing Boolean functions Abstract   A normal form system (NFS) for representing Boolean functions is thought of as a set of stratified terms over a fixed set of connectives. For a fixed NFS A, the complexity of a Boolean function f with respect to A is the minimum of the sizes of terms in A that represent f. This induces a preordering of NFSs: an NFS A is polynomially as efficient as an NFS B if there is a polynomial P with nonnegative integer coefficients such that the complexity of any Boolean function f with respect to A is at most the value of P in the complexity of f with respect to B. In this paper we study monotonic NFSs, i.e., NFSs whose connectives are increasing or decreasing in each argument. We describe the monotonic NFSs that are optimal, i.e., that are minimal with respect to the latter preorder. We show that these minimal monotonic NFSs are all equivalent. Moreover, we address some natural questions, e.g.: does optimality depend on the arity of connectives? Does it depend on the number of connectives used? We show that optimal monotonic NFSs are exactly those that use a single connective or one connective and the negation. Finally, we show that optimality does not depend on the arity of the connectives.

Polynomially Ambiguous Probabilistic Automata on Restricted Languages We consider the computability and complexity of decision questions for Probabilistic Finite Automata (PFA) with subexponential ambiguity. We show that the emptiness problem for strict and nonstrict cutpoints of polynomially ambiguous commutative PFA remains undecidable, implying that the problem is undecidable when inputs are from a letter monotonic language. We show that the problem remains undecidable over a binary input alphabet when the input word is over a bounded language, in the noncommutative case. In doing so, we introduce a new technique based upon the Turakainen construction of a PFA from a Weighted Finite Automata which can be used to generate PFA of lower dimensions and of subexponential ambiguity. We also study freenessinjectivity problems for polynomially ambiguous PFA and study the border of decidability and tractability for various cases.