Techniques for the analysis of monotone Boolean networks
Monotone Boolean networks are one the most widely studied restricted forms of combinational networks. This dissertation examines the complexity of such networks realising single output monotone Boolean functions and develops recent results on their relation to unrestricted networks. Two standard analytic techniques are considered: the inductive gate elimination argument, and replacement rules. In Chapters (3) and (4) the former method is applied to obtain new lower bounds on the monotone network complexity of threshold functions. In Chapter (5) a complete characterisation of all replacement rules, valid when computing some monotone Boolean functions, is given. The latter half of the dissertation concentrates on the relation between the combinational and monotone network complexity of monotone functions, and extends works of Berkowitz and Wegener on “slice functions”. In Chapter (6) the concept of “pseudo-complementation”, the replacement of instances of negated variables by monotone functions, without affecting computational behaviour, is defined. Pseudo-complements are show to exist for all monotone Boolean functions and using these a generalisation of slice function is proposed. Chapter (7) examines the slice functions of some NP-complete predicates. For the predicates considered, it is shown that the “canonical” slice has polynomial network complexity, and that the “central” slice is also NP-complete. This result permits a reformulation of the P = NP? Question in terms of monotone network complexity. Finally, Chapter (8) examines the existence of gaps for the combinational and monotone network complexity measures. A natural series of classes of monotone Boolean functions is defined and it is shown that for the “hardest” members of each class there is no asymptotic gap between these measures.