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Boole's expansion theorem

It has been called the "fundamental theorem of Boolean algebra".^[1] 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 engineering contexts (especially in BDDs), the expansion is interpreted as a if-then-else, with the variable ${\displaystyle x}$ being the condition and the cofactors being the branches (${\displaystyle F_{x}}$ when ${\displaystyle x}$ is true and respectively ${\displaystyle F_{x'}}$ when ${\displaystyle x}$ is false).^[2]

Statement of the theorem[edit]

A more explicit way of stating the theorem is:

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

Variations and implications[edit]

XOR-Form

The statement also holds when the disjunction "+" is replaced by the XOR operator:

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

Dual form

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

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

Repeated application for each argument leads to the Sum of Products (SoP) canonical form of the Boolean function ${\displaystyle f}$. For example for ${\displaystyle n=2}$ that would be

${\displaystyle {\begin{aligned}f(X_{1},X_{2})&=X_{1}\cdot f(1,X_{2})+X_{1}'\cdot f(0,X_{2})\\&=X_{1}X_{2}\cdot f(1,1)+X_{1}X_{2}'\cdot f(1,0)+X_{1}'X_{2}\cdot f(0,1)+X_{1}'X_{2}'\cdot f(0,0)\end{aligned}}}$

Likewise, application of the dual form leads to the Product of Sums (PoS) canonical form (using the distributivity lawof${\displaystyle +}$ over ${\displaystyle \cdot }$):

${\displaystyle {\begin{aligned}f(X_{1},X_{2})&=(X_{1}+f(0,X_{2}))\cdot (X_{1}'+f(1,X_{2}))\\&=(X_{1}+X_{2}+f(0,0))\cdot (X_{1}+X_{2}'+f(0,1))\cdot (X_{1}'+X_{2}+f(1,0))\cdot (X_{1}'+X_{2}'+f(1,1))\end{aligned}}}$

Properties of cofactors[edit]

Linear properties of cofactors:

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

If${\displaystyle F=H'}$ then ${\displaystyle F_{x}=H'_{x}}$

If${\displaystyle F=G\cdot H}$ then ${\displaystyle F_{x}=G_{x}\cdot H_{x}}$

If${\displaystyle F=G+H}$ then ${\displaystyle F_{x}=G_{x}+H_{x}}$

If${\displaystyle F=G\oplus H}$ then ${\displaystyle F_{x}=G_{x}\oplus H_{x}}$

Characteristics of unate functions:

IfF is a unate function and...

IfF is positive unate then ${\displaystyle F=x\cdot F_{x}+F_{x'}}$

IfF is negative unate then ${\displaystyle F=F_{x}+x'\cdot F_{x'}}$

Operations with cofactors[edit]

Boolean difference:

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

${\displaystyle {\frac {\partial F}{\partial x}}=F_{x}\oplus F_{x'}}$

Universal quantification:

The universal quantification of F is defined as:

${\displaystyle \forall xF=F_{x}\cdot F_{x'}}$

Existential quantification:

The existential quantification of F is defined as:

${\displaystyle \exists xF=F_{x}+F_{x'}}$

History[edit]

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),^[3] and it was "widely applied by Boole and other nineteenth-century logicians".^[4]

Claude Shannon mentioned this expansion, among other Boolean identities, in a 1949 paper,^[5] 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.^[6][4]

Application to switching circuits[edit]

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.

References[edit]

See also[edit]

• Reed–Muller expansion

External links[edit]

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