Boole's expansion theorem

Boole's expansion theorem, often referred to as the Shannon expansion or decomposition, is the identity: $F=x\cdot F_{x}+x'\cdot F_{x}'$, where $F$ is any Boolean function and $F_{x}$and $F_{x}'$ are $F$ with the argument $x$ equal to $1$ and to $0$ respectively.

The terms $F_{x}$ and $F_{x}'$ are sometimes called the positive and negative Shannon cofactors, respectively, of $F$ with respect to $x$. These are functions, computed by restrict operator, $restrict(F,x,0)$ and $restrict(F,x,1)$ (see valuation (logic) and partial application).

It has been called the "fundamental theorem of Boolean algebra". Besides its theoretical importance, it paved the way for binary decision diagrams, satisfiability solvers, and many other techniques relevant to computer engineering and formal verification of digital circuits.

Statement of the theorem

A more explicit way of stating the theorem is-

$f(X_{1},X_{2},\dots ,X_{n})=X_{1}\cdot f(1,X_{2},\dots ,X_{n})+X_{1}'\cdot f(0,X_{2},\dots ,X_{n})$

Proof for the statement follows from direct use of mathematical induction, from the observation that $f(X_{1})=X_{1}.1+X_{1}'.0$ and expanding a 2-ary and n-ary Boolean functions identically.

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 , noted information-theorist and communications pioneer, mentioned this expansion, among other Boolean identities, in a 1948 paper, and showed the switching network interpretations of the identity. In the literature of computer design and switching theory, it is often 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.

External links

• Shannon’s Decomposition Example with multiplexers.
• Optimizing Sequential Cycles Through Shannon Decomposition and Retiming (PDF) Paper on application.