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This paper presents algorithms that relate to the problem of finding aminimum-cost sum-of-primes representation of a Boolean function f when the costfunction C is positive and additive. A set of primes whose sum equals f iscalled a basis for f, so a solution to the problem is a minimum-cost basis.  The algorithms construct the following canonical partition of the completeset of primes and identify the members of sets 1, 2, and 3:  (1) Essential Primes, which must be part of any basis for f,  (2) Unnecessary Primes that cannot be part of a minimum-cost basis for f forany positive additive cost function,  (3) Unique disjoint sets of primes, PS1,...,PSN with associated "covering"tables TS1,..., TSN such that any minimum-cost basis consists of the union ofthe sets  Essential Primes, QS1(C), ..., QSN(C) where QSi(C) is contained in PSi andQSi(C) is a minimum-cost "cover" for PSi. Covering is defined by operationCascade(QS, TS), which has the property that QS covers PS if and only ifCascade(QS, TS) is empty.  The key to the results is the study of objects called Ancestor Sets. TheAncestor Theorem proves that if A is an Ancestor Set for f, every minimum-costbasis includes a minimum-cost cover for the set of primes PS in Ancestor Set Aand a minimum-cost cover for the set of primes that are not in A (and are notcovered by the the union of the Essentials with PS). The PSi in the partitionare the sets of primes in canonical disjoint Independent Ancestor Sets Ai,which are easy to generate when the calculation of the primes (and theirconsensus combinations) is within computational scope. The paper also presentsa condition under which QSi(C) can be easily determined, and another conditionsuch that PSi can be broken into disjoint pieces that can be minimizedseparately.

A Canonical Partition of the Primes of Logic Functions