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Wikipedia / Boole's expansion theorem

Boole's expansion theorem, often referred to as the Shannon expansion or Shannon decomposition, is the identity is any Boolean function respectively.

are sometimes called the positive and negative Shannon cofactors.

The theorem has been called the "fundamental theorem of Boolean algebra". Besides its theoretical importance, it paved the way for binary decision diagrams (BDDs), satisfiability solvers, and many other techniques relevant to computer engineering and formal verification of digital circuits. In such contexts, particularly in BDDs, the expansion is interpreted as an if-then-else is false).

Statement of the theorem

A more explicit way of stating the theorem is:

Variations and implications

  1. XOR-Form : The statement also holds when the disjunction "+" is replaced by the XOR operator:

  2. Dual form: There is a dual form of the Shannon expansion (which does not have a related XOR form):

Repeated application for each argument leads to the Sum of Products that would be

Likewise, application of the dual form leads to the Product of Sums (PoS) canonical form (using the distributivity law):

Properties of cofactors

  1. Linear properties of cofactors:

  2. For a Boolean function F which is made up of two Boolean functions G and H the following are true:

  3. Characteristics of unate functions:

  4. If F is a unate function and...

  5. If F

  6. If F

Operations with cofactors

  1. Boolean difference:

  2. The Boolean difference or Boolean derivative of the function F with respect to the literal x is defined as:

  3. Universal quantification:

  4. The universal quantification of F is defined as:

  5. Existential quantification:

  6. The existential quantification of F is defined as:

History

George Boole presented this expansion as his Proposition II, "To expand or develop a function involving any number of logical symbols", in his Laws of Thought (1854), and it was "widely applied by Boole and other nineteenth-century logicians".

Claude Shannon mentioned this expansion, among other Boolean identities, in a 1949 paper, and showed the switching network interpretations of the identity. In the literature of computer design and switching theory, the identity is often incorrectly attributed to Shannon.

Application to switching circuits

  1. Binary decision diagrams follow from systematic use of this theorem

  2. Any Boolean function can be implemented directly in a switching circuit using a hierarchy of basic multiplexer by repeated application of this theorem.

See also

References