Match the tool to the artifact

This overview explains how matrices, evaluators, decision diagrams, solvers, counters, and simplifiers serve different requests. Choosing a tool before naming the output can leave the caller with the wrong artifact or the wrong count. A direct evaluator answers a specified point. A satisfying-assignment procedure finds an input making F true. An equivalence procedure establishes that F and the displayed De Morgan expression agree over the full universe. These are different services, even when they use the same Boolean rule. A decision diagram for or can test A first. If A is one, the answer is already one; otherwise it follows a B test. This compact path structure can answer a point query without writing all four truth outputs. Other tasks require additional traversal or bookkeeping. A full-vector method delivers every output in order. A counting method can return three without enumerating a public list of every witness. The caller's required output determines whether either is an acceptable replacement for the other. SAT asks whether there is any input that makes a Boolean rule true. A satisfying input is a witness. That question differs from evaluating one input chosen in advance. BDD means binary decision diagram: yes-or-no variable tests form paths that can share later branches. A miter is the exclusive-or of two outputs; it is one exactly where they disagree. A logic cover lists accepting terms. For A or the conjunction of B and C, the two terms are A, and B and C. Logic minimization seeks a simpler equivalent description. The required output determines which tool is suitable. A complete truth layout, a reusable evaluator, a decision diagram, a witness, a count and a simplified expression are different outputs. Start by naming the deliverable. For a feature rule, a point-validity query can evaluate one complete selection. SAT can seek a satisfying selection; an equivalence miter can seek a counterexample to agreement. A BDD represents the function under an ordering and supports further operations. Espresso-style minimization asks for a reduced logic cover. Symbolic algebra asks for a symbolic transformation. Packed evaluation can enumerate a complete ordered truth vector. Scalar counting and projected counting deserve their own branches. If auxiliary inputs give several witnesses for one selected feature choice, a full-assignment count and a distinct projected count differ. This is an interface map, not a claim that every optional backend is installed or universally fastest. For or, there are three complete satisfying assignments. Each specifies both A and B. Counting these full assignments gives three. Project each full assignment onto A. The two assignments with A one become the same selected answer, so deduplication leaves zero and one. The projected count is two. A tool returning the full-model count has not automatically returned the projected count. Similarly, one witness is not all witnesses. The toolbox becomes easier to use when its entries are described by artifacts and exact contracts rather than broad claims of solving Boolean problems. Start with the caller's required artifact. For projected counting, count each selected assignment once even when it has several auxiliary witnesses.