Boole's expansion theorem

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

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

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

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

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 law of ${\displaystyle +}$ 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

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:

If F is a unate function and...

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

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

Operations with cofactors

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

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

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

1. Rosenbloom, Paul Charles (1950). The Elements of Mathematical Logic. p. 5.
2. G. D. Hachtel and F. Somenzi (1996), Logic Synthesis and Verification Algorithms, p. 234
3. Boole, George (1854). An Investigation of the Laws of Thought: On which are Founded the Mathematical Theories of Logic and Probabilities. p. 72.
4. Brown, Frank Markham (2012) [2003, 1990]. Boolean Reasoning - The Logic of Boolean Equations (reissue of 2nd ed.). Mineola, New York: Dover Publications, Inc. p. 42. ISBN 978-0-486-42785-0.
5. Shannon, Claude (January 1949). "The Synthesis of Two-Terminal Switching Circuits" (PDF). Bell System Technical Journal. 28: 59–98 [62]. doi:10.1002/j.1538-7305.1949.tb03624.x. ISSN 0005-8580.
6. Perkowski, Marek A.; Grygiel, Stanislaw (1995-11-20), "6. Historical Overview of the Research on Decomposition", A Survey of Literature on Function Decomposition, Version IV, Functional Decomposition Group, Department of Electrical Engineering, Portland University, Portland, Oregon, USA, p. 21, CiteSeerX 10.1.1.64.1129 (188 pages)

See also

• Reed–Muller expansion

External links

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